Expositio libri Posteriorum Analyticorum


Thomas Aquinas

translated by

Fabian R. Larcher, O.P.

re-edited and html-formated by Joseph Kenny, O.P.


Preface by James A. Weiseipl, O.P.

Foreword of Thomas Aquinas


  1. The need for pre-existent knowledge in all learning (71a1-10)
  2. Extent and order of the pre-existent knowledge required for obtaining science (71a11-24)
  3. Pre-existent knowledge of the conclusion (71a24-b9)
  4. Nature of the demonstrative syllogism (71b8-72a8)
  5. First and immediate propositions (72a8-24)
  6. Knowledge of immediate principles (72a25-b4)
  7. Discussion of two errors-exclusion of the first one (72b5-24)
  8. The second error is excluded by showing that circular demonstration is not acceptable (72b-73a20)
  9. How something is said to be predicated of all (73a21-34)
  10. How something is said to be predicated per se of a thing (73a34-b26)
  11. How something is said to be predicated as commensurately universal (73b27-74a3)
  12. How error occurs in taking the universal (74a4-b4)
  13. Demonstration proceeds from necessary things (74b5-75a17)
  14. Demonstration bears upon and proceeds from things which are per se (75a18-37)
  15. Demonstration does not skip from one genus to an alien genus (75a38-b20)
  16. Demonstration is not of perishable but of eternal matters (75b21-36)
  17. Demonstration does not proceed from common principles, but from principles proper to the thing demonstrated (75b37-76a25)
  18. Difference between principles and non-principles, common and proper principles (76a26-b22)
  19. How common principles differ from one another (76b23-77a9)
  20. Relation between demonstrative sciences and common principles (77a10-35)
  21. Of the questions, responses and disputations peculiar to each science (77a36-77b15)
  22. Each science has its own deceptions and areas of ignorance (77b16-78a21)
  23. How demonstration “quia” and “propter quid” differ in a same science. Demonstration “quia” through an effect (78a22-b13)
  24. How there is demonstration “quia” through things not immediately connected (78b13-34)
  25. How demonstration “quia” differs from demonstration “propter quid” when the former pertains to one science and the latter to another (78b34-79a16)
  26. Demonstrative syllogisms best made in the first figure. On mediate and immediate negative propositions (79a17-b22)
  27. How ignorance or deception bearing on first and immediate things can be induced by syllogism and lead one to suppose something to be which is not (79b23-80a8)
  28. How by syllogizing in the first or second figure a false negative is concluded contrary to an immediate affirmative (80a8-b16)
  29. Syllogism of ignorance in regard to mediate propositions (80b17-81a37)
  30. Cause of simple negative ignorance (81a38-b9)
  31. Three questions about proceeding to infinity in confirming demonstrations (81b10-81b20)
  32. Solution of some of these doubts hinges upon solution of others of these doubts (82a21-b34)
  33. That one does not proceed to infinity in essential predicates is shown “logically” (82b34-83a35)
  34. Logical reasons why one does not proceed to infinity in predicates (83a36-84a7)
  35. That there is not an infinite process upward or downward in predicates is shown analytically (84a8-b2)
  36. Certain corollaries from preceding lectures (84b3-85a11)
  37. Whether universal demonstration is stronger than particular demonstration (85a12-b21)
  38. Universal demonstration is stronger than particular demonstration (85b22-86a32)
  39. Affirmative demonstration is stronger than negative (86a32-b40)
  40. Negative ostensive demonstration is stronger than demonstration leading to the impossible (87a1-30)
  41. Comparison of science to science from the standpoint of certainty and of unity and diversity (87a31-b17)
  42. Science is not concerned with things caused by fortune or with things learned through sense-perception (87b19-88a17)
  43. The principles of all syllogisms are not the same (88a18-b29)
  44. Science compared with other modes of knowing (88b30-89b20)


  1. Each of the four questions which pertain to science is one way or another a question of the middle (89b2l-90a35)
  2. Whether the definition which signifies the quod quid of a thing can be demonstrated (90a36-91a3)
  3. Whether the quod quid signified by the definition can be demonstrated by taking convertible terms (91a12-b11)
  4. Whether quod quid can be demonstrated by the method of division (91b12-92a6)
  5. Whether a quod quid can be demonstrated by taking that which is required for a quod quid (92a6-b3)
  6. Whether quod quid can be shown by demonstration or definition (92b3-39)
  7. The two ways, logical and demonstrative, of manifesting quod quid (93a1-b21)
  8. To attain quod quid through demonstration is not possible in all cases. Relation of definition to demonstration (93b22-94a19)
  9. Propter quid can be manifested in four genera of causes (94a20-95a9)
  10. How something is demonstrated through a cause not simultaneous with its caused. How a cause not simultaneous with its effect is taken as middle in demonstrating (95a10-b1)
  11. How a cause which is not simultaneous with its effect may be taken as a middle in demonstration (95b1-37)
  12. How in things that come to be reciprocally, a cause which is not simultaneous with the effect is taken as middle in a demonstration. How one demonstrates through cause differently in things that occur always and in things that occur as a general rule (95b38-96a20)
  13. Characteristics which should be present in the items which constitute the definition signifying the essence of a thing (96a22-b14)
  14. Dividing the genus to investigate which items should be put in a definition (95b15-97a6)
  15. Two errors are excluded. What is really required for constituting a definition according to the method of division? (97a6-b6)
  16. How to search for the definition of a thing by examining things similar to it and dissimilar (97b7-40)
  17. How to investigate the why in special problems. How certain problems agree as to propter quid, either because their middles have a kind of unity or are subordinated (98a1-34)
  18. Co-existence of cause and caused (98a35-b40)
  19. Whether upon unity of cause follows unity of effect, and vice versa. How cause and effect follow upon one another (99a1-b18)
  20. How the first principles of demonstration are known by us (99b18-100b17)


The purpose of logic is to provide an analytic guide to the discovery of demonstrated truth and all its various approximations throughout the philosophical sciences. In the words of St. Albert the Great, logic “teaches the principles by which one can arrive at the knowledge of things unknown through that which is known” (De Praedicab., tr. I, c. 5, ed. Borgnet 1, 8b). St. Thomas defines logic as an art “directive of the acts of reason themselves so that man may proceed orderly, easily and without error in the very act of reason itself” (Foreword). Logic is thus a construct based on the natural processes of the mind invented for a very specific use, namely, scientific reasoning. Because it is a construct, logic is said to deal with “second intentions,” that is, deliberate constructs of the mind, existing solely in the mind, of ideas based upon the way human beings know reality (“first intentions”), such as predicables, subject, predicate, major premise, minor premise, middle term and conclusion. The analysis and construction of this guide is the scientific, or theoretical, aspect of logic. Under this consideration, logic is itself a science and it is this aspect that modem logicians seem to be interested in. Nevertheless the purpose of this construct is that it be used by thinkers who want to get on with the discovery of truth in the various sciences. In this way the whole of logic is a methodology, solidly established by analysis, to guide the mind in its quest for answers to problems raised in scientific inquiry. The general name given to all of Aristotle’s logical treatises is Organon, the instrument. For this reason Boethius, the 6th century translator of most of Aristotle’s Organon, says that logic is “not so much a science as an instrument of science” (Comm. super Porphyry, ed. 2a, 1, c. 3; see St. Thomas, In Boeth. de Trin., q. 5, a. I ad 2; St. Albert, Post Anal. I, tr. I, c. 1, ed. Borgnet II, 2b).

It must be noted, however, that logic is only a general methodology common to all scientific knowledge (See J.A. Weisheipl, “The Evolution of Scientific Method,” The Logic of Science, ed. V.E. Smith—New York: St. John’s Univ. 1964—59-86). There is over and above this a particular logic peculiar to each field of knowledge. That is to say, the proper method of natural philosophy is not at all identical with that of mathematics, metaphysics or moral philosophy. Logic, or general methodology, must be understood before any of the particular sciences are investigated and organized systematically. This, at least, was the common view accepted by all scholastic thinkers, even though this was not the actual procedure followed in medieval universities (see J.A. Weisheipl, “Classification of the Sciences in Medieval Thought,” Mediaeval Studies, 27 (1965)89).

In studying methodology, or the common logic of all the sciences, Aristotle and those after him followed a logical order which considered problems arising from each step of logical thinking. The scholastics thought they had found this order in the various books of logic. Thus according to St. Thomas (Foreword) the predicables (in the Isagogy of Porphyry) and the categories (in the De praedicamentis of Aristotle) deal with universals that are begotten by the first act of the mind. Propositions, or enunciations (in the Peri hermeneias of Aristotle) deal with constructs of various types of judgement in the second act of the mind. These two areas of logical investigation are prior to the analysis of reasoning itself, the third act of the mind. St. Thomas recognized that there are two types of analysis, or resolution, to be considered: the formal structure of reasoning, which Aristotle discusses in the Prior Analytics, and the material structure of the premises, which can be of three kinds, namely necessary and scientific (considered in the Posterior Analytics), probable and dialectical (considered in the Topics), and erroneous and false (considered in the Sophistici Elenchi). Of all these branches of logic, the most important is the Posterior Analytics, the only logical book commented upon in full by St. Thomas Aquinas. St. Albert clearly states that the Posterior Analytics is the apex, the most perfect and only absolutely desirable (simpliciter desiderabile) study among the logical works of Aristotle (Post. Anal., 1, tr. 1, cap. 1, ed. Borgnet II, 2b). And the Leonine editors of the works of St. Thomas state that “the posterior analytics deal with demonstration and thus are the ultimate goal of the whole science of logic” (Praef. ed. Leon., 1, p. 131).

No one has ever doubted that the Posterior Analytics is an extremely difficult work to understand. Even Themistius, paraphrasing the Greek text, found much to complain about (Paraphrasis in lib. Post., praef.). According to John of Salisbury, after the text was translated into Latin, there was scarcely a master willing to expound it because of its extreme subtlety and obscurity; “there are almost as many stumbling blocks as there are chapters” (Metalogicon IV, c. 6, ed. Webb 171). However, John blames most of this on the bungling mistakes of scribes and he proceeds to give the Latin West the first paraphrase of Aristotle’s difficult work. Part of the difficulty seems to be that this is an early work of Aristotle, for the terminology is not yet fixed and especially in the First Book, Aristotle seems to approach the same point from many directions, giving the reader the impression that many different points are being made. The best guides for understanding Aristotle are St. Thomas Aquinas, St. Albert the Great, Averroes and Robert Grosseteste.

When reading St. Thomas’ commentary one must not only read the text of Aristotle first, but one should have a pencil and sufficient paper to outline the text as understood by St. Thomas. Division of the text was always one of the basic tools of the scholastic method. Therefore it is important to keep in mind this outline in order to understand the point about to be made and to appreciate it in the context of the work as a whole. Clearly Aristotle himself wrote according to a systematic order, and it is up to the reader to appreciate this order.

Since St. Thomas did not know Greek, he had to rely on one of the many Latin translations of the Posterior Analytics available to him. At the time St. Thomas wrote his commentary, around 1270, there were four Latin translations from the Greek and two from the Arabic. Even though it seems that Boethius himself translated the work, the Posterior Analytics had to come into the Latin West anew in the 12th century as part of the logica nova. The common text in the Middle Ages was the version made by James of Venice before 1159; it was the “vulgate text” (Arist. Lat., IV.2) in use during the second half of the 12th century and the earlier part of the 13th century. A very influential version from the Arabic was made by the translator, probably Michael Scott, of the works of Averroes together with the Commentator’s views between 1220 and 1240. St. Thomas was undoubtedly familiar with these two translations, but he most likely relied on the revised version made by William of Moerbeke in the second half of the 13th century (Arist. Lat., IV. 4; cf. De Rubeis, Diss. XXIII, c. 1-2, ed. Leon., 1, cclix-cclxii).

The Posterior Analytics of Aristotle possesses a remarkable unity from beginning to end. The first chapter of Book I is a propaedeutic to the entire work; it poses the fundamental problem concerning the possibility of learning, that is, of demonstrative knowledge. Its point of departure is the problem posed by Plato in the Meno (80 D-86) where Socrates attempts to inquire into the nature of virtue, a subject about which he admittedly does not have full knowledge. Meno, intervenes and objects that all inquiry is impossible, for “a man cannot inquire either about that which he knows, or about that which he does not know; for if he knows, he has no need to inquire; and if not, he cannot, for he does not know the very subject about which he is to inquire.” Either we already know what we seek to learn, and this is not learning, or we do not know what we are seeking, and hence cannot know when we have found it. Plato solves this dilemma by his doctrine of remembering ideas already innate in the mind. The Sophists and nominalists, of the Academy took an opposite view and claimed that all learning is simply an aggregation of individual observations. In other words, the Sophists maintained that there can be no demonstrations, but only the acquisition of a totally new fact. Aristotle took a middle course between these two extremes that would have all knowledge in act or no knowledge in act by his ingenious doctrine of potentiality. Instead of saying that all knowledge is actually in the mind or actually not in the mind, Aristotle insists that all knowledge is potentially in the mind and the business of learning is to draw this potentiality into actuality. Basically it is the same solution Aristotle offers in the Physics to explain the possibility of real change.

In the very question posed for inquiry there is already some knowledge in the mind from which inquiry begins. The all important starting point for inquiry is the question or problem posed. Already we have some idea, if only tentative, of the subject of inquiry, and some knowledge of the predicate; otherwise the question would never have arisen. The purpose of inquiry is to find the definitive medium or middle term that will provide an answer to the question raised. This middle term must be one or all of the physical causes in reality; the mind will not rest until it has found a causal reason for the conclusion. However there are many different kinds of scientific questions that can be raised: whether something exists (an sit), what is it (quid sit), does it have this or that property (quia sit), and why is this so (propter quid). For Aristotle and for St. Thomas only a true, objective, invariable cause can produce demonstrations worthy of the name scientific. This cause or middle term cannot be found outside the area in question, for this would give only a probable view. In other words, if a question is raised concerning the physical world, then only an answer found within natural philosophy will do. One cannot, in this case, appeal to harmony, morals or metaphysics for the right answer. The cause must be found within the context of the question. The answer is not found despite the question or problem, but because of it.

Nothing could be farther from the truth than to think of all demonstrative knowledge as “deductive.” This is only rarely the case. Most scientific inquiry requires the reverse process of analysis or breaking down. Once a middle term, a true medium of demonstration, has been found in whole or in part, the result may be expressed in the form of a syllogism that can be tested according to all the rules described in the Prior Analytics and Sophistici Elenchi. Thus the syllogism is not a means of discovery, but rather a means of exposition of the truth acquired by analysis. In fact the syllogism itself can be expressed in a definition that explicitly states the reason. Aristotle calls such a definition a statement “which differs from the syllogism only in position.”

Although the Posterior Analytics is a scientific work that can be studied and understood in its own right, it cannot be fully understood until one can see this kind of process at work in the various Aristotelian sciences. The scholastics themselves did not grasp the significance of this work until they could see it at work in the other writings of Aristotle. The Physics and Ethics of Aristotle in particular helped to instruct the scholastic in its use. Only then could Albert the Great and St. Thomas apply this methodology to such new branches as theology. St. Thomas’ Summa theologiae is the crowning glory of the use that can be made by applying the methodology to a new realm of knowledge. The very first question of the Summa is a masterpiece of Aristotelian methodology.

Although the present English translation of St. Thomas’ commentary may seem to many to be excessively literal, it has the merit of following the procedure of William of Moerbeke, who rendered, apparently at the request of St. Thomas, a literal translation from the Greek lest any nuance be lost. It is hoped that those who are able will also consult the Latin text in difficult passages.

James A. Weisheipl, O.P.
Pontifical Institute of Mediaeval Studies Toronto, Canada


Sicut dicit Aristoteles in principio metaphysicae, hominum genus arte et rationibus vivit: in quo videtur philosophus tangere quoddam hominis proprium quo a caeteris animalibus differt. Alia enim animalia quodam naturali instinctu ad suos actus aguntur; homo autem rationis iudicio in suis actionibus dirigitur. Et inde est quod ad actus humanos faciliter et ordinate perficiendos diversae artes deserviunt. Nihil enim aliud ars esse videtur, quam certa ordinatio rationis quomodo per determinata media ad debitum finem actus humani perveniant. As the Philosopher says in Metaphysics I (980b26), “the human race lives by art and reasonings.” In this statement the Philosopher seems to touch upon that property whereby man differs from the other animals. For the other animals are prompted to their acts by a natural impulse, but man is directed in his actions by a judgment of reason. And this is the reason why there are various arts devoted to the ready and orderly performance of human acts. For an art seems to be nothing more than a definite and fixed procedure established by reason, whereby human acts reach their due end through appropriate means.
Ratio autem non solum dirigere potest inferiorum partium actus, sed etiam actus sui directiva est. Hoc enim est proprium intellectivae partis, ut in seipsam reflectatur: nam intellectus intelligit seipsum et similiter ratio de suo actu ratiocinari potest. Si igitur ex hoc, quod ratio de actu manus ratiocinatur, adinventa est ars aedificatoria vel fabrilis, per quas homo faciliter et ordinate huiusmodi actus exercere potest; eadem ratione ars quaedam necessaria est, quae sit directiva ipsius actus rationis, per quam scilicet homo in ipso actu rationis ordinate, faciliter et sine errore procedat. Now reason is not only able to direct the acts of the lower powers but is also director of its own act: for what is peculiar to the intellective part of man is its ability to reflect upon itself. For the intellect knows itself. In like manner reason is able to reason about its own act. Therefore just as the art of building or carpentering, through which man is enabled to perform manual acts in an easy and orderly manner, arose from the fact that reason reasoned about manual acts, so in like manner an art is needed to direct the act of reasoning, so that by it a man when performing the act of reasoning might proceed in an orderly and easy manner and without error.
Et haec ars est logica, idest rationalis scientia. Quae non solum rationalis est ex hoc, quod est secundum rationem (quod est omnibus artibus commune); sed etiam ex hoc, quod est circa ipsum actum rationis sicut circa propriam materiam. And this art is logic, i.e., the science of reason. And it concerns reason not only because it is according to reason, for that is common to all arts, but also because it is concerned with the very act of reasoning as with its proper matter. Therefore it seems to be the art of the arts, because it directs us in the act of reasoning, from which all arts proceed.
Et ideo videtur esse ars artium, quia in actu rationis nos dirigit, a quo omnes artes procedunt. Oportet igitur logicae partes accipere secundum diversitatem actuum rationis. Consequently one should view the parts of logic according to the diversity among the acts of reason.
Sunt autem rationis tres actus: quorum primi duo sunt rationis, secundum quod est intellectus quidam. Una enim actio intellectus est intelligentia indivisibilium sive incomplexorum, secundum quam concipit quid est res. Et haec operatio a quibusdam dicitur informatio intellectus sive imaginatio per intellectum. Et ad hanc operationem rationis ordinatur doctrina, quam tradit Aristoteles in libro praedicamentorum. Secunda vero operatio intellectus est compositio vel divisio intellectus, in qua est iam verum vel falsum. Et huic rationis actui deservit doctrina, quam tradit Aristoteles in libro perihermeneias. Tertius vero actus rationis est secundum id quod est proprium rationis, scilicet discurrere ab uno in aliud, ut per id quod est notum deveniat in cognitionem ignoti. Et huic actui deserviunt reliqui libri logicae. Now there are three acts of the reason, the first two of which belong to reason regarded as an intellect. One action of the intellect is the understanding of indivisible or uncomplex things, and according to this action it conceives what a thing is. And this operation is called by some the informing of the intellect, or representing by means of the intellect. To this operation of the reason is ordained the doctrine which Aristotle hands down in the book of Predicaments, [i.e., Categories]. The second operation of the intellect is its act of combining or dividing, in which the true or the false are for the first time present. And this act of reason is the subject of the doctrine which Aristotle hands down in the book entitled On Interpretation. But the third act of the reason is concerned with that which is peculiar to reason, namely, to advance from one thing to another in such a way that through that which is known a man comes to a knowledge of the unknown. And this act is considered in the remaining books of logic.
Attendendum est autem quod actus rationis similes sunt, quantum ad aliquid, actibus naturae. Unde et ars imitatur naturam in quantum potest. In actibus autem naturae invenitur triplex diversitas. In quibusdam enim natura ex necessitate agit, ita quod non potest deficere. In quibusdam vero natura ut frequentius operatur, licet quandoque possit deficere a proprio actu. Unde in his necesse est esse duplicem actum; unum, qui sit ut in pluribus, sicut cum ex semine generatur animal perfectum; alium vero quando natura deficit ab eo quod est sibi conveniens, sicut cum ex semine generatur aliquod monstrum propter corruptionem alicuius principii. It should be noted that the acts of reason are in a certain sense not unlike the acts of nature: hence so far as it can, art imitates nature. Now in the acts of nature we observe a threefold diversity. For in some of them nature acts from necessity, i.e., in such a way that it cannot fail; in others, nature acts so as to succeed for the most part, although now and then it fails in its act. Hence in this latter case there must be a twofold act: one which succeeds in the majority of cases, as when from seed is generated a perfect animal; the other when nature fails in regard to what is appropriate to it, as when from seed something monstrous is generated owing to a defect in some principle.
Et haec etiam tria inveniuntur in actibus rationis. Est enim aliquis rationis processus necessitatem inducens, in quo non est possibile esse veritatis defectum; et per huiusmodi rationis processum scientiae certitudo acquiritur. Est autem alius rationis processus, in quo ut in pluribus verum concluditur, non tamen necessitatem habens. Tertius vero rationis processus est, in quo ratio a vero deficit propter alicuius principii defectum; quod in ratiocinando erat observandum. These three are found also in the acts of the reason. For there is one process of reason which induces necessity, where it is not possible to fall short of the truth; and by such a process of reasoning the certainty of science is acquired. Again, there is a process of reason in which something true in most cases is concluded but without producing necessity. But the third process of reason is that in which reason fails to reach a truth because some principle which should have been observed in reasoning was defective.
Pars autem logicae, quae primo deservit processui, pars iudicativa dicitur, eo quod iudicium est cum certitudine scientiae. Et quia iudicium certum de effectibus haberi non potest nisi resolvendo in prima principia, ideo pars haec analytica vocatur, idest resolutoria. Certitudo autem iudicii, quae per resolutionem habetur, est, vel ex ipsa forma syllogismi tantum, et ad hoc ordinatur liber priorum analyticorum, qui est de syllogismo simpliciter; vel etiam cum hoc ex materia, quia sumuntur propositiones per se et necessariae, et ad hoc ordinatur liber posteriorum analyticorum, qui est de syllogismo demonstrativo. Now the part of logic which is devoted to the first process is called the judicative part, because it leads to judgments possessed of the certitude of science. And because a certain and sure judgment touching effects cannot be obtained except by analyzing them into their first principles, this part is called analytical, i.e., resolvent. Furthermore, the certitude obtained by such an analysis of a judgment is derived either from the mere form of the syllogism—and to this is ordained the book of the Prior Analytics which treats of the syllogism as such—or from the matter along with the form, because the propositions employed are per se and necessary [cf. infra, Lectures 10, 13]—and to this is ordained the book of the Posterior Analytics which is concerned with the demonstrative syllogism.
Secundo autem rationis processui deservit alia pars logicae, quae dicitur inventiva. Nam inventio non semper est cum certitudine. Unde de his, quae inventa sunt, iudicium requiritur, ad hoc quod certitudo habeatur. Sicut autem in rebus naturalibus, in his quae ut in pluribus agunt, gradus quidam attenditur (quia quanto virtus naturae est fortior, tanto rarius deficit a suo effectu), ita et in processu rationis, qui non est cum omnimoda certitudine, gradus aliquis invenitur, secundum quod magis et minus ad perfectam certitudinem acceditur. Per huiusmodi enim processum, quandoque quidem, etsi non fiat scientia, fit tamen fides vel opinio propter probabilitatem propositionum, ex quibus proceditur: quia ratio totaliter declinat in unam partem contradictionis, licet cum formidine alterius, et ad hoc ordinatur topica sive dialectica. Nam syllogismus dialecticus ex probabilibus est, de quo agit Aristoteles in libro topicorum. To the second process of reason another part of logic called investigative is devoted. For investigation is not always accompanied by certitude. Hence in order to have certitude a judgment must be formed, bearing on that which has been investigated. But just as in the works of nature which succeed in the majority of cases certain levels are achieved—because the stronger the power of nature the more rarely does it fail to achieve its effect—so too in that process of reason which is not accompanied by complete certitude certain levels are found accordingly as one approaches more or less to complete certitude. For although science is not obtained by this process of reason, nevertheless belief or opinion is sometimes achieved (on account of the provability of the propositions one starts with), because reason leans completely to one side of a contradiction but with fear concerning the other side. The Topics or dialectics is devoted to this. For the dialectical syllogism which Aristotle treats in the book of Topics proceeds from premises which are provable.
Quandoque vero, non fit complete fides vel opinio, sed suspicio quaedam, quia non totaliter declinatur ad unam partem contradictionis, licet magis inclinetur in hanc quam in illam. Et ad hoc ordinatur rhetorica. Quandoque vero sola existimatio declinat in aliquam partem contradictionis propter aliquam repraesentationem, ad modum quo fit homini abominatio alicuius cibi, si repraesentetur ei sub similitudine alicuius abominabilis. Et ad hoc ordinatur poetica; nam poetae est inducere ad aliquod virtuosum per aliquam decentem repraesentationem. Omnia autem haec ad rationalem philosophiam pertinent: inducere enim ex uno in aliud rationis est. At times, however, belief or opinion is not altogether achieved, but suspicion is, because reason does not lean to one side of a contradiction unreservedly, although it is inclined more to one side than to the other. To this the Rhetoric is devoted. At other times a mere fancy inclines one to one side of a contradiction because of some representation, much as a man turns in disgust from certain food if it is described to him in terms of something disgusting. And to this is ordained the Poetics. For the poet’s task is to lead us to something virtuous by some excellent description. And all these pertain to the philosophy of the reason, for it belongs to reason to pass from one thing to another.
Tertio autem processui rationis deservit pars logicae, quae dicitur sophistica, de qua agit Aristoteles in libro elenchorum. The third process of reasoning is served by that part of logic which is called sophistry, which Aristotle treats in the book On Sophistical Refutations.


Lectio 1
Caput 1
Πᾶσα διδασκαλία καὶ πᾶσα μάθησις διανοητικὴ ἐκ προϋπαρχούσης γίνεται γνώσεως. φανερὸν δὲ τοῦτο θεωροῦσιν ἐπὶ πασῶν·

αἵ τε γὰρ μαθηματικαὶ τῶν ἐπιστημῶν διὰ τούτου τοῦ τρόπου παραγίνονται καὶ τῶν ἄλλων ἑκάστη τεχνῶν.

ὁμοίως δὲ καὶ περὶ τοὺς λόγους οἵ τε διὰ συλλογισμῶν καὶ οἱ δι' ἐπαγωγῆς· ἀμφότεροι γὰρ διὰ προγινωσκομένων ποιοῦνται τὴν διδασκαλίαν, οἱ μὲν λαμβάνοντες ὡς παρὰ ξυνιέντων, οἱ δὲ δεικνύντες τὸ καθόλου διὰ τοῦ δῆλον εἶναι τὸ καθ' ἕκαστον.

ὡς δ' αὔτως καὶ οἱ ῥητορικοὶ συμπείθουσιν· ἢ γὰρ διὰ παραδειγμάτων, ὅ ἐστιν ἐπαγωγή, ἢ δι' ἐνθυμημάτων, ὅπερ ἐστὶ συλλογισμός.

a1. All instruction given or received by way of argument proceeds from pre-existent knowledge. This becomes evident upon a survey of all the species of such instruction

a3. The mathematical sciences and all other speculative disciplines are acquired in this way,

a4. and so are the two forms of dialectical reasoning, syllogistic and inductive; for each of these latter make use of old knowledge to impart new, the syllogism assuming an audience that accepts its premisses, induction exhibiting the universal as implicit in the clearly known particular.

a8. Again, the persuasion exerted by rhetorical arguments is in principle the same, since they use either example, a kind of induction, or enthymeme, a form of syllogism.

Aliis igitur partibus logicae praetermissis, ad praesens intendendum est circa partem iudicativam, prout traditur in libro posteriorum analyticorum. Qui dividitur in partes duas: in prima, ostendit necessitatem demonstrativi syllogismi, de quo est iste liber; in secunda, de ipso syllogismo demonstrativo determinat; ibi: scire autem opinamur et cetera. Leaving aside the other parts of logic, we shall fix our attention on the judicative part as it is presented in the book of Posterior Analytics which is divided into two parts. In the first he shows the need for the demonstrative syllogism, with which this book is concerned. In the second part he comes to a decision concerning that syllogism (71b8) [Lect. 4).
Necessitas autem cuiuslibet rei ordinatae ad finem ex suo fine sumitur; finis autem demonstrativi syllogismi est acquisitio scientiae; unde, si scientia acquiri non posset per syllogismum vel argumentum, nulla esset necessitas demonstrativi syllogismi. Posuit autem Plato quod scientia in nobis non causatur ex syllogismo, sed ex impressione formarum idealium in animas nostras, ex quibus etiam effluere dicebat formas naturales in rebus naturalibus, quas ponebat esse participationes quasdam formarum a materia separatarum. Ex quo sequebatur quod agentia naturalia non causabant formas in rebus inferioribus, sed solum materiam praeparabant ad participandum formas separatas. Et similiter ponebat quod per studium et exercitium non causatur in nobis scientia; sed tantum removentur impedimenta, et reducitur homo quasi in memoriam eorum, quae naturaliter scit ex impressione formarum separatarum. Now the need for anything directed to an end is caused by that end. But the end of the demonstrative syllogism is the attainment of science. Hence if science could not be achieved by syllogizing or arguing, there would be no need for the demonstrative syllogism. Plato, as a matter of fact, held that science in us is not the result of a syllogism but of an impression upon our minds of ideal forms from which, he said, are also derived the natural forms in natural things, which he supposed were participations of forms separated from matter. From this it followed that natural agents were not the causes of forms in natural things but merely prepared the matter for participating in the separated forms. In like fashion he postulated that science in us is not caused by study and exercise, but only that obstacles are removed and man is brought to recall things which he naturally understands in virtue of an imprint of separated forms.
Sententia autem Aristotelis est contraria quantum ad utrumque. Ponit enim quod formae naturales reducuntur in actum a formis quae sunt in materia, scilicet a formis naturalium agentium. Et similiter ponit quod scientia fit in nobis actu per aliquam scientiam in nobis praeexistentem. Et hoc est fieri in nobis scientiam per syllogismum aut argumentum quodcumque. Nam ex uno in aliud argumentando procedimus. But Aristotle’s view is opposed to this on two counts. For he maintains that natural forms are made actual by forms present in matter, i.e., by the forms of natural agents. He further maintains that science is made actual in us by other knowledge already existing in us. This means that it is formed in us through a syllogism or some type of argument. For in arguing we proceed from one thing into another.
Ad ostendendum igitur necessitatem demonstrativi syllogismi, praemittit Aristoteles quod cognitio in nobis acquiritur ex aliqua cognitione praeexistenti. Duo igitur facit: primo, ostendit propositum; secundo, docet modum praecognitionis; ibi: dupliciter autem et cetera. Therefore, in order to show the need for demonstrative syllogism Aristotle begins by stating that some of our knowledge is acquired from knowledge already existing. Hence he does two things. First, he states his thesis. Secondly, he explains the character of prior knowledge (71a11) [Lect. 2]. Concerning the first he does two things.
Circa primum duo facit. Primo, inducit universalem propositionem propositum continentem, scilicet quod acceptio cognitionis in nobis fit ex aliqua praeexistenti cognitione. Et ideo dicit: omnis doctrina et omnis disciplina, non autem omnis cognitio, quia non omnis cognitio ex priori cognitione dependet: esset enim in infinitum abire. Omnis autem disciplinae acceptio ex praeexistenti cognitione fit. Nomen autem doctrinae et disciplinae ad cognitionis acquisitionem pertinet. Nam doctrina est actio eius, qui aliquid cognoscere facit; disciplina autem est receptio cognitionis ab alio. Non autem accipitur hic doctrina et disciplina secundum quod se habent ad acquisitionem scientiae tantum, sed ad acquisitionem cognitionis cuiuscumque. Quod patet, quia manifestat hanc propositionem etiam in disputativis et rhetoricis disputationibus, per quas non acquiritur scientia. Propter quod etiam non dicit ex praeexistenti scientia vel intellectu, sed universaliter cognitione. Addit autem intellectiva ad excludendum acceptionem cognitionis sensitivae vel imaginativae. Nam procedere ex uno in aliud rationis est solum. First (71a1), he asserts a universal proposition containing his thesis, namely, that the production of knowledge in us is caused from knowledge already existing; hence he says, “Every doctrine and every discipline...” He does not say, “all knowledge,” because not all knowledge depends on previous knowledge, for that would involve an infinite process: but the acquisition of every discipline comes from knowledge already possessed. For the names “doctrine” and “discipline” pertain to the learning process, doctrine being the action exerted by the one who makes us know, and discipline the reception of knowledge from another. Furthermore, “doctrine” and “discipline” are not taken here as pertaining only to the acquisition of scientific knowledge but to the acquiring of any knowledge. That this is so is evidenced by the fact that he explains the proposition even in regard to dialectical and rhetorical disputations, neither of which engenders science. Hence this is another reason why he did not say, “from pre-existent science or intuition,” but “knowledge” universally. However he does add, “intellectual,” in order to preclude knowledge acquired by sense or imagination. For reason alone proceeds from one thing into another.
Secundo, cum dicit: mathematicae enim etc., manifestat propositionem praemissam per inductionem. Et primo, in demonstrativis in quibus acquiritur scientia. In his autem principaliores sunt mathematicae scientiae, propter certissimum modum demonstrationis. Consequenter autem sunt et omnes aliae artes, quia in omnibus est aliquis modus demonstrationis, alias non essent scientiae. Then (71a3) he employs induction to prove his thesis; and first of all in regard to those demonstrations in which scientific knowledge is acquired. Of these the best are the mathematical sciences because of their most certain manner of demonstrating. After them come the other arts, because some manner of demonstrating is found in all of them; otherwise they would not be sciences.
Secundo, cum dicit: similiter autem etc., manifestat idem in orationibus disputativis sive dialecticis, quae utuntur syllogismo et inductione: in quorum utroque proceditur ex aliquo praecognito. Nam in syllogismo accipitur cognitio alicuius universalis conclusi ab aliis universalibus notis. In inductione autem concluditur universale ex singularibus, quae sunt manifesta quantum ad sensum. Secondly (71a4), he proves the same thing in regard to disputative, i.e., dialectical, arguments, because they employ syllogism and induction, in each of which the process starts from something already known. For in a syllogism the knowledge of some universal conclusion is obtained from other universals already known; in induction, however, a universal is concluded from singulars made known in sense-perception.
Tertio, cum dicit: similiter autem rhetoricae etc., manifestat idem in rhetoricis, in quibus persuasio fit per enthymema aut per exemplum; non autem per syllogismum vel inductionem completam, propter incertitudinem materiae circa quam versatur, scilicet circa actus singulares hominum, in quibus universales propositiones non possunt assumi vere. Et ideo utitur loco syllogismi, in quo necesse est esse aliquam universalem, aliquo enthymemate; et similiter loco inductionis, in qua concluditur universale, aliquo exemplo, in quo proceditur a singulari, non ad universale, sed ad singulare. Unde patet quod, sicut enthymema est quidam syllogismus detruncatus, ita exemplum est quaedam inductio imperfecta. Si ergo in syllogismo et inductione proceditur ex aliquo praecognito, oportet idem intelligi in enthymemate et exemplo. Thirdly (71a8), he manifests the same thing in rhetorical arguments, in which persuasion is produced through an enthymeme or example but not through a syllogism or complete induction because of the uncertainty attending the matters discussed, namely, the individual acts of men in which universal propositions cannot be truthfully assumed. Therefore, in place of a syllogism in which there must be something universal, an enthymeme is employed in which it is not necessary to have something universal. Similarly, in place of induction in which a universal is concluded, an example is employed in which one goes from the singular not to the universal but to the singular. Hence it is clear that just as the enthymeme is an abridged syllogism, so an example is an incomplete induction. Therefore, if in the case of the syllogism and induction one proceeds from knowledge already existing, the same must be granted in the case of the enthymeme and example.

Lectio 2
Caput 1 cont.
διχῶς δ' ἀναγκαῖον προγινώσκειν· τὰ μὲν γάρ, ὅτι ἔστι, προϋπολαμβάνειν ἀναγκαῖον, τὰ δέ, τί τὸ λεγόμενόν ἐστι, ξυνιέναι δεῖ, τὰ δ' ἄμφω, οἷον ὅτι μὲν ἅπαν ἢ φῆσαι ἢ ἀποφῆσαι ἀληθές, ὅτι ἔστι, τὸ δὲ τρίγωνον, ὅτι τοδὶ σημαίνει, τὴν δὲ μονάδα ἄμφω, καὶ τί σημαίνει καὶ ὅτι ἔστιν· οὐ γὰρ ὁμοίως τούτων ἕκαστον δῆλον ἡμῖν. 71a11. The pre-existent knowledge required is of two kinds. In some cases admission of the fact must be assumed, in others comprehension of the meaning of the term used, and sometimes both assumptions are essential. Thus, we assume that every predicate can be either truly affirmed or truly denied of any subject, and that 'triangle' means so and so; as regards 'unit' we have to make the double assumption of the meaning of the word and the existence of the thing. The reason is that these several objects are not equally obvious to us.
Ἔστι δὲ γνωρίζειν τὰ μὲν πρότερον γνωρίσαντα, τῶν δὲ καὶ ἅμα λαμβάνοντα τὴν γνῶσιν, οἷον ὅσα τυγχάνει ὄντα ὑπὸ τὸ καθόλου οὗ ἔχει τὴν γνῶσιν. ὅτι μὲν γὰρ πᾶν τρίγωνον ἔχει δυσὶν ὀρθαῖς ἴσας, προῄδει· ὅτι δὲ τόδε τὸ ἐν τῷ ἡμικυκλίῳ τρίγωνόν ἐστιν, ἅμα ἐπαγόμενος ἐγνώρισεν. (ἐνίων γὰρ τοῦτον τὸν τρόπον ἡ μάθησίς ἐστι, καὶ οὐ διὰ τοῦ μέσου τὸ ἔσχατον γνωρίζεται, ὅσα ἤδη τῶν καθ' ἕκαστα τυγχάνει ὄντα καὶ μὴ καθ' ὑποκειμένου τινός.) a16. Recognition of a truth may in some cases contain as factors both previous knowledge and also knowledge acquired simultaneously with that recognition - knowledge, this latter, of the particulars actually falling under the universal and therein already virtually known. For example, the student knew beforehand that the angles of every triangle are equal to two right angles; but it was only at the actual moment at which he was being led on to recognize this as true in the instance before him that he came to know 'this figure inscribed in the semicircle' to be a triangle. For some things (viz. the singulars finally reached which are not predicable of anything else as subject) are only learnt in this way, i.e. there is here no recognition through a middle of a minor term as subject to a major.
Postquam ostendit philosophus quod omnis disciplina ex praeexistenti fit cognitione nunc ostendit quis sit modus praecognitionis. Et circa hoc duo facit: primo, determinat modum praecognitionis quantum ad illa quae oportet praecognoscere ut habeatur cognitio conclusionis, cuius scientia quaeritur; secundo, determinat modum praecognitionis ipsius conclusionis, cuius scientia per demonstrationem quaeritur; ibi: antequam sit inducere et cetera. In praecognitione autem duo includuntur, scilicet cognitio et cognitionis ordo. Primo ergo, determinat modum praecognitionis quantum ad cognitionem ipsam; secundo, quantum ad cognitionis ordinem; ibi: est autem cognoscere et cetera. After showing that every discipline is developed from knowledge already existing, the Philosopher shows what is the extent of this preexisting knowledge. Concerning this he does two things. First, he determines the extent of pre-existing knowledge in regard to the things that must be known in order to attain knowledge of the conclusion, of which scientific knowledge is sought. Secondly, he determines the extent of pre-existing knowledge of the conclusion, of which scientific knowledge is sought through demonstration (71a24) [L.3]. Now two things are included in pre-existing knowledge, namely, the knowledge and the order of the knowledge. First, therefore, he determines the extent of pre-existing knowledge so far as the knowledge itself is concerned. Secondly, so far as the order of the knowledge is concerned (71a16).
Circa primum sciendum est quod id cuius scientia per demonstrationem quaeritur est conclusio aliqua in qua propria passio de subiecto aliquo praedicatur: quae quidem conclusio ex aliquibus principiis infertur. Et quia cognitio simplicium praecedit cognitionem compositorum, necesse est quod, antequam habeatur cognitio conclusionis, cognoscatur aliquo modo subiectum et passio. Et similiter oportet quod praecognoscatur principium, ex quo conclusio infertur, cum ex cognitione principii conclusio innotescat. In regard to the first it should be noted that the object of which scientific knowledge is sought through demonstration is some conclusion in which a proper attribute is predicated of some subject, which conclusion is inferred from the principles. And because the knowledge of simple things precedes the knowledge of compound things, it is necessary -that the subject and the proper attribute be somehow known before knowledge of the conclusion is obtained. In like manner it is required that the principle be known from which the conclusion is inferred, for the conclusion is made known from a knowledge of the principle.
Horum autem trium, scilicet, principii, subiecti et passionis est duplex modus praecognitionis, scilicet, quia est et quid est. Ostensum est autem in VII metaphysicae quod complexa non definiuntur. Hominis enim albi non est aliqua definitio et multo minus enunciationis alicuius. Unde cum principium sit enunciatio quaedam, non potest de ipso praecognosci quid est, sed solum quia verum est. De passione autem potest quidem sciri quid est, quia, ut in eodem libro ostenditur, accidentia quodammodo definitionem habent. Passionis autem esse et cuiuslibet accidentis est inesse subiecto: quod quidem demonstratione concluditur. Non ergo de passione praecognoscitur quia est, sed quid est solum. Subiectum autem et definitionem habet et eius esse a passione non dependet; sed suum esse proprium praeintelligitur ipsi esse passionis in eo. Et ideo de subiecto oportet praecognoscere et quid est et quia est: praesertim cum ex definitione subiecti et passionis sumatur medium demonstrationis. Now the extent of pre-existent knowledge of these three items, i.e., of the principle, of the subject, and of the proper attribute, is limited to knowing two things about them, namely, that each is and what each is. But, as stated in Metaphysics VII, complex things are not defined. For there is no definition of “white man,” much less of an enunciation [proposition]. Hence since a principle is an enunciation, there cannot be preexisting knowledge of what it is but only of the fact that it is true. But in regard to the proper attribute, it is possible to know what it is, because, as is pointed out in the same book, accidents do have some sort of definition. Now the being of a proper attribute and of any accident is being in a subject; and this fact is concluded by the demonstration. Consequently, it is not known beforehand that the proper attribute exists, but only what it is. The subject, too, has a definition; moreover, its being does not depend on the proper attribute—rather its own being is known before one knows the proper attribute to be in it. Consequently, it is necessary to know both what the subject is and that it is, especially since the medium of demonstration is taken from the definition of the subject of the proper attribute.
Propter hoc ergo dicit philosophus quod dupliciter necessarium est praecognoscere: quia duo sunt quae praecognoscuntur de his, quorum praecognitionem habemus, scilicet quia est et quid est. Et quod alia sunt de quibus necesse est primo cognoscere quia sunt, sicut principia de quibus postea exemplificat, ponens in exemplo primum omnium principiorum, scilicet quod de unoquoque est affirmatio vel negatio vera. This, therefore, is why the Philosopher says (71a11) that it is necessary to know beforehand in two ways; because two items are known beforehand concerning things of which we have pre-existing knowledge, namely, that it is and what it is. [Then he goes on to say] that there are some things concerning which it is necessary first to know that they are, such as principles, concerning which he then gives examples, citing as one example the first of all principles, namely, “There is true affirmation or negation about everything.”
Alia vero sunt, de quibus oportet praeintelligere quid est quod dicitur, idest quid significatur per nomen, scilicet de passionibus. Et non dicit quid est simpliciter, sed quid est quod dicitur, quia antequam sciatur de aliquo an sit, non potest sciri proprie de eo quid est: non entium enim non sunt definitiones. Unde quaestio, an est, praecedit quaestionem, quid est. Sed non potest ostendi de aliquo an sit, nisi prius intelligatur quid significatur per nomen. Propter quod etiam philosophus in IV metaphysicae, in disputatione contra negantes principia docet incipere a significatione nominum. Exemplificat autem de triangulo, de quo oportet praescire quoniam nomen eius hoc significat, quod scilicet in sua definitione continetur. Again, there are other things, namely, proper attributes, concerning which it is necessary to know what is said to be predicated, i.e., what is signified by their name. And he does not say unqualifiedly, “what it is,” but “what is said to be predicated,” because one cannot properly know of something what it is before it is known that it is. For there are no definitions of non-beings. Hence the question, whether it is, precedes the question, what it is. But “whether a thing is” cannot be shown unless it is known beforehand what is signified by its name. On this account the Philosopher teaches in Metaphysics IV that in disputing against those who deny principles one must begin with the meanings of names. An example of this is “triangle,” concerning which one must know beforehand that its name signifies such and such, namely what is contained in its definition.
Cum enim accidentia quodam ordine ad substantiam referantur, non est inconveniens id quod est accidens in respectu ad aliquid, esse etiam subiectum in respectu alterius. Sicut superficies est accidens substantiae corporalis: quae tamen superficies est primum subiectum coloris. Id autem quod est ita subiectum, quod nullius est accidens, substantia est. Unde in illis scientiis, quarum subiectum est aliqua substantia, id quod est subiectum nullo modo potest esse passio, sicut est in philosophia prima, et in scientia naturali, quae est de subiecto mobili. But since accidents are referred to their subjects in a definite order, it is not impossible for something which is an accident in relation to one thing to be a subject in relation to something else: for example, a surface is an accident in relation to a bodily substance, but in relation to color it is the first subject. However, that which is a subject in such a way as never to be an accident of anything else is a substance. Hence in those sciences whose subject is a substance, that which is the subject can never be a proper attribute, as in first philosophy and in natural science, which treats of mobile being.
In illis autem scientiis, quae sunt de aliquibus accidentibus, nihil prohibet id, quod accipitur ut subiectum respectu alicuius passionis, accipi etiam ut passionem respectu anterioris subiecti. Hoc tamen non in infinitum procedit. Est enim devenire ad aliquod primum in scientia illa, quod ita accipitur ut subiectum, quod nullo modo ut passio; sicut patet in mathematicis scientiis, quae sunt de quantitate continua vel discreta. Supponuntur enim in his scientiis ea quae sunt prima in genere quantitatis; sicut unitas, et linea, et superficies et alia huiusmodi. Quibus suppositis, per demonstrationem quaeruntur quaedam alia, sicut triangulus aequilaterus, quadratum in geometricis et alia huiusmodi. Quae quidem demonstrationes quasi operativae dicuntur, ut est illud, super rectam lineam datam triangulum aequilaterum constituere. Quo adinvento, rursus de eo aliquae passiones probantur, sicut quod eius anguli sunt aequales aut aliquid huiusmodi. Patet igitur quod triangulus in primo modo demonstrationis se habet ut passio, in secundo se habet ut subiectum. Unde philosophus hic exemplificat de triangulo ut est passio, non ut est subiectum, cum dicit quod de triangulo oportet praescire quoniam hoc significat. But in those sciences which bear upon accidents, nothing prevents a same thing from being taken as a subject in reference to one proper attribute, and as an attribute in reference to a more basic subject. Nevertheless, this must not develop into an infinite process, for one must arrive at something which is first in that science and which is taken as a subject in such a way that it is never taken as a proper attribute, as is clear in the mathematical sciences, which treat of continuous or discrete quantity. For in these sciences those things are postulated which are first in the genus of quantity; for example, unit and line and surface and the like. Once these are postulated, certain other things are sought through demonstration, such as the equilateral triangle and the square and so on in geometry. In these cases the demonstrations are said to be, as it were, operational, as when it is required to construct an equilateral triangle on a given straight line. But once it has been constructed, certain proper attributes are proved about it; for example, that its angles are equal, or something of that sort. It is clear, therefore, that in the first type of demonstration “triangle” behaves as a proper attribute, and in the second type as a subject. Hence the Philosopher is using “triangle” as a proper attribute and not as a subject when he says by way of example, “We must assume that triangle means so and so” (71a14).
Dicit etiam quod quaedam sunt de quibus oportet praescire utrunque, scilicet quid est et quia est. Et exemplificat de unitate quae est principium in omni genere quantitatis. Etsi enim aliquo modo sit accidens respectu substantiae, tamen in scientiis mathematicis, quae sunt de quantitate, non potest accipi ut passio, sed ut subiectum tantum, cum in hoc genere nihil habeat prius. He says, ‘furthermore, that there are certain things about which we must know beforehand both what each is and whether it is. And he uses the example of “one,” which is the principle in every genus of quantity. For although it is somehow an accident in reference to substance, yet in the mathematical sciences, which treat of quantity, it cannot be taken as a proper attribute but only as a subject, since in this genus [quantity] it has nothing prior to it.
Rationem autem huiusmodi diversitatis ostendit, quia non est similis modus manifestationis praedictorum, scilicet principii, passionis et subiecti. Non enim est eadem ratio cognitionis in ipsis: nam principia cognoscuntur per actum componentis et dividentis; subiectum autem et passio per actum apprehendentis quod quid est. Quod quidem non similiter competit subiecto et passioni: cum subiectum definiatur absolute, quia in definitione eius non ponitur aliquid, quod sit extra essentiam ipsius; passio autem definitur cum dependentia ad subiectum, quod in eius definitione ponitur. Unde, ex quo non eodem modo cognoscuntur, non est mirum si eorum diversa praecognitio sit. The reason for this difference is shown by the fact that the manner in which the aforesaid, namely, principle, proper attribute and subject, are manifested is not the same. For the way in which they are known is not the same: for principles are known through the act of composing and dividing, but subject and proper attribute by the act of apprehending the essence. And this, too, does not belong in similar fashion to a subject and to a proper attribute, since a subject is defined absolutely, for nothing outside its essence is mentioned in its definition; but a proper attribute is defined with dependence on the subject which is mentioned in its definition. Therefore, since they are not known in the same way, it is not surprising if they are not foreknown in the same way.
Deinde cum dicit: est autem cognoscere etc., determinat modum praecognitionis ex parte ipsius ordinis, quem praecognitio importat. Est enim aliquid prius altero et secundum tempus et secundum naturam. Et hic duplex ordo in praecognitione considerandus est. Aliquid enim praecognoscitur sicut prius notum tempore. Et de his dicit quod alia contingit cognoscere aliquem cognoscentem ea prius tempore, quam illa quibus praecognosci dicuntur. Quaedam vero cognoscuntur simul tempore, sed prius natura. Et de his dicit quod quorundam praecognitorum simul tempore est accipere notitiam, et illorum quibus praecognoscuntur. Then (7a16) he determines the extent of foreknowledge on the part of the order which foreknowing implies. For something is prior to another both in the order of time and in the order of nature. And this twofold order must be considered in regard to pre-existent knowing. For something is known before something else in the sense of being known prior in time. Concerning such things he says that someone could know certain things by knowing them prior to the time when he knows the things to which they are said to be foreknown. But certain others are known at one and the same time, although one is prior by nature to the other. Concerning these he says that one acquires a knowledge of some of these foreknown things at the same time that knowledge of the things to which they are foreknown is acquired.
Quae autem sint ista manifestat subdens quod huiusmodi sunt quaecumque continentur sub aliquibus universalibus, quorum habent cognitionem, idest de quibus notum est ea sub talibus universalibus contineri. Et hoc ulterius manifestat per exemplum. Cum enim ad conclusionem inferendam duae propositiones requirantur, scilicet maior et minor, scita propositione maiori, nondum habetur conclusionis cognitio. Maior ergo propositio praecognoscitur conclusioni non solum natura, sed tempore. Rursus autem si in minori propositione inducatur sive assumatur aliquid contentum sub universali propositione, quae est maior, de quo manifestum non sit quod sub hoc universali contineatur, nondum habetur conclusionis cognitio, quia nondum erit certa veritas minoris propositionis. Si autem in minori propositione assumatur terminus, de quo manifestum sit quod continetur sub universali in maiori propositione, patet veritas minoris propositionis: quia id quod accipitur sub universali habet eius cognitionem, et sic statim habetur conclusionis cognitio. Ut si sic demonstraret aliquis, omnis triangulus habet tres angulos aequales duobus rectis, ista cognita, nondum habetur conclusionis cognitio: sed cum postea assumitur, haec figura descripta in semicirculo, est triangulus, statim scitur quod habet tres angulos aequales duobus rectis. Si autem non esset manifestum quod haec figura in semicirculo descripta est triangulus, nondum statim inducta assumptione sciretur conclusio; sed oporteret ulterius aliud medium quaerere, per quod demonstraretur hanc figuram esse triangulum. He indicates what these are when he adds that they are the things contained under certain universals of which we have knowledge, i.e., of which it is known that they are contained under such universals. Then he clarifies this with an example. For since two propositions are needed for inferring a conclusion, namely, a major and a minor; when the major proposition is known, the conclusion is not yet known. Therefore, the major proposition is known before the conclusion not only in nature but in time. Further, if in the minor proposition something is introduced or employed which is contained under the universal proposition which is the major, but it is not evident that it is contained under this universal, then a knowledge of the conclusion is not yet possessed, because the truth of the minor proposition will not yet be certain. But if in the minor proposition a term is taken about which it is clear that it is contained under the universal in the major proposition, the truth of the minor proposition is clear, because that which is taken under the universal shares in the same knowledge, and so the knowledge of the conclusion is had at once. Thus, suppose that someone should begin to demonstrate by stating that every triangle has three angles equal to two right angles. When this is known, the knowledge of the conclusion is not yet known. But when it is later assumed that this figure inscribed in a semicircle is a triangle, he knows at once that it has three angles equal to two right angles. However, if it were not clear that this figure inscribed in the semicircle is a triangle, the conclusion would not be known as soon as the minor was stated; rather, it would be necessary to search for a middle through which to demonstrate that this figure is a triangle.
Exemplificans ergo philosophus de his quae cognoscuntur ante conclusionem prius tempore, dicit quod aliquis per demonstrationem conclusionis cognitionem accipiens, hanc propositionem praescivit etiam secundum tempus, scilicet, quod omnis triangulus habet tres angulos duobus rectis aequales. Sed inducens hanc assumptionem, scilicet, quod hoc quod est in semicirculo sit triangulus, simul, scilicet tempore, cognovit conclusionem, quia hoc inductum habet notitiam universalis, sub quo continetur, ut non oporteat ulterius medium quaerere. Et ideo subdit quod, quorundam est hoc modo disciplina, idest eorum accipitur cognitio per se, et non oportet ea cognoscere per aliquod aliud medium, quod sit ultimum in resolutione, qua mediata ad immediata reducuntur. Vel potest legi sic: quod ultimum, idest extremum, quod accipitur sub universali medio, non oportet ut cognoscatur esse sub illo universali per aliquod aliud medium. Et quae sint ista, quae semper habent cognitionem sui universalis, manifestat subdens quod huiusmodi sunt singularia, quae non dicuntur de aliquo subiecto: cum inter singularia et speciem nullum medium possit inveniri. In giving this example of things which are known at a time prior to the conclusion the Philosopher says that a person obtaining a knowledge of the conclusion through demonstration foreknew this proposition even according to time, namely, that every triangle has three angles equal to two right angles. But inducing this assumption, namely, that this figure in the semicircle is a triangle, he knew the conclusion at the same time, because this induction shares in the evidence of the universal under which it is contained, so that there is no need to search for another middle. He adds, therefore, that “some things are only learnt in this way” (71a23), i.e., learnt in virtue of themselves, so that it is not necessary to learn them through some other middle which is the ultimate reached by analysis in which the mediate is reduced to the immediate. Or it can be read in such a way that the “ultimate,” i.e., the extreme, which is subsumed under the universal middle does not need a further middle to show that it is contained under that universal. And he manifests what those things are which always share the knowledge of their universal, saying that they are the singulars, which are not predicated of any subject, since no middle can be found between singulars and their species.

Lectio 3
Caput 1 cont.
πρὶν δ' ἐπαχθῆναι ἢ λαβεῖν συλλογισμὸν τρόπον μέν τινα ἴσως φατέον ἐπίστασθαι, τρόπον δ' ἄλλον οὔ. ὃ γὰρ μὴ ᾔδει εἰ ἔστιν ἁπλῶς, τοῦτο πῶς ᾔδει ὅτι δύο ὀρθὰς ἔχει ἁπλῶς; ἀλλὰ δῆλον ὡς ὡδὶ μὲν ἐπίσταται, ὅτι καθόλου ἐπίσταται, ἁπλῶς δ' οὐκ ἐπίσταται. a24. Before he was led on to recognition or before he actually drew a conclusion, we should perhaps say that in a manner he knew, in a manner not. If he did not in an unqualified sense of the term know the existence of this triangle, how could he know without qualification that its angles were equal to two right angles? No: clearly he knows not without qualification but only in the sense that he knows universally.
εἰ δὲ μή, τὸ ἐν τῷ Μένωνι ἀπόρημα συμβήσεται· ἢ γὰρ οὐδὲν μαθήσεται ἢ ἃ οἶδεν. a28. If this distinction is not drawn, we are faced with the dilemma in the Meno: either a man will learn nothing or what he already knows;
οὐ γὰρ δή, ὥς γέ τινες ἐγχειροῦσι λύειν, λεκτέον. ἆρ' οἶδας ἅπασαν δυάδα ὅτι ἀρτία ἢ οὔ; φήσαντος δὲ προήνεγκάν τινα δυάδα ἣν οὐκ ᾤετ' εἶναι, ὥστ' οὐδ' ἀρτίαν. λύουσι γὰρ οὐ φάσκοντες εἰδέναι πᾶσαν δυάδα ἀρτίαν οὖσαν, ἀλλ' ἣν ἴσασιν ὅτι δυάς. a30. for we cannot accept the solution which some people offer. A man is asked, 'Do you, or do you not, know that every pair is even?' He says he does know it. The questioner then produces a particular pair, of the existence, and so a fortiori of the evenness, of which he was unaware. The solution which some people offer is to assert that they do not know that every pair is even, but only that everything which they know to be a pair is even:
καίτοι (71b.) ἴσασι μὲν οὗπερ τὴν ἀπόδειξιν ἔχουσι καὶ οὗ ἔλαβον, ἔλαβον δ' οὐχὶ παντὸς οὗ ἂν εἰδῶσιν ὅτι τρίγωνον ἢ ὅτι ἀριθμός, ἀλλ' ἁπλῶς κατὰ παντὸς ἀριθμοῦ καὶ τριγώνου· οὐδεμία γὰρ πρότασις λαμβάνεται τοιαύτη, ὅτι ὃν σὺ οἶδας ἀριθμὸν ἢ ὃ σὺ οἶδας εὐθύγραμμον, ἀλλὰ κατὰ παντός. b1. yet what they know to be even is that of which they have demonstrated evenness, i.e. what they made the subject of their premiss, viz. not merely every triangle or number which they know to be such, but any and every number or triangle without reservation. For no premiss is ever couched in the form 'every number which you know to be such', or 'every rectilinear figure which you know to be such': the predicate is always construed as applicable to any and every instance of the thing.
ἀλλ' οὐδέν (οἶμαι) κωλύει, ὃ μανθάνει, ἔστιν ὡς ἐπίστασθαι, ἔστι δ' ὡς ἀγνοεῖν· ἄτοπον γὰρ οὐκ εἰ οἶδέ πως ὃ μανθάνει, ἀλλ' εἰ ὡδί, οἷον ᾗ μανθάνει καὶ ὥς. b5. On the other hand, I imagine there is nothing to prevent a man in one sense knowing what he is learning, in another not knowing it. The strange thing would be, not if in some sense he knew what he was learning, but if he were to know it in that precise sense and manner in which he was learning it.
Postquam ostendit philosophus quomodo oportet praecognoscere quaedam alia, antequam de conclusione cognitio sumatur, nunc vult ostendere quomodo ipsam conclusionem contingit praecognoscere, antequam cognitio sumatur de ea per syllogismum vel inductionem. Et circa hoc duo facit. Having shown the manner in which certain other things must be known before knowledge of the conclusion is obtained, the Philosopher now wishes to show how we know even the conclusion beforehand, i.e., before knowledge of it is obtained through a syllogism or induction. Concerning this he does two things:
Primo namque determinat veritatem, dicens quod, antequam inducatur inductio vel syllogismus ad faciendam cognitionem de aliqua conclusione, illa conclusio quodammodo scitur, et quodammodo non: simpliciter enim nescitur, sed scitur solum secundum quid. Sicut si debeat probari ista conclusio, triangulus habet tres angulos aequales duobus rectis; antequam demonstraretur, ille qui per demonstrationem accipit scientiam eius, nescivit simpliciter, sed scivit secundum quid. Unde quodammodo praescivit, simpliciter autem non. Cuius quidem ratio est, quia, sicut iam ostensum est, oportet principia conclusioni praecognoscere. Principia autem se habent ad conclusiones in demonstrativis, sicut causae activae in naturalibus ad suos effectus (unde in II physicorum propositiones syllogismi ponuntur in genere causae efficientis). Effectus autem, antequam producatur in actu, praeexistit quidem in causis activis virtute, non autem actu, quod est simpliciter esse. Et similiter antequam ex principiis demonstrativis deducatur conclusio, in ipsis quidem principiis praecognitis praecognoscitur conclusio virtute, non autem actu: sic enim in eis praeexistit. Et sic patet quod non praecognoscitur simpliciter, sed secundum quid. First (71a24), he establishes the truth of the fact, saying that before an induction or syllogism is formed to beget knowledge of a conclusion, that conclusion is somehow known and somehow not known: for, absolutely speaking, it is not known; but in a qualified sense, it is known. Thus, if the conclusion that a triangle has three angles equal to two right angles has to be proved, the one who obtains science of this fact through demonstration already knew it in some way before it was demonstrated; although absolutely speaking, he did not know it. Hence in one sense he already knew it, but in the full sense he did not. And the reason is that, as has been pointed out, the principles of the conclusion must be known beforehand. Now the principles in demonstrative matters are to the conclusion as efficient causes in natural things are to their effects; hence in Physics II the propositions of a syllogism are set in the genus of efficient cause. But an effect, before it is actually produced, pre-exists virtually in its efficient causes but not actually, which is to exist absolutely. In like manner, before it is drawn out of its demonstrative principles, the conclusion is pre-known virtually, although not actually, in its self-evident principles. For that is the way it pre-exists in them. And so it is clear that it is not pre-known in the full sense, but in some sense.
Secundo; ibi: si vero non etc., excludit ex veritate determinata quandam dubitationem, quam Plato ponit in libro Menonis, sic intitulato ex nomine sui discipuli. Est autem dubitatio talis. Inducit enim quendam, omnino imperitum artis geometricae, interrogatum ordinate de principiis, ex quibus quaedam geometrica conclusio concluditur, incipiendo ex principiis per se notis: ad quae omnia ille, ignarus geometriae, id quod verum est respondit: et sic deducendo quaestiones usque ad conclusiones per singula verum respondit. Ex hoc igitur vult habere, quod etiam illi, qui videntur imperiti aliquarum artium, antequam de eis instruantur, earum notitiam habent. Et sic sequitur quod vel homo nihil addiscat, vel addiscat ea quae prius novit. Secondly (71a28), in virtue of this established fact, he settles a doubt which Plato maintained in the book, Meno, which gets its title from the name of his disciple. The doubt is presented in the following manner: A person utterly ignorant of the art of geometry is questioned in an orderly Way concerning the per se known principles from which a geometric conclusion is concluded. By starting with principles that are per se known, to each of which this person ignorant of geometry gives a true answer, Aid leading him thus by questions to the conclusion, he gives the true Answer step by step. From this, therefore, he would have it that even those who seem to be ignorant of certain arts really have a knowledge of them before being instructed in them. And so it follows that either a man learns nothing or he learns what he already knew.
Circa hoc ergo quatuor facit. Primo enim, proponit quod praedicta dubitatio vitari non potest, nisi supposita praedeterminata veritate, scilicet quod conclusio, quam quis addiscit per demonstrationem vel inductionem, erat nota non simpliciter, sed secundum quod est virtute in suis principiis: de quibus aliquis, ignarus scientiae, interrogatus, veritatem respondere potest. Secundum vero Platonis sententiam, conclusio erat praecognita simpliciter; unde non addiscebatur de novo, sed potius per deductionem aliquam rationis in memoriam reducebatur. Sicut etiam de formis naturalibus Anaxagoras ponit, quod ante generationem praeexistebant in materia simpliciter. Aristoteles vero ponit quod praeexistunt in potentia et non simpliciter. In dealing with this problem he [Aristotle] does four things. First, he suggests that it cannot be settled unless we grant the truth established above, namely, that the conclusion which a person learns through demonstration or induction was already known, not absolutely, but as it was virtually known in its principles concerning which a person ignorant of a science can give true answers. However, according to Plato’s theory the conclusion was pre-known absolutely, so that no one learns afresh but is led to recall by some rational process of deduction. This is similar to Anaxagoras’ position on natural forms, namely, that before they are generated, they already pre-existed in the matter absolutely, whereas Aristotle says that they pre-exist in potency and not absolutely.
Secundo, cum dicit: non enim sicut etc., ponit falsam quorundam obviationem ad dubitationem Platonis: qui scilicet dicebant quod conclusio antequam demonstraretur vel quocunque modo addisceretur, nullo modo erat cognita. Poterat enim eis obiici secundum dubitationem Platonis hoc modo: si quis interrogaret ab aliquo imperito: nunquid tu scis quod omnis dualitas par est? Et dicente eo, idest concedente se scire, afferret quandam dualitatem, quam ille interrogatus non opinaretur esse, puta illam dualitatem quae est tertia pars senarii; concluderetur quod sciret tertiam partem senarii esse numerum parem: quod erat ei incognitum, sed per demonstrationem inductam addiscit. Et sic videtur sequi quod vel non hoc addisceret vel addisceret quod prius scivit. Ut igitur hanc dubitationem evitarent, solvebant dicentes, quod ille qui interrogatus respondit se scire quod omnis dualitas sit numerus par, non dixit se cognoscere omnem dualitatem simpliciter, sed illam, quam scivit esse dualitatem. Unde cum ista dualitas, quae est allata, fuerit ab eo penitus ignorata, nullo modo scivit quod haec dualitas esset numerus par. Et sic sequitur quod apud cognoscentem principia nullo modo conclusio sit praecognita, nec simpliciter, nec secundum quid. Secondly (7100), he shows that the way some have answered Plato’s problem is false, namely, by saying that a conclusion is not in any sense known before it is demonstrated or learned by some method or other. For they might face the following objection based on Plato’s problem: If an unlearned person were asked by someone, “Do you know that every duo (pair) is an even number?” and, if upon answering that he does know this, he were presented with a duo which the person interrogated did not know existed, for example, the duo which is one third of six, the conclusion would be that he knew one third of six to be an even number, a fact which had not been known by him but which he learned through the demonstration proposed to him. And so it seems to follow that he either did not freshly learn this or that he learned what he already knew. To avoid this dilemma, they would answer that the person who was questioned and who answered that he knew every duo to be an equal number did not say that he knew every duo absolutely, but those he knew to be duo’s. Hence, since that duo which was proposed was utterly unknown to him, he did not in any sense know that this duo was an even number. And so it follows that when one knows the principles, the conclusion is not in any sense pre-known, either absolutely or in a qualified sense.
Tertio, ibi: et etiam sciunt etc., improbat hanc solutionem hoc modo. Illud scitur de quo demonstratio habetur, vel de quo de novo accipitur demonstratio. Et hoc dicitur propter addiscentem, qui incipit scire. Addiscentes autem non accipiunt demonstrationem de omni dualitate de qua sciunt, sed de omni simpliciter, et similiter de omni numero aut de omni triangulo. Non ergo verum est quod sciat de omni numero, quem scit esse numerum, aut de omni dualitate, quam scit esse dualitatem, sed de omni simpliciter. Quod autem non sciat de omni numero, quem scit esse numerum, sed de omni simpliciter, probat; ibi: neque enim una propositio etc.: conclusio cum praemissis convenit in terminis: nam subiectum et praedicatum conclusionis sunt maior et minor extremitas in praemissis; sed in praemissis non accipitur aliqua propositio de numero aut de recta linea, cum hac additione, quam tu nosti, sed simpliciter de omni; neque ergo conclusio demonstrationis est cum additione praedicta, sed simpliciter de omni. Thirdly (71bl), he refutes this solution in the following way: That is known, concerning which a demonstration is had, or concerning which a demonstration is for the first time received. And this is said on account of those learners who begin to know scientifically. But learners do not obtain a demonstration touching every duo they happen to know but every duo absolutely; and the same applies to every number or every triangle. Therefore, it is not true that he knows something about every number which he knows to be a number, or of every duo which he knows to be a duo, but he knows it about every one absolutely. And that he knows it not only of every number he happens to know is a number, but of every number absolutely, is proved at (71b5) on the ground that the conclusion agrees with the premises in its terms. For the subject and predicate of the conclusion are the major and minor extremes in the premises. But in the premises no proposition concerning number or straight line is stated with the addition, “which you know,” but it is stated of all without qualification. Neither, therefore, is the conclusion of the demonstration asserted with the aforesaid qualification, but it is asserted of all without reservation.
Quarto, ibi: sed nihil etc., ponit veram solutionem dubitationis praedictae secundum praedeterminatam veritatem, dicens quod illud quod quis addiscit, nihil prohibet primo quodammodo scire et quodammodo ignorare. Non enim est inconveniens si aliquis quodammodo praesciat id quod addiscit; sed esset inconveniens si hoc modo praecognosceret, secundum quod addiscit. Addiscere enim proprie est scientiam in aliquo generari. Quod autem generatur, ante generationem neque fuit omnino non ens neque omnino ens, sed quodammodo ens et quodammodo non ens: ens quidem in potentia, non ens vero actu: et hoc est generari, reduci de potentia in actum. Unde nec id quod quis addiscit erat omnino prius notum, ut Plato posuit, nec omnino ignotum, ut secundum solutionem supra improbatam ponebatur; sed erat notum potentia sive virtute in principiis praecognitis universalibus, ignotum autem actu, secundum propriam cognitionem. Et hoc est addiscere, reduci de cognitione potentiali, seu virtuali, aut universali, in cognitionem propriam et actualem. Fourthly (71b5), he presents the true solution of the problem under discussion in terms of the truth already established, saying that there is nothing to prevent a person from somehow knowing and somehow not knowing a fact before he learns it. For it is not a paradox if one somehow already knows what he learns, but it would be, if he already knew it in the same way that he knows it when he has learned it. For learning is, properly speaking, the generation of science in someone. But that which is generated was not, prior to its generation, a being absolutely, but somehow a being and somehow non-being: for it was a being in potency, although actually non-being. And this is what being generated consists in, namely, in being converted from potency to act. In like fashion, that which a person learns was not previously known absolutely, as Plato preferred; but neither was it absolutely unknown, as they maintained whose answer was refuted above. Rather it was known in potency, i.e., virtually, in the pre-known universal principles; however, it was not actually known in the sense of specific knowledge. And this is what learning consists in, namely, in being brought from potential or virtual or universal knowledge to specific and actual knowledge.

Lectio 4
Caput 2
Ἐπίστασθαι δὲ οἰόμεθ' ἕκαστον ἁπλῶς, ἀλλὰ μὴ τὸν σοφιστικὸν τρόπον τὸν κατὰ συμβεβηκός, b8. We suppose ourselves to possess unqualified scientific knowledge of a thing, as opposed to knowing it in the accidental way in which the sophist knows,
ὅταν τήν τ' αἰτίαν οἰώμεθα γινώσκειν δι' ἣν τὸ πρᾶγμά ἐστιν, ὅτι ἐκείνου αἰτία ἐστί, καὶ μὴ ἐνδέχεσθαι τοῦτ' ἄλλως ἔχειν. b10. when we think that we know the cause on which the fact depends, as the cause of that fact and of no other, and, further, that the fact could not be other than it is.
δῆλον τοίνυν ὅτι τοιοῦτόν τι τὸ ἐπίστασθαί ἐστι· καὶ γὰρ οἱ μὴ ἐπιστάμενοι καὶ οἱ ἐπιστάμενοι οἱ μὲν οἴονται αὐτοὶ οὕτως ἔχειν, οἱ δ' ἐπιστάμενοι καὶ ἔχουσιν, b12. Now that scientific knowing is something of this sort is evident - witness both those who falsely claim it and those who actually possess it, since the former merely imagine themselves to be, while the latter are also actually, in the condition described.
ὥστε οὗ ἁπλῶς ἔστιν ἐπιστήμη, τοῦτ' ἀδύνατον ἄλλως ἔχειν. b14. Consequently the proper object of unqualified scientific knowledge is something which cannot be other than it is.
Εἰ μὲν οὖν καὶ ἕτερος ἔστι τοῦ ἐπίστασθαι τρόπος, ὕστερον ἐροῦμεν, b16. There may be another manner of knowing as well - that will be discussed later.
φαμὲν δὲ καὶ δι' ἀποδείξεως εἰδέναι. b17. What I now assert is that at all events we do know by demonstration.
ἀπόδειξιν δὲ λέγω συλλογισμὸν ἐπιστημονικόν· b18. By demonstration I mean a syllogism productive of scientific knowledge,
ἐπιστημονικὸν δὲ λέγω καθ' ὃν τῷ ἔχειν αὐτὸν ἐπιστάμεθα. b19. a syllogism, that is, the grasp of which is eo ipso such knowledge.
εἰ τοίνυν ἐστὶ τὸ ἐπίστασθαι οἷον ἔθεμεν, ἀνάγκη καὶ τὴν ἀποδεικτικὴν ἐπιστήμην ἐξ ἀληθῶν τ' εἶναι καὶ πρώτων καὶ ἀμέσων καὶ γνωριμωτέρων καὶ προτέρων καὶ αἰτίων τοῦ συμπεράσματος· b20. Assuming then that my thesis as to the nature of scientific knowing is correct, the premisses of demonstrated knowledge must be true, primary, immediate, better known than and prior to the conclusion, which is further related to them as effect to cause.
οὕτω γὰρ ἔσονται καὶ αἱ ἀρχαὶ οἰκεῖαι τοῦ δεικνυμένου. b22. Unless these conditions are satisfied, the basic truths will not be 'appropriate' to the conclusion.
συλλογισμὸς μὲν γὰρ ἔσται καὶ ἄνευ τούτων, ἀπόδειξις δ' οὐκ ἔσται· οὐ γὰρ ποιήσει ἐπιστήμην. b23. Syllogism there may indeed be without these conditions, but such syllogism, not being productive of scientific knowledge, will not be demonstration.
ἀληθῆ μὲν οὖν δεῖ εἶναι, ὅτι οὐκ ἔστι τὸ μὴ ὂν ἐπίστασθαι, οἷον ὅτι ἡ διάμετρος σύμμετρος. b24. The premisses must be true: for that which is non-existent cannot be known - we cannot know, e.g. that the diagonal of a square is commensurate with its side.
ἐκ πρώτων δ' ἀναποδείκτων, ὅτι οὐκ ἐπιστήσεται μὴ ἔχων ἀπόδειξιν αὐτῶν· τὸ γὰρ ἐπίστασθαι ὧν ἀπόδειξις ἔστι μὴ κατὰ συμβεβηκός, τὸ ἔχειν ἀπόδειξίν ἐστιν. b27. The premisses must be primary and indemonstrable; otherwise they will require demonstration in order to be known, since to have knowledge, if it be not accidental knowledge, of things which are demonstrable, means precisely to have a demonstration of them.
αἴτιά τε καὶ γνωριμώτερα δεῖ εἶναι καὶ πρότερα, αἴτια μὲν ὅτι τότε ἐπιστάμεθα ὅταν τὴν αἰτίαν εἰδῶμεν, καὶ πρότερα, εἴπερ αἴτια, καὶ προγινωσκόμενα οὐ μόνον τὸν ἕτερον τρόπον τῷ ξυνιέναι, ἀλλὰ καὶ τῷ εἰδέναι ὅτι ἔστιν. πρότερα δ' ἐστὶ καὶ γνωριμώτερα διχῶς· οὐ γὰρ ταὐτὸν πρότερον τῇ φύσει καὶ πρὸς ἡμᾶς πρότερον, (72a.) οὐδὲ γνωριμώτερον καὶ ἡμῖν γνωριμώτερον. λέγω δὲ πρὸς ἡμᾶς μὲν πρότερα καὶ γνωριμώτερα τὰ ἐγγύτερον τῆς αἰσθήσεως, ἁπλῶς δὲ πρότερα καὶ γνωριμώτερα τὰ πορρώτερον. ἔστι δὲ πορρωτάτω μὲν τὰ καθόλου μάλιστα, ἐγγυτάτω δὲ τὰ καθ' ἕκαστα· καὶ ἀντίκειται ταῦτ' ἀλλήλοις. ἐκ πρώτων δ' ἐστὶ τὸ ἐξ ἀρχῶν οἰκείων· ταὐτὸ γὰρ λέγω πρῶτον καὶ ἀρχήν. b29. The premisses must be the causes of the conclusion, better known than it, and prior to it; its causes, since we possess scientific knowledge of a thing only when we know its cause; prior, in order to be causes; antecedently known, this antecedent knowledge being not our mere understanding of the meaning, but knowledge of the fact as well. Now 'prior' and 'better known' are ambiguous terms, for there is a difference between what is prior and better known in the order of being and what is prior and better known to man. I mean that objects nearer to sense are prior and better known to man; objects without qualification prior and better known are those further from sense. Now the most universal causes are furthest from sense and particular causes are nearest to sense, and they are thus exactly opposed to one another. In saying that the premisses of demonstrated knowledge must be primary, I mean that they must be the 'appropriate' basic truths, for I identify primary premiss and basic truth.
Postquam ostendit philosophus necessitatem syllogismi demonstrativi, hic iam incipit de ipso syllogismo demonstrativo determinare. Et dividitur in duas partes: in prima, determinat de syllogismo demonstrativo; in secunda, de medio ex quo syllogismus demonstrativus procedit: et hoc in secundo libro; ibi: quaestiones sunt aequales numero et cetera. Prima dividitur in duas: in prima, determinat de syllogismo demonstrativo absolute; in secunda, comparando demonstrationem demonstrationi; ibi: cum autem demonstratio sit alia quidem universalis et cetera. Prima in duas dividitur: in prima, determinat de syllogismo demonstrativo; in secunda, ostendit quod non sit in demonstrationibus in infinitum procedere; ibi: est autem omnis syllogismus per tres terminos et cetera. Prima dividitur in duas: in prima, determinat de syllogismo demonstrativo per quem acquirimus scientiam: in secunda, ostendit quomodo etiam in nobis per syllogismum acquiritur aliqua ignorantia; ibi: ignorantia autem secundum negationem et cetera. Circa primum tria facit: primo, determinat de syllogismo demonstrativo, ostendendo quid sit; secundo, determinat de materia syllogismi demonstrativi, ostendendo quae et qualia sint, ex quibus est; ibi: quoniam autem impossibile est etc.; tertio, determinat de forma ipsius, ostendendo in qua figura praecipue fiat; ibi: figurarum autem magis faciens scire et cetera. Circa primum tria facit: primo, ostendit de demonstrativo syllogismo, quid est; secundo, notificat quaedam, quae in definitione syllogismi demonstrativi ponuntur; ibi: est autem principium demonstrationis etc.; tertio, excludit quosdam errores, qui ex praemissis circa demonstrationem oriri possunt; ibi: quibusdam quidem igitur et cetera. After indicating the need for the demonstrative syllogism, the Philosopher now begins to settle questions concerning the demonstrative syllogism itself. And his treatment is divided into two parts. In the first he determines concerning the demonstrative syllogism. In the second he determines concerning the middle from which the demonstrative syllogism proceeds (89b21) [Book II]. The first is divided into two parts. In the first he determines concerning the demonstrative syllogism in itself. In the second he compares demonstration to demonstration (85a12) [L. 37]. The first is divided into two parts. In the first he determines concerning the demonstrative syllogism. In the second he shows that one does not proceed to infinity in demonstrations (81b10) [L. 31] The first is divided into two parts. In the first he determines concerning the demonstrative syllogism through which we acquire science. In the second he shows how we also acquire ignorance through a syllogism (79b23) [L. 27]. Concerning the first he does three things. First, he determines concerning the demonstrative syllogism by showing what it is. Secondly, he determines concerning the matter of the demonstrative syllogism, pointing out the nature and character of the matter out of which it is formed (73a21) [L. 9]. Thirdly, he determines concerning the form of the syllogism, pointing out the figure in which it is chiefly presented (79a17) [L. 26]. Concerning the first he does three things. First, he shows what the demonstrative syllogism is. Secondly, he clarifies certain terms that appear in the definition of the demonstrative syllogism (72a8) [L. 5]. Thirdly, he excludes certain errors that could arise from his doctrine on the nature of demonstration (755) [L. 7].
Circa primum sciendum est quod in omnibus quae sunt propter finem, definitio quae est per causam finalem, est ratio definitionis, quae est per causam materialem, et medium probans ipsam: propter hoc enim oportet ut domus fiat ex lapidibus et lignis, quia est operimentum protegens nos a frigore et aestu. Sic igitur Aristoteles de demonstratione dat hic duas definitiones: quarum una sumitur a fine demonstrationis, qui est scire; et ex hac concluditur altera, quae sumitur a materia demonstrationis. Unde circa hoc tria facit: primo, definit ipsum scire; secundo, definit demonstrationem per finem eius, qui est ipsum scire; ibi: dicimus autem scire etc.; tertio, ex utraque definitione concludit definitionem demonstrationis, quae sumitur per comparationem materiae demonstrationis; ibi: si igitur est scire ut posuimus et cetera. In regard to the first it should be noted that in all things which exist for an end, the definition which employs a final cause is both the explanation of the definition which expresses the material cause, and is the middle which proves the latter. For the reason why a house should made of stone and wood is that it is a structure protecting us from the cold and heat. Along these lines, therefore, he gives two definitions demonstration, one of which is expressed in terms of the end of demonstration, which is to know in a scientific manner. And from this one is concluded the other, which is drawn from the matter of a demonstration. Hence he does three things in regard to this. First, he defines what it is to know in a scientific manner. Secondly, he defines demonstration in terms of its end, which is to know in a scientific manner (71b18). Thirdly, from these two definitions he concludes to that definition of demonstration which is expressed in terms of the matter of demonstration (71b19).
Circa primum quinque facit. Primo enim, determinat cuiusmodi scire sit, quod definire intendit. Circa quod sciendum est quod aliquid dicimur scire simpliciter, quando scimus illud in seipso. Dicimur scire aliquid secundum quid, quando scimus illud in alio, in quo est, vel sicut pars in toto, sicut si scientes domum, diceremur scire parietem; vel sicut accidens in subiecto, sicut si scientes Coriscum, diceremur scire venientem; vel sicut effectus in causa, sicut dictum est supra quod conclusionem praescimus in principiis; vel quocunque simili modo. Et hoc est scire per accidens, quia scilicet scito aliquo per se, dicimur scire illud quod accidit ei quocunque modo. Intendit igitur philosophus definire scire simpliciter, non autem scire secundum accidens. Hic enim modus sciendi est sophisticus. Utuntur enim sophistae tali modo arguendi: cognosco Coriscum; Coriscus est veniens; ergo cognosco venientem. Concerning the first he does five things. First (71b8), he determines what the scientific knowing, which he intends to define, bears upon. And in regard to this it should be recognized that we are said to know something in a scientific manner absolutely, when we know it in itself. On the other hand, we are said to know something in a scientific manner qualifiedly, when we know it in something else in which it exists either as a part in a whole (as we are said to know a wall through knowing the house), or as an accident in its subject (as in knowing Coriscus we are said to know who is coming toward us), or as an effect in its cause (as in the example given earlier, we know the conclusion in the principles), or indeed in any fashion similar to these. To know in these ways is to know incidentally, because we are said to know that which is somehow accidental to what is known of itself. However, what the Philosopher intends to define here is scientific knowing in the strict sense and not according to an accident. For this form of knowing is sophistical, since Sophists use a form of argument typified by the following: “I know Coriscus; Coriscus is coming toward me: therefore, I know the person coming toward me.”
Secundo, cum dicit: cum causam arbitramur etc., ponit definitionem ipsius scire simpliciter. Circa quod considerandum est quod scire aliquid est perfecte cognoscere ipsum, hoc autem est perfecte apprehendere veritatem ipsius: eadem enim sunt principia esse rei et veritatis ipsius, ut patet ex II metaphysicae. Oportet igitur scientem, si est perfecte cognoscens, quod cognoscat causam rei scitae. Si autem cognosceret causam tantum, nondum cognosceret effectum in actu, quod est scire simpliciter, sed virtute tantum, quod est scire secundum quid et quasi per accidens. Et ideo oportet scientem simpliciter cognoscere etiam applicationem causae ad effectum. Quia vero scientia est etiam certa cognitio rei; quod autem contingit aliter se habere, non potest aliquis per certitudinem cognoscere; ideo ulterius oportet quod id quod scitur non possit aliter se habere. Quia ergo scientia est perfecta cognitio, ideo dicit: cum causam arbitramur cognoscere; quia vero est actualis cognitio per quam scimus simpliciter, addit: et quoniam illius est causa; quia vero est certa cognitio, subdit: et non est contingere aliter se habere. Then (71b10) he presents the definition of scientific knowing in the strict sense. Apropos of this it should be noted that to know something scientifically is to know it completely, which means to apprehend its truth perfectly. For the principles of a thing’s being are the same as those of its truth, as is stated in Metaphysics II. Therefore, the scientific knower, if he is to know perfectly, must know the cause of the thing known; hence he says, “when we think that we know the cause” (71b10). But if he were to know the cause by itself, he would not yet know the effect actually—which would be to know it absolutely—but only virtually, which is the same as knowing in a qualified sense and incidentally. Consequently, one who knows scientifically in the full sense must know the application of the cause to the effect; hence he adds, “as the cause of that fact” (71b11). Again, because science is also sure and certain knowledge of a thing, whereas a thing that could be otherwise cannot be known with certainty, it is further required that what is scientifically known could not be otherwise. To repeat: because science is perfect knowledge, lie says, “Men we think that we know the cause”; but because the knowledge through which we know scientifically in the full sense is actual, he adds, “as the cause of that fact.” Finally, because it is certain knowledge, he adds, “and that the fact could not be other than it is (71b11).”
Tertio, ibi: manifestum igitur etc., manifestat positam definitionem per hoc, quod tam scientes, quam non scientes, existimantes tamen se scire, hoc modo accipiunt scire sicut dictum est: non scientes enim qui existimant se scire, opinantur sic se habere in cognoscendo, sicut dictum est; scientes autem vere sic se habent. Est autem haec recta manifestatio definitionis. Definitio enim est ratio, quam significat nomen, ut dicitur in IV metaphysicae; significatio autem nominis accipienda est ab eo, quod intendunt communiter loquentes per illud nomen significare: unde et in II topicorum dicitur quod nominibus utendum est, ut plures utuntur. Si quis etiam recte consideret, hac notificatione magis ostenditur quid significet nomen, quam directe aliquid significetur: non enim notificat scientiam, de qua proprie posset definitio assignari, cum sit species alicuius generis, sed notificat ipsum scire. Unde et a principio dixit: scire autem opinamur etc., et non dixit: scire est aliquid tale vel tale. Thirdly (71b12), he explains the definition he laid down, appealing to the fact that both those who know scientifically and those who do not know in that way but believe that they do, take scientific knowing to be as above described. For those who do not know in a scientific manner but believe that they do, are convinced that they know in the manner described, whereas those who know in a scientific manner do know in the manner described. Furthermore, this is the proper way to manifest a definition. For a definition is the notion which a name signifies, as it is stated in Metaphysics IV. But the signification of a name must be based on what is generally meant by those who employ the name. Hence it is stated in Topics II that names must be used as the majority of people use them. Again, careful consideration would indicate that this explanation seems rather to show what the name signifies than to signify something directly. For he does not explain science, concerning which a definition could, properly speaking, be formed, since it is a species of some genus; rather he explains scientific knowing. Hence at the very beginning he said, “We suppose ourselves to possess unqualified scientific knowledge” (71b8) and not that scientific knowledge is such and such.
Quarto, ibi: quare cuius etc., concludit quoddam corollarium ex definitione posita, scilicet quod illud, de quo simpliciter habetur scientia, oportet esse necessarium, scilicet quod non contingat aliter se habere. Fourthly (71b14), he draws a corollary from the definition, namely, that that of which there is unqualified scientific knowledge must be something necessary, i.e., which cannot be otherwise.
Quinto, ibi: si quidem etc., respondet tacitae quaestioni, utrum scilicet sit aliquis alius modus sciendi a praedicto. Quod promittit se in sequentibus dicturum: est enim scire etiam per effectum, ut infra patebit. Dicimur etiam aliquo modo scire ipsa principia indemonstrabilia, quorum non est accipere causam. Sed proprius et perfectus sciendi modus est qui praedictus est. Fifthly (71b16), he answers a tacit question, namely, whether there is another way of knowing scientifically in addition to the way described here. And he promises to discuss this later. For it is possible to know scientifically through an effect, as will be explained below (cf. L. 23). Furthermore, there is a sense in which we are said to know scientifically the indemonstrable principles to which no cause is ascribed. But the proper and perfect manner of knowing scientifically is the one we have described.
Deinde, cum dicit: dicimus autem etc., definit syllogismum demonstrativum per comparationem ad finem suum, qui est scire. Circa quod tria facit. Primo, ponit quod scire est finis syllogismi demonstrativi sive effectus eius, cum scire nihil aliud esse videatur, quam intelligere veritatem alicuius conclusionis per demonstrationem. Then (71b17) he defines the demonstrative syllogism in terms of its end, which is to know in a scientific manner. In regard to this he does three things. First, he asserts that scientific knowing is the end of a demonstrative syllogism or is its effect, since to know scientifically seems to b nothing less than to understand the truth of a conclusion through demonstration.
Secundo, ibi: demonstrationem autem etc., definit syllogismum demonstrativum per huiusmodi finem: dicens quod demonstratio est syllogismus scientialis, idest faciens scire. Secondly (71b18), he defines demonstration in terms of the end, saying that a demonstration is a sciential syllogism, i.e., producing scientific knowledge.
Tertio, exponit hoc quod dixerat scientialem; ibi: sed scientialem etc., dicens quod scientialis syllogismus dicitur, secundum quem scimus, in quantum ipsum habemus, ne forte aliquis syllogismum scientialem intelligeret, quo aliqua scientia uteretur. Thirdly (71b18), he explains, “sciential,” saying that a sciential syllogism is one according to which we know scientifically insofar as we understand it, and not in the sense of a syllogism yielding knowledge to be put to use.
Deinde, cum dicit: si igitur est scire etc., concludit ex praedictis definitionem syllogismi demonstrativi ex materia sumptam. Et circa hoc duo facit: primo, concludit; secundo, manifestat eam; ibi: verum quidem igitur oportet esse et cetera. Then (71b19) he concludes from the foregoing a definition of the demonstrative syllogism that is based on its matter. Concerning this he does two things. First, he concludes it. Secondly, he clarifies it (71b24).
Circa primum tria facit. Primo, ponit consequentiam, qua demonstrationis materialis definitio concluditur ex praemissis, dicens quod si scire hoc significat quod diximus, scilicet, causam rei cognoscere etc., necesse est quod demonstrativa scientia, idest quae per demonstrationem acquiritur, procedat ex propositionibus veris, primis et immediatis, idest quae non per aliquod medium demonstrantur, sed per seipsas sunt manifestae (quae quidem immediatae dicuntur, in quantum carent medio demonstrante; primae autem in ordine ad alias propositiones, quae per eas probantur); et iterum ex notioribus, et prioribus, et causis conclusionis. Concerning the first he does three things. First (71b20), he sets forth the consequent in which the material definition of demonstration is concluded from the premises laid down above. And he says that if scientific knowing is what we have stated it to be, namely, knowing the cause of a thing, etc., then it is necessary that demonstrative science, i.e., science acquired through demonstration, proceed from propositions which are true, first, and immediate, i.e., not demonstrated by some other mid but clear in virtue of themselves (they are called “immediate,” inasmuch as they do not have a middle demonstrating them, but “first,” in relation to other propositions which are proved through them); and which, furthermore, are better known than, prior to, and causes of, the conclusion.
Secundo, ibi: sic enim erunt etc., excusat se ab additione alterius particulae, quae videbatur apponenda: quod scilicet demonstratio ex propriis principiis procederet. Sed ipse dicit quod hoc intelligitur per ea, quae dicta sunt. Nam si propositiones demonstrationis sunt causae conclusionis, necesse est quod sint propria principia eius: oportet enim causas esse proportionatas effectibus. Secondly (71b22), he justifies himself for not adding another element which, it might seem, should be added, namely, that demonstration proceeds from proper principles. But he says that this is understood in virtue of the elements he did state. For since the propositions of a demonstration are causes of the conclusion, they must be its proper principles. For effects require proportionate causes.
Tertio, ibi: syllogismus quidem etc., manifestat praemissae consequentiae necessitatem, dicens quod licet syllogismus non requirat praemissas conditiones in propositionibus, ex quibus procedit, requirit tamen eas demonstratio: aliter enim non faceret scientiam. Thirdly (71b23), he manifests the necessity of the aforesaid consequence, saying that although a syllogism does not require these conditions in the premises from which it concludes, a demonstration does require them, for otherwise it would not produce science.
Deinde, cum dicit: verum quidem etc., manifestat positam definitionem, manifestans etiam quod immediate dixerat, scilicet quod nisi praemissae conditiones demonstrationi adessent, scientiam facere non posset. Primo ergo, ostendit quod semper procedit ex veris ad hoc quod scientiam faciat: quia quod non est, non est scire; sicut diametrum esse symmetrum, idest commensurabilem lateri quadrati (dicuntur enim quantitates incommensurabiles, quarum non potest accipi aliqua mensura communis; et huiusmodi quantitates sunt, quarum non est proportio ad invicem sicut numeri ad numerum; quod de necessitate contingit de diametro quadrati et eius latere, ut patet ex X Euclidis). Quod autem non est verum, non est: nam esse et esse verum convertuntur. Oportet ergo id quod scitur esse verum. Et sic conclusionem demonstrationis, quae facit scire, oportet esse veram, et per consequens eius propositiones: non enim contingit verum sciri ex falsis, etsi concludi possit ex eis, ut infra ostendet. Then (71b24) he explains this definition as well as the subsequent statement that unless these conditions are fulfilled in a demonstration it cannot beget science. First, therefore, he shows that a demonstration must proceed from true principles in order to beget science, because there cannot be scientific knowledge of that which does not exist, for example, that the diagonal is symmetrical, i.e., commensurable with the side of the square. For those quantities are said to be incommensurable which lack a common measuring unit. These are quantities whose ratio to one another cannot be expressed in terms of one number to another number. That this is the case with the diagonal of a square and its side is plain from Euclid’s sixteenth proposition. Now what is not true does not exist, for to be and to be true are convertible. Therefore, anything scientifically known must be true. Consequently, the conclusion of a demonstration which does beget scientific knowing must be true, and a fortiori its premises. For the true cannot be known in a scientific way from the false, although something true can follow as a conclusion from something false, as he will show later (cf. Lectio 13).
Secundo, ibi: ex primis autem etc., ostendit quod demonstratio sit ex primis et immediatis, sive indemonstrabilibus. Non enim contingit aliquem habere scientiam, nisi habeat demonstrationem eorum, quorum potest esse demonstratio, et hoc dico per se, et non per accidens. Et hoc ideo dicit, quia possibile esset scire aliquam conclusionem, non habentem demonstrationem praemissorum, etiam si essent demonstrabilia: quia sciret eam per alia principia; et hoc esset secundum accidens. Secondly (71b27), he shows that the demonstration is composed of first and immediate or indemonstrable principles. For no one can possess scientific knowledge unless he possesses the demonstration of things that can be demonstrated—“and I am speaking per se and not per accidens.” He says this because it would be possible to know some conclusion without having a demonstration of the premises, even were they demonstrable; because one would know it through other principles, and this would be accidental.
Detur ergo quod aliquis demonstrator syllogizet ex demonstrabilibus, sive mediatis: aut ergo habet illorum demonstrationem, aut non habet: si non habet, ergo non scit praemissa, et ita nec conclusionem propter praemissa; si autem habet, cum in demonstrationibus non sit abire in infinitum, ut infra ostendet, tandem erit devenire ad aliqua immediata et indemonstrabilia. Et sic oportet quod demonstratio ex immediatis procedat, vel statim, vel per aliqua media. Unde et in primo libro topicorum dicitur quod demonstratio est ex primis et veris, aut ex his quae per ea fidem sumpserunt. Suppose, therefore, that a demonstrator syllogizes from demonstrable, i.e., mediate, premises. Now he either possesses a demonstration of those premises or he does not. If he does not, then he does not know the premises in a scientific way; nor consequently, the conclusion because of the premises. But if he does possess their demonstration, then, since one may not proceed to infinity in demonstrations, principles immediate and indemonstrable must be reached. And so it is required that demonstration proceed from principles that are immediate either straightway or through middles. Hence it is stated in Topics I that demonstration is composed of first and true statements or of statements made credible by these.
Tertio, ibi: causas quoque etc., probat quod demonstrationis propositiones sint causae conclusionis, quia tunc scimus, cum causas cognoscimus. Et ex hoc concludit ulterius quod sint priores et notiores, quia omnis causa est naturaliter prior et notior suo effectu. Oportet autem quod causa conclusionis demonstrativae sit notior, non solum quantum ad cognitionem quid est, sed etiam quantum ad cognitionem quia est. Non enim ad demonstrandum quod eclipsis solis est, sufficit scire quod est lunae interpositio, sed oportet etiam scire quod luna interponitur inter solem et terram. Et quia prius et notius dicitur dupliciter, scilicet quoad nos, et secundum naturam; dicit consequenter quod ea, ex quibus procedit demonstratio, sunt priora et notiora simpliciter et secundum naturam, et non quoad nos. Thirdly (71b29), he proves that the propositions of a demonstration are the causes of the conclusion, because we know in a scientific manner when we know the causes. And in virtue of this he shows that they are prior and better known, because every cause is by nature prior and better known than its effect. However, the cause of a demonstrated conclusion must be better known not only with respect to the knowledge of what it is, but also with respect to the knowledge that it is. For in order to demonstrate that there is an eclipse of the sun, it is not enough to know, that the moon is interposed; in addition it is necessary to know that the moon is interposed between the sun and the earth. Again, because prior and better known are taken in two ways, namely, in reference to us and according to nature, he says that the things from which a demonstration, proceeds are prior and better known absolutely and according to nature, and not in reference to us.
Et ad huius expositionem dicit quod priora et notiora simpliciter sunt illa, quae sunt remota a sensu ut universalia. Priora autem et notiora quoad nos sunt proxima sensui, scilicet singularia, quae opponuntur universalibus, sive oppositione prioris et posterioris, sive oppositione propinqui et remoti. To elucidate this he says that “those things are prior and better known absolutely,” which are farthest from sense, as are universals; but “the prior and better known in reference to us” are nearest to sense, namely, the singulars, which are opposed to universals in the way that the prior and the later are opposite, or in the way that the nearest and the farthest are opposite.
Videtur autem contrarium huius haberi in I Physic., ubi dicitur quod universalia sunt priora quoad nos, et posteriora secundum naturam. Sed dicendum est quod hic loquitur de ordine singularis ad universale simpliciter, quorum ordinem oportet accipere secundum ordinem cognitionis sensitivae et intellectivae in nobis. Cognitio autem sensitiva est in nobis prior intellectiva, quia intellectualis cognitio ex sensu procedit in nobis. Unde et singulare est prius et notius quoad nos quam universale. In I autem Physic. non ponitur ordo universalis ad singulare simpliciter, sed magis universalis ad minus universale, ut puta, animalis ad hominem, et sic oportet quod quoad nos, universalius sit prius et magis notum. In omni enim generatione, quod est in potentia est prius tempore et posterius natura, quod autem est completum in actu est prius natura et posterius tempore. Cognitio autem generis est quasi potentialis, in comparatione ad cognitionem speciei, in qua actu sciuntur omnia essentialia rei. Unde in generatione scientiae nostrae prius est cognoscere magis commune quam minus commune. However, it seems that the contrary of this is found in Physics I where it is stated that universals are prior in reference to us and later according to nature. But it should be said that there [in the Posterior Analytics] he is speaking of the order of singular to universal absolutely; and this order must be taken according to the order of sensitive and intellectual knowledge in us. Now in us sensitive knowledge is prior to intellectual, because intellectual knowledge in us proceeds from sense. For this reason the singular is prior and better known in relation to us than the universal. But in Physics I he is not speaking of the order of the universal to the singular absolutely but of the order of the more universal to the less universal, for example of animal to man. In this case the more universal is prior and better known in reference to us. For in every instance of generation, that which is in potency is prior in time but is later according to nature; whereas that which is complete in act is prior by nature but later in time. Now one’s knowledge of a genus is, as it were, potential in comparison to one’s knowledge of the species in which all the essentials of a thing are actually known. Hence, too, in the generation of our science, knowledge of the more common precedes knowledge of the less common.
Item, in libro Physic. dicitur quod innata est nobis via ex nobis notioribus; non ergo demonstratio fit ex his quae sunt priora simpliciter, sed quoad nos. Sed dicendum quod hic loquitur secundum quod id quod est in sensu est notius quoad nos, eo quod est in intellectu. Ibi autem secundum quod id quod est notius quoad nos, est etiam in intellectu. Ex singularibus autem quae sunt in sensu, non sunt demonstrationes, sed ex universalibus tantum, quae sunt in intellectu. Again, in the Physics it is stated that it is natural for us to proceed from what is better known to us. Therefore, it seems that a demonstration is composed not of things that are prior absolutely but in reference to But it must be said that here he is speaking according to the fact that what is in the sense is better known in reference to us than what is in the intellect; but there he was speaking according to the fact that what is better known in reference to us is also in the intellect. But demonstrations do not proceed from singulars which are in the sense but only from universals, which are in the intellect.
Vel dicendum quod in omni demonstratione, oportet quod procedatur ex his, quae sunt notiora quoad nos, non tamen singularibus, sed universalibus. Non enim aliquid potest fieri nobis notum, nisi per id quod est magis notum nobis. Quandoque autem id quod est magis notum quoad nos, est etiam magis notum simpliciter et secundum naturam; sicut accidit in mathematicis, in quibus, propter abstractionem a materia, non fiunt demonstrationes nisi ex principiis formalibus. Et in talibus fiunt demonstrationes ex his quae sunt notiora simpliciter. Item, quandoque id quod est notius quoad nos non est notius simpliciter, sicut accidit in naturalibus, in quibus essentiae et virtutes rerum, propter hoc quod in materia sunt, sunt occultae, sed innotescunt nobis per ea, quae exterius de ipsis apparent. Unde in talibus fiunt demonstrationes ut plurimum per effectus, qui sunt notiores quoad nos, et non simpliciter. Nunc autem non loquitur de hoc modo demonstrationum, sed de primo. Or it might be said that in every demonstration one must proceed from things better known to us, provided they are not singulars but universals. For something is made known to us only by that which is more known to us. But sometimes that which is more known in reference to us is also more known absolutely and according to nature, as happens in mathematics where on account of abstraction from matter the demonstrations proceed from formal principles alone. In this case the demonstrations proceed from things which are more known absolutely. But sometimes that which is more known in reference to us is not more known absolutely, as happens in natural sciences where the essences and powers of things are hidden, because they are in matter, but are disclosed to us through the things which appear outwardly. Hence in these sciences the demonstrations are for the most part made through effects which are better known in reference to us but not absolutely. But he is not now speaking of this form of demonstration, but of the first.
Quia vero in hac manifestatione hoc etiam omiserat manifestare, quod demonstratio esset ex propriis principiis, consequenter subdit quod hoc habetur etiam ex praemissis. Per hoc enim quod dicitur quod demonstratio est ex primis, habetur quod sit ex propriis principiis, sicut et superius dictum est. Idem enim videtur esse primum et principium: nam primum in unoquoque genere et maximum est causa omnium eorum, quae sunt post, ut dicitur in II methaphysicae. Finally, because in his explanation he neglected to point out that demonstration should proceed from proper principles, he hastens to add that this fact is easily ascertainable from what he did say. For from the fact that he stated that demonstration is from things which are first, it follows that it is from proper principles, as he stated above. For “first” and “principle” seem to be the same: for that which is first and highest in each genus is the cause of all the things that are after it, as it is stated in Metaphysics II.

Lectio 5
Caput 2 cont.
ἀρχὴ δ' ἐστὶν ἀποδείξεως πρότασις ἄμεσος, ἄμεσος δὲ ἧς μὴ ἔστιν ἄλλη προτέρα. a8. A ‘basic truth’ in a demonstration is an immediate proposition. An immediate proposition is one which has no other proposition prior to it.
πρότασις δ' ἐστὶν ἀποφάνσεως τὸ ἕτερον μόριον, ἓν καθ' ἑνός, a9. A proposition is either part of an enunciation, i.e. it predicates a single attribute of a single subject
διαλεκτικὴ μὲν ἡ ὁμοίως λαμβάνουσα ὁποτερονοῦν, ἀποδεικτικὴ δὲ ἡ ὡρισμένως θάτερον, ὅτι ἀληθές. a10. If a proposition is dialectical, it assumes either part indifferently; if it is demonstrative, it lays down one part to the definite exclusion of the other because that part is true.
ἀπόφανσις δὲ ἀντιφάσεως ὁποτερονοῦν μόριον, ἀντίφασις δὲ ἀντίθεσις ἧς οὐκ ἔστι μεταξὺ καθ' αὑτήν, μόριον δ' ἀντιφάσεως τὸ μὲν τὶ κατὰ τινὸς κατάφασις, τὸ δὲ τὶ ἀπὸ τινὸς ἀπόφασις. a11. The term 'enunciation' denotes either part of a contradiction indifferently. A contradiction is an opposition which of its own nature excludes a middle. The part of a contradiction which conjoins a predicate with a subject is an affirmation; the part disjoining them is a negation.
Ἀμέσου δ' ἀρχῆς συλλογιστικῆς θέσιν μὲν λέγω ἣν μὴ ἔστι δεῖξαι, μηδ' ἀνάγκη ἔχειν τὸν μαθησόμενόν τι· ἣν δ' ἀνάγκη ἔχειν τὸν ὁτιοῦν μαθησόμενον, ἀξίωμα· ἔστι γὰρ ἔνια τοιαῦτα· τοῦτο γὰρ μάλιστ' ἐπὶ τοῖς τοιούτοις εἰώθαμεν ὄνομα λέγειν. a15. I call an immediate basic truth of syllogism a 'thesis' when, though it is not susceptible of proof by the teacher, yet ignorance of it does not constitute a total bar to progress on the part of the pupil: one which the pupil must know if he is to learn anything whatever is an axiom. I call it an axiom because there are such truths and we give them the name of axioms par excellence.
θέσεως δ' ἡ μὲν ὁποτερονοῦν τῶν μορίων τῆς ἀντιφάσεως λαμβάνουσα, οἷον λέγω τὸ εἶναί τι ἢ τὸ μὴ εἶναί τι, ὑπόθεσις, ἡ δ' ἄνευ τούτου ὁρισμός. ὁ γὰρ ὁρισμὸς θέσις μέν ἐστι· τίθεται γὰρ ὁ ἀριθμητικὸς μονάδα τὸ ἀδιαίρετον εἶναι κατὰ τὸ ποσόν· ὑπόθεσις δ' οὐκ ἔστι· τὸ γὰρ τί ἐστι μονὰς καὶ τὸ εἶναι μονάδα οὐ ταὐτόν. a19. If a thesis assumes one part or the other of an enunciation, i.e. asserts either the existence or the non-existence of a subject, it is a hypothesis; if it does not so assert, it is a definition. Definition is a 'thesis' or a 'laying something down', since the arithmetician lays it down that to be a unit is to be quantitatively indivisible; but it is not a hypothesis, for to define what a unit is is not the same as to affirm its existence.
Quia superius philosophus dixerat quod demonstratio est ex primis et immediatis, et haec ab ipso nondum manifestata erant, ideo intendit ista notificare. Et dividitur in partes tres: in prima, ostendit quid sit propositio immediata; in secunda vero ostendit quod oporteat huiusmodi propositiones esse notiores conclusione; ibi: quoniam autem oportet credere et scire etc.; in tertia, excludit quosdam errores, qui ex praedictis occasionem habebant; ibi: quibusdam quidem igitur et cetera. Circa primum duo facit: primo, ostendit quid sit propositio immediata; secundo, dividit ipsam; ibi: immediati autem principii et cetera. Because the Philosopher had stated above that demonstration is from “first and immediate principles,” but had not yet identified them, he now sets out to identify them. And this is divided into three parts. In the first part he shows what an immediate proposition is. In the second part he shows that such propositions must be better known than the conclusion (72a25) [L. 6]. In the third part he excludes certain errors which arose from the foregoing (72b5) [L. 7]. Concerning the first he does two things. First, he shows what an immediate principle is. Secondly, he divides them (72a15).
Circa primum hoc modo procedit. Primo namque, resumit quod supra dictum erat, scilicet quod principium demonstrationis sit propositio immediata: nam et supra dixerat quod demonstratio est ex primis et immediatis. With respect to the first he proceeds this way. First (72a8), he recall, what has been said above, namely, that a principle of demonstration I, an immediate proposition, for he had also stated above that a demonstration is composed of things which are first and immediate.
Secundo, ibi: immediata autem etc., definit immediatam propositionem, et dicit quod immediata propositio est qua non est altera prior. Cuius quidem notificationis ratio ex praedictis apparet. Dictum est enim supra quod demonstratio est ex prioribus. Quandocunque igitur aliqua propositio est mediata, idest habens medium per quod demonstretur praedicatum de subiecto, oportet quod priores ea sint propositiones ex quibus demonstratur: nam praedicatum conclusionis per prius inest medio quam subiecto; cui etiam per prius inest medium quam praedicatum. Relinquitur ergo quod illa propositio, qua non est altera prior, sit immediata. Secondly (ibid.), he defines the immediate proposition and says that a immediate proposition is one which has no other one prior to it. the reason underlying this description is clear from what has been said. For it has been said above that demonstration is composed of things that, are prior. Accordingly, whenever a proposition is mediate, i.e., has a middle through which the predicate is demonstrated of its subject, it is required that there be prior propositions by which this one is demonstrated. For the predicate of a conclusion is present in the middle previously to being present in the subject; in which, however, the middle is present before the predicate is. Therefore, it follows that that proposition which does not have some other one prior to it is immediate.
Tertio, ibi: propositio autem est etc., ostendit quid sit propositio, quae ponitur in definitione immediatae propositionis. Et circa hoc tria facit. Thirdly (720), he shows what is the nature of the proposition which is mentioned in the definition of an immediate proposition. Concerning this he does three things:
Primo namque definit propositionem simpliciter, dicens quod propositio est altera pars enunciationis, in qua praedicatur unum de uno. Habet enim enunciatio duas partes, scilicet affirmationem et negationem. Oportet autem quod omnis syllogizans alteram earum proponat, non autem utramque: hoc enim est proprium eius, qui a principio quaestionem movet. Unde per hoc separatur propositio a problemate. Sicut etiam in uno syllogismo non concluditur nisi unum, ita oportet quod propositio, quae est syllogismi principium, sit una. Una autem est in qua est unum de uno. Unde per hoc quod philosophus dicit unum de uno, separatur propositio ab enunciatione, quae dicitur plures, in qua plura de uno vel unum de pluribus praedicatur. First, he defines absolutely what a proposition is, saying that it is one or the other part of an enunciation in which one thing is predicated of one thing. For the enunciation has two parts, namely, affirmation and negation. For anyone who syllogizes must propose one or the other of these parts but not both, for this latter procedure is characteristic of one who first raises a question. (Hence it is on this basis that proposition is distinguished from a problem). For just as one and only one thing is concluded in one syllogism, so the proposition which is a principle of the syllogism should be one—and it is one if one thing is stated of one thing. Hence in asserting that it is “one of one,” he distinguishes the proposition from the enunciation which is said to be “of several,” whether sundry things are said of one thing or of one thing sundry.
Secundo, ibi: dialectica etc., ponit differentiam inter dialecticam propositionem et demonstrativam, dicens quod cum propositio accipiat alteram partem enunciationis, dialectica indifferenter accipit quancunque earum. Habet enim viam ad utranque partem contradictionis, eo quod ex probabilibus procedit. Unde etiam et in proponendo accipit utramlibet partem contradictionis et quaerendo proponit. Demonstrativa autem propositio accipit alteram partem determinate, quia nunquam habet demonstrator viam, nisi ad verum demonstrandum. Unde etiam semper proponendo accipit veram partem contradictionis. Propter hoc etiam non interrogat, sed sumit, qui demonstrat, quasi notum. Secondly (72a10), he lays down the difference between the dialectical and the demonstrative proposition, saying that whereas the demonstrative proposition takes one definite side of a question, the dialectical takes either side indifferently. For since dialectic begins with the probable, it can lead to each side of a contradiction. Hence when it lays down its propositions, it employs both parts of a contradiction and presents them in the form of a question [Is an animal that walks on its feet a man, or not?]. But a demonstrative proposition takes one side definitively, because a demonstrator never has any other alternative but to demonstrate the truth. Hence in forming its propositions he always assumes the true side of a contradiction [An animal which walks on two feet is a man, is it not?]. On this account he does not ask but posits something as known in the demonstration.
Tertio, ibi: enunciatio autem etc., definit enunciationem quae ponitur in definitione propositionis, dicens quod enunciatio complectitur utranque partem contradictionis, ut ex dictis patet. Thirdly (72a11), he defines the term, “enunciation,” which appeared in the definition of a proposition, saying that an enunciation embraces both sides of a contradiction, as is clear from what has been said.
Quid autem sit contradictio consequenter ostendit, dicens quod contradictio est oppositio, cuius non est medium secundum se. Quamvis enim in privatione et habitu, et in contrariis immediatis non sit medium circa determinatum subiectum, tamen est medium simpliciter, nam lapis neque caecus, neque videns est, et albedo neque par, neque impar est. Et hoc etiam quod habent de immediatione circa determinatum subiectum, habent in quantum aliquid participant contradictionis: nam privatio est negatio in subiecto determinato. Et alterum etiam contrariorum habet aliquid privationis. Sed contradictio simpliciter in omnibus caret medio; et hoc non habet ab alio, sed ex seipsa; et propter hoc dicit quod eius non est medium secundum se. Exponit etiam consequenter quid sit pars contradictionis: est enim contradictio oppositio affirmationis et negationis. Unde altera pars eius est affirmatio, quae praedicat aliquid de aliquo; altera vero negatio, quae removet aliquid ab aliquo. Then he shows what contradiction is, saying that contradiction is a form of opposition between whose parts there is of itself no middle. For although between privation and possession and between immediate contraries there is no middle in a given subject, nevertheless, absolutely speaking, there is one; for a stone is neither blind nor seeing, and something white is neither even nor odd. Furthermore, whatever immediacy they have in relation to a definite subject is traced to their participation in contradiction, for privation is negation in a definite subject; and of two things that are immediately contrary, one has some of the marks of privation. But contradiction in the full sense lacks a middle in all cases. And this belongs to it of its very nature and not in virtue of something else. Hence he says that of itself it has no medium.
Deinde cum dicit: immediati autem etc. dividit immediatum principium. Et circa hoc duo facit: primo dividit; secundo subdividit; ibi: positiones autem quaedam et cetera. He then explains what the parts of a contradiction are. For contradiction is an opposition of affirmation and negation; hence one of its parts is affirmation, which asserts something of something, and the other is negation, which denies something of something. Then (72a15) he divides immediate principle. Concerning this he does two things. First, he divides. Secondly, he subdivides (72a19).
Dicit ergo primo quod immediatum principium syllogismi duplex est. Unum est quod dicitur positio, quam non contingit demonstrare et ex hoc immediatum dicitur; neque tamen aliquem docendum, idest qui doceri debet in demonstrativa scientia, necesse est habere, idest mente concipere sive ei assentire. Aliud vero est, quod dicitur dignitas vel maxima propositio, quam necesse est habere in mente et ei assentire quemlibet, qui doceri debet. Et manifestum est quod quaedam principia talia sunt ut probatur in IV metaphysicae de hoc principio, quod affirmatio et negatio non sunt simul vera, cuius contrarium nullus mente credere potest etsi ore proferat. Et in talibus utimur nomine praedicto, scilicet dignitatis vel maximae propositionis, propter huiusmodi principiorum certitudinem ad manifestandum alia. He says therefore first (7205), that there are two types of immediate principles of a syllogism: the first is called a “position” [thesis] and is said to be immediate because one does not demonstrate (neither is it required that the student, i.e., the one being instructed in the demonstrative science, have it, i.e., advert to it or assent to it); the other is called a “dignity” or “maxim,” which anyone who is to be instructed must have in his mind and assent to. That there are such principles is clear from Metaphysics IV, where it is proved that one such is the principle that “affirmation and negation are not simultaneously true,” for no one can believe the contrary of this in his mind ‘ even though he should state it orally. To such principles we give the aforesaid name of “dignity” or “maxim” on account of their certainty in manifesting other things.
Ad huius autem divisionis intellectum sciendum est quod quaelibet propositio, cuius praedicatum est in ratione subiecti, est immediata et per se nota, quantum est in se. Sed quarundam propositionum termini sunt tales, quod sunt in notitia omnium, sicut ens, et unum, et alia quae sunt entis, in quantum ens: nam ens est prima conceptio intellectus. Unde oportet quod tales propositiones non solum in se, sed etiam quoad omnes, quasi per se notae habeantur. Sicut quod, non contingit idem esse et non esse; et quod, totum sit maius sua parte: et similia. Unde et huiusmodi principia omnes scientiae accipiunt a metaphysica, cuius est considerare ens simpliciter et ea, quae sunt entis. To clarify this division it should be noted that any proposition whose predicate is included within the notion of its subject is immediate and known in virtue of itself as it stands. However, in the case of some of these propositions the terms are such that they are understood by everyone, as being and one and those other notions that are characteristic of being precisely as being: for being is the first concept in the intellect. Hence it is necessary that propositions of this kind be held as known in virtue of themselves not only as they stand but also in reference to us. Examples of these are the propositions that “It does not occur that the same thing is and is not” and that “The whole is greater than its part,” and others like these. Hence all the sciences take principles of this kind from metaphysics whose task it is to consider being absolutely and the characteristics of being.
Quaedam vero propositiones sunt immediatae, quarum termini non sunt apud omnes noti. Unde, licet praedicatum sit de ratione subiecti, tamen quia definitio subiecti non est omnibus nota, non est necessarium quod tales propositiones ab omnibus concedantur. Sicut haec propositio: omnes recti anguli sunt aequales, quantum est in se, est per se nota sive immediata, quia aequalitas cadit in definitione anguli recti. Angulus enim rectus est, quem facit linea recta super aliam rectam cadens, ita quod ex utraque parte anguli reddantur aequales. Et ideo, cum quadam positione recipiuntur huiusmodi principia. On the other hand, there are some immediate propositions whose terms are not known by everyone. Hence, although their predicate may be included in the very notion of their subject, yet because the definition of the subject is not known to everyone, it is not necessary that such propositions be conceded by everyone. (Thus the proposition, “All right angles are equal,” is in itself a proposition which is immediate and known in virtue of itself, because equality appears in the definition of a right angle. For a right angle is one which a straight line form when it meets another straight line in such a way that the angles on each side are equal). Therefore, such principles are received as being posited or laid down.
Est et alius modus, quo aliquae propositiones suppositiones dicuntur. Sunt enim quaedam propositiones, quae non possunt probari nisi per principia alterius scientiae; et ideo oportet quod in illa scientia supponantur, licet probentur per principia alterius scientiae. Sicut a puncto ad punctum rectam lineam ducere, supponit geometra et probat naturalis; ostendens quod inter quaelibet duo puncta sit linea media. There is yet another way, and according to it certain propositions are called “suppositions.” For there are some propositions which can be proved only by the principles of some other science; therefore, they must be supposed in the one science, although they are proved by the principles of the other science. Thus the geometer supposes that he can draw one straight line from one point to another, but the philosopher of nature proves it by showing that there is one straight line between any two points.
Deinde cum dicit: positionis autem quae est etc., subdividit alterum membrum primae divisionis, scilicet positionem: dicens quod quaedam positio est, quae accipit aliquam partem enunciationis, scilicet affirmationem vel negationem; quod significat cum dicit: ut dico aliquid esse aut non esse. Et haec positio suppositio dicitur, quia tanquam veritatem habens supponitur. Alia autem positio est, quae non significat esse vel non esse, sicut definitio, quae positio dicitur. Ponitur enim ab arithmetico definitio unitatis, tanquam quoddam principium, scilicet quod unitas est indivisibile secundum quantitatem. Sed tamen definitio non dicitur suppositio: illud enim proprie supponitur, quod verum vel falsum significat. Et ideo subdit quod non idem est, quod quid est unitas, quod neque verum, neque falsum significat, et esse unitatem, quod significat verum vel falsum. Then (72a19) he subdivides a member of the original division, namely, “position,” and says that there is one type of position which takes one side of an enunciation, namely, either affirmation or negation. He refers to this type when he says, “i.e., asserts either the existence or non-existence of a subject.” Such a position is called a “supposition” or “hypothesis,” because it is accepted as having truth. Another type of position is the one which does not signify existence or non-existence: in this way a definition is a position. For the definition of “one” is laid down in arithmetic as a principle, namely, that “one is the quantitatively indivisible.” Nevertheless a definition is not called a supposition, for a supposition, strictly speaking, is a statement which signifies the true or the false. Consequently, he adds that “the definition of ‘one,”’ inasmuch as it signifies neither the true nor the false, “is not the same as ‘to be one,”’ which does signify the true or the false.
Sed potest quaeri: cum definitio non sit propositio significans esse vel non esse, quomodo ponatur in subdivisione immediatae propositionis. Sed dicendum quod in subdivisione non resumit immediatam propositionem ad subdividendum, sed immediatum principium. Principium autem syllogismi dici potest non solum propositio, sed etiam definitio. Vel potest dici quod licet definitio in se non sit propositio in actu, est tamen in virtute propositio quia cognita definitione, apparet definitionem de subiecto vere praedicari. Now it might be asked how it is that definition is set down as a member of the subdivision of immediate proposition, if a definition is not a proposition signifying either existence or non-existence. One might answer that in this subdivision he was not subdividing immediate proposition but, immediate principle. Or one might answer that although a definition as such is not an actual proposition, it is one virtually, because once a definition is known, it becomes clear that it is truly predicated of the subject.

Lectio 6
Caput 2 cont.
Ἐπεὶ δὲ δεῖ πιστεύειν τε καὶ εἰδέναι τὸ πρᾶγμα τῷ τοιοῦτον ἔχειν συλλογισμὸν ὃν καλοῦμεν ἀπόδειξιν, ἔστι δ' οὗτος τῷ ταδὶ εἶναι ἐξ ὧν ὁ συλλογισμός, ἀνάγκη μὴ μόνον προγινώσκειν τὰ πρῶτα, ἢ πάντα ἢ ἔνια, ἀλλὰ καὶ μᾶλλον· a25. Now since the required ground of our knowledge — i.e. of our conviction — of a fact is the possession of such a syllogism as we call demonstration, and the ground of the syllogism is the facts constituting its premisses, we must not only know the primary premisses — some if not all of them — beforehand, but know them better than the conclusion:
αἰεὶ γὰρ δι' ὃ ὑπάρχει ἕκαστον, ἐκείνῳ μᾶλλον ὑπάρχει, οἷον δι' ὃ φιλοῦμεν, ἐκεῖνο φίλον μᾶλλον. ὥστ' εἴπερ ἴσμεν διὰ τὰ πρῶτα καὶ πιστεύομεν, κἀκεῖνα ἴσμεν τε καὶ πιστεύομεν μᾶλλον, ὅτι δι' ἐκεῖνα καὶ τὰ ὕστερα. a28. for the cause of an attribute's inherence in a subject always itself inheres in the subject more firmly than that attribute; e.g. the cause of our loving anything is dearer to us than the object of our love. So since the primary premisses are the cause of our knowledge — i.e. of our conviction — it follows that we know them better — that is, are more convinced of them — than their consequences, precisely because of our knowledge of the latter is the effect of our knowledge of the premisses.
οὐχ οἷόν τε δὲ πιστεύειν μᾶλλον ὧν οἶδεν ἃ μὴ τυγχάνει μήτε εἰδὼς μήτε βέλτιον διακείμενος ἢ εἰ ἐτύγχανεν εἰδώς. a33. Now a man cannot believe in anything more than in the things he knows, unless he has either actual knowledge of it or something better than actual knowledge. But we are faced with this paradox if a student whose belief rests on demonstration has not prior knowledge;
συμβήσεταιδὲ τοῦτο, εἰ μή τις προγνώσεται τῶν δι' ἀπόδειξιν πιστευόντων· μᾶλλον γὰρ ἀνάγκη πιστεύειν ταῖς ἀρχαῖς ἢ πάσαις ἢ τισὶ τοῦ συμπεράσματος. a36. a man must believe in some, if not in all, of the basic truths more than in the conclusion. Moreover, if a man sets out to acquire the scientific knowledge that comes through demonstration,
τὸν δὲ μέλλοντα ἕξειν τὴν ἐπιστήμην τὴν δι' ἀποδείξεως οὐ μόνον δεῖ τὰς ἀρχὰς μᾶλλον γνωρίζειν καὶ μᾶλλον αὐταῖς πιστεύειν ἢ τῷ δεικνυμένῳ, (72b.) ἀλλὰ μηδ' ἄλλο αὐτῷ πιστότερον εἶναι μηδὲ γνωριμώτερον τῶν ἀντικειμένων ταῖς ἀρχαῖς ἐξ ὧν ἔσται συλλογισμὸς ὁ τῆς ἐναντίας ἀπάτης, εἴπερ δεῖ τὸν ἐπιστάμενον ἁπλῶς ἀμετάπειστον εἶναι. a38. he must not only have a better knowledge of the basic truths and a firmer conviction of them than of the connexion which is being demonstrated: more than this, nothing must be more certain or better known to him than these basic truths in their character as contradicting the fundamental premisses which lead to the opposed and erroneous conclusion. For indeed the conviction of pure science must be unshakable.
Postquam ostendit philosophus quae sunt immediata principia, hic de eorum cognitione determinat. Et circa hoc duo facit: primo, ostendit quod immediata principia sunt magis nota conclusione; secundo, ostendit quod etiam falsitas contrariorum debet esse notissima; ibi: non solum autem et cetera. Circa primum tria facit. After showing what immediate principles are, the Philosopher now determines concerning our knowledge of them. Apropos of this he does two things. First, he shows that immediate principles are better known than the conclusion. Secondly, that the falsity of their contraries ought to be most evident (72a38). Concerning the first he does three things.
Primo, proponit intentum dicens quod quia nos credimus alicui rei conclusae et scimus eam per hoc quod habemus syllogismum demonstrativum, et hoc quidem est in quantum scimus syllogismum demonstrativum; necesse est non solum praecognoscere prima principia conclusioni, sed etiam ea magis cognoscere, quam conclusionem. First (72a25), he states his proposition and says that because we give our assent to a thing which has been concluded and we know it scientifically precisely because we have a demonstrative syllogism (and this insofar as we know the demonstrative syllogism in a scientific way), it is necessary not only to know the first principles of the conclusion beforehand, but also to know them better than we know the conclusion.
Addit autem aut omnia, aut quaedam; quia quaedam principia probatione indigent, ad hoc quod sint nota, et antequam probentur, non sunt magis nota conclusione. Sicut quod, angulus exterior trianguli valeat duos angulos intrinsecos sibi oppositos, antequam probetur, ita ignotum est, sicut quod, triangulus habeat tres angulos aequales duobus rectis. Quaedam vero principia sunt quae, statim proposita, sunt magis nota conclusione. Vel aliter: quaedam conclusiones sunt quae sunt notissimae, utpote per sensum acceptae, sicut quod, sol eclipsetur. Unde principium per quod probatur non est simpliciter magis notum, scilicet quod, luna interponatur inter solem et terram, licet sit magis notum in via rationis procedentis ex causa in effectum. Vel aliter: hoc ideo dicit, quia etiam supra dixerat quod quaedam principia tempore prius cognoscuntur quam conclusio, quaedam vero simul tempore nota fiunt cum conclusione. He adds, “either all or some,” because some principles require proof in order to be known; so that before they are proved, they are not better known than the conclusion. Thus the fact that an exterior angle of a triangle is equal to its two opposite interior angles is, until proved, as unknown as the fact that a triangle has three angles equal to two right angles. But there are other principles which, once they are posited, are better known than the conclusion. Or, in another way, there are some conclusions which are most evident; for example, those based on sense perception, as that the sun is eclipsed. Hence the principle through which this is proved is not better known absolutely—the principle being that the moon is between the sun and the earth—although it is better known within the reasoning process that goes from cause to effect. Or, in another way, he says this because he had said above that in the order of time certain principles are known before the conclusion, but others are known along with the conclusion at the same moment of time.
Secundo; ibi: semper enim propter quod etc., probat propositum dupliciter. Primo ratione ostensiva; sic: propter quod unumquodque et illud est magis; sicut si amamus aliquem, propter alterum; ut si magistrum propter discipulum, discipulum magis amamus. Sed conclusiones scimus et eis credimus propter principia. Ergo magis scimus principia et magis eis credimus, quam conclusioni. Secondly (72a28), he proves his proposition in two ways: first, with an ostensive argument, thus: That in virtue of which something is so, is itself more so; for example, if we love someone because of someone else, as a master because of his disciple, we love the disciple more. But we know conclusions and give our assent to them because of the principles. Therefore, we know the principles with more conviction and give them stronger assent than the conclusion.
Attendendum est autem circa hanc rationem quod causa semper est potior effectu suo. Quando ergo causa et effectus conveniunt in nomine, tunc illud nomen magis praedicatur de causa quam de effectu; sicut ignis est magis calidus quam ea, quae per ignem calefiunt. Quandoque vero causa et effectus non conveniunt in nomine et tunc licet nomen effectus non conveniat causae, tamen convenit ei aliquid dignius; sicut etsi in sole non sit calor, est tamen in eo virtus quaedam, quae est principium caloris. Apropos of this reason it should be noted that a cause is always more noble than its effect. When, therefore, cause and effect have the same name, that name is said principally of the cause rather than of the effect; thus fire is primarily called hot rather than things heated by fire. But sometimes the name of the effect is not attributed to the cause. In that case, although the name the effect has does not belong to the cause, nevertheless, something more noble belongs to it. For example, although the sun does not possess heat, nevertheless, there is in it a certain power which is the principle of heat.
Deinde, cum dicit: non potest autem credere etc., probat idem ratione ducente ad impossibile, quae talis est. Principia praecognoscuntur conclusioni, ut supra habitum est; et sic quando principia cognoscuntur nondum conclusio est cognita. Si igitur principia non essent magis nota quam conclusio, sequeretur quod homo vel plus, vel aequaliter cognosceret ea, quae non novit quam ea quae novit. Hoc autem est impossibile. Ergo impossibile est quod principia non sint magis nota quam conclusio. Then (72a33) he proves the same thing with a principle which leads to an impossibility. He reasons thus: Principles are known prior to the conclusion, as has been shown above; consequently, when the principles are known, the conclusion is not yet known. If, therefore, the principles were not more known than the conclusion, it would follow that a man would know things he does not know either as well as or better than the things he does know. But this is impossible. Therefore, it is also impossible that the principles not be better known than the conclusion.
Littera sic exponitur: neque sciens, neque alius melius dispositus in cognoscendo quam sciens, si contingeret aliquem esse talem (quod dicit propter intelligentem principia, de quo adhuc non est manifestum), non potest magis credere quae non contingunt, scilicet sciri ab eo, his, quae iam scit. Accidet autem hoc, nisi aliquis de numero credentium conclusionem per demonstrationem, praecognoverit, idest magis cognoverit principia. In Graeco planius habetur sic: non est autem possibile credere magis his, quae novit, quae non existit nec sciens, neque melius dispositus quam si contigerit sciens. Phrase by phrase this is explained in the following manner: “A man who knows scientifically or even one who knows in a way superior to this, if such there be,” (he says this, having in mind the person who has the intuition of principles, a state he has not yet explained), “cannot give more credence to things he does not know than to things he does know. But this will be the case if one who assents to a conclusion obtained through demonstration did not foreknow,” i.e., did not know the principles better. In Greek it is stated more clearly: “But no one, whether he has scientific knowledge or that form of knowledge which is better than the scientific (if there be such), can believe anything more firmly than the things he knows.”
Tertio, ibi: magis enim necesse est etc., exponit quod dixerat, dicens quod hoc, quod dictum est, quod magis necesse est credere principiis aut omnibus aut quibusdam quam conclusioni, intelligendum est de illo, qui debet accipere scientiam per demonstrationem. Si enim aliunde conclusio esset nota, sicut per sensum, nihil prohiberet principia non esse magis nota conclusione in via illa. Thirdly (7206), he clarifies what he had said, saying that his statement to the effect that it is more necessary to believe the principles (either all or some) than the conclusion should be understood as referring to a person who is to acquire a discipline through demonstration. For if the conclusion were more known through some other source, such as sense-perception, nothing would preclude the principles not being better known than the conclusion in that case.
Deinde, cum dicit: non solum oportet etc., ostendit quod non solum oportet magis cognoscere principia quam conclusionem demonstrandam, sed etiam nihil debet esse certius quam quod opposita principiis sint falsa. Et hoc ideo, quia oportet scientem non esse incredibilem principiis, sed firmissime eis assentire. Quicunque autem dubitat de falsitate unius oppositorum, non potest firmiter inhaerere opposito: quia semper formidat de veritate alterius oppositi. Then (72a38) he shows that it is not only necessary to know the principles more than the demonstrative conclusion, but nothing should be more certain than the fact that the opposites of the principles are false. And this because the scientific knower must not disbelieve the principles, but assent to them most firmly. But anyone who doubts the falseness of one of two opposites cannot assent firmly to the other, because he will always fear that the opposite one might be true.

Lectio 7
Caput 3
Ἐνίοις μὲν οὖν διὰ τὸ δεῖν τὰ πρῶτα ἐπίστασθαι οὐ δοκεῖ ἐπιστήμη εἶναι, τοῖς δ' εἶναι μέν, πάντων μέντοι ἀπόδειξις εἶναι· ὧν οὐδέτερον οὔτ' ἀληθὲς οὔτ' ἀναγκαῖον. b5. Some hold that, owing to the necessity of knowing the primary premisses, there is no scientific knowledge. Others think there is, but that all truths are demonstrable. Neither doctrine is either true or a necessary deduction from the premisses.
οἱ μὲν γὰρ ὑποθέμενοι μὴ εἶναι ὅλως ἐπίστασθαι, οὗτοι εἰς ἄπειρον ἀξιοῦσιν ἀνάγεσθαι ὡς οὐκ ἂν ἐπισταμένους τὰ ὕστερα διὰ τὰ πρότερα, ὧν μὴ ἔστι πρῶτα, ὀρθῶς λέγοντες· ἀδύνατον γὰρ τὰ ἄπειρα διελθεῖν. εἴ τε ἵσταται καὶ εἰσὶν ἀρχαί, ταύτας ἀγνώστους εἶναι ἀποδείξεώς γε μὴ οὔσης αὐτῶν, ὅπερ φασὶν εἶναι τὸ ἐπίστασθαι μόνον· εἰ δὲ μὴ ἔστι τὰ πρῶτα εἰδέναι, οὐδὲ τὰ ἐκ τούτων εἶναι ἐπίστασθαι ἁπλῶς οὐδὲ κυρίως, ἀλλ' ἐξ ὑποθέσεως, εἰ ἐκεῖνα ἔστιν. b8. The first school, assuming that there is no way of knowing other than by demonstration, maintain that an infinite regress is involved, on the ground that if behind the prior stands no primary, we could not know the posterior through the prior (wherein they are right, for one cannot traverse an infinite series): if on the other hand — they say — the series terminates and there are primary premisses, yet these are unknowable because incapable of demonstration, which according to them is the only form of knowledge. And since thus one cannot know the primary premisses, knowledge of the conclusions which follow from them is not pure scientific knowledge nor properly knowing at all, but rests on the mere supposition that the premisses are true.
οἱ δὲ περὶ μὲν τοῦ ἐπίστασθαι ὁμολογοῦσι· δι' ἀποδείξεως γὰρ εἶναι μόνον· ἀλλὰ πάντων εἶναι ἀπόδειξιν οὐδὲν κωλύειν· ἐνδέχεσθαι γὰρ κύκλῳ γίνεσθαι τὴν ἀπόδειξιν καὶ ἐξ ἀλλήλων. b15. The other party agree with them as regards knowing, holding that it is only possible by demonstration, but they see no difficulty in holding that all truths are demonstrated, on the ground that demonstration may be circular and reciprocal.
Ἡμεῖς δέ φαμεν οὔτε πᾶσαν ἐπιστήμην ἀποδεικτικὴν εἶναι, ἀλλὰ τὴν τῶν ἀμέσων ἀναπόδεικτον (καὶ τοῦθ' ὅτι ἀναγκαῖον, φανερόν· εἰ γὰρ ἀνάγκη μὲν ἐπίστασθαι τὰ πρότερα καὶ ἐξ ὧν ἡ ἀπόδειξις, ἵσταται δέ ποτε τὰ ἄμεσα, ταῦτ' ἀναπόδεικτα ἀνάγκη εἶναι)- ταῦτά τ' οὖν οὕτω λέγομεν, καὶ οὐ μόνον ἐπιστήμην ἀλλὰ καὶ ἀρχὴν ἐπιστήμης εἶναί τινά φαμεν, ᾗ τοὺς ὅρους γνωρίζομεν. b18. Our own doctrine is that not all knowledge is demonstrative: on the contrary, knowledge of the immediate premisses is independent of demonstration. (The necessity of this is obvious; for since we must know the prior premisses from which the demonstration is drawn, and since the regress must end in immediate truths, those truths must be indemonstrable.) Such, then, is our doctrine, and in addition we maintain that besides scientific knowledge there is its originative source which enables us to recognize the definitions.
Postquam determinavit philosophus de cognitione principiorum demonstrationis, hic excludit errores ex praedeterminata veritate exortos. Et circa hoc tria facit: primo, ponit errores; secundo, rationes errantium; ibi: supponentes quidem igitur etc., tertio, excludit rationum radices; ibi: nos autem dicimus et cetera. After determining about the knowledge of the principles of demonstration, the Philosopher now excludes the errors which have arisen from these determinations. Concerning this he does three things. First, he states the errors. Secondly, the reasons they erred (72b8). Thirdly, he removes the roots of these reasons (72b 18).
Dicit ergo primo quod ex una veritate superius determinata duo errores contrarii sunt exorti. Determinatum est enim supra quod oportet principia demonstrationis scire, immo quod etiam ea magis scire oportet. Sed primum solum sufficit ad propositum. Propter hoc autem videtur quibusdam quod nullius rei sit scientia; quibusdam autem videtur quod sit quidem scientia, sed quod omnium possit haberi scientia per demonstrationem. Neutrum autem horum est verum, nec necessario consequitur ex rationibus eorum. He says therefore first (72b5), that two contrary errors have arisen from one of the truths established above. For it has been established above that the principles of demonstration must be known and must be even better known. But the first of these is sufficient for our purpose. For some, basing themselves on this first statement, have come to believe that there is no science of anything, whereas others believe that there is science, even to the extent of believing that there is science of everything through demonstration. But neither of these positions is true and neither follows necessarily from their reasons.
Deinde, cum dicit: supponentes etc., ponit rationes quibus in praedictos errores incidunt. Et primo ponit rationem dicentium quod non est scientia. Quae talis est. Principia demonstrationis aut procedunt in infinitum aut est status in eis. Si proceditur in infinitum, non est in eis accipere prima: quia infinita non est transire, ut ad primum veniatur: et ita non est primum cognoscere. Et in hoc recte argumentantur. Nam posteriora non possunt cognosci, ignoratis primis. Then (72b8) he presents the reasons why they have fallen into these errors. And first of all he presents the reason given by those who say that there is no science, and it is this: The principles of demonstration either proceed to infinity or there is a halt somewhere. But if there is a process to infinity, nothing in that process can be taken as being first, because one cannot exhaust an infinite series and reach what is first. Consequently, it is not possible to know what is first. (They are correct in thus arguing, for the later things cannot be known unless the prior ones are known).
Si autem stetur in principiis, oportet quod prima nesciantur; si scire solum est per demonstrationem: non enim prima habent aliqua priora, per quae demonstrentur. Si autem prima ignorentur, oportet et posteriora iterum non scire simpliciter nec proprie: sed solum sub hac conditione, si principia sunt. Non enim potest per aliquid ignotum aliquid cognosci, nisi sub hac conditione, si illud primum quod ignotum est, sit. Ergo sequitur utroque modo, sive principia stent, sive in eis procedatur in infinitum, quod nullius rei est scientia. On the other hand, if there is a halt in the principles, then even so, the first things are still not known, if the only way to know scientifically is through demonstration. For first things do not have prior principles through which they are demonstrated. But if the first things are not known, it follows again that the later things are not known in the strict and proper sense, but only on condition that there are principles. For it is not possible for something to be known in virtue of something not known, except on condition that that unknown be a principle. So in either case, whether the principles stop or go on to infinity, it follows that there is no science of anything.
Secundo, cum dicit: quidam autem etc., ponit rationem dicentium omnium esse scientiam per demonstrationem: quia praemissae radici, scilicet quod non esset scire nisi per demonstrationem, addebant aliam, scilicet quod posset circulariter demonstrari. Sic enim sequebatur quod etiamsi in principiis demonstrationis esset status, prima tamen principia erat scire per demonstrationem: quia illa prima dicebant demonstrari ex posterioribus. Nam circulariter demonstrare est demonstrare ex invicem, idest ut quod primo fuit principium, postmodum fiat conclusio et e converso. Secondly (72b15), he presents the reasoning of those who say that there is science of everything through demonstration, because to their there is science of everything through demonstration, because to their basic premise-the only way to know scientifically is by demonstration—they added another, namely, that one may demonstrate circularly. From these premises it followed that even if a limit is reached in the series of the principles of demonstration, the first principles are still known through demonstration, because, they said, those principles were demonstrated by previous ones. For a circular demonstration is one which is reciprocal, i.e., something which was first a principle is later a conclusion, and vice versa.
Deinde, cum dicit: nos autem dicimus etc., excludit falsas radices praedictarum rationum. Et primo, hoc quod supponebant quod non esset scire nisi per demonstrationem. Secundo, hoc quod dicebant quod contingeret circulariter demonstrari; ibi: circulo quoque et cetera. Then (72b18) he cuts away the false bases of these arguments. First, their supposition that the only way to know scientifically is by demonstration. Secondly, their statement that it is legitimate to demonstrate circularly (72b25).
Dicit ergo primo quod non omnis scientia est demonstrativa, idest per demonstrationem accepta; sed immediatorum principiorum est scientia indemonstrabilis, idest non per demonstrationem accepta. Sciendum est tamen quod hic Aristoteles large accipit scientiam pro qualibet certitudinali cognitione, et non secundum quod scientia dividitur contra intellectum, prout dicitur quod, scientia est conclusionum et intellectus principiorum. He says therefore first (72b18), that not all scientific knowledge is demonstrative, i.e., obtained through demonstration, but the scientific knowledge of immediate principles is indemonstrable, i.e., not obtained by demonstration. However, it should be noted that Aristotle is here taking science in a wide sense to include any knowledge that is certain, and not in the sense in which science is set off against understanding, according to the dictum that science deals with conclusions and understanding [intuition] with principles.
Quod autem hoc necessarium sit, scilicet quod certa cognitio aliquorum habeatur sine demonstratione, sic probat. Necesse est scire priora, ex quibus est demonstratio; sed haec aliquando contingit reducere in aliqua immediata: alias oporteret dicere quod inter duo extrema, scilicet subiectum et praedicatum essent infinita media in actu; immo plus, quod non esset aliqua duo accipere, inter quae non essent infinita media. Qualitercunque autem media assumantur, est accipere aliquid alteri immediatum. Immediata autem, cum sint priora, oportet esse indemonstrabilia. Et ita patet quod necesse est aliquorum scientiam habere sine demonstratione. But that it is necessary for some things to be held as certain without demonstration he proves in the following way: It is necessary that the prior things from which a demonstration proceeds be known in a scientific way. Furthermore, these must be ultimately reduced to something immediate; otherwise one would be forced to admit that there is an actual infinitude of middles between two extremes—in this case between the subject and predicate. Again, one would have to admit that no two extremes could be found between which there would not be an infinitude of middles. But as it is, the middles are such that it is possible to find two things which are immediate. But immediate principles, being prior, must be indemonstrable. Thus it is clear that it is necessary for some things to be scientifically known without demonstration.
Si ergo quaeratur quomodo immediatorum habeatur scientia, respondet quod non solum eorum est scientia, immo eorum cognitio est principium quoddam totius scientiae. Nam ex cognitione principiorum derivatur cognitio conclusionum, quarum proprie est scientia. Ipsa autem principia immediata non per aliquod medium extrinsecum cognoscuntur, sed per cognitionem propriorum terminorum. Scito enim quid est totum et quid est pars, cognoscitur quod omne totum est maius sua parte: quia in talibus propositionibus, ut supra dictum est, praedicatum est de ratione subiecti. Et ideo rationabiliter cognitio horum principiorum est causa cognitionis conclusionum: quia semper, quod est per se, est causa eius, quod est per aliud. Therefore, if someone were to ask how the science of immediate principles is possessed, the answer would be that not only are they known in a scientific manner, but knowledge of them is the source of an science. For one passes from the knowledge of principles to a demonstration of conclusion on which science, properly speaking, bears. But those immediate principles are not made known through an additional middle but through an understanding of their own terms. For as soon as it is known what a whole is and what a part is, it is known that every whole is greater than its part, because in such a proposition, as has been stated above, the predicate is included in the very notion of the subject. And therefore it is reasonable that the knowledge of these principles is the cause of the knowledge of conclusions, because always, that which exists in virtue of itself is the cause of that which exists in virtue of something else.

Lectio 8
Caput 3 cont.
κύκλῳ τε ὅτι ἀδύνατον ἀποδείκνυσθαι ἁπλῶς, δῆλον, εἴπερ ἐκ προτέρων δεῖ τὴν ἀπόδειξιν εἶναι καὶ γνωριμωτέρων· ἀδύνατον γάρ ἐστι τὰ αὐτὰ τῶν αὐτῶν ἅμα πρότερα καὶ ὕστερα εἶναι, εἰ μὴ τὸν ἕτερον τρόπον, οἷον τὰ μὲν πρὸς ἡμᾶς τὰ δ' ἁπλῶς, ὅνπερ τρόπον ἡ ἐπαγωγὴ ποιεῖ γνώριμον. εἰ δ' οὕτως, οὐκ ἂν εἴη τὸ ἁπλῶς εἰδέναι καλῶς ὡριμένον, ἀλλὰ διττόν· ἢ οὐχ ἁπλῶς ἡ ἑτέρα ἀπόδειξις, γινομένη γ' ἐκ τῶν ἡμῖν γνωριμωτέρων. b25. Now demonstration must be based on premisses prior to and better known than the conclusion; and the same things cannot simultaneously be both prior and posterior to one another: so circular demonstration is clearly not possible in the unqualified sense of 'demonstration', but only possible if 'demonstration' be extended to include that other method of argument which rests on a distinction between truths prior to us and truths without qualification prior, i.e. the method by which induction produces knowledge. But if we accept this extension of its meaning, our definition of unqualified knowledge will prove faulty; for there seem to be two kinds of it. Perhaps, however, the second form of demonstration, that which proceeds from truths better known to us, is not demonstration in the unqualified sense of the term.
συμβαίνει δὲ τοῖς λέγουσι κύκλῳ τὴν ἀπόδειξιν εἶναι οὐ μόνον τὸ νῦν εἰρημένον, ἀλλ' οὐδὲν ἄλλο λέγειν ἢ ὅτι τοῦτ' ἔστιν εἰ τοῦτ' ἔστιν· οὕτω δὲ πάντα ῥᾴδιον δεῖξαι. δῆλον δ' ὅτι τοῦτο συμβαίνει τριῶν ὅρων τεθέντων. τὸ μὲν γὰρ διὰ πολλῶν ἢ δι' ὀλίγων ἀνακάμπτειν φάναι οὐδὲν διαφέρει, δι' ὀλίγων δ' ἢ δυοῖν. b33. The advocates of circular demonstration are not only faced with the difficulty we have just stated: in addition their theory reduces to the mere statement that if a thing exists, then it does exist — an easy way of proving anything. That this is so can be clearly shown by taking three terms, for to constitute the circle it makes no difference whether many terms or few or even only two are taken.
ὅταν γὰρ τοῦ Α ὄντος ἐξ ἀνάγκης ᾖ τὸ Β, τούτου δὲ τὸ Γ, τοῦ Α ὄντος ἔσται τὸ Γ. b38. Thus by direct proof, if A is, B must be; if B is, C must be; therefore if A is, C must be.
εἰ δὴ τοῦ Α ὄντος ἀνάγκη τὸ Β εἶναι, τούτου δ' (73a.) ὄντος τὸ Α (τοῦτο γὰρ ἦν τὸ κύκλῳ), κείσθω τὸ Α ἐφ' οὗ τὸ Γ. τὸ οὖν τοῦ Β ὄντος τὸ Α εἶναι λέγειν ἐστὶ τὸ Γ εἶναι λέγειν, τοῦτο δ' ὅτι τοῦ Α ὄντος τὸ Γ ἔστι· τὸ δὲ Γ τῷ Α τὸ αὐτό. ὥστε συμβαίνει λέγειν τοὺς κύκλῳ φάσκοντας εἶναι τὴν ἀπόδειξιν οὐδὲν ἕτερον πλὴν ὅτι τοῦ Α ὄντος τὸ Α ἔστιν. οὕτω δὲ πάντα δεῖξαι ῥᾴδιον. a1. Since then — by the circular proof — if A is, B must be, and if B is, A must be, A may be substituted for C above. Then 'if B is, A must be'='if B is, C must be', which above gave the conclusion 'if A is, C must be': but C and A have been identified. Consequently the upholders of circular demonstration are in the position of saying that if A is, A must be — a simple way of proving anything.
Οὐ μὴν ἀλλ' οὐδὲ τοῦτο δυνατόν, πλὴν ἐπὶ τούτων ὅσα ἀλλήλοις ἕπεται, ὥσπερ τὰ ἴδια. ἑνὸς μὲν οὖν κειμένου δέδεικται ὅτι οὐδέποτ' ἀνάγκη τι εἶναι ἕτερον (λέγω δ' ἑνός, ὅτι οὔτε ὅρου ἑνὸς οὔτε θέσεως μιᾶς τεθείσης), ἐκ δύο δὲ θέσεων πρώτων καὶ ἐλαχίστων ἐνδέχεται, εἴπερ καὶ συλλογίσασθαι. ἐὰν μὲν οὖν τό τε Α τῷ Β καὶ τῷ Γ ἕπηται, καὶ ταῦτ' ἀλλήλοις καὶ τῷ Α, οὕτω μὲν ἐνδέχεται ἐξ ἀλλήλων δεικνύναι πάντα τὰ αἰτηθέντα ἐν τῷ πρώτῳ σχήματι, ὡς δέδεικται ἐν τοῖς περὶ συλλογισμοῦ. δέδεικται δὲ καὶ ὅτι ἐν τοῖς ἄλλοις σχήμασιν ἢ οὐ γίνεται συλλογισμὸς ἢ οὐ περὶ τῶν ληφθέντων. τὰ δὲ μὴ ἀντικατηγορούμενα οὐδαμῶς ἔστι δεῖξαι κύκλῳ, ὥστ' ἐπειδὴ ὀλίγα τοιαῦτα ἐν ταῖς ἀποδείξεσι, φανερὸν ὅτι κενόν τε καὶ ἀδύνατον τὸ λέγειν ἐξ ἀλλήλων εἶναι τὴν ἀπόδειξιν καὶ διὰ τοῦτο πάντων ἐνδέχεσθαι εἶναι ἀπόδειξιν. a6. Moreover, even such circular demonstration is impossible except in the case of attributes that imply one another, viz. 'peculiar' properties. Now, it has been shown that the positing of one thing — be it one term or one premiss — never involves a necessary consequent: two premisses constitute the first and smallest foundation for drawing a conclusion at all and therefore a fortiori for the demonstrative syllogism of science. If, then, A is implied in B and C, and B and C are reciprocally implied in one another and in A, it is possible, as has been shown in my writings on the syllogism, to prove all the assumptions on which the original conclusion rested, by circular demonstration in the first figure. But it has also been shown that in the other figures either no conclusion is possible, or at least none which proves both the original premisses. Propositions the terms of which are not convertible cannot be circularly demonstrated at all, and since convertible terms occur rarely in actual demonstrations, it is clearly frivolous and impossible to say that demonstration is reciprocal and that therefore everything can be demonstrated.
Postquam philosophus exclusit unam falsam radicem, ostendens quod non omnis scientia est per demonstrationem, hic excludit aliam, ostendens quod non contingit circulariter demonstrare. After excluding one false basis by showing that not all science depends on demonstration, the Philosopher now excludes another by showing that it is not possible to demonstrate circularly.
Ad cuius evidentiam sciendum est quod circularis syllogismus dicitur, quando ex conclusione et altera praemissarum conversa concluditur reliqua. Sicut si fiat talis syllogismus: To understand this it should be noted that a demonstration is circular when the conclusion and one of the premises (in converted form) of a syllogism are used to prove the other premise. For example, we might form the following syllogism:
omne animal rationale mortale est risibile;
omnis homo est animal rationale mortale;
ergo omnis homo est risibilis:
Every rational mortal animal is risible;
Every man is a rational mortal animal:
Therefore, every man is risible.
assumatur autem conclusio tanquam principium, et adiungatur ei minor conversa, hoc modo:
omnis homo est risibilis;
omne animal rationale mortale est homo;
sequitur quod, omne animal rationale mortale sit risibile: quae erat maior primi syllogismi.
Now if the conclusion were to be used as one principle and the minor in converted form as the other, we would get:
Every man is risible;
Every rational mortal animal is a man:
Therefore, every rational mortal animal is risible—which was the major of the first syllogism.
Ostendit autem Aristoteles per tres rationes quod non contingit circulariter demonstrare. Quarum prima est. In circulari syllogismo idem fit et principium et conclusio. Principium autem demonstrationis est prius et notius conclusione: ut supra ostensum est. Sequitur ergo quod idem sit prius et posterius respectu unius et eiusdem, et notius et minus notum. Hoc autem est impossibile. Ergo impossibile est circulariter demonstrare. Accordingly, he presents three arguments to show that it is not possible to demonstrate circularly. The first of these (72b25) is this: In a circular syllogism the same thing is at once a conclusion and a principle. But a principle of a demonstration is prior to and better known than the conclusion, as has been shown above. Therefore, it follows that a same thing is both prior to and subsequent to one same thing, and also more known and less known. But this is impossible. Therefore, it is impossible to demonstrate circularly.
Sed posset aliquis dicere quod idem potest esse prius et posterius alio et alio modo, scilicet ut hoc sit prius quoad nos, et illud sit prius simpliciter. Sicut singularia sunt priora quoad nos, et posteriora simpliciter; universalia vero e converso. Hoc autem modo inductio facit notum, scilicet altero modo a demonstratione. Nam demonstratio procedit ex prioribus simpliciter: inductio autem ex prioribus quoad nos. But someone might say that a same thing can be both prior and subsequent, although not in the same way. For example, this might be prior in reference to us, but that prior absolutely. Thus singulars are prior in reference to us and subsequent absolutely: and conversely for universals. Again, induction makes something known in one way and demonstration in another way. For demonstration proceeds from things that are prior absolutely, but induction from things that are prior in reference to us.
Sed si sic fieret demonstratio circularis, ut scilicet primo concluderetur ex prioribus simpliciter, postea vero ex prioribus quoad nos; sequeretur quod non esset bene determinatum superius quid est scire. Dictum est enim quod scire est causam rei cognoscere. Et ideo ostensum est quod oportet demonstrationem, quae facit scire, ex prioribus simpliciter procedere. Si autem demonstratio, nunc ex prioribus simpliciter, nunc ex prioribus quoad nos procederet; oporteret etiam quod scire non solum esset causam rei cognoscere, sed dupliciter diceretur quia esset etiam quoddam scire per posteriora. Aut ergo oportebit sic dicere, aut oportebit dicere quod altera demonstratio, quae fit ex nobis notioribus, non sit simpliciter demonstratio. Now if a circular demonstration were so constructed that something is first concluded from things that are absolutely prior, and then from things that are prior in reference to us, it would follow that our doctrine on scientific knowing was not well established. For we stated that to know scientifically is to know the cause of a thing. From this it followed that a demonstration which causes scientific knowledge must proceed from the absolutely prior. But if demonstration were at one time to proceed from the absolutely prior and at another time from things which are prior in reference to us, we would be forced to admit that scientific knowing is not confined to knowing the cause of a thing, but that there is another, namely, that form of knowing which proceeds from what is later. Therefore, one must either admit both or admit that the second form, namely, the demonstration which proceeds from what is better known to us is not a demonstration in the absolute sense.
Ex his autem apparet quare dialecticus syllogismus potest esse circularis. Procedit enim ex probabilibus. Probabilia autem dicuntur, quae sunt magis nota vel sapientibus vel pluribus. Et sic dialecticus syllogismus procedit ex his quae sunt magis nobis nota. Contingit autem idem esse magis et minus notum quoad diversos; et ideo nihil prohibet syllogismum dialecticum fieri circularem. Sed demonstratio fit ex notioribus simpliciter; et ideo, ut dictum est, non potest fieri demonstratio circularis. The aforesaid also reveals why a dialectical syllogism can be circular. For it proceeds from things which are probable. But things are said to be probable if they are better known to the wise or to a great number of persons. Consequently, a dialectical syllogism proceeds from things that are better known to us. However, it happens that a same thing is better known to some and less known to others. Consequently, there is nothing to hinder a dialectical syllogism from being circular. But a demonstration is formed from things that are absolutely prior. Therefore, as we have already stated, there cannot be circular demonstration.
Secundam rationem ponit ibi: accidit autem etc.: quae talis est. Si demonstratio sit circularis, sequitur quod in demonstratione probetur idem per idem; ut si dicamus: si est hoc, est hoc. Sic autem facile est cuilibet demonstrare omnia. Hoc enim poterit facere quilibet, tam sciens quam ignorans. Et sic per demonstrationem non acquiritur scientia: quod est contra definitionem demonstrationis. Then he sets forth the second argument (72b33) and it is this: If there were circular demonstration, it would follow that a same thing is demonstrated by the same thing, as if I were to say: If it is this, it is this. In this way it is easy for anyone to demonstrate everything, for anyone, wise or ignorant, will be able to do this. Accordingly, science is not acquired through demonstration. But this is against the definition of demonstration. Therefore, there cannot be circular demonstration.
Non ergo potest esse demonstratio circularis. Veritatem autem primae consequentiae hoc modo ostendit. Primo enim, dicit quod manifestum est quod accidit, circulari demonstratione facta, hoc quod prius dictum est, scilicet quod idem probetur per idem, si quis sumat tres terminos. Reflexionem autem fieri per multos, aut per paucos terminos nihil differt (nominat autem hic reflexionem processum, qui fit in demonstratione circulari a principio ad conclusionem, et iterum a conclusionem ad principium.) Huiusmodi ergo reflexio quantum ad vim argumentandi, sive fiat per multa, sive per pauca, non differt. Nec differt de paucis, aut de duobus. Eadem enim virtus arguendi est, si quis sic procedat: si est a, est b, et si est b, est c, et si est c, est d; et iterum reflectat dicens: si est d, est c, et si est c, est b, et si est b, est a; sicut si statim a principio reflecteret dicens: si est a, est b, et si est b, est a. Dicit autem per duos terminos, cum supra dixerit, tribus terminis positis, quia in deductione, quam faciet, utetur tertio termino, qui sit idem cum primo. He proves the truth of the first consequence in the following way: It is obvious, first of all, that with a circular demonstration the same thing is proved by a same thing, as has been stated above, i.e., if only three terms are employed; although it makes no difference whether the reflexion be made with fewer terms or more. (By reflexion he means the process whereby one goes from principle to conclusion in a demonstration, and then from conclusion to principle). In such a reflexion it makes no difference, so far as the force of the argument is concerned, whether it involves several or fewer terms or even two. For an argument has the same force if one proceeds thus: “If it is A, it is B, and if it is B, it is C, and if it is C, it is D,” and then by reflecting continues, “If it is D, it is C, and if it is C, it is B, and if it is B, it is A”; or if he proceeds by reflecting at the very start, saying: “If it is A, it is B, and if it is B, it is A.” (Although he spoke above of three terms, he restricted himself to two terms in this example, because in the deduction he is about to make he will use a third term, which is the same as the first).
Deinde, cum dicit: cum enim etc., proponit formam argumentandi in tribus terminis, hoc modo: si sit a, est b; et si est b, est c; ergo si est a, ex necessitate est c. Then (72b38) he gives the form of the argument in three terms, namely: “If it is A, it is B, and if it is B, it is C; therefore, if it is A, it is of necessity C.”
Deinde, cum dicit: si igitur cum sit a etc., per formam argumentandi praemissam ostendit quod in circulari demonstratione concluditur idem per idem, sumptis duobus terminis tantum. Erit enim dicere: si est a, est b; et reflectendo (quod est circulariter demonstrare): si est b, est a. Ex quibus duobus sequitur, secundum formam arguendi praemissam, si est a, est a. Then (73a1) he shows by the aforesaid form of arguing that in a circular demonstration a same thing is proved by a same thing, using only two terms. For it consists in saying, “If it is A, it is B,” and then reflecting, “If it is B, it is A”—which is a circular demonstration. Now according to the above given form it follows from these two, that “if it is A, it is A.”
Quod sic patet. Sicut enim in prima deductione, quae fiebat per tres terminos, ad b sequebatur c; ita in deductione reflexa duorum terminorum, ad b sequitur a. Ponatur ergo quod idem significet a in secunda deductione reflexa, quod c significabat in prima directa, quae est per tres terminos. Igitur dicere in secunda deductione, si est b, est a, est hoc ipsum quod erat dicere in prima deductione, si est b, est c. Sed cum dicebatur in prima deductione, si est b, est c, sequebatur, si est a, est c. Ergo in deductione circulari sequitur, si est a, est a: quia c cum a idem ponitur. Et ita leve erit demonstrare omnia, ut dictum est. That it does follow is obvious: for just as in the first deduction which involved three terms’ C followed from B, so in the reflex deduction of two terms, A followed from B. Let us suppose, then, that the A of the second deduction, i.e., the reflex, signifies the same thing that C signified in the first, i.e., in the direct deduction which was composed of three terms. Therefore, to state in the second deduction that “if it is B it is A” is to state the same thing as was stated in the first deduction, namely, that “if it is B, it is C.” But when it was stated in the first deduction that “if it is B, it is C,” it followed that “if it is A, it is C.” Therefore, in the circular deduction it follows that “if it is A, it is A,” since C is assumed to be the same as A. In this way, it will be easy to demonstrate all things, as has been said.
Tertiam rationem ponit; ibi: at vero nec hoc possibile est et cetera. Quae talis est. Ponentes omnia posse sciri per demonstrationem, quia demonstratio est circularis, necesse habent dicere quod omnia possunt demonstrari demonstratione circulari; et ita necesse habent dicere quod in demonstratione circulari ex conclusione possit concludi utraque praemissarum. Hoc autem non potest fieri nisi in his, quae ad se invicem convertuntur, idest convertibilia sunt, sicut propria. Sed non omnia sunt huiusmodi. Ergo vanum est dicere quod omnia possunt demonstrari, propter hoc quod demonstratio est circularis. Then he presents the third argument (73a6) which is this: Those who suppose that everything can be known through demonstration on the ground that demonstration is circular, must grant that anything can be demonstrated by a circular demonstration and, as a consequence, grant that in a circular demonstration each of the premises can be concluded from the conclusion. However, the only cases in which this can be done are those in which mutual conversion is possible, i.e., in things that are convertible, as properties. But not all things are so related. Therefore, it is ridiculous to say that everything can be demonstrated on the ground that there are such things as circular demonstrations.
Quod autem oporteat in demonstratione circulari omnia esse convertibilia secundum positionem istorum, sic ostendit. Ostensum est in libro priorum quod, uno posito, non sequitur ex necessitate aliud; nec posito uno termino, nec posita una propositione tantum. Nam omnis syllogismus est ex tribus terminis et duabus propositionibus ad minus. Oportet ergo accipere tres terminos convertibiles in demonstratione circulari, scilicet a, b, c; ita quod a insit omni b et omni c, et haec scilicet b et c inhaereant sibi invicem, ita quod omne b sit c, et omne c sit b, et iterum haec insint ipsi a, ita quod omne a sit b, et omne a sit c. Et ita se habentibus terminis, contingit monstrare in prima figura ex alterutris, idest circulariter, omnia quaesita, idest conclusionem ex duabus praemissis et utramlibet praemissarum ex conclusione et altera conversa, sicut ostensum est in his, quae sunt de syllogismo, idest in libro priorum, in quo agitur de syllogismo simpliciter. Now the reason is obvious why in a circular demonstration all the propositions must be convertible. For it has been shown in the book of Prior Analytics that if one thing is laid down, another does not follow of necessity, whether the thing laid down be one term or one proposition. For every syllogism must start with three terms and two propositions as a minimum. Therefore, in a circular demonstration three terms which are convertible must be taken, namely, A, B, C, such that A is in every B and in every C, and these, namely, B and C, must inhere in each other, so that every B is C and every C is B, and also inhere in A so that every A is B and every A is C. And so, the terms being thus related, it is possible, when using the first figure, to derive any one from any two circularly, i.e., the conclusion from two premises and each premise from the conclusion and the remaining premise, as we pointed out in the Prior Analytics, where we treated the syllogism formally.
Quod sic patet. Sumantur tres termini convertibiles, scilicet risibile, animal rationale mortale, et homo, et syllogizetur sic: The way it is done is this: take the three convertible terms, “risible,” “rational mortal animal” and “man,” and form the syllogism:
omne animal rationale mortale est risibile;
omnis homo est animal rationale mortale;
ergo omnis homo est risibilis.
Every rational mortal animal is risible;
Every man is a rational mortal animal:
Therefore, every man is risible.
Et ex hac conclusione potest iterum concludi tam maior quam minor. Maior sic: Then from the conclusion it is possible to conclude both the major and the minor; the major thus:
omnis homo est risibilis;
omne animal rationale mortale est homo;
ergo omne animal rationale mortale est risibile.
Every man is risible;
But every rational mortal animal is a man:
Therefore, every rational mortal animal is risible
Minor sic: and the minor thus:
omne risibile est animal rationale mortale;
omnis homo est risibilis;
ergo omnis homo est animal rationale mortale.
Every risible is a rational mortal animal;
But every man is risible:
Therefore, every man is a rational mortal animal.
Ostensum est autem in libro priorum quod in aliis figuris, scilicet in secunda et tertia, aut non fit circularis syllogismus, scilicet per quem ex conclusione possit syllogizari utraque praemissarum: aut si fiat, non erit ex acceptis, sed ex aliis propositionibus, quae non sumuntur in primo syllogismo. However, it has also been proved in the Prior Analytics that in figures other than the first, namely, in the second and third, one cannot form a circular syllogism, i.e., one through which each of the premises can be syllogized from the conclusion; or if one is formed, it is done not by using the premises already used but by using propositions other than those which appear in the first syllogism.
Quod sic patet. In secunda figura non est conclusio nisi negativa: oportet autem alteram praemissarum esse negativam, et alteram affirmativam. Non enim ex duabus negativis potest aliquid concludi, nec ex duabus affirmativis potest concludi negativa. Non est ergo possibile quod ex conclusione et praemissa negativa concludatur affirmativa. Si ergo debet affirmativa probari, oportet quod per alias propositiones probetur, quae non sunt sumptae. Similiter in tertia figura non est conclusio nisi particularis. Oportet autem alteram praemissarum ad minus esse universalem. Si autem in praemissis sit aliqua particularis, non potest concludi universalis. Unde non potest esse quod in tertia figura ex conclusione syllogizetur utralibet praemissarum. That this is so is obvious. For the second figure always yields a negative conclusion. Consequently, one premise must be affirmative and the other negative. However, it is true that if both are negative, nothing can be concluded; and if both are affirmative, a negative conclusion cannot follow. Therefore, it is not possible to use the negative conclusion and the negative premise to obtain the affirmative premise as a conclusion. Hence, if this affirmative is to be proved, it must be proved through propositions other than the ones originally used. Again, in the third figure the only conclusion ever obtained is particular. However, at least one premise must be universal; furthermore, if either premise is particular, a universal cannot be concluded. Hence it cannot occur that in the third figure each of the premises can be syllogized from the conclusion and the remaining premise.
Et eadem ratione apparet quod nec in prima figura talis circularis syllogismus potest fieri, per quem utraque praemissarum concludatur, nisi in primo modo, in quo solo concluditur universalis affirmativa. Nec etiam in hoc modo potest fieri talis syllogismus circularis, per quem utraque praemissarum concludatur, nisi sumantur tres termini aequales, idest convertibiles. Quod ex hoc patet. Oportet enim ex conclusione, et altera praemissarum conversa, concludere reliquam: sicut dictum est. Non autem potest utraque praemissarum converti, cum utraque sit universalis affirmativa, nisi in terminis convertibilibus. For the same reasons it is obvious that such a circular syllogism (through which each premise could be concluded) cannot be formed in the first figure except in the first mode, which is the only one that concludes to a universal affirmative. Furthermore, even in this mode the only case in which a circular syllogism could be formed such that each of the premises could be concluded, is when the three terms employed are equal, i.e., convertible. The proof is this: The premise must be concluded from the conclusion and the converse of the other premise, as has been stated. But such a conversion of each premise is impossible (for each is universal), except when the terms happen to be equal.

Lectio 9
Caput 4
Ἐπεὶ δ' ἀδύνατον ἄλλως ἔχειν οὗ ἔστιν ἐπιστήμη ἁπλῶς, ἀναγκαῖον ἂν εἴη τὸ ἐπιστητὸν τὸ κατὰ τὴν ἀποδεικτικὴν ἐπιστήμην· ἀποδεικτικὴ δ' ἐστὶν ἣν ἔχομεν τῷ ἔχειν ἀπόδειξιν. ἐξ ἀναγκαίων ἄρα συλλογισμός ἐστιν ἡ ἀπόδειξις. a2l. Since the object of pure scientific knowledge cannot be other than it is, the truth obtained by demonstrative knowledge will be necessary. And since demonstrative knowledge is only present when we have a demonstration, it follows that demonstration is an inference from necessary premisses. So we must consider what are the premisses of demonstration — i.e. what is their character:
ληπτέον ἄρα ἐκ τίνων καὶ ποίων αἱ ἀποδείξεις εἰσίν. πρῶτον δὲ διορίσωμεν τί λέγομεν τὸ κατὰ παντὸς καὶ τί τὸ καθ' αὑτὸ καὶ τί τὸ καθόλου. a25. and as a preliminary, let us define what we mean by an attribute 'true in every instance of its subject', an 'essential' attribute, and a 'commensurate and universal' attribute.
Κατὰ παντὸς μὲν οὖν τοῦτο λέγω ὃ ἂν ᾖ μὴ ἐπὶ τινὸς μὲν τινὸς δὲ μή, μηδὲ ποτὲ μὲν ποτὲ δὲ μή, οἷον εἰ κατὰ παντὸς ἀνθρώπου ζῷον, εἰ ἀληθὲς τόνδ' εἰπεῖν ἄνθρωπον, ἀληθὲς καὶ ζῷον, καὶ εἰ νῦν θάτερον, καὶ θάτερον, καὶ εἰ ἐν πάσῃ γραμμῇ στιγμή, ὡσαύτως. a28. I call 'true in every instance' what is truly predicable of all instances — not of one to the exclusion of others — and at all times, not at this or that time only; e.g. if animal is truly predicable of every instance of man, then if it be true to say 'this is a man', 'this is an animal' is also true, and if the one be true now the other is true now. A corresponding account holds if point is in every instance predicable as contained in line.
σημεῖον δέ· καὶ γὰρ τὰς ἐνστάσεις οὕτω φέρομεν ὡς κατὰ παντὸς ἐρωτώμενοι, ἢ εἰ ἐπί τινι μή, ἢ εἴ ποτε μή. a32. There is evidence for this in the fact that the objection we raise against a proposition put to us as true in every instance is either an instance in which, or an occasion on which, it is not true.
Postquam philosophus ostendit quid sit syllogismus demonstrativus, in parte ista incipit ostendere ex quibus et qualibus sit. Et circa hoc tria facit; primo, continuat se ad praecedentia; secundo, interponit quaedam quae sunt necessaria ad praecognoscendum; ibi: primum autem determinabimus etc.; tertio, determinat propositum, scilicet ex quibus sit syllogismus demonstrativus; ibi: si igitur est demonstrativa et cetera. After showing what a demonstrative syllogism is, the Philosopher in this section begins to show the nature and characteristics of the things that comprise a demonstration. Concerning this he does three things. First, he connects this with what has already been established. Secondly, he explains certain matters that must be understood first (73a25). Thirdly, he establishes what he had in mind, namely, to show what and of what sort are the things that constitute a syllogism (74b5) [L. 12].
Dicit ergo primo, quod quia dictum est supra, quod impossibile est aliter se habere in definitione eius quod est scire, necessarium erit id quod scitur secundum demonstrationem. Quid autem sit quod est secundum demonstrationem scire exponit, dicens quod demonstrativa scientia est quam habemus in habendo demonstrationem, idest quam ex demonstratione acquirimus. Et sic habetur quod demonstrationis conclusio sit necessaria. He says therefore first (73a21), that since the definition of scientific knowledge given above spoke of that which cannot be otherwise, that which is scientifically known through demonstration will be necessary. Then he explains what it is to know something in a scientific way through demonstration, saying that demonstrative science is “what we possess in having a demonstration,” i.e., what we acquire through demonstration. Consequently, it follows that the conclusion of a demonstration is necessary.
Quamvis autem necessarium possit syllogizari ex contingentibus, non tamen de necessario potest haberi scientia per medium contingens, ut infra probabitur. Et quia conclusio demonstrationis non solum est necessaria, sed etiam per demonstrationem scita, ut dictum est, sequitur quod demonstrativus syllogismus sit ex necessariis. Et ideo accipiendum est ex quibus necessariis et qualibus sint demonstrationes. Now although the necessary could be syllogized from the contingent, it is not possible through a contingent middle to obtain scientific knowledge of the necessary, as will be proved later. Furthermore, because the conclusion. of a demonstration is not only necessary, but, as has been said, is known through demonstration, it follows that a demonstrative syllogism proceeds from necessary things. Consequently, we must establish from what and from what sort of necessary things a demonstration proceeds.
Deinde cum dicit: primum autem etc., interponit ea, quae sunt praeintelligenda ad cognoscendum de his de quibus tractaturus est. Et circa hoc duo facit. Then (73a25) he interjects certain things that must be understood as preliminaries to the matters to be discussed. Apropos of these he does two things:
Primo, dicit de quo est intentio, dicens quod antequam determinetur in speciali ex quibus et qualibus sit demonstratio, primo determinandum est quid intelligatur cum dicimus de omni, et per se, et universale. Cognoscere enim ista est necessarium ad sciendum ex quibus sit demonstratio. Hoc namque oportet observari in demonstrationibus. Oportet enim in propositionibus demonstrationis aliquid universaliter praedicari, quod significat dici de omni, et per se, et etiam primo, quod significat universale. Haec autem tria se habent ex additione ad invicem. Nam omne quod per se praedicatur, etiam universaliter praedicatur; sed non e converso. Similiter omne quod primo praedicatur, praedicatur per se, sed non convertitur. Unde etiam apparet ratio ordinis istorum. First, he states his intention (73a25), saying that before determining specifically the nature and characteristics of the things that form a demonstrative syllogism, we must indicate what is meant when we say, “of all,” and “per se,” i.e., in virtue of itself, and “commensurately universal.” For if we are to understand the nature of the things that form a demonstration, we must know what these terms mean, because they describe things that must be observed in demonstrations. For in the propositions of a demonstration it is required that something be predicated universally~ which he signifies by the term “said of all”—and “per se,” i.e., in virtue of itself, and “first”—which he signifies by the words, “commensurately universal.” But these three things are related by adding something to the previous one. For whatever is predicated per se is predicated universally [i.e., of all], but not vice versa. Again, whatever is predicated first is predicated per se, but not vice versa. This, therefore, shows why they are arranged as they are.
Differentia etiam et numerus istorum trium apparet ex hoc, quod aliquid praedicari dicitur de omni sive universaliter per comparationem ad ea, quae continentur sub subiecto. Tunc enim dicitur aliquid de omni, ut habetur in libro priorum, quando nihil est sumere sub subiecto, de quo praedicatum non dicatur. Per se autem dicitur aliquid praedicari, per comparationem ad ipsum subiectum: quia ponitur in eius definitione, vel e converso, ut infra patebit. Primo vero dicitur aliquid praedicari de altero per comparationem ad ea, quae sunt priora subiecto et continentia ipsum. Nam habere tres angulos etc., non praedicatur primo de isoscele: quia prius praedicatur de priori, scilicet de triangulo. But why there are three and wherein they differ are explained by the fact that something is said to be predicated “of all” or universally in relation to things contained under the subject. For, as it is stated in Prior Analytics, something is said “of all,” when there is nothing under the extension of ,the subject that does not receive the given predicate. But it is in relation to the subject that something is said to be predicated per se, because the subject is mentioned when this predicate is defined, or vice versa, as will be explained below. Finally, something is said to be predicated of another thing “first” in relation to items that are prior to the, subject and embrace or include it, as the more universal includes the less. Thus to have three angles equal to two right angles is not predicated “first” of isosceles, because it is previously predicated of something prior to isosceles, namely, of triangle.
Secundo, ibi: de omni quidem etc., determinat propositum. Et dividitur in tres partes. Primo, ostendit quid sit dici de omni; secundo, quid sit dici per se; ibi: per se autem etc.; tertio, quid sit universale; ibi: universale autem dico et cetera. Secondly (73a28), he establishes his proposition. And his treatment is divided into three parts. First, he shows what is meant by “said of all.” Secondly, what is meant by “said per se,” i.e., in virtue of itself (73a34) [L. 10]. Thirdly, what is meant by “commensurately universal” (73b27) [L. II].
Circa primum duo facit. Primo, ostendit quid sit dici de omni. Ad quod sciendum est quod dici de omni, prout hic sumitur, addit supra dici de omni, prout sumitur in libro priorum. Nam in libro priorum accipitur dici de omni communiter, prout utitur eo et dialecticus et demonstrator. Et ideo non plus ponitur in definitione eius, quam quod praedicatum insit cuilibet eorum quae continentur sub subiecto. Hoc autem contingit vel ut nunc, et sic utitur quandoque dici de omni dialecticus; vel simpliciter et secundum omne tempus, et sic solum utitur eo demonstrator. Concerning the first he does two things. First (73a28), he states what it is to be “said of all.” And it should be noted that the phrase, “said of all,” is taken here in a sense somewhat different from the sense it has in Prior Analytics, where it is taken in a very general sense so as to accommodate both the dialectician and the demonstrator. Therefore, no more is mentioned in its definition than that the predicate be found in each of the things included under its subject. But that might be verified only at a given moment—which is the sense in which the dialectician sometimes uses it; or it might be verified absolutely and at all times—which is the sense to which the demonstrator must always limit himself.
Et ideo in definitione dici de omni, duo ponit: quorum unum est, ut nihil sit sumere sub subiecto cui praedicatum non insit. Et hoc significat cum dicit: non in quodam quidem sic, in quodam autem non. Aliud est, quod non sit accipere aliquod tempus, in quo praedicatum subiecto non conveniat. Et hoc designat cum dicit: neque aliquando sic, aliquando non; et ponit exemplum. Sicut de omni homine praedicatur animal; et de quocunque verum est dicere quod sit homo, verum est dicere quod sit animal, et quandocunque est homo, est animal. Et similiter se habet de linea et de puncto: nam punctum est in linea qualibet et semper. Accordingly, two things are mentioned in the definition of “said of all”: one is that there is nothing within the extension of the subject that the predicate does not apply to. And he indicates this when he says, “not of one to the exclusion of others”; the other is that there is no time in which the predicate does not belong. And this he indicates when he says, “not at this or that time only.” And he gives the example of “man” and “animal,” saying that “animal” is predicated of every man; and of anything of which it is true to say that it is a man, it is true to say that it is an animal, and whenever it is a man, it is an animal. The same is true between line and point: for a point is in every line and always in every line.
Secundo; ibi: signum autem etc., manifestat positam definitionem per signum ab instantiis sumptum. Non enim fertur instantia contra universalem propositionem, nisi quia deficit aliquid eorum, quae per eam significantur. Cum autem interrogamur, an aliquid praedicetur de omni in demonstrativis, dupliciter ferimus instantias; vel quia in quodam eorum quae continentur sub subiecto non est verum; vel quia aliquando non est verum. Unde manifestum est quod dici de omni utrunque praedictorum significat. Secondly (73a32), he explains this definition, using as evidence the techniques employed in rebuttals. For a universal proposition is not rebutted unless one or other of things it states is not verified. For when we are asked whether something is said “of all” in a demonstration, we can say, “No,” for two reasons, i.e., either because it is not true of each instance of the subject, or because now and then it is not true. Hence it is clear that “being said of all” signifies each of these.

Lectio 10
Caput 4 cont.
Καθ' αὑτὰ δ' ὅσα ὑπάρχει τε ἐν τῷ τί ἐστιν, οἷον τριγώνῳ γραμμὴ καὶ γραμμῇ στιγμή (γὰρ οὐσία αὐτῶν ἐκ τούτων ἐστί, καὶ ἐν τῷ λόγῳ τῷ λέγοντι τί ἐστιν ἐνυπάρχει), a34. Essential attributes are (1) such as belong to their subject as elements in its essential nature (e.g. line thus belongs to triangle, point to line; for the very being or 'substance' of triangle and line is composed of these elements, which are contained in the formulae defining triangle and line):
καὶ ὅσοις τῶν ὑπαρχόντων αὐτοῖς αὐτὰ ἐν τῷ λόγῳ ἐνυπάρχουσι τῷ τί ἐστι δηλοῦντι, οἷον τὸ εὐθὺ ὑπάρχει γραμμῇ καὶ τὸ περιφερές, καὶ τὸ περιττὸν καὶ ἄρτιον ἀριθμῷ, καὶ τὸ πρῶτον καὶ σύνθετον, καὶ ἰσόπλερον (73b.) καὶ ἑτερόμηκες· καὶ πᾶσι τούτοις ἐνυπάρχουσιν ἐν τῷ λόγῳ τῷ τί ἐστι λέγοντι ἔνθα μὲν γραμμὴ ἔνθα δ' ἀριθμός. ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων τὰ τοιαῦθ' ἑκάστοις καθ'αὑτὰ λέγω, ὅσα δὲ μηδετέρως ὑπάρχει, συμβεβηκότα, οἷον τὸ μουσικὸν ἢ λευκὸν τῷ ζῴῳ. a38. (2) such that, while they belong to certain subjects, the subjects to which they belong are contained in the attribute's own defining formula. Thus straight and curved belong to line, odd and even, prime and compound, square and oblong, to number; and also the formula defining any one of these attributes contains its subject — e.g. line or number as the case may be. Extending this classification to all other attributes, I distinguish those that answer the above description as belonging essentially to their respective subjects; whereas attributes related in neither of these two ways to their subjects I call accidents or 'coincidents'; e.g. musical or white is a 'coincident' of animal.
ἔτι ὃ μὴ καθ' ὑποκειμένου λέγεται ἄλλου τινός, οἷον τὸ βαδίζον ἕτερόν τι ὂν βαδίζον ἐστὶ καὶ τὸ λευκὸν, ἡ δ' οὐσία, καὶ ὅσα τόδε τι σημαίνει, οὐχ ἕτερόν τι ὄντα ἐστὶν ὅπερ ἐστίν. τὰ μὲν δὴ μὴ καθ' ὑποκειμένου καθ' αὑτὰ λέγω, τὰ δὲ καθ' ὑποκειμένου συμβεβηκότα. b5. Further (a) that is essential which is not predicated of a subject other than itself: e.g. 'the walking [thing]' walks and is white in virtue of being something else besides; whereas substance, in the sense of whatever signifies a 'this somewhat', is not what it is in virtue of being something else besides. Things, then, not predicated of a subject I call essential; things predicated of a subject I call accidental or 'coincidental'.
ἔτι δ' ἄλλον τρόπον τὸ μὲν δι' αὑτὸ ὑπάρχον ἑκάστῳ καθ' αὑτό, τὸ δὲ μὴ δι' αὑτὸ συμβεβηκός, οἷον εἰ βαδίζοντος ἤστραψε, συμβεβηκός· οὐ γὰρ διὰ τὸ βαδίζειν ἤστραψεν, ἀλλὰ συνέβη, φαμέν, τοῦτο. εἰ δὲ δι' αὑτό, καθ' αὑτό, οἷον εἴ τι σφαττόμενον ἀπέθανε, καὶ κατὰ τὴν σφαγήν, ὅτι διὰ τὸ σφάττεσθαι, ἀλλ' οὐ συνέβη σφαττόμενον ἀποθανεῖν. b10. In another sense again (b) a thing consequentially connected with anything is essential; one not so connected is 'coincidental'. An example of the latter is 'While he was walking it lightened': the lightning was not due to his walking; it was, we should say, a coincidence. If, on the other hand, there is a consequential connexion, the predication is essential; e.g. if a beast dies when its throat is being cut, then its death is also essentially connected with the cutting, because the cutting was the cause of death, not death a 'coincident' of the cutting.
τὰ ἄρα λεγόμενα ἐπὶ τῶν ἁπλῶς ἐπιστητῶν καθ' αὑτὰ οὕτως ὡς ἐνυπάρχειν τοῖς κατηγορουμένοις ἢ ἐνυπάρχεσθαι δι' αὑτά τέ ἐστι καὶ ἐξ ἀνάγκης. οὐ γὰρ ἐνδέχεται μὴ ὑπάρχειν ἢ ἁπλῶς ἢ τὰ ἀντικείμενα, οἷον γραμμῇ τὸ εὐθὺ ἢ τὸ καμπύλον καὶ ἀριθμῷ τὸ περιττὸν ἢ τὸ ἄρτιον. ἔστι γὰρ τὸ ἐναντίον ἢ στέρησις ἢ ἀντίφασις ἐν τῷ αὐτῷ γένει, οἷον ἄρτιον τὸ μὴ περιττὸν ἐν ἀριθμοῖς ᾗ ἕπεται. ὥστ' εἰ ἀνάγκη φάναι ἢ ἀποφάναι, ἀνάγκη καὶ τὰ καθ' αὑτὰ ὑπάρχειν. b16. So far then as concerns the sphere of connexions scientifically known in the unqualified sense of that term, all attributes which (within that sphere) are essential either in the sense that their subjects are contained in them, or in the sense that they are contained in their subjects, are necessary as well as consequentially connected with their subjects. For it is impossible for them not to inhere in their subjects either simply or in the qualified sense that one or other of a pair of opposites must inhere in the subject; e.g. in line must be either straightness or curvature, in number either oddness or evenness. For within a single identical genus the contrary of a given attribute is either its privative or its contradictory; e.g. within number what is not odd is even, inasmuch as within this sphere even is a necessary consequent of not-odd. So, since any given predicate must be either affirmed or denied of any subject, essential attributes must inhere in their subjects of necessity.
Τὸ μὲν οὖν κατὰ παντὸς καὶ καθ' αὑτὸ διωρίσθω τὸν τρόπον τοῦτον· b25. Thus, then, we have established the distinction between the attribute which is 'true in every instance' and the 'essential' attribute.
Postquam determinavit philosophus de dici de omni, hic determinat de per se. Et circa hoc duo facit: primo, ostendit quot modis dicitur aliquid per se; secundo, ostendit qualiter his modis demonstrator utatur; ibi: quae ergo dicuntur et cetera. After determining about “said of all,” the Philosopher now determines about “said per se” [i.e., said in virtue of itself] and does three things. First, he shows the number of ways something is said per se. Secondly, how the demonstrator makes use of these ways (73b16). Thirdly, he summarizes (73b25).
Circa primum sciendum est quod haec praepositio per designat habitudinem causae; designat etiam interdum et situm, sicut cum dicitur aliquis esse per se, quando est solitarius. Causae autem habitudinem designat, aliquando quidem formalis; sicut cum dicitur quod corpus vivit per animam. Quandoque autem habitudinem causae materialis; sicut cum dicitur quod corpus est coloratum per superficiem: quia scilicet proprium subiectum coloris est superficies. Designat etiam habitudinem causae extrinsecae et praecipue efficientis; sicut cum dicitur quod aqua calescit per ignem. Sicut autem haec praepositio per designat habitudinem causae, quando aliquid extrinsecum est causa eius, quod attribuitur subiecto; ita quando subiectum vel aliquid eius est causa eius, quod attribuitur ei, et hoc significat per se. In regard to the first it should be noted that this preposition per [“in virtue of” or “by”] denotes a causal relationship, although sometimes it also signifies a state, as when someone is said to be per se, i.e., by himself, when he is alone. But when it designates a relationship to a cause, sometimes the cause is formal, as when it is stated that the body lives in virtue of the soul; sometimes the relationship is to a material cause, as when it is stated that a body is colored in virtue of its surface, i.e., because the surface is the subject of color; again, it might even designate a relationship to an extrinsic cause, particularly an efficient cause, as when it is said that water is made hot in virtue of fire. But just as this preposition per designates a relationship to a cause, when something extrinsic is the cause of that which is attributed to the subject, so also when the subject or something pertaining to the subject is the cause of that which is attributed to the subject. This latter is what per se, i.e., in virtue of itself, signifies.
Primus ergo modus dicendi per se est, quando id, quod attribuitur alicui, pertinet ad formam eius. Et quia definitio significat formam et essentiam rei, primus modus eius quod est per se est, quando praedicatur de aliquo definitio vel aliquid in definitione positum (et hoc est quod dicit quod per se sunt quaecunque insunt in eo, quod quid est, idest in definitione indicante quid est), sive ponatur in recto sive in obliquo. Sicut in definitione trianguli ponitur linea; unde linea per se inest triangulo: et similiter in definitione lineae ponitur punctum; unde punctum per se inest lineae. Rationem autem quare ista ponantur in definitione subiungit dicens: substantia, idest essentia, quam significat definitio ipsorum, idest trianguli et lineae, est ex his, idest ex linea et punctis. Quod non est intelligendum quod linea ex punctis componatur, sed quod punctum sit de ratione lineae, sicut linea de ratione trianguli. Et hoc dicit ad excludendum ea, quae sunt partes materiae et non speciei, quae non ponuntur in definitione, sicut semicirculus non ponitur in definitione circuli, nec digitus in definitione hominis, ut dicitur in VII metaphysicae. Therefore, the first way of saying something per se (73a34) is when that which is attributed to a subject pertains to its form. And because the form and essence of a thing are signified by its definition, the first mode of that which is per se is when the definition itself or something expressed in the definition is predicated of the thing defined. This is what he means when he says, “Essential attributes are such as belong to their subject as elements in its essential nature,” i.e., included in the definition which indicates what it is, whether those elements are stated in the nominative case or in one of the oblique cases. Thus, “line” is stated in the definition of triangle. Hence “line” is in triangle per se. Again, in the definition of line, “point” is mentioned; hence “point” is per se in line. And the reason why they are mentioned in the definition is stated when he says, “for the very being or substance” [i.e., the essence, which the definition signifies] “of triangle and line is composed of these elements,” namely, of lines and points. However, this does not mean that a line is formed out of points, but that “point” is involved in the very notion of line, just as “line” is involved in the very notion of triangle. And he asserts this in order to exclude things which are part of a thing’s matter and not of its species: thus, “semicircle” is not mentioned in the definition of circle, or “finger” in the definition of man, as it is stated in Metaphysics VII.
Et subiungit quod quaecumque universaliter insunt in ratione dicente quid est, per se attribuuntur alicui. He states further that all those items which are found universally in the definition expressing what a thing is are attributed to it per se.
Secundus modus dicendi per se est, quando haec praepositio per designat habitudinem causae materialis, prout scilicet id, cui aliquid attribuitur, est propria materia et proprium subiectum ipsius. Oportet autem quod proprium subiectum ponatur in definitione accidentis: quandoque quidem in obliquo, sicut cum accidens in abstracto definitur, ut cum dicimus, quod simitas est curvitas nasi; quandoque vero in recto, ut cum accidens definitur in concreto, ut cum dicimus quod simus est nasus curvus. Cuius quidem ratio est, quia cum esse accidentis dependeat a subiecto, oportet etiam quod definitio eius significans esse ipsius contineat in se subiectum. Unde secundus modus dicendi per se est, quando subiectum ponitur in definitione praedicati, quod est proprium accidens eius. The second mode of saying per se is when this preposition per implies a relationship of material cause, in the sense that that to which something, is attributed is its proper matter and subject. For it is required, when defining an accident, to mention its proper subject in one of the oblique, cases: thus when an accident is defined abstractly, we say that “aquilinity” is a curvature of a nose,” but when it is defined concretely, the subject is put in the nominative case, so that we say that “the aquiline is a curved nose.” Now the reason for this is that since the being of an accident depends on its subject, its definition—which signifies its being—must mention that subject. Hence it is the second mode of saying per se, when the subject is mentioned in the definition of a predicate which is a proper accident of the subject.
Et hoc est quod dicit, et per se dicuntur quibuscunque eorum, idest de numero eorum, quae insunt ipsis, idest subiectis accidentium, ipsa subiecta insunt in ratione demonstrante quid est ipsum accidens, idest in definitione accidentis. Sicut rectum et circulare insunt lineae per se: nam linea ponitur in definitione eorum. Et eadem ratione par et impar per se insunt numero, quia numerus in eorum definitione ponitur: nam par est numerus medium habens. Et similiter primum et compositum per se praedicantur de numero, et numerus in definitione eorum ponitur. Est enim primum in numeris, numerus qui nullo alio numero mensuratur, sed sola unitate, ut septenarius. Compositus autem numerus est, quem etiam alius numerus mensurat, sicut novenarius. Et similiter isopleuros, idest aequilaterum, et scalenon, idest trium inaequalium laterum et altera parte longius, per se insunt triangulo, et triangulus ponitur in definitione eorum. Et ideo subiungit quod, subiecta quae insunt omnibus praemissis accidentibus in ratione dicente quid est, idest in definitione, sicut alicui praedictorum accidentium inest linea, alicui vero numerus, et similiter in aliis, unicuique, inquam, ipsorum subiectorum, per se inesse dico suum accidens. And this is what he means when he states (73a38), “essential attributes are those such that while they belong to certain subjects,” i.e., to subjects of accidents, “the subjects to which they belong are contained in the attribute’s own defining formula,” i.e., in the expression which describes what the accident is, i.e., in the definition of the accident. “Thus straight and curved belong to line per se.” For “line” is mentioned in their definition. For the same reason “odd” and “even” belong per se to number, because “number” is mentioned in their definition. Again, prime and compound are predicated per se of number, and “number” is mentioned in their definition. (For a prime number, for example, seven, is one which is exactly divisible by no other number but “1”; but a compound number, for example, nine, is one which is exactly divisible by some number greater than “1.” Again, “isoplural,” i.e., equilateral, and scalene, i.e., having three unequal sides, belong per se to triangle, and “triangle” is mentioned in their definition. Accordingly, he adds that their respective subjects belong to each of the aforesaid accidents and are mentioned in the expression which states what each is, i.e., in the definition: thus “line” belongs to some of them, and “number” to others.
Quae vero praedicata neutraliter insunt, idest neque ita quod ponantur in definitione subiectorum, neque subiecta in definitione eorum, sunt accidentia, idest per accidens praedicantur, sicut musicum et album praedicantur de animali per accidens. In each of these subjects that have been mentioned, I say that its accident is in it per se. But those predicates which are neutral, i.e., of such a nature as not to be mentioned in the definition of their subjects, nor the subjects in their definition, are accidents, i.e., are predicated per accidens: for example, “musical” and “white” are predicated per accidens of animal.
Deinde cum dicit: amplius quod non etc., ponit alium modum eius, quod est per se, prout per se significat aliquid solitarium, sicut dicitur quod per se est aliquod particulare, quod est in genere substantiae, quod non praedicatur de aliquo subiecto. Et huius ratio est, quia cum dico, ambulans vel album, non significo ambulans vel album, quasi aliquid per se solitarium existens, cum intelligatur aliquid aliud esse quod sit ambulans vel album. Sed in his, quae significant hoc aliquid, scilicet in primis substantiis, hoc non contingit. Cum enim dicitur Socrates vel Plato, non intelligitur quod sit aliquid alterum, quam id quod vere ipsa sunt, quod scilicet sit subiectum eorum. Sic igitur hoc modo quae non praedicantur de subiecto sunt per se, quae vero dicuntur de subiecto, scilicet sicut in subiecto existentia, accidentia sunt. Nam quae dicuntur de subiecto, sicut universalia de inferioribus, non semper accidentia sunt. Then (73b5) he sets down another mode of that which is per se, i.e., the sense in which it signifies something in isolation. Thus something which is a singular in the genus of substance and which is not predicated of any subject is said to be per se. The reason for this is that when I say, “walking” or “white,” I do not signify either of them as something isolated or apart, since something else which is walking or white is understood. But this is not the case with terms which signify a “this something,” i.e., with terms that signify first substance. For when I say, “Socrates” or “Plato,” it is not to be supposed that there is something else, over and above what they really are, which would be their subject. Therefore, things which are thus not predicated of any subject are per se, but things which are predicated of a subject, as being in the subject, are accidents. However, not all things predicated of a subject, as universals of their inferiors, are accidents.
Sciendum est autem quod iste modus non est modus praedicandi, sed modus existendi. Unde etiam in principio non dixit, per se dicuntur, sed, per se sunt. It should be noted, however, that this mode is not a mode of predicating, but a mode of existing; hence at the very start he said that they exist per se and not that they are said per se.
Deinde cum dicit: item alio modo etc., ponit quartum modum, secundum quod haec praepositio per designat habitudinem causae efficientis vel cuiuscunque alterius. Et ideo dicit quod quidquid inest unicuique propter seipsum, per se dicitur de eo; quod vero non propter seipsum inest alicui, per accidens dicitur, sicut cum dico: hoc ambulante coruscat. Non enim propter id quod ambulat, coruscavit; sed hoc dicitur secundum accidens. Si vero quod praedicatur insit subiecto propter seipsum, per se inest, ut si dicamus quod interfectum interiit: manifestum est enim quod propter id quod illud interfectum est, interiit, et non est accidens quod interfectum interierit. Then (73b10) he gives the fourth mode, according to which the preposition per designates a relationship of efficient cause or of any other. Consequently, he says that whatever is attributed to a thing because of itself, is said of it per se; but whatever is not so attributed is said per accidens, as when I say, “While he was walking, it lightened.” For it is not the fact that he walks that causes lightning, but this is said by coincidence. Butt if the predicate is in the subject because of itself, it is per se, as when we say, “Slaughtered, it died.” For it is obvious that because something was slaughtered, it died, and it is not a mere coincidence that something slaughtered should die.
Deinde cum dicit: quae ergo dicuntur etc., ostendit qualiter utatur praedictis modis demonstrator. Ubi notandum est quod cum scientia proprie sit conclusionum, intellectus autem principiorum, proprie scibilia dicuntur conclusiones demonstrationis, in quibus passiones praedicantur de propriis subiectis. Propria autem subiecta non solum ponuntur in definitione accidentium, sed etiam sunt causae eorum. Unde conclusiones demonstrationum includunt duplicem modum dicendi per se, scilicet secundum et quartum. Then (73b16) he shows how the demonstrator uses the aforesaid modes. But first it should be noted that, since science bears on conclusions, and understanding [intuition] bears on principles, the scientifically knowable are, properly speaking, the conclusions of a demonstration wherein proper attributes are predicated of their appropriate subjects. Now the appropriate subjects are not only placed in the definition of attributes, but they are also their causes. Hence the conclusions of demonstrations involve two modes of predicating per se, namely, the second and the fourth.
Et hoc est quod dicit quod illa quae praedicantur in simpliciter scibilibus, hoc est, in conclusionibus demonstrationum, sic sunt per se, sicut inesse praedicantibus, idest sicut quando subiecta insunt in definitione accidentium, quae de eis praedicantur, aut inesse propter ipsa, idest quando praedicata insunt subiecto propter ipsum subiectum, quod est causa praedicati. And this is what he means when he says that the predications “in the scientifically knowable in the strict sense,” i.e., in the conclusions of demonstrations are per se in the sense of something contained in the predicates, i.e., in the way that subjects are contained in the definition of accidents which are predicated of the former; or are present on account of them, i.e., in the way that predicates are in a subject by reason of the subject itself, which is the cause of the predicate.
Et consequenter ostendit quod huiusmodi scibilia sunt necessaria: quia non contingit quin proprium accidens praedicetur de subiecto. Sed hoc est duobus modis. Quandoque quidem simpliciter, sicut cum unum accidens convertitur cum subiecto, ut habere tres cum triangulo, et risibile cum homine. Quandoque autem duo opposita sub disiunctione accepta ex necessitate subiecto insunt, ut lineae aut rectum aut obliquum, et numero par aut impar. Cuius rationem ostendit, quia contrarium, privatio et contradictio sunt in eodem genere. Nam privatio nihil aliud est, quam negatio in subiecto determinato. Quandoque etiam contrarium aequiparatur negationi in aliquo genere, sicut in numeris idem est impar quod non par secundum consequentiam. Sicut ergo necesse est affirmare vel negare, ita necesse est alterum eorum, quae per se insunt, proprio inesse subiecto. Then he shows that such scientifically knowable things are necessary, because it is impossible for a proper accident not to be predicated of its subject. But this can occur in two ways: sometimes it is absolute, as when the accident is convertible with its subject, as “having three angles equal to two right angles” is convertible with triangle, and “risible” with man. At other times, two opposites stated disjunctively are of necessity in the subject, as “straight or oblique” in line, and “odd or even” in number. He shows that the reason for this is the fact that contrariety, privation and contradiction are in the same genus. For privation is nothing more, than a negation in a determinate subject. Again, a contrary is equivalent to a negation in some genus, as in the genus of numbers, odd is the same as “not even” by way of consequence. Therefore, just as it is necessary, either to affirm or deny, so it is necessary that one of two things that belong per se, be in its proper subject.
Deinde epilogat dicens: de omni quidem et cetera. Quod est planum. Then (73b25) he summarizes, and the text is clear.

Lectio 11
Caput 4 cont.
καθόλου δὲ λέγω ὃ ἂν κατὰ παντός τε ὑπάρχῃ καὶ καθ' αὑτὸ καὶ ᾗ αὐτό. b27. I term 'commensurately universal' an attribute which belongs to every instance of its subject, and to every instance essentially and as such;
φανερὸν ἄρα ὅτι ὅσα καθόλου, ἐξ ἀνάγκης ὑπάρχει τοῖς πράγμασιν. b28. from which it clearly follows that all commensurate universals inhere necessarily in their subjects.
τὸ καθ' αὑτὸ δὲ καὶ ᾗ αὐτὸ ταὐτόν, οἷον καθ' αὑτὴν τῇ γραμμῇ ὑπάρχει στιγμὴ καὶ τὸ εὐθύ (καὶ γὰρ ᾗ γραμμή), καὶ τῷ τριγώνῳ ᾗ τρίγωνον δύο ὀρθαί (καὶ γὰρ καθ' αὑτὸ τὸ τρίγωνον δύο ὀρθαῖς ἴσον). b29. The essential attribute, and the attribute that belongs to its subject as such, are identical. E.g. point and straight belong to line essentially, for they belong to line as such; and triangle as such has two right angles, for it is essentially equal to two right angles.
τὸ καθόλου δὲ ὑπάρχει τότε, ὅταν ἐπὶ τοῦ τυχόντος καὶ πρώτου δεικνύηται. b32. An attribute belongs commensurately and universally to a subject when it can be shown to belong to any random instance of that subject and when the subject is the first thing to which it can be shown to belong.
οἷον τὸ δύο ὀρθὰς ἔχειν οὔτε τῷ σχήματί ἐστι καθόλου (καίτοι ἔστι δεῖξαι κατὰ σχήματος ὅτι δύο ὀρθὰς ἔχει, ἀλλ' οὐ τοῦ τυχόντος σχήματος, οὐδὲ χρῆται τῷ τυχόντι σχήματι δεικνύς· τὸ γὰρ τετράγωνον σχῆμα μέν, οὐκ ἔχει δὲ δύο ὀρθαῖς ἴσας)- τὸ δ' ἰσοσκελὲς ἔχει μὲν τὸ τυχὸν δύο ὀρθαῖς ἴσας, ἀλλ' οὐ πρῶτον, ἀλλὰ τὸ τρίγωνον πρότερον. ὃ τοίνυν τὸ τυχὸν πρῶτον δείκνυται δύο ὀρθὰς ἔχον ἢ ὁτιοῦν ἄλλο, τούτῳ πρώτῳ (74a.) ὑπάρχει καθόλου, b34. Thus, e.g. (1) the equality of its angles to two right angles is not a commensurately universal attribute of figure. For though it is possible to show that a figure has its angles equal to two right angles, this attribute cannot be demonstrated of any figure selected at haphazard, nor in demonstrating does one take a figure at random — a square is a figure but its angles are not equal to two right angles. On the other hand, any isosceles triangle has its angles equal to two right angles, yet isosceles triangle is not the primary subject of this attribute but triangle is prior. So whatever can be shown to have its angles equal to two right angles, or to possess any other attribute, in any random instance of itself and primarily — that is the first subject to which the predicate in question belongs commensurately and universally,
καὶ ἡ ἀπόδειξις καθ' αὑτὸ τούτου καθόλου ἐστί, τῶν δ' ἄλλων τρόπον τινὰ οὐ καθ' αὑτό, οὐδὲ τοῦ ἰσοσκελοῦς οὐκ ἔστι καθόλου ἀλλ' ἐπὶ πλέον. a1. and the demonstration, in the essential sense, of any predicate is the proof of it as belonging to this first subject commensurately and universally: while the proof of it as belonging to the other subjects to which it attaches is demonstration only in a secondary and unessential sense. Nor again (2) is equality to two right angles a commensurately universal attribute of isosceles; it is of wider application.
Postquam philosophus determinavit de dici de omni et per se, hic determinat de universali. Et dividitur in duas partes: in prima, ostendit quid sit universale; in secunda, ostendit quomodo in acceptione universalis contingit errare; ibi: oportet autem non latere et cetera. Circa primum duo facit: primo, ostendit quid sit universale; secundo, ostendit quomodo demonstrator universali utatur; ibi: demonstratio autem per se et cetera. Circa primum duo facit: primo, ostendit quod universale continet in se et dici de omni et per se; secundo, ostendit quid supra ea addat; ibi: universale autem et cetera. After determining about “said of all” and “said per se,” the Philosopher here determines concerning the “universal.” This treatment falls into two parts. In the first he shows what the universal is. Secondly, how error occurs in our understanding of it (74a4) [L. 12]. Concerning the first he does two things. First, he shows what the universal is. Secondly, how the demonstrator uses the universal (74a1). Concerning the first he does two things. First, he shows that the universal contains within itself the attributes of “being said of all” and of “being said per se.” Secondly, he shows what the universal adds to them (73b33).
Ad evidentiam autem eorum, quae hic dicuntur, sciendum est quod universale non hoc modo hic accipitur, prout omne quod praedicatur de pluribus universale dicitur, secundum quod Porphyrius determinat de quinque universalibus; sed dicitur hic universale secundum quandam adaptationem vel adaequationem praedicati ad subiectum, cum scilicet neque praedicatum invenitur extra subiectum, neque subiectum sine praedicato. To understand what is being said here it should be noted that “universal” is not to be taken here in the sense that anything predicated of several is a universal, as when Porphyry treats of the five universals; rather “universal” is taken here according to a certain correspondence or commensurateness of the subject with the predicate, so that the predicate is not found outside the subject nor is the subject without the predicate.
His autem visis, sciendum est quod circa primum tria facit. Primo dicit quod universale, scilicet praedicatum, est quod et de omni est, idest universaliter praedicatur de subiecto, et etiam per se, scilicet inest ei, idest convenit subiecto secundum quod ipsum subiectum est. Multa enim universaliter de aliquibus praedicantur, quae non conveniunt eis per se, et secundum quod ipsa. Sicut omnis lapis coloratus est; non tamen secundum quod lapis, sed secundum quod est superficiem habens. With this in mind, it should be noted that he does three things with respect to the first point. First (73b27), he says that the universal, namely, the predicate, is both verified of all, i.e., is predicated universally of its subject, and is said per se, i.e., is in and belongs to the subject according to the essential nature of the subject. For many things are said universally of certain things to which they do not belong per se and as such. Thus, every stone is colored, but not precisely as stone, but as it has a surface.
Secundo; ibi: manifestum igitur etc., infert quoddam corollarium ex dictis, dicens quod, ex quo universale est, quod per se inest; quae autem per se insunt ex necessitate insunt, ut supra ostensum est; manifestum est quod universalia praedicata, prout hic sumuntur, ex necessitate insunt rebus, de quibus praedicantur. Secondly (73b28), he draws a corollary from this and says that since the universal is something which is per se in a thing, and since it has been shown that whatever things are in something per se are in it of necessity, it is obvious that universal predicates, as they are being taken here, are necessarily present in the things of which they are predicated.
Tertio; ibi: per se autem etc., ne aliquis crederet aliud esse quod in definitione universalis dixerat per se, et secundum quod ipsum est, ostendit quod per se et secundum quod ipsum est, idem est. Sicut lineae per se inest punctum primo modo, et rectitudo secundo modo: nam utrunque inest ei secundum quod linea est. Et e converso triangulo secundum quod triangulus est insunt duo recti, idest quod valet duos rectos, quia per se triangulo inest. Thirdly (73b29), lest anyone suppose that “per se” and “precisely as such,” both of which were mentioned in the definition of the universal, are different, he shows that they are the same. Thus, “point” is per se in line in the first way, and “straight” in the second way. For each is in line precisely as it is a line. In like manner, “two right angles” belongs to triangle precisely as triangle, i.e., its angles are equal to two right angles, which is per se in triangle.
Deinde cum dicit: universale autem etc., ostendit quid addat universale supra dici de omni et per se. Et circa hoc duo facit. Then (73b33) he shows what “universal” adds to the notions, “being said of all” and “being said per se.” In regard to this he does two things:
Primo, dicit quod tunc est universale praedicatum, cum non solum in quolibet est de quo praedicatur, sed et primo demonstratur inesse ei, de quo praedicatur. First, he says that a predicate is “universal,” when it is not only in each thing of which it is asserted, but it is demonstrated to be first or primarily in the thing which receives that predicate.
Secundo; ibi: ut duos rectos habere etc., manifestat per exemplum, dicens quod habere tres angulos aequales duobus rectis, non inest cuilibet figurae universaliter: licet hoc de figura demonstretur, quia de triangulo demonstratur qui est figura; sed tamen non cuilibet figurae inest, nec demonstrator in sua demonstratione utitur qualibet figura. Quadrangulus enim figura quaedam est, sed non habet tres duobus rectis aequales. Isosceles autem, idest triangulus duorum aequalium laterum, habet quidem universaliter tres angulos aequales duobus rectis, sed non convenit primo isosceli, sed prius triangulo, quia isosceli convenit, in quantum est triangulus. Quod igitur primo demonstratur habere duos rectos, aut quodcunque aliud huiusmodi, huic primo inest praedicatum universale, sicut triangulo. Secondly (73b34), he clarifies this with an example and says that “having three angles equal to two right angles” is not found in just any figure in general, although this could be demonstrated of some figure, because it is demonstrated of triangle, which is a figure; yet it is not found in any random figure, nor is just any figure used when it is demonstrated. For a rhombus is a figure, but it does not have three angles equal to two right angles. But an isosceles, i.e., a triangle with two equal sides, always has its three angles equal to two right angles. Nevertheless, isosceles is not the primary thing to which this belongs, for it belongs basically to triangle, and belongs to isosceles precisely as it is a triangle. Therefore, whatever is demonstrated basically to have its three angles equal to two right angles (or whatever else be thus demonstrated), the universal predicate is present in it primarily, as in triangle.
Deinde cum dicit: et demonstratio etc., ostendit qualiter demonstrator universali utatur, et dicit quod demonstratio est per se huius universalis: sed aliorum est quodammodo et non per se. Demonstrator enim demonstrat passionem de proprio subiecto: et si demonstret de aliquo alio, hoc non est nisi in quantum pertinet ad illud subiectum. Sicut passionem trianguli probat de figura et isoscele, in quantum quaedam figura triangulus est, et triangulus quidam isosceles est. Quod autem non primo inest isosceli habere tres, hoc non est quia non universaliter praedicetur de eo, sed quia est frequentius, idest in pluribus quam isosceles, cum hoc commune sit omni triangulo. Then (74a1) he shows how a demonstrator uses the “universal,” saying that demonstration is concerned per se with such a universal, but with other things qualifiedly and not per se. For a demonstrator demonstrates a proper attribute of its proper subject; and if he demonstrates it of anything else, he does so only insofar as it pertains to that subject. Thus, he proves that some property of triangle belongs to a figure and to an isosceles precisely as some figure is a triangle, and as the isosceles is a triangle. But the reason why “having three” is not in isosceles primarily is not because it is not predicated of it universally, but because it is found more frequently, i.e., in more things than in isosceles, since this is common to every triangle.

Lectio 12
Caput 5
Δεῖ δὲ μὴ λανθάνειν ὅτι πολλάκις συμβαίνει διαμαρτάνειν καὶ μὴ ὑπάρχειν τὸ δεικνύμενον πρῶτον καθόλου, ᾗ δοκεῖ δείκνυσθαι καθόλου πρῶτον. a4. We must not fail to observe that we often fall into error because our conclusion is not in fact primary and commensurately universal in the sense in which we think we prove it so.
ἀπατώμεθα δὲ ταύτην τὴν ἀπάτην, ὅταν ἢ μηδὲν ᾖ λαβεῖν ἀνώτερον παρὰ τὸ καθ' ἕκαστον [ἢ τὰ καθ' ἕκαστα], ἢ ᾖ μέν, ἀλλ' ἀνώνυμον ᾖ ἐπὶ διαφόροις εἴδει πράγμασιν, ἢ τυγχάνῃ ὂν ὡς ἐν μέρει ὅλον ἐφ' ᾧ δείκνυται· τοῖς γὰρ ἐν μέρει ὑπάρξει μὲν ἡ ἀπόδειξις, καὶ ἔσται κατὰ παντός, ἀλλ' ὅμως οὐκ ἔσται τούτου πρώτου καθόλου ἡ ἀπόδειξις. λέγω δὲ τούτου πρώτου, ᾗ τοῦτο, ἀπόδειξιν, ὅταν ᾖ πρώτου καθόλου. a6. We make this mistake (1) when the subject is an individual or individuals above which there is no universal to be found: (2) when the subjects belong to different species and there is a higher universal, but it has no name: (3) when the subject which the demonstrator takes as a whole is really only a part of a larger whole; for then the demonstration will be true of the individual instances within the part and will hold in every instance of it, yet the demonstration will not be true of this subject primarily and commensurately and universally. When a demonstration is true of a subject primarily and commensurately and universally, that is to be taken to mean that it is true of a given subject primarily and as such.
εἰ οὖν τις δείξειεν ὅτι αἱ ὀρθαὶ οὐ συμπίπτουσι, δόξειεν ἂν τούτου εἶναι ἡ ἀπόδειξις διὰ τὸ ἐπὶ πασῶν εἶναι τῶν ὀρθῶν. οὐκ ἔστι δέ, εἴπερ μὴ ὅτι ὡδὶ ἴσαι γίνεται τοῦτο, ἀλλ' ᾗ ὁπωσοῦν ἴσαι. a13. Case (3) may be thus exemplified. If a proof were given that perpendiculars to the same line are parallel, it might be supposed that lines thus perpendicular were the proper subject of the demonstration because being parallel is true of every instance of them. But it is not so, for the parallelism depends not on these angles being equal to one another because each is a right angle, but simply on their being equal to one another.
καὶ εἰ τρίγωνον μὴ ἦν ἄλλο ἢ ἰσοσκελές, ᾗ ἰσοσκελὲς ἂν ἐδόκει ὑπάρχειν. a17. An example of (1) would be as follows: if isosceles were the only triangle, it would be thought to have its angles equal to two right angles qua isosceles.
καὶ τὸ ἀνάλογον ὅτι καὶ ἐναλλάξ, ᾗ ἀριθμοὶ καὶ ᾗ γραμμαὶ καὶ ᾗ στερεὰ καὶ ᾗ χρόνοι, ὥσπερ ἐδείκνυτό ποτε χωρίς, ἐνδεχόμενόν γε κατὰ πάντων μιᾷ ἀποδείξει δειχθῆναι· ἀλλὰ διὰ τὸ μὴ εἶναι ὠνομασμένον τι ταῦτα πάντα ἓν, ἀριθμοί μήκη χρόνοι στερεά, καὶ εἴδει διαφέρειν ἀλλήλων, χωρὶς ἐλαμβάνετο. νῦν δὲ καθόλου δείκνυται· οὐ γὰρ ᾗ γραμμαὶἢ ᾗ ἀριθμοὶ ὑπῆρχεν, ἀλλ' ᾗ τοδί, ὃ καθόλου ὑποτίθενται ὑπάρχειν. a18. An instance of (2) would be the law that proportionals alternate. Alternation used to be demonstrated separately of numbers, lines, solids, and durations, though it could have been proved of them all by a single demonstration. Because there was no single name to denote that in which numbers, lengths, durations, and solids are identical, and because they differed specifically from one another, this property was proved of each of them separately. To-day, however, the proof is commensurately universal, for they do not possess this attribute qua lines or qua numbers, but qua manifesting this generic character which they are postulated as possessing universally.
διὰ τοῦτο οὐδ' ἄν τις δείξῃ καθ' ἕκαστον τὸ τρίγωνον ἀποδείξει ἢ μιᾷ ἢ ἑτέρᾳ ὅτι δύο ὀρθὰς ἔχει ἕκαστον, τὸ ἰσόπλευρον χωρὶς καὶ τὸ σκαληνὲς καὶ τὸ ἰσοσκελές, οὔπω οἶδε τὸ τρίγωνον ὅτι δύο ὀρθαῖς, εἰ μὴ τὸν σοφιστικὸν τρόπον, οὐδὲ καθ' ὅλου τριγώνου, οὐδ' εἰ μηδὲν ἔστι παρὰ ταῦτα τρίγωνον ἕτερον. οὐ γὰρ ᾗ τρίγωνον οἶδεν, οὐδὲ πᾶν τρίγωνον, ἀλλ' ἢ κατ' ἀριθμόν· κατ' εἶδος δ' οὐ πᾶν, καὶ εἰ μηδὲν ἔστιν ὃ οὐκ οἶδεν. a25. Hence, even if one prove of each kind of triangle that its angles are equal to two right angles, whether by means of the same or different proofs; still, as long as one treats separately equilateral, scalene, and isosceles, one does not yet know, except sophistically, that triangle has its angles equal to two right angles, nor does one yet know that triangle has this property commensurately and universally, even if there is no other species of triangle but these. For one does not know that triangle as such has this property, nor even that 'all' triangles have it — unless 'all' means 'each taken singly': if 'all' means 'as a whole class', then, though there be none in which one does not recognize this property, one does not know it of 'all triangles'.
Πότ' οὖν οὐκ οἶδε καθόλου, καὶ πότ' οἶδεν ἁπλῶς; δῆλον δὴ ὅτι εἰ ταὐτὸν ἦν τριγώνῳ εἶναι καὶ ἰσοπλεύρῳ ἢ ἑκάστῳ ἢ πᾶσιν. εἰ δὲ μὴ ταὐτὸν ἀλλ' ἕτερον, ὑπάρχει δ' ᾗ τρίγωνον, οὐκ οἶδεν. a33. When, then, does our knowledge fail of commensurate universality, and when it is unqualified knowledge? If triangle be identical in essence with equilateral, i.e. with each or all equilaterals, then clearly we have unqualified knowledge: if on the other hand it be not, and the attribute belongs to equilateral qua triangle; then our knowledge fails of commensurate universality.
πότερον δ' ᾗ τρίγωνον ἢ ᾗ ἰσοσκελὲς ὑπάρχει; καὶ πότε κατὰ τοῦθ' ὑπάρχει πρῶτον; καὶ καθόλου τίνος ἡ ἀπόδειξις; δῆλον ὅτι ὅταν ἀφαιρουμένων ὑπάρχῃ πρώτῳ. οἷον τῷ ἰσοσκελεῖ χαλκῷ τριγώνῳ ὑπάρξουσι δύο ὀρθαί, ἀλλὰ καὶ τοῦ χαλκοῦν εἶναι ἀφαιρεθέντος (74b.) καὶ τοῦ ἰσοσκελές. ἀλλ' οὐ τοῦ σχήματος ἢ πέρατος. ἀλλ' οὐ πρώτων. τίνος οὖν πρώτου; εἰ δὴ τριγώνου, κατὰ τοῦτο ὑπάρχει καὶ τοῖς ἄλλοις, καὶ τούτου καθόλου ἐστὶν ἡ ἀπόδειξις. a35. 'But', it will be asked, 'does this attribute belong to the subject of which it has been demonstrated qua triangle or qua isosceles? What is the point at which the subject to which it belongs is primary? (i.e. to what subject can it be demonstrated as belonging commensurately and universally?)' Clearly this point is the first term in which it is found to inhere as the elimination of inferior differentiae proceeds. Thus the angles of a brazen isosceles triangle are equal to two right angles: but eliminate brazen and isosceles and the attribute remains. 'But' — you may say — 'eliminate figure or limit, and the attribute vanishes.' True, but figure and limit are not the first differentiae whose elimination destroys the attribute. 'Then what is the first?' If it is triangle, it will be in virtue of triangle that the attribute belongs to all the other subjects of which it is predicable, and triangle is the subject to which it can be demonstrated as belonging commensurately and universally.
Postquam notificavit Aristoteles quid sit universale, hic ostendit quomodo in acceptione universalis errare contingat. Et circa hoc tria facit: primo, dicit quod aliquando circa hoc peccare contingit; secundo, assignat quot modis; ibi: oberramus etc.; tertio, dat documentum quomodo possit cognosci utrum vere acceptum sit universale; ibi: utrum autem secundum quod et cetera. After specifying what the universal is, the Philosopher here shows how one might err in understanding the universal. In regard to this he does three things. First, he says that sometimes one might err in this matter. Secondly, he tells in how many ways (74a6). Thirdly, he gives the criterion for knowing whether the universal is being employed correctly (74a35).
Dicit ergo primo quod ad hoc, quod non accidat in demonstratione peccatum, oportet non latere quod multoties videtur demonstrari universale, non autem demonstratur. He says therefore first (74a4), that in order to avoid mistakes in demonstrating, one should be aware of the fact that quite often something universal seems to be demonstrated, which is not being demonstrated.
Deinde cum dicit: oberramus autem etc., assignat modos quibus circa hoc errare contingit. Et circa hoc duo facit. Secondly (74a6), he indicates the ways in which this mistake can occur. And in regard to this he does two things.
Primo, enumerat ipsos modos, dicens quod tripliciter contingit decipi circa acceptionem universalis. Primo quidem, cum nihil aliud sit accipere sub aliquo communi cui primo competit universale, quam hoc singulare, cui inconvenienter assignatur. Sicut si sensibile, quod primo et per se inest animali, assignaretur ut universale primum homini, nullo alio animali existente. Unde notandum quod singulare hic large accipitur pro quolibet inferiori, sicut si species dicatur singulare sub genere contentum. Vel potest dici quod non est possibile invenire aliquod genus, cuius una tantum sit species. Genus enim dividitur in species per oppositas differentias; oportet autem, si unum contrariorum invenitur in natura, et reliquum inveniri, ut patet per philosophum in II de caelo et mundo; et ideo si una species invenitur, invenitur et alia. Una autem species dividitur in diversa individua per divisionem materiae. Contingit autem totam materiam alicui speciei proportionatam, sub uno individuo comprehendi, et tunc non est nisi unum individuum sub una specie. Unde et signanter de singulari mentionem facit. First, he enumerates these ways and says that there are three possible errors in understanding a universal. The first is likely to occur when under some common genus there is nothing else to take as the thing to which the universal initially applies than this singular, to which it is incorrectly applied. For example, if man were the only animal existing, and “sensible,” which is initially and per se in animal, were to be assigned as a primary universal to man. (It should be noted that singular is being used here in a wide sense for any inferior, in the way that a species might be called a singular contained under a genus). Or we might say that it is not possible to find a genus with only one species: for a genus is divided into species through opposing differences. But if one contrary is found in nature, so must the other, as the Philosopher explains in On the Heavens II. Therefore, if one species is found, another will be found. However, one species is divided into distinct individuals by the division of matter. But it sometimes happens that all the matter proportionate to a given species is comprehended under one individual, so that in that case there is only one individual under one species. Hence it is significant that he did say, “singular.”
Secundus modus est, quando est quidem accipere sub aliquo communi multa inferiora, sed tamen illud est commune innominatum, quod invenitur in rebus differentibus specie. Sicut si animali non esset nomen impositum, et sensibile, quod est proprium animalis, assignaretur ut universale primum his quae sub animali continentur, vel divisim vel coniunctim. The second way is when it is possible to take several inferiors under something common which is verified in things that differ in species, but that common item has no name. For example, if “animality” had no name, and “sensibility,” which is proper to animal, were to be assigned to the inferiors of animal (either collectively or distributively) as their primary universal.
Tertius modus est, quando illud de quo demonstratur aliquid, ut universale primum, se habet ad id quod demonstratur de eo, sicut totum ad partem. Sicut si posse videre assignaretur animali ut universale primum. Non enim omne animal potest videre. Inest enim his, quae sunt in parte, idest quae particulariter et non universaliter alicui subiecto conveniunt, demonstratio, idest quod demonstrari possint, et erit quidem demonstratio de omni, non tamen respectu huius de quo demonstratur. Posse enim videre demonstratur quidem de aliquo universaliter, non tamen universaliter de animali, sicut de eo cui primo insit. Et exponit quid sit primum, secundum quod demonstratio fertur, quod est universale primum. The third way is when that of which something is demonstrated to be its primary universal is related to what is demonstrated of it, as a whole is related to a part. For example, if the power to see were assigned as a primary universal to animal: for not every animal can see. In this case the demonstration, i.e., what should have been demonstrated, is “in those things which are in part,” i.e., in some but not all of the things included under the subject; furthermore, it will be a demonstration “of all,” but not of all that the demonstration mentions. For the power to see can indeed be demonstrated universally of something, but not of animal universally, as of that to which it belongs primarily. And he explains why he says, “primarily,” namely, because a demonstration bears on what is both universal and first.
Secundo; ibi: si igitur etc., subiungit exempla ad praedictos modos, et primo ad tertium, dicens quod, si quis demonstret de lineis rectis quod non intercidant, idest non concurrant, videbitur huiusmodi esse demonstratio, scilicet universalis primi, propter hoc quod non concurrere inest aliquibus lineis rectis. Non autem ita quod hoc fiat, nisi lineae rectae sint aequales, idest aeque distantes. Sed si lineae fuerint aequales, idest aeque distantes, tunc non concurrere convenit eis in quolibet, quia universaliter verum est quod lineae rectae aeque distantes, etiam si in infinitum protrahantur, in neutram partem concurrent. Secondly (74a13), he gives examples of each of these ways. First, of the third way, saying that if someone were to demonstrate of two straight lines that they do not intersect, i.e., that they do not meet, it might seem that we have a demonstration of this sort, i.e., one that bears on a primary universal, on the ground that “not to meet” is true of certain straight lines, and not that this happens only because the straight lines are equal, i.e., equally distant. But if the lines should be equal, i.e., equidistant, then “not to meet” belongs to any and all of them, because it is universally true that lines which are straight and equally distant, even should they be lengthened ad infinitum, will not meet.
Secundo; ibi: et si triangulus etc., ponit exemplum ad primum modum, dicens quod si non esset alius triangulus, quam isosceles, qui est triangulus duorum aequalium laterum, quod est trianguli in quantum huiusmodi, videretur esse isoscelis secundum quod est isosceles: nec tamen hoc esset verum. Secondly (74a17), he gives an example of the first way, saying that if there were no triangle but the isosceles which is a triangle having two equal sides, it might seem that what is true of triangle as triangle should be true of isosceles as isosceles. But this would not be so.
Tertio; ibi: et proportionale etc., exemplificat de secundo modo. Et videtur hoc ultimo ponere, quia circa hoc diutius immoratur. Et circa hoc tria facit: primo, ponit exemplum; secundo, inducit quoddam corollarium ex dictis; ibi: propter hoc nec si aliquis etc.; tertio, assignat rationem dictorum; ibi: quando igitur non novit et cetera. Thirdly (74a18), he gives an example of the second way. Now it seems that he saved this for last because he wished to spend more time on it. Concerning it he does three things. First, he gives the example. Secondly, he draws a corollary from what he has said (74a25). Thirdly, he gives a reason for the aforesaid (74a33).
Circa primum sciendum est quod proportio est habitudo unius quantitatis ad alteram, sicut sex ad tria se habent in proportione dupla. Proportionalitas vero est collatio duarum proportionum. Quae, si sit disiuncta, habet quatuor terminos; ut hic: sicut se habent quatuor ad duo, ita sex ad tria: si vero sit coniuncta, habet tres terminos: nam uno utitur ut duobus; ut hic: sicut se habent octo ad quatuor, ita quatuor ad duo. Concerning the first it should be noted that a ratio is a relation of one quantity to another, as 6 is related to 3 in the ratio of 2 to 1. The com. paring of one ratio to another is a proportion, which, if it is disjoint, has four terms: for example, as 4 is to 2, so 6 is to 3. But if it is joint, it has three terms, one of which is used twice: for example, as 8 is to 4, so 4 is to 2.
Patet autem quod in proportione duo termini se habent ut antecedentia; duo vero ut consequentia; ut hic: sicut se habent quatuor ad duo, ita se habent sex ad tria; sex et quatuor sunt antecedentia: tria vero et duo sunt consequentia. Permutata ergo proportio est quando antecedentia invicem conferuntur, et consequentia similiter. Ut si dicam: sicut se habent quatuor ad duo, ita se habent sex ad tria; ergo sicut se habent quatuor ad sex, ita se habent duo ad tria. Now it is obvious that in a proportion two terms are antecedents and two are consequents. For example, in the proportion that 6 is to 3 as 4 is to 2, 6 and 4 are the antecedents, 3 and 2 the consequents. Again, a proportion is alternated by bringing the antecedents together and the consequents together. For example, when I say: As 4 is to 2, so 6 is to 3; therefore, as 4 is to 6, so 2 is to 3.
Dicit ergo quod esse proportionale commutabiliter convenit numeris, et lineis, et firmis, idest corporibus, et temporibus. Sicut autem de singulis determinatum est aliquando seorsum, de numeris quidem in arithmetica, de lineis et firmis in geometria, de temporibus in naturali philosophia vel astrologia, ita contingens est, quod de omnibus praedictis commutatim proportionari una demonstratione demonstretur. Sed ideo commutatim proportionari, de singulis horum seorsum demonstratur, quia non est nominatum illud commune, in quo omnia ista sunt unum. Etsi enim quantitas omnibus his communis sit, tamen sub se et alia, praeter haec, comprehendit, sicut orationem et quaedam, quae sunt quantitates per accidens. What he is saying, therefore, is that alternate proportion is verified in lines, numbers, solids, (i.e., bodies), and times. But just as this is established separately for each of them, namely, for numbers in arithmetic, for lines and solids in geometry, and for times in natural philosophy or astronomy, so it is possible to prove it of all of them in a single demonstration. But the reason why a separate demonstration was employed to prove alternation for each was that the one feature they had in common was unnamed. For although quantity is a feature common to all of them, quantity includes other things besides them; for example, speech and other things that are accidentally quantitative.
Vel melius dicendum quod commutatim proportionari non convenit quantitati in quantum est quantitas, sed in quantum est comparata alteri quantitati secundum proportionalitatem quandam. Et ideo dixerat etiam in principio proportionale esse quod commutabiliter est. Omnibus autem istis, in quantum sunt proportionalia, non est nomen commune positum. Cum autem demonstratur commutatim proportionari de singulis praedictorum divisim, non demonstratur universale. Non enim commutatim proportionari inest numeris et lineis, secundum quod huiusmodi, sed secundum quoddam commune. Demonstrantes autem de lineis seorsum vel de numeris ponunt hoc, quod est commutabiliter proportionari, esse quasi quoddam universale praedicatum lineae secundum quod linea est, aut numeri secundum quod numerus. Or, better still, it was because alternate proportion does not belong to quantity precisely as quantity, but as compared to another quantity according to a fixed ratio. That is why at the very start he spoke of alternate proportion. But there is no general name for the aforesaid things precisely as they are proportional. Furthermore, when alternate proportion is demonstrated separately for each of them, it is not a universal that is being demonstrated. For “to be alternately proportional” is not in numbers and lines according to what each of them is, but according to something common. Besides, those who demonstrate alternate proportion of lines look upon this attribute as a universal predicate of line precisely as it is a line, or of number precisely as it is number.
Deinde cum dicit: propter hoc nec si aliquis etc., inducit quoddam corollarium ex dictis, dicens quod eadem ratione, qua non demonstratur universale cum de singulis speciebus aliquid demonstratur, quod est universale praedicatum communis innominati; nec etiam demonstratur universale modo praedicto, si sit commune nomen positum. Sicut si aliquis aut eadem demonstratione aut diversa demonstret de unaquaque specie trianguli, quod habet duos rectos, seorsum scilicet de isoscele et seorsum de gradato, idest de triangulo trium laterum inaequalium, non tamen propter hoc cognovit quod triangulus tres angulos habeat aequales duobus rectis, nisi sophistico modo, idest per accidens: quia non cognovit de triangulo secundum quod est triangulus, sed secundum quod est aequilaterus, aut duorum aequalium laterum, aut trium inaequalium. Then (74a25), he draws a corollary from the above and says that for the same reason that something universal is not being demonstrated when a common unnamed attribute is proved to be a universal predicate of each species, so too when something is demonstrated this way of a common attribute which does have a name. For example, if someone uses the same demonstration for each species of triangle and proves that each has three angles equal to two right angles, or if he uses different demonstrations, say one for isosceles and another for scalene, even then he does not on that account know that a triangle has three angles equal to two right angles except in a sophistical way, i.e., per accidens: for he does not know it of triangle as triangle, but as equilateral or as having two equal sides or as having three unequal sides.
Neque etiam demonstrans cognovit universale trianguli, idest habet cognitionem de triangulo in universali, etiamsi nullus alius triangulus esset praeter illos, de quibus cognovit. Et hoc ideo, quia non cognovit de triangulo secundum quod est triangulus, sed sub ratione specierum eius. Unde neque cognovit, per se loquendo, omnem triangulum: quia et si secundum numerum cognovit omnem triangulum (si nullus est, quem non novit), tamen secundum speciem non cognovit omnem. Tunc enim cognoscitur aliquid universaliter secundum speciem, quando cognoscitur secundum rationem speciei. Secundum numerum autem et non universaliter, quando cognoscitur secundum multitudinem contentorum sub specie. Nec est differentia quantum ad hoc si comparemus species ad individua vel genera ad species. Nam triangulus est genus aequilateri et isoscelis. Furthermore, one who demonstrates in this way does not know the universal of triangle, i.e., he does not have a knowledge of triangle in a universal way (even if there happens to be no other triangle besides those of which he knew this: and this because he had not the knowledge of triangle as triangle but under the aspect of its various species). Again, strictly speaking, he did not know every triangle: for although he knew every triangle according to number (if there was none he did not know), yet according to species he did not know each one. For something is known universally according to species, when it is known according to the notion of the species, but according to number and not universally, when it is known according to the multifarious things contained under the species. And in this matter there is no difference whether we compare species to individuals or genera to species. For triangle is the genus of equilateral and isosceles.
Deinde cum dicit: quando igitur non novit etc., assignat rationem praedictorum, quaerens quando aliquis cognoscat universaliter et simpliciter, ex quo praedicto modo cognoscens non cognoscit universaliter. Et respondet manifestum esse quod, si eadem esset ratio trianguli in communi et uniuscuiusque specierum eius seorsum acceptae aut omnium simul acceptarum, tunc universaliter et simpliciter nosceret de triangulo, quando sciret de aliqua specie eius vel de omnibus simul. Si vero non est eadem ratio, tunc non erit idem cognoscere triangulum in communi et singulas species eius; sed est alterum. Et cognoscendo de speciebus, non cognoscitur de triangulo secundum quod est triangulus. Then (74a33) he assigns the reason for the aforesaid, first asking when does one know universally and absolutely, if one who knew in the aforesaid way did not know universally. And he answers that obviously if the essence of triangle in general and of each of its species (each taken separately or all taken together) were the same, then one would know universally and absolutely about triangle, when he knew about any one species of it separately or about all of them together. But if the essence is not the same, then it will not be the same but something different to know triangle and to know its several species. And by knowing something of the species one does not know it of triangle precisely as triangle.
Deinde cum dicit: utrum autem etc., dat documentum quo proprie possit accipi universale, dicens quod, utrum aliquid sit trianguli secundum quod est triangulus, aut isoscelis, secundum quod est isosceles, et quando id cuius est demonstratio sit primum et universale, secundum hoc, idest secundum aliquod subiectum positum; manifestum est ex hoc quod dicam. Then (74a35) he gives the rule on how the universal can be properly understood, saying that whether something belongs to triangle precisely as triangle or to isosceles precisely as isosceles, and when it is that what is demonstrated is first and universal in the given subject precisely as this, i.e., precisely according to the subject—all this will be clear from what I shall say.
Quandocumque enim, remoto aliquo, adhuc remanet illud quod assignatur universale, sciendum est quod non est primum universale illius. Sicut, remoto isoscele vel aeneo triangulo, remanet quod habeat tres angulos, scilicet duobus rectis aequales. Unde patet quod habere tres angulos aequales duobus rectis non est universale primum, neque isoscelis, neque aenei trianguli. Remota autem figura non remanet habere tres, nec etiam, remoto termino, qui est superius ad figuram, cum figura sit, quae termino vel terminis clauditur; sed tamen non primo convenit neque figurae, neque termino, quia non convenit eis universaliter. For whenever some item is removed from the subject and the original universal still applies to what remains, then it is not the first universal of that subject. For example, upon the removal of isosceles or of brazen there still remains the triangle which has three angles equal to two right angles; hence the possession of three angles equal to two right angles is not a first universal of isosceles or of brazen triangle. But if the item “figure” be removed, nothing which has three angles equal to two right angles remains, nor again if you remove “bounded” which is more universal than figure, since a figure is something enclosed by a bound or bounds. Nevertheless, the attribute in question [namely, having three angles equal to two right angles] does not belong first to figure or to bounded, because it does not belong to either of them universally.
Cuius ergo erit primo? Manifestum est quod trianguli, quia secundum triangulum inest aliis, tam superioribus, quam inferioribus: ideo enim competit figurae habere tres, quia triangulus est quaedam figura; et similiter isosceli, quia triangulus est, et de triangulo habere tres universaliter demonstratur. Unde eius est universale primum. To what then will it be first? Obviously to triangle, because it belongs to the others (both superiors and inferiors) precisely as they are triangles For it belongs to figure to have three angles equal to two right angles only because a triangle is some figure, and similarly to isosceles, only because it is a triangle, and it is of triangle that “having three...” is demonstrated. Hence, it is to triangle that it belongs as a first universal

Lectio 13
Caput 6
Εἰ οὖν ἐστιν ἡ ἀποδεικτικὴ ἐπιστήμη ἐξ ἀναγκαίων ἀρχῶν (ὃ γὰρ ἐπίσταται, οὐ δυνατὸν ἄλλως ἔχειν), b5. Demonstrative knowledge must rest on necessary basic truths; for the object of scientific knowledge cannot be other than it is.
τὰ δὲ καθ' αὑτὰ ὑπάρχοντα ἀναγκαῖα τοῖς πράγμασιν (τὰ μὲν γὰρ ἐν τῷ τί ἐστιν ὑπάρχει· τοῖς δ' αὐτὰ ἐν τῷ τί ἐστιν ὑπάρχει κατηγορουμένοις αὐτῶν, ὧν θάτερον τῶν ἀντικειμένων ἀνάγκη ὑπάρχειν), φανερὸν ὅτι ἐκ τοιούτων τινῶν ἂν εἴη ὁ ἀποδεικτικὸς συλλογισμός· ἅπαν γὰρ ἢ οὕτως ὑπάρχει ἢ κατὰ συμβεβηκός, τὰ δὲ συμβεβηκότα οὐκ ἀναγκαῖα. b6. Now attributes attaching essentially to their subjects attach necessarily to them: for essential attributes are either elements in the essential nature of their subjects, or contain their subjects as elements in their own essential nature. (The pairs of opposites which the latter class includes are necessary because one member or the other necessarily inheres.) It follows from this that premisses of the demonstrative syllogism must be connexions essential in the sense explained: for all attributes must inhere essentially or else be accidental, and accidental attributes are not necessary to their subjects.
Ἢ δὴ οὕτω λεκτέον, ἢ ἀρχὴν θεμένοις ὅτι ἡ ἀπόδειξις ἀναγκαίων ἐστί, καὶ εἰ ἀποδέδεικται, οὐχ οἷόν τ' ἄλλως ἔχειν· ἐξ ἀναγκαίων ἄρα δεῖ εἶναι τὸν συλλογισμόν. ἐξ ἀληθῶν μὲν γὰρ ἔστι καὶ μὴ ἀποδεικνύντα συλλογίσασθαι, ἐξ ἀναγκαίων δ' οὐκ ἔστιν ἀλλ' ἢ ἀποδεικνύντα· τοῦτο γὰρ ἤδη ἀποδείξεώς ἐστιν. b13. We must either state the case thus, or else premise that the conclusion of demonstration is necessary and that a demonstrated conclusion cannot be other than it is, and then infer that the conclusion must be developed from necessary premisses. For though you may reason from true premisses without demonstrating, yet if your premisses are necessary you will assuredly demonstrate — in such necessity you have at once a distinctive character of demonstration.
σημεῖον δ' ὅτι ἡ ἀπόδειξις ἐξ ἀναγκαίων, ὅτι καὶ τὰς ἐνστάσεις οὕτω φέρομεν πρὸς τοὺς οἰομένους ἀποδεικνύναι, ὅτι οὐκ ἀνάγκη, ἂν οἰώμεθα ἢ ὅλως ἐνδέχεσθαι ἄλλως ἢ ἕνεκά γε τοῦ λόγου. b18. That demonstration proceeds from necessary premisses is also indicated by the fact that the objection we raise against a professed demonstration is that a premiss of it is not a necessary truth — whether we think it altogether devoid of necessity, or at any rate so far as our opponent's previous argument goes.
δῆλον δ' ἐκ τούτων καὶ ὅτι εὐήθεις οἱ λαμβάνειν οἰόμενοι καλῶς τὰς ἀρχάς, ἐὰν ἔνδοξος ᾖ ἡ πρότασις καὶ ἀληθής, οἷον οἱ σοφισταὶ ὅτι τὸ ἐπίστασθαι τὸ ἐπιστήμην ἔχειν. οὐ γὰρ τὸ ἔνδοξον ἡμῖν ἀρχή ἐστιν, ἀλλὰ τὸ πρῶτον τοῦ γένους περὶ ὃ δείκνυται· καὶ τἀληθὲς οὐ πᾶν οἰκεῖον. b22. This shows how naive it is to suppose one's basic truths rightly chosen if one starts with a proposition which is (1) popularly accepted and (2) true, such as the sophists' assumption that to know is the same as to possess knowledge. For (1) popular acceptance or rejection is no criterion of a basic truth, which can only be the primary law of the genus constituting the subject matter of the demonstration; and (2) not all truth is 'appropriate'.
ὅτι δ' ἐξ ἀναγκαίων εἶναι δεῖ τὸν συλλογισμόν, φανερὸν καὶ ἐκ τῶνδε. εἰ γὰρ ὁ μὴ ἔχων λόγον τοῦ διὰ τί οὔσης ἀποδείξεως οὐκ ἐπιστήμων, εἴη δ' ἂν ὥστε τὸ Α κατὰ τοῦ Γ ἐξ ἀνάγκης ὑπάρχειν, τὸ δὲ Β τὸ μέσον, δι' οὗ ἀπεδείχθη, μὴ ἐξ ἀνάγκης, οὐκ οἶδε διότι. οὐ γάρ ἐστι τοῦτο διὰ τὸ μέσον· τὸ μὲν γὰρ ἐνδέχεται μὴ εἶναι, τὸ δὲ συμπέρασμα ἀναγκαῖον. b27. A further proof that the conclusion must be the development of necessary premisses is as follows. Where demonstration is possible, one who can give no account which includes the cause has no scientific knowledge. If, then, we suppose a syllogism in which, though A necessarily inheres in C, yet B, the middle term of the demonstration, is not necessarily connected with A and C, then the man who argues thus has no reasoned knowledge of the conclusion, since this conclusion does not owe its necessity to the middle term; for though the conclusion is necessary, the mediating link is a contingent fact.
ἔτι εἴ τις μὴ οἶδε νῦν ἔχων τὸν λόγον καὶ σῳζόμενος, σῳζομένου τοῦ πράγματος, μὴ ἐπιλελησμένος, οὐδὲ πρότερον ᾔδει. φθαρείη δ' ἂν τὸ μέσον, εἰ μὴ ἀναγκαῖον, ὥστε ἕξει μὲν τὸν λόγον σῳζόμενος σῳζομένου τοῦ πράγματος, οὐκ οἶδε δέ. οὐδ' ἄρα πρότερον ᾔδει. εἰ δὲ μὴ ἔφθαρται, ἐνδέχεται δὲ φθαρῆναι, τὸ συμβαῖνον ἂν εἴη δυνατὸν καὶ ἐνδεχόμενον. ἀλλ' ἔστιν ἀδύνατον οὕτως ἔχοντα εἰδέναι. b32. Or again, if a man is without knowledge now, though he still retains the steps of the argument, though there is no change in himself or in the fact and no lapse of memory on his part; then neither had he knowledge previously. But the mediating link, not being necessary, may have perished in the interval; and if so, though there be no change in him nor in the fact, and though he will still retain the steps of the argument, yet he has not knowledge, and therefore had not knowledge before. Even if the link has not actually perished but is liable to perish, this situation is possible and might occur. But such a condition cannot be knowledge.
(75a.) Ὅταν μὲν οὖν τὸ συμπέρασμα ἐξ ἀνάγκης ᾖ, οὐδὲν κωλύει τὸ μέσον μὴ ἀναγκαῖον εἶναι δι' οὗ ἐδείχθη (ἔστι γὰρ τὸ ἀναγκαῖον καὶ μὴ ἐξ ἀναγκαίων συλλογίσασθαι, ὥσπερ καὶ ἀληθὲς μὴ ἐξ ἀληθῶν)· ὅταν δὲ τὸ μέσον ἐξ ἀνάγκης, καὶ τὸ συμπέρασμα ἐξ ἀνάγκης, ὥσπερ καὶ ἐξ ἀληθῶν ἀληθὲς ἀεί (ἔστω γὰρ τὸ Α κατὰ τοῦ Β ἐξ ἀνάγκης, καὶ τοῦτο κατὰ τοῦ Γ· ἀναγκαῖον τοίνυν καὶ τὸ Α τῷ Γ ὑπάρχειν)· ὅταν δὲ μὴ ἀναγκαῖον ᾖ τὸ συμπέρασμα, οὐδὲ τὸ μέσον ἀναγκαῖον οἷόν τ' εἶναι (ἔστω γὰρ τὸ Α τῷ Γ μὴ ἐξ ἀνάγκης ὑπάρχειν, τῷ δὲ Β, καὶ τοῦτο τῷ Γ ἐξ ἀνάγκης· καὶ τὸ Α ἄρα τῷ Γ ἐξ ἀνάγκης ὑπάρξει· ἀλλ' οὐχ ὑπέκειτο). a1. When the conclusion is necessary, the middle through which it was proved may yet quite easily be non-necessary. You can in fact infer the necessary even from a non-necessary premiss, just as you can infer the true from the not true. On the other hand, when the middle is necessary the conclusion must be necessary; just as true premisses always give a true conclusion. Thus, if A is necessarily predicated of B and B of C, then A is necessarily predicated of C. But when the conclusion is nonnecessary the middle cannot be necessary either. Thus: let A be predicated non-necessarily of C but necessarily of B, and let B be a necessary predicate of C; then A too will be a necessary predicate of C, which by hypothesis it is not.
Ἐπεὶ τοίνυν εἰ ἐπίσταται ἀποδεικτικῶς, δεῖ ἐξ ἀνάγκης ὑπάρχειν, δῆλον ὅτι καὶ διὰ μέσου ἀναγκαίου δεῖ ἔχειν τὴν ἀπόδειξιν· ἢ οὐκ ἐπιστήσεται οὔτε διότι οὔτε ὅτι ἀνάγκη ἐκεῖνο εἶναι, ἀλλ' ἢ οἰήσεται οὐκ εἰδώς, ἐὰν ὑπολάβῃ ὡς ἀναγκαῖον τὸ μὴ ἀναγκαῖον, ἢ οὐδ' οἰήσεται, ὁμοίως ἐάν τε τὸ ὅτι εἰδῇ διὰ μέσων ἐάν τε τὸ διότι καὶ δι' ἀμέσων. a13. To sum up, then: demonstrative knowledge must be knowledge of a necessary nexus, and therefore must clearly be obtained through a necessary middle term; otherwise its possessor will know neither the cause nor the fact that his conclusion is a necessary connexion. Either he will mistake the non-necessary for the necessary and believe the necessity of the conclusion without knowing it, or else he will not even believe it — in which case he will be equally ignorant, whether he actually infers the mere fact through middle terms or the reasoned fact and from immediate premisses.
Postquam determinavit philosophus de dici de omni, et per se, et universali quibus utimur in demonstratione, hic iam incipit ostendere ex quibus demonstratio procedit. Et dividitur in duas partes: in prima, ostendit ex quibus procedat demonstratio propter quid; in secunda, ex quibus procedat demonstratio quia; ibi: sed quia differt et propter quid et cetera. Prima in duas: in prima, ostendit qualia sint ex quibus demonstratio procedit; in secunda, docet quae sint demonstrationis principia; ibi: quid quidem igitur prima significent et cetera. Prima in tres: in prima, ostendit quod demonstratio est ex necessariis; in secunda, quod est ex his, quae sunt per se; ibi: accidentium autem etc.; in tertia, quod procedat ex principiis propriis; ibi: non ergo est ex alio genere et cetera. Circa primum duo facit: primo, ostendit quod demonstratio procedat ex necessariis; secundo, probat quaedam quae supposuerat; ibi: quod autem oporteat ex necessariis et cetera. Circa primum tria facit: primo, continuat se ad praecedentia; secundo, probat propositum; ibi: quae autem sunt per se etc.; tertio, infert ex dictis quandam conclusionem; ibi: manifestum autem ex his et cetera. Having finished his treatment of that which is said “of all” and “per se” and “universally,” all of which we use in demonstration, the Philosopher now begins to discuss the items from which demonstration proceeds. And there are two parts. In the first he shows what demonstration of the reasoned fact [demonstration propter quid] proceeds from. In the second what demonstration of the fact [demonstration quia] proceeds from (78a22) [L. 23]. The first is divided into two parts. In the first he shows the sort of things from which a demonstration proceeds. In the second, what the principles of demonstration are (76a26) [L. 18]. The first is divided into three parts. In the first he shows that demonstration issues from necessary things. In the second that it issues from things that are per se (76a26) [L. 14]. In the third that it proceeds from proper principles (75a38) [L. 15]. Concerning the first he does two things. First, he shows that demonstration should proceed from necessary things. Secondly, he proves certain things he had presupposed (74b27). In regard to the first he does three things. First, he makes a connection with what went before. Secondly, he proves his proposition (74b6). Thirdly, he draws a conclusion from the aforesaid (74b22).
Dicit ergo primo, ex praedictis inferens, quod si est demonstrativa scientia, idest si scientia per demonstrationem acquiritur, oportet quod sit ex necessariis principiis. Cuius illationis necessitas ex hoc apparet, quia quod scitur impossibile est aliter se habere, ut habitum est in definitione eius quod est scire. He says therefore first (74b5), as an inference from what has already been established, that if there be demonstrative science, i.e., if science is acquired through demonstration, it must issue from necessary principles. The necessity of this inference is clear, because that which is known scientifically cannot be otherwise than it is, as was pointed out in the definition of scientific knowing.
Deinde cum dicit: quae autem per se sunt etc., ostendit quod demonstratio sit ex necessariis; et primo, per rationem; secundo, per signum; ibi: signum autem est et cetera. Then (74b6) he shows that demonstration issues from necessary things. First, with a reason. Secondly, with a sign (74b19).
Circa primum ponit duas rationes: quarum prima talis est. Ea, quae per se praedicantur, necessario insunt. Et hoc manifestat in duobus modis per se. In primo quidem, quia ea, quae per se praedicantur, insunt in eo quod quid est, idest in definitione subiecti. Quod autem ponitur in definitione alicuius, necessario praedicatur de eo. In secundo vero, quia quaedam sunt subiecta, quae ponuntur in quod quid est praedicantibus de ipsis, idest in definitione suorum praedicatorum. Quae quidem si sint opposita, necesse est quod alterum eorum subiecto insit; sicut par vel impar numero, ut superius ostensum est. Sed manifestum est quod ex quibusdam principiis huiusmodi, scilicet per se, fit syllogismus demonstrativus: quod probat per hoc, quod omne quod praedicatur, aut praedicatur per se aut per accidens; et ea, quae praedicantur per accidens, non sunt necessaria: ex his autem, quae sunt per accidens, non fit demonstratio, sed magis sophisticus syllogismus. Unde relinquitur quod demonstratio sit ex necessariis. With respect to the first he gives two reasons. The first (74b6) is this: Things predicated per se are in a thing necessarily. (And he manifests this in two of the modes of predicating per se: in the first mode, because the things predicated per se are “in that which is what,” i.e., in the definition of the subject. But whatever is put in the definition of anything is predicated of it necessarily. In the second mode, because certain subjects are put in “that which is the what of things predicated of them,” i.e., in the definition of their predicates. And if these are opposites, one or other of them is necessarily in the subject: thus, either “odd” or “even” is in number, as we showed above). Now it is clear that it is from principles of this kind, namely, per se, that a demonstrative syllogism proceeds. (And this is proved by the fact that whatever is predicated, is predicated either per se or per accidens, and that things predicated per accidens are not necessary). Therefore, since it is not a demonstration but rather a sophistic syllogism that issues from things that are per accidens, it follows that demonstration proceeds from necessary things.
Sciendum autem est quod cum in demonstratione probetur passio de subiecto per medium, quod est definitio, oportet quod prima propositio, cuius praedicatum est passio et subiectum est definitio, quae continet principia passionis, sit per se in quarto modo; secunda autem, cuius subiectum est ipsum subiectum et praedicatum ipsa definitio, in primo modo. Conclusio vero, in qua praedicatur passio de subiecto, est per se in secundo modo. Furthermore, it should be noted that, since in a demonstration a proper attribute is proved of a subject through a middle which is the definition, it is required that the first proposition (whose predicate is the proper attribute, and whose subject is the definition which contains the principles of the proper attribute) be per se in the fourth mode, and that the second proposition (whose subject is the subject itself and the predicate its definition) must be in the first mode. But the conclusion, in which the proper attribute is predicated of the subject, must be per se in the second mode.
Secundam rationem ponit; ibi: aut igitur sic etc.: quae talis est. Demonstratio circa necessarium est et demonstratum, idest demonstrationis conclusio, non potest aliter se habere. Et hoc accipiendum est tanquam principium ad ostendendum propositum, scilicet quod demonstratio ex necessariis procedat; cuius quidem principii veritas ex praemissis apparet, ut iam dictum est. Ex hoc autem principio sic argumentatur. Conclusio necessaria non potest sciri nisi ex principiis necessariis; sed demonstratio facit scire conclusionem necessariam; ergo oportet quod sit ex principiis necessariis. Then (74b14) he sets forth the second reason and it is this: “Demonstration is concerned with the necessary and the demonstrated,” i.e., the conclusion of the demonstration cannot be other than it is. This statement is to be taken as the principle proving our proposition that demonstration proceeds from necessary things. And from what has been said, the truth of this principle is obvious. From this principle one argues thus: a necessary conclusion cannot be scientifically known save from necessary principles; but a demonstration makes a necessary conclusion scientifically known: therefore, it must proceed from necessary principles.
In quo differt demonstratio ab aliis syllogismis: sufficit enim in aliis syllogismis quod syllogizetur ex veris. Nec est aliquod aliud genus syllogismi, in quo oporteat ex necessariis procedere, sed in demonstratione tantum oportet hoc observare. Et hoc est proprium demonstrationis, scilicet ex necessariis semper procedere. Herein lies the difference between a demonstration and other syllogisms. For in the latter it is enough if one syllogizes from true principles. Nor is there any other type of syllogism in which it is required to proceed from the necessary: in a demonstration alone must this be observed. And this is proper to a demonstration, i.e., to proceed from the necessary.
Deinde cum dicit: signum autem etc., probat idem per signum hoc modo. Contra rationem aliquam non infertur instantia, nisi per hoc quod deficit aliquid eorum, quae in ratione illa observanda sunt; sed contra eum, qui opinatur se demonstrare, ferimus instantiam quod non sit necesse ea, ex quibus procedit, esse vera: sive opinemur ea contingere aliter se habere, sive talem instantiam feramus rationis, idest disputationis causa; ergo demonstratio debet procedere ex necessariis. Then (74b18) he proves the same thing in the following way, using a sign: Suit is brought against an argument solely on the ground that something is wanting that should have been observed in that argument. But against one who believes that he has demonstrated, we bring the charge that the things from which he proceeded are not necessarily true, whether we are convinced that they could be otherwise than stated, or whether we bring a charge of reason, i.e., for the sake of disputing. Therefore, demonstration should proceed from necessary things.
Deinde cum dicit: manifestum autem etc., infert conclusionem ex dictis: dicens quod manifestum est, ex hoc quod oportet demonstrationem ex necessariis concludere, quod stulti sunt illi, qui opinati sunt bene se principia demonstrationis accipere, si solum propositio accepta sit probabilis vel vera, ut sophistae faciunt, idest illi, qui apparent scientes et non sunt. Nam scire non est nisi per hoc quod scientia habetur, scilicet ex demonstratione; ex hoc autem quod aliquid est probabile vel improbabile non habetur quod sit primum vel non primum: sed tamen oportet illud circa quod fit demonstratio esse primum in genere aliquo et esse verum. Non tamen omne primum accipit demonstrator, sed primum proprium illi generi, circa quod demonstrat; sicut arithmeticus non accipit primum, quod est circa magnitudinem, sed circa numerum. Then (74b22) he draws a conclusion from the foregoing, saying that “this,” namely, the fact that a demonstration must conclude from necessary things, “shows” that they are obtuse who assumed that they rightly chose the principles of demonstration, if the proposition they chose was probable or true, as the Sophists do, i.e., those who appear wise but are not. For scientific knowing consists in nothing less than having science, namely, from demonstration. But when something is probable or improbable, it is not certain whether it is first or not first, whereas that on which a demonstration bears must be first in some genus and be true. This does not mean that a demonstrator may take anything that is first: it must be first in that genus with respect to which he is demonstrating. Thus, arithmetic does not choose what is first in regard to magnitude, in regard to number.
Attendendum est autem quod sophistae non sumuntur hic sicut in libro elenchorum, qui procedunt ex his quae videntur probabilia et non sunt, aut videntur syllogizare, non tamen syllogizant. Sicut enim tales sophistae dicuntur, idest apparentes et non existentes, in quantum deficiunt a dialectica argumentatione; ita dialecticae argumentationes si appareant demonstrative probare et non probent, sophisticae sunt, in quantum videntur sua argumentatione scientes, et non sunt. It should be noted that Sophists are not taken here as they are in the book of Sophistical Refutations, i.e., who proceed from things that seem probable but are not, or seem to syllogize but do not. For just as some are called Sophists, i.e., seeming to be wise but not really so, inasmuch as they appear by their arguments to be scientific knowers but are not, so dialectical arguments, if they seem to prove demonstratively but do not, are sophistic.
Deinde cum dicit: quod autem ex necessariis etc., ostendit quod supposuerat. Et circa hoc duo facit: primo, ostendit quod conclusio necessaria non potest sciri ex principiis non necessariis; secundo, quod licet non possit sciri necessarium ex non necessariis, tamen syllogizari potest; ibi: cum quidem igitur conclusio et cetera. Then (74b27) he proves something he had presupposed. And concerning this he does two things. First, he shows that a necessary conclusion cannot be scientifically known from non-necessary principles. Secondly, that although the necessary cannot be scientifically known from the non-necessary, it can nevertheless be syllogized (75al).
Primum ostendit duabus rationibus, quarum prima talis est. Si quis non habeat rationem propter quid ostendentem, non efficitur sciens, etiam demonstratione habita: quia scire est causam rei cognoscere, ut supra dictum est. Sed ratio, quae infert conclusionem necessariam ex non necessariis principiis, non ostendit propter quid. He proves the first with two reasons, one of which (74b27) is this: If one does not have an argument which shows the cause why [i.e., propter quid], he does not know scientifically, even though a syllogism be had; because to know scientifically is to know the cause of a thing, as we stated above. But an argument which infers a necessary conclusion from non-necessary principles does not show the cause why.
Quod exemplificat in terminis communibus. Ponatur enim quod haec conclusio sit necessaria: omne c est a; et demonstretur per hoc medium b, quod non sit necessarium medium, sed contingens, puta quod haec propositio sit contingens, omne b est a, vel omne c est b, aut utraque; constat quod per hoc medium contingens, quod est b, non potest sciri de conclusione necessaria, quae est, omne c est a, propter quid. Quod sic probatur. Remota causa propter quam est aliquid, oportet quod removeatur effectus; sed hoc medium cum sit contingens, contingit removeri, conclusionem autem removeri non contingit cum sit necessaria; relinquitur ergo quod non potest sciri conclusio necessaria per medium contingens. To clarify this he gives an example, using general terms. Assume that the conclusion, “Every C is A,” is necessary, and that it is demonstrated by a middle, B, which is not a necessary middle but a contingent one, so that, for example, one of the propositions, “Every B is A,” and “Every C is B,” or both, are contingent. Now it is obvious that through this contingent middle, namely, B, one cannot obtain scientific knowledge propter quid of the necessary conclusion, “Every C is A.” For if the cause on which something depends is removed, the effect must be removed. But this middle, being contingent, can be withdrawn, whereas the conclusion, since it is necessary, cannot be withdrawn. It follows, therefore, that a necessary conclusion cannot be scientifically known through a contingent middle.
Secundam rationem ponit; ibi: amplius si etc., quae talis est. Si aliquis nunc nescit, cum tamen habeat eandem rationem, quam prius habuit, et salvatus est, idest non desiit esse, salva re, idest etiam re scita non corrupta, et iterum ipse non est oblitus; manifestum est quod etiam neque prius scivit. In hoc autem philosophus innuit quatuor modos, quibus aliquis amittit scientiam, quam prius habuit. Unus modus est quando excidit a mente eius ratio, per quam prius sciebat. Alius modus est per corruptionem ipsius scientis. Tertius per corruptionem ipsius rei scitae, sicut si sciam te sedere, dum sedes, te non sedente, haec scientia perit. Quartus est per oblivionem. Unde nullo istorum modorum existente, si aliquis modo nesciat aliquid, nec prius scivit. Sed ille, qui habet conclusionem necessariam per medium contingens, corrupto medio contingenti, nescit, medio non existente, et tamen eandem rationem habet, et salvus est, et salva est res, et non est oblitus. Ergo neque prius scivit, quando medium non erat corruptum. The second reason (74b32) is this: “If a man is without scientific knowledge now, then even though he possesses the same argument he had up to now, and he has been kept sound,” i.e., has not ceased to be, “and the thing known has not changed; and if he has not forgotten, then obviously he has not known scientifically.” In this passage the Philosopher mentions four ways in which a person loses the science he once had: one way is when there slips from his mind the argument through which he formerly knew scientifically. Another is through the destruction of the knower. A third is by a change occurring in the thing known: thus if, while you are sitting, I know that you are sitting, this knowledge perishes when you are not sitting. The fourth way is by forgetting. Hence if none of these ways has occurred, then, if a person does not know a thing scientifically now, he never did know it. Now one who holds a necessary conclusion in virtue of a contingent middle, no longer knows it scientifically when the middle ceases to be, for the middle no longer exists, even though he retains the same argument and the fact remains the same, and he has not forgotten anything. Therefore, even when the middle had not ceased to be, he did not know scientifically.
Quod autem medium, quod est contingens, corrumpatur, probat, quia id quod non est necessarium, oportet quod aliquando corrumpatur. Si autem dicatur quod medium nondum est corruptum: quia tamen non est necessarium, manifestum est quod contingit ipsum corrumpi. Posito autem contingenti, illud quod accidit non est impossibile, sed possibile et contingens. Quod autem sequebatur erat impossibile, scilicet, quod aliquis scientiam haberet alicuius, quod postea nesciret, manentibus conditionibus supra positis: quod tamen sequitur ex hoc quod est medium esse corruptum; quod et si non sit verum, est tamen contingens, ut dictum est. But that a contingent middle is liable to perish he proves by the fact that whatever is not necessary must at some time perish. And even though the middle has not perished, nevertheless, because it is not necessary, it is obvious that it is liable to perish. Now when something contingent is put forward, its result is not impossible but possible and contingent. But what followed in our case was impossible, namely, that one should have had science of something and later not have had it, all the conditions described above prevailing. And this follows from the fact that a contingent middle has perished, which, although it may not be true [that it has], is nevertheless liable to, as has been said.
Deinde cum dicit: cum quidem igitur etc., ostendit quod licet per medium contingens non possit sciri conclusio necessaria, tamen potest syllogizari conclusio necessaria ex medio non necessario. Dicit ergo quod nihil prohibet, cum conclusio necessaria est, medium non necessarium esse per quod ostenditur, syllogismo tamen dialectico, non demonstrativo, qui facit scire. Contingit enim necessarium syllogizari ex non necessariis, sicut contingit syllogizari verum ex non veris: non tamen contingit e converso; quia cum medium est necessarium et conclusio necessaria erit, sicuti ex veris praemissis semper concluditur verum. Then (75a1) he shows that although a necessary conclusion cannot be scientifically known through a contingent middle, nevertheless a necessary conclusion can be syllogized from a non-necessary middle. Hence he says that nothing hinders a necessary conclusion from being obtained through a non-necessary middle, as witness the case in which something is syllogized with a dialectical syllogism, although not with a demonstrative syllogism which causes science. For the necessary happens to be syllogized from the non-necessary, just as the true happens to be syllogized from the untrue—although the converse does not occur. For when the middle is necessary, the conclusion too will be necessary; similarly, from true premises something true is concluded.
Quod autem ex necessariis semper concludatur necessarium, sic probat: sit enim a de b ex necessitate, idest sit haec propositio necessaria: omne b est a; et hoc de c, idest sit haec etiam necessaria: omne c est b; ex his autem duabus necessariis sequitur tertia necessaria, scilicet conclusio, quod, omne c est a. Ostensum est enim in libro priorum quod ex duabus propositionibus de necessitate sequitur conclusio de necessitate. But that the necessary is always concluded from necessary premises he proves thus: “Let A be necessary of B,” i.e., let this proposition, “Every B is A,” be necessary, “and the latter of C,” i.e., let “Every C is B” also be necessary. Now from these two necessary things follows a third necessary thing, namely, the conclusion, “Every C is A.” For it has been proved in the Prior Analytics that “from two propositions of necessity follows a’ conclusion of necessity.”
Ostendit etiam consequenter quod si conclusio non esset necessaria, nec medium posset esse necessarium. Ponatur enim quod haec conclusio, omne c est a, sit non necessaria, praemissae autem duae sint necessariae; secundum id quod praeostensum est, sequitur quod conclusio sit necessaria, cum tamen contrarium sit positum, scilicet quod conclusio sit non necessaria. Then by way of consequence he shows that if the conclusion were not necessary, the middle could not be necessary. For example, suppose that this conclusion, “Every C is A,” is not necessary, but that the two premises are necessary. According to our previous doctrine it follows that the conclusion is necessary—which is contrary to the supposition in our example, namely, that the conclusion not be necessary.
Deinde cum dicit: quoniam igitur etc., infert conclusionem principaliter intentam ex omnibus praedictis, dicens quod quia oportet necessarium esse aliquid, si demonstratione sciatur, manifestum est ex praemissis quod oportet demonstrationem haberi per medium necessarium: alioquin nesciretur quod conclusio sit necessaria, neque propter quid, neque quia, cum necessarium non possit sciri per non necessarium, ut ostensum est. Sed si aliquis habeat rationem per medium non necessarium, dupliciter potest esse dispositus. Aut enim cum ipse sit non sciens, opinabitur tamen se scire, si accipiat in sua opinione medium non necessarium, tanquam necessarium; aut etiam non opinabitur se scire, si scilicet credat non se habere medium necessarium. Et hoc universaliter intelligendum est, tam de scientia quia, qua scitur aliquid per mediata, quam de scientia propter quid, qua scitur aliquid per immediata. Horum autem differentia posterius ostendetur. Then (75a13) he infers from the aforesaid the conclusion originally intended, saying that since a thing must be necessary if it is made known by way of demonstration, it is clear from the foregoing that a demonstration must rest on a necessary middle. For otherwise it would not be scientifically known that the conclusion is necessary, neither propter quid nor quia, since the necessary cannot be known through the non-necessary, as we have shown. But if someone rests on an argument based on a non-necessary middle, he will be in one of two states. For since he does not actually know in a scientific way, he will either believe that he does know in a scientific way, if he assumes a non-necessary middle as necessary, or he will not presume that he knows in a scientific way, i.e., if he believes that he does not have a necessary middle. And this is to be universally understood both of scientific knowledge quia, in which something is known through mediate principles, and of science propter quid, in which something is known through immediate principles. The difference between these two will be explained later.

Lectio 14
Caput 6 cont.
Τῶν δὲ συμβεβηκότων μὴ καθ' αὑτά, ὃν τρόπον διωρίσθη τὰ καθ' αὑτά, οὐκ ἔστιν ἐπιστήμη ἀποδεικτική. οὐ γὰρ ἔστιν ἐξ ἀνάγκης δεῖξαι τὸ συμπέρασμα· τὸ συμβεβηκὸς γὰρ ἐνδέχεται μὴ ὑπάρχειν· περὶ τοῦ τοιούτου γὰρ λέγω συμβεβηκότος. a18. Of accidents that are not essential according to our definition of essential there is no demonstrative knowledge; for since an accident, in the sense in which I here speak of it, may also not inhere, it is impossible to prove its inherence as a necessary conclusion.
καίτοι ἀπορήσειεν ἄν τις ἴσως τίνος ἕνεκα ταῦτα δεῖ ἐρωτᾶν περὶ τούτων, εἰ μὴ ἀνάγκη τὸ συμπέρασμα εἶναι· οὐδὲν γὰρ διαφέρει εἴ τις ἐρόμενος τὰ τυχόντα εἶτα εἴπειεν τὸ συμπέρασμα. a2l. A difficulty, however, might be raised as to why in dialectic, if the conclusion is not a necessary connexion, such and such determinate premisses should be proposed in order to deal with such and such determinate problems. Would not the result be the same if one asked any questions whatever and then merely stated one's conclusion?
δεῖ δ' ἐρωτᾶν οὐχ ὡς ἀναγκαῖον εἶναι διὰ τὰ ἠρωτημένα, ἀλλ' ὅτι λέγειν ἀνάγκη τῷ ἐκεῖνα λέγοντι, καὶ ἀληθῶς λέγειν, ἐὰν ἀληθῶς ᾖ ὑπάρχοντα. a24. The solution is that determinate questions have to be put, not because the replies to them affirm facts which necessitate facts affirmed by the conclusion, but because these answers are propositions which if the answerer affirm, he must affirm the conclusion and affirm it with truth if they are true.
Ἐπεὶ δ' ἐξ ἀνάγκης ὑπάρχει περὶ ἕκαστον γένος ὅσα καθ' αὑτὰ ὑπάρχει καὶ ᾗ ἕκαστον, φανερὸν ὅτι περὶ τῶν καθ' αὑτὰ ὑπαρχόντων αἱ ἐπιστημονικαὶ ἀποδείξεις καὶ ἐκ τῶν τοιούτων εἰσίν. τὰ μὲν γὰρ συμβεβηκότα οὐκ ἀναγκαῖα, ὥστ' οὐκ ἀνάγκη τὸ συμπέρασμα εἰδέναι διότι ὑπάρχει, οὐδ' εἰ ἀεὶ εἴη, μὴ καθ' αὑτὸ δέ, οἷον οἱ διὰ σημείων συλλογισμοί. τὸ γὰρ καθ' αὑτὸ οὐ καθ' αὑτὸ ἐπιστήσεται, οὐδὲ διότι (τὸ δὲ διότι ἐπίστασθαί ἐστι τὸ διὰ τοῦ αἰτίου ἐπίστασθαι), δι' αὑτὸ ἄρα δεῖ καὶ τὸ μέσον τῷ τρίτῳ καὶ τὸ πρῶτον τῷ μέσῳ ὑπάρχειν. a28. Since it is just those attributes within every genus which are essential and possessed by their respective subjects as such that are necessary it is clear that both the conclusions and the premisses of demonstrations which produce scientific knowledge are essential. For accidents are not necessary: and, further, since accidents are not necessary one does not necessarily have reasoned knowledge of a conclusion drawn from them (this is so even if the accidental premisses are invariable but not essential, as in proofs through signs; for though the conclusion be actually essential, one will not know it as essential nor know its reason); but to have reasoned knowledge of a conclusion is to know it through its cause. We may conclude that the middle must be consequentially connected with the minor, and the major with the middle.
Postquam ostendit philosophus quod demonstratio est de necessariis et ex necessariis, consequenter ostendit quod est de his, quae sunt per se et ex his, quae sunt per se. Et circa hoc tria facit: primo, ostendit quod demonstratio est de his, quae sunt per se, idest quod conclusiones demonstrationis sunt per se; secundo, movet dubitationem et solvit; ibi: et tamen opponet etc.; tertio, ostendit quod demonstratio est ex his, quae sunt per se, idest quod principia demonstrationis oportet per se esse; ibi: quoniam autem ex necessitate et cetera. After showing that demonstration proceeds from necessary things, the Philosopher then shows that it is concerned with things that are per se and proceeds from things that are per se. In regard to this he does three things. First, he shows that demonstration is concerned with things that are per se, i.e., that the conclusions of demonstrations are per se. Secondly, he raises a problem (75a21). Thirdly, he shows that demonstration proceeds from things that are per se, i.e., that the principles of demonstrations must be per se (75a28).
Dicit igitur primo quod demonstrativa scientia non potest esse accidentium, quae non sunt per se, sicut determinatum est per se superius, scilicet quod accidens per se est in cuius definitione ponitur subiectum; sicut par aut impar est per se accidens numeri. Album autem animalis non est per se accidens: quia animal non ponitur in eius definitione. He says therefore first (75a18) that demonstrative science cannot bear on accidents that are not per se in the way that per se was explained above, namely, that a per se accident is one in whose definition the subject is mentioned, as “even” or “odd” is a per se accident of number. But “white” is not a per se accident of animal, because animal is not mentioned in its definition.
Quod autem de huiusmodi accidentibus, quae non sunt per se, non possit esse demonstratio, sic probat. Accidens, quod non est per se, contingit non inesse (de hoc enim accidente loquimur); si ergo demonstratio fieret de accidente, quod non est per se, sequeretur quod conclusio demonstrationis non esset necessaria: cuius contrarium supra ostensum est. That there cannot be demonstration bearing on accidents that are not per se he proves in the following way: An accident which is not per se might happen not to be present (and this is the accident under discussion). Therefore, if a demonstration were to bear on an accident which is not per se, it would follow that the conclusion of the demonstration would not be necessary: which is contrary to what has been established.
Quod autem accidens, quod non est per se, non necessario insit, ex hoc potest haberi. Si enim aliquod accidens ex necessitate et semper insit subiecto, oportet quod causam habeat in subiecto, qua posita, non possit accidens non inesse. Quod quidem contingit dupliciter. Uno modo, quando ex principiis speciei accidens causatur; et tale accidens dicitur per se passio vel proprium. Alio modo quando accidens causatur ex principiis individui; et hoc est accidens inseparabile. Omne autem accidens, quod causatur ex principiis subiecti, si debeat definiri, oportet quod subiectum ponatur in sua definitione: nam unumquodque definitur ex propriis principiis; et sic oportet omne accidens, quod ex necessitate inest subiecto, esse accidens per se. Illa ergo quae non sunt per se, non ex necessitate insunt. That an accident which is not per se does not inhere necessarily can be obtained from the following: If an accident inheres necessarily and always in a subject, it must have its cause in the subject—in which case the accident cannot but inhere. Now this can occur in two ways: in one way, when it is caused from the principles of the species, and such an accident is called a per se attribute or property; in another way, when it is caused from the principles of the individual, and this is called an inseparable accident. In either case every accident which is caused from the principles of the subject must, if it is defined, be defined in such a way as to mention the subject in its definition: for a thing is defined in terms of its proper principles. Thus it is clear that an accident which necessarily inheres in its subject must be a per se accident. Therefore, those that are not per se do not inhere of necessity.
Videtur autem quod Aristoteles utatur demonstratione circulari, quam supra improbavit. Ostenderat enim supra quod demonstratio necessariorum est ex hoc quod est eorum quae sunt per se; nunc autem e converso ostendit quod demonstratio est eorum quae sunt per se, quia est necessariorum. Sed dicendum quod supra Aristoteles non solum ostendit demonstrationem esse necessariorum propter hoc, quod est eorum quae sunt per se, sed ex definitione eius quod est scire; et hic fuit verus demonstrationis modus. Quod autem ostendit demonstrationem esse necessariorum propter hoc, quod est eorum quae sunt per se, non est vera demonstratio, sed est ostensio ad hominem, apud quem notum est quod demonstratio sit eorum quae sunt per se. It seems, however, that Aristotle is using the circular demonstration, which he previously repudiated. For he had proved earlier that a demonstration is concerned with necessary things, because it is concerned with, things which are per se; but now he shows that demonstration is concerned with things which are per se, because it is concerned with things which are necessary. The answer is that above Aristotle proved that demonstration is concerned with necessary things not only because it is concerned with things which are per se but also because of the definition of scientific knowing—and this was a true way to demonstrate. But the proof that demonstration is concerned with necessary things because it is concerned with things which are per se, is not a demonstration but an indication directed to a person who already knows that demonstration is concerned with things which are per se.
Deinde cum dicit: et tamen opponet etc., movet dubitationem quamdam. Et circa hoc duo facit. Primo, ponit dubitationem dicens quod potest aliquis forte opponere: si conclusio, quae sequitur ex contingentibus vel ex his quae sunt per accidens, non est necessaria; quare de contingentibus fit interrogatio sive de his quae sunt per accidens, ut ex iis datis procedatur ad conclusionem, cum tamen in syllogismo requiratur quod conclusio ex necessitate accidat. Et quod interrogatio fiat de contingentibus vel ex his, quae sunt per accidens, manifestat per hoc quod subdit: nihil enim differt, si aliquis interrogatus contingentia, postea dicat conclusionem. Quasi dicat: ita potest inferri conclusio ex contingentibus interrogatis et concessis, sicut ex necessariis: utrisque enim eadem forma syllogizandi est. Then (75a21) he raises a problem in regard to which he does two things. First, he states the problem, saying that someone may perhaps, object that if the conclusion which follows from contingent things or things which are per accidens is not necessary, why are inquiries made concerning contingent things or things which are per accidens, so that from such data one may reach a conclusion, since it is required of the’ syllogism that the conclusion follow of necessity. That inquiries touching contingent things or things which are per accidens do occur is evident from his next statement, namely, that it makes no difference whether one inquires into contingent things and then gives his conclusion. As if to say: A conclusion can be drawn from contingent things that have been investigated and verified, just as from necessary things. For each employs the same syllogistic form.
Secundo; ibi: oportet autem etc., solvit dicens quod non ita interrogatur de praemissis contingentibus, quasi conclusio sit necessaria absolute propter interrogata, idest propter praemissa contingentia; sed quia necesse est dicenti praemissa conclusionem dicere, et dicere vera in conclusione, si vera sunt, quae praemissa sunt: quasi dicat quod licet ex praemissis contingentibus non sequatur conclusio necessaria necessitate absoluta, sequitur tamen secundum quod est ibi necessitas consequentiae, secundum quod conclusio sequitur ex praemissis. Secondly (75a24), he solves the difficulty and says that questions solved by contingent premises are not of such a nature that the conclusion is necessary absolutely, on the basis of the things investigated, i.e., in virtue of the contingent premises, but because it is necessary for the one who admits the premises to admit the conclusion and to admit truth in the conclusion, if the things premised are true. It is as though he were saying that although a conclusion which is necessary with absolute necessity does not follow from contingent premises, yet it follows with the necessity of consequence according to which the conclusion follows from premises.
Deinde cum dicit: quoniam autem etc., ostendit quod demonstratio sit ex his, quae sunt per se, tali ratione. Demonstratio est ex necessariis et de necessariis. Et hoc ideo, quia est scientifica, idest faciens scire. Ea autem, quae non sunt per se, non sunt necessaria: sunt enim per accidens et huiusmodi non sunt necessaria, ut dictum est. Sed illa sunt ex necessitate circa unumquodque genus, quaecunque sunt per se et conveniunt unicuique secundum quod unumquodque est. Relinquitur ergo quod demonstratio non possit esse nisi ex his, quae sunt per se et de talibus. Then (75a28) he shows that demonstration proceeds from things which are per se, using the following argument: Demonstration proceeds from necessary things and bears on necessary things—and this because it is scientific, i.e., makes one know scientifically. But things which are not per se are not necessary, for they are per accidens and, being such, are not necessary, as has been stated above. On the other hand, those things bear with necessity on a genus which are per se and belong to it according to what it is. It follows, therefore, that demonstration cannot be from anything or of anything but what is per se.
Ulterius autem ostendit quod etiam si praemissa essent semper, et necessaria, et vera, et non per se, non tamen sciretur de conclusione propter quid; sicut patet in syllogismis, qui fiunt per signa, in quibus conclusionem, quae est per se, non scit aliquis per se, neque propter quid. Sicut si aliquis probaret quod omne elementum est corruptibile, per hoc quod videtur tempore antiquari, esset quidem probatio per signum, non autem per se, neque propter quid: quia propter quid scire est per causam scire. Oportet ergo medium esse causam eius, quod in demonstratione concluditur. Et hoc manifestum est ex praemissis: quia oportet et medium inesse tertio propter ipsum, idest per se, et similiter primum medio. Primum autem et tertium vocat duas extremitates. He further shows that even if the premises were always both necessary and true but not per se, one would not know the cause why [propter quid] of the conclusion. This is clear in syllogisms which prove through signs, for although the conclusion be per se, one does not know it per se nor propter quid. For example, if someone were to prove that every element is corruptible on the ground that it is seen to grow old. This would be a proof through a sign but neither per se nor propter quid, because to know propter quid one must know through the cause. Therefore, the middle must be the cause of that which is concluded in the demonstration. And this is obvious from the premises: for the middle must inhere in the third causatively, i.e., per se, and likewise the first in the middle. Here he calls the two extremes the first and the third.

Lectio 15
Caput 7
Οὐκ ἄρα ἔστιν ἐξ ἄλλου γένους μεταβάντα δεῖξαι, οἷον τὸ γεωμετρικὸν ἀριθμητικῇ. a38. It follows that we cannot in demonstrating pass from one genus to another. We cannot, for instance, prove geometrical truths by arithmetic.
τρία γάρ ἐστι τὰ ἐν ταῖς ἀποδείξεσιν, ἓν μὲν τὸ ἀποδεικνύμενον, τὸ συμπέρασμα (τοῦτο δ' ἐστὶ τὸ ὑπάρχον γένει τινὶ καθ' αὑτό), ἓν δὲ τὰ ἐξιώματα (ἀξιώματα δ' ἐστὶν ἐξ ὧν)· τρίτον τὸ γένος τὸ ὑποκείμενον, (75b.) οὗ τὰ πάθη καὶ τὰ καθ' αὑτὰ συμβεβηκότα δηλοῖ ἡ ἀπόδειξις. a39. For there are three elements in demonstration: (1) what is proved, the conclusion — an attribute inhering essentially in a genus; (2) the axioms, i.e. axioms which are premisses of demonstration; (3) the subject — genus whose attributes, i.e. essential properties, are revealed by the demonstration.
ἐξ ὧν μὲν οὖν ἡ ἀπόδειξις, ἐνδέχεται τὰ αὐτὰ εἶναι· ὧν δὲ τὸ γένος ἕτερον, ὥσπερ ἀριθμητικῆς καὶ γεωμετρίας, οὐκ ἔστι τὴν ἀριθμητικὴν ἀπόδειξιν ἐφαρμόσαι ἐπὶ τὰ τοῖς μεγέθεσι συμβεβηκότα, εἰ μὴ τὰ μεγέθη ἀριθμοί εἰσι· τοῦτο δ' ὡς ἐνδέχεται ἐπί τινων, ὕστερον λεχθήσεται. ἡ δ' ἀριθμητικὴ ἀπόδειξις ἀεὶ ἔχει τὸ γένος περὶ ὃ ἡ ἀπόδειξις, καὶ αἱ ἄλλαι ὁμοίως. b2. The axioms which are premisses of demonstration may be identical in two or more sciences: but in the case of two different genera such as arithmetic and geometry you cannot apply arithmetical demonstration to the properties of magnitudes unless the magnitudes in question are numbers. How in certain cases transference is possible I will explain later. Arithmetical demonstration and the other sciences likewise possess, each of them, their own genera;
ὥστ' ἢ ἁπλῶς ἀνάγκη τὸ αὐτὸ εἶναι γένος ἢ πῇ, εἰ μέλλει ἡ ἀπόδειξις μεταβαίνειν. b8. so that if the demonstration is to pass from one sphere to another, the genus must be either absolutely or to some extent the same.
ἄλλως δ' ὅτι ἀδύνατον, δῆλον· ἐκ γὰρ τοῦ αὐτοῦ γένους ἀνάγκη τὰ ἄκρα καὶ τὰ μέσα εἶναι. εἰ γὰρ μὴ καθ' αὑτά, συμβεβηκότα ἔσται. b10. If this is not so, transference is clearly impossible, because the extreme and the middle terms must be drawn from the same genus: otherwise, as predicated, they will not be essential and will thus be accidents.
διὰ τοῦτο τῇ γεωμετρίᾳ οὐκ ἔστι δεῖξαι ὅτι τῶν ἐναντίων μία ἐπιστήμη, ἀλλ' οὐδ' ὅτι οἱ δύο κύβοι κύβος· οὐδ' ἄλλῃ ἐπιστήμῃ τὸ ἑτέρας, ἀλλ' ἢ ὅσα οὕτως ἔχει πρὸς ἄλληλα ὥστ' εἶναι θάτερον ὑπὸ θάτερον, οἷον τὰ ὀπτικὰ πρὸς γεωμετρίαν καὶ τὰ ἁρμονικὰ πρὸς ἀριθμητικήν. b13. That is why it cannot be proved by geometry that opposites fall under one science, nor even that the product of two cubes is a cube. Nor can the theorem of any one science be demonstrated by means of another science, unless these theorems are related as subordinate to superior (e.g. as optical theorems to geometry or harmonic theorems to arithmetic).
οὐδ' εἴ τι ὑπάρχει ταῖς γραμμαῖς μὴ ᾗ γραμμαὶ καὶ ᾗ ἐκ τῶν ἀρχῶν τῶν ἰδίων, οἷον εἰ καλλίστη τῶν γραμμῶν ἡ εὐθεῖα ἢ εἰ ἐναντίως ἔχει τῇ περιφερεῖ· οὐ γὰρ ᾗ τὸ ἴδιον γένος αὐτῶν, ὑπάρχει, ἀλλ' ᾗ κοινόν τι. b17. Geometry again cannot prove of lines any property which they do not possess qua lines, i.e. in virtue of the fundamental truths of their peculiar genus: it cannot show, for example, that the straight line is the most beautiful of lines or the contrary of the circle; for these qualities do not belong to lines in virtue of their peculiar genus, but through some property which it shares with other genera.
Postquam ostendit philosophus quod demonstratio est ex his quae sunt per se, hic concludit quod demonstratio est ex principiis propriis, non extraneis, neque ex communibus. Et dividitur in duas partes: in prima, ostendit quod demonstratio procedit ex propriis principiis; in secunda, determinat quae sint principia propria et quae communia; ibi: difficile autem et cetera. Prima in duas: in prima, ostendit quod demonstratio non procedit ex principiis extraneis; in secunda, ostendit quod non procedit ex principiis communibus; ibi: quoniam autem manifestum est et cetera. Prima in duas: in prima, ex praemissis ostendit quod demonstratio non est ex principiis extraneis; in secunda, ex praemissis etiam ostendit quod demonstrationes non sunt de rebus corruptibilibus, sed de sempiternis; ibi: manifestum autem et si sint propositiones et cetera. Circa primum tria facit: primo, proponit intentum; secundo, probat propositum; ibi: tria enim sunt etc.; tertio, concludit intentum; ibi: propter hoc geometriae et cetera. After showing that demonstration is from things that are per se, the Philosopher here concludes that demonstration is from principles which are proper and not from extraneous or common principles. His treatment falls into two parts. In the first he shows that demonstration proceeds from proper principles. In the second he establishes which principles are proper and which common (76a26) [L. 18]. The first is divided into two parts. In the first he shows that demonstration does not proceed from extraneous principles. In the second that it does not proceed from common principles (75b37) [L. 17]. The first is divided into two parts. In the first, from the preceding, he shows that demonstration is not from extraneous principles. In the second, also from the preceding, he shows that demonstrations do not bear on corruptible things but on eternal (75b2I) [L. 16]. Concerning the first he does three things. First, he states his proposition. Secondly, he proves it (75a39). Thirdly, he concludes to what he intended (75b13).
Dicit ergo primo quod, ex quo demonstratio est ex his quae sunt per se, manifestum est quod non contingit demonstrare descendentem vel procedentem ex alio genere in aliud genus, sicut non contingit quod geometria ex propriis principiis demonstret aliquid descendens in arithmeticam. He says therefore first (75a38) that inasmuch as demonstration is from things that are per se, it is plain that demonstration does not consist in descending or skipping from one genus to another, as geometry, demonstrating from its own principles, does not descend to something in arithmetic.
Deinde cum dicit: tria enim etc., propositum probat. Et circa hoc tria facit. Primo, praemittit quae sint necessaria ad demonstrationem, dicens quod in demonstrationibus tria sunt. Unum est, quod demonstratur, scilicet conclusio, quae quidem continet in se id, quod per se inest alicui generi: per demonstrationem enim concluditur propria passio de proprio subiecto. Aliud autem sunt dignitates, ex quibus demonstratio procedit. Tertium autem est genus subiectum, cuius proprias passiones et per se accidentia demonstratio ostendit. Then (75a39) he proves this proposition and does three things. First, he lays down the things that are necessary for a demonstration, saying that “three things are necessary in demonstrations. One is that which is demonstrated,” namely, the conclusion which, as a matter of fact, contains within itself that which inheres in its genus per se. For through demonstration a proper attribute is concluded of its proper subject. Another is the dignities [maxims or axioms] from which demonstration proceeds. The third is the generic subject whose proper attributes and per se accidents demonstration reveals.
Secundo; ibi: ex quibus igitur etc., ostendit quid praedictorum trium possit esse commune diversis scientiis et quid non, dicens quod horum trium unum, scilicet dignitates, ex quibus demonstratio procedit, contingit esse idem in diversis demonstrationibus et etiam in diversis scientiis: sed in illis scientiis, quarum est diversum genus subiectum, sicut in arithmetica, quae est de numeris, et geometria, quae est de magnitudinibus, non contingit quod demonstratio, quae procedit ex principiis unius scientiae, puta arithmeticae, descendat ad subiecta alterius scientiae, sicut ad magnitudines, quae sunt subiecta geometriae; nisi forte subiectum unius scientiae contineatur sub subiecto alterius, sicut si magnitudines contineantur sub numeris (quod quidem qualiter contingat, scilicet subiectum unius scientiae contineri sub subiecto alterius, posterius dicetur). Magnitudines enim sub numeris non continentur, nisi forte secundum quod magnitudines numeratae sunt. Subiecta etiam diversarum demonstrationum sive scientiarum diversa sunt. Arithmetica enim demonstratio semper habet genus proprium circa quod demonstrat. Et aliae scientiae similiter. Secondly (75b3), he shows which of the three aforesaid can be common to various sciences and which cannot, saying that one of the three, namely, the dignities, from which demonstration proceeds, happens to be the same in diverse demonstrations and even in diverse sciences. But in those sciences whose respective generic subject is diverse, as in arithmetic which is concerned with numbers, and in geometry which is concerned with magnitudes, it does not occur that a demonstration which starts with the principles of one science, say, of arithmetic, descends to the subjects of another science, say to magnitudes, which pertain to geometry; unless perchance the subject of one science should be contained under the subject of the other, for example, if magnitudes should be contained under, numbers. (How this occurs, namely, how the subject of one science may be contained under the subject of another, will be discussed later). For magnitudes are not contained under numbers except perhaps in the sense that magnitudes are numbered. In any case, the subjects of diverse demonstrations or sciences are diverse. For an arithmetical demonstration always has its proper genus in respect to which it demonstrates, and so do the other sciences.
Tertio; ibi: quare aut simpliciter etc., probat propositum. Et circa hoc duo facit. Primo, inducit principale propositum per modum conclusionis, eo quod ex praemissis haberi potest, dicens: quare manifestum est quod necesse est, aut esse simpliciter idem genus, circa quod sumuntur principia et conclusiones, et sic non est descensus, neque transitus de genere in genus: aut si debet demonstratio descendere ab uno genere in aliud, oportet esse unum genus sic, idest quodammodo. Aliter enim impossibile est quod demonstretur aliqua conclusio ex aliquibus principiis, cum non sit idem genus vel simpliciter vel secundum quid. Sciendum est autem quod simpliciter idem genus accipitur, quando ex parte subiecti non sumitur aliqua differentia determinans, quae sit extranea a natura illius generis; sicut si quis per principia verificata de triangulo procedat ad demonstrandum aliquid circa isoscelem vel aliquam aliam speciem trianguli. Secundum quid autem est unum genus, quando assumitur circa subiectum aliqua differentia extranea a natura illius generis; sicut visuale est extraneum a genere lineae, et sonus est extraneus a genere numeri. Numerus ergo simpliciter, qui est genus subiectum arithmeticae, et numerus sonorum, qui est genus subiectum musicae, non sunt unum genus simpliciter. Similiter autem nec linea simpliciter, quam considerat geometra, et linea visualis, quam considerat perspectivus. Unde patet quod quando ea, quae sunt lineae simpliciter, applicantur ad lineam visualem, fit quodammodo descensus in aliud genus: non autem quando ea, quae sunt trianguli, applicantur ad isoscelem. Thirdly (75b8), he proves his proposition and does two things. First he brings in the main intent after the manner of a conclusion on the ground that it can be obtained from the aforesaid, saying: Hence it is clear that it is necessary either that the genus with which the principles and conclusions deal be absolutely the same (in which case there is no descent or skipping from one genus to another); or if the demonstration is to descend from one genus to another, the two must be one thus, i.e., somehow. For otherwise it is impossible for a conclusion to be demonstrated from principles, since there is not the same genus either absolutely or in a qualified sense. However, it should be noted that a genus absolutely the same is being taken when on the part of the subject no determinate difference is admitted which is alien to the nature of that genus: for example, if someone using principles verified of triangle should proceed to demonstrate something about isosceles or about any other species of triangle. But a genus is one in a qualified sense when a difference alien to the nature of the genus is admitted of the subject, as “visual” is alien to the genus of line, and “sound” to the genus of number. Therefore, number absolutely, which is the generic subject of arithmetic, and sonant number, which is the generic subject of music, are not one genus absolutely speaking. The same goes for line absolutely, which geometry considers, and the visual line considered in optics. Hence it is clear that when matters pertaining absolutely to the line are applied to the visual line, a descent is being made to another genus—which is not the case when matters pertaining to triangle are applied to isosceles.
Secundo; ibi: ex eodem enim genere etc., ostendit propositum hoc modo. Oportet in demonstratione eiusdem generis esse media et extrema. Extrema autem in conclusione continentur. Nam maior extremitas in conclusione est praedicatum; minor vero extremitas subiectum; medium autem in praemissis continetur. Oportet igitur principia et conclusiones circa idem genus sumi. Cum autem huic coniunxerimus quod diversae scientiae sint circa diversa genera subiecta; ex necessitate sequitur quod ex principiis unius scientiae non concludatur aliquid in alia scientia, quae non sit sub ea posita. Secondly (75b10), he manifests the proposition in this way. In demonstration the middles and the extremes must belong to the same genus. Now the extremes are contained in the conclusion: for the major extreme in the conclusion is its predicate, and the minor its subject. But the middle is contained in the premises. It is required, therefore, that the principles and conclusion be taken with respect to the same genus. When we add to this the fact that diverse sciences are of necessity concerned with subjects generically diverse, it follows that from the principles of one science, something in another science not under it may not be concluded.
Quod autem in demonstratione oporteat media et extrema unius generis esse, sic probat. Detur enim quod medium sit alterius generis ab extremis, sicut si extrema sint triangulus et habere tres angulos aequales duobus rectis. Manifestum est quod passio conclusa de triangulo, per se inest ei; non autem per se inest aeneo. Et si e contrario passio per se inesset aeneo, puta sonorum esse, vel aliquid huiusmodi, palam est quod per accidens inesset triangulo. That the middles and extremes in a demonstration must be of one genus he proves thus: Suppose that a middle belongs to a genus other than that of the extremes which might be, for example, “triangle” and “have three angles equal to two right angles.” Now a proper attribute concluded of triangle is in it per se but is not in “brazen” per se; or, conversely, if the proper attribute is per se in “brazen,” say, “high-sounding” or the like, it is obviously in triangle per accidens.
Unde patet quod oportet omnino, si subiectum conclusionis et medium sint penitus alterius generis, quod passio vel non per se insit medio vel non per se insit subiecto: et ita oportet quod alteri eorum insit per accidens. Et si quidem insit medio per accidens, erit per accidens in praemissis; si autem subiecto, erit in conclusione: et hoc ex parte passionis. Sed utroque modo oportebit per accidens esse in praemissis, quantum ad hoc quod subiectum accipitur sub medio: sicut si triangulus accipiatur sub aeneo aut e converso. Ostensum est autem quod in demonstrationibus tam conclusio, quam praemissae sunt per se et non per accidens. Oportet ergo in demonstrationibus medium et extrema eiusdem generis esse. This example makes it plain that if the subject of the conclusion and the middle are in entirely different genera, then the proper attribute is either not in the middle per se or else not in the subject per se. Consequently, it is in one of them per accidens. If it is in the middle per accidens, something will be in the premises per accidens. But if it is in the subject per accidens, something will be per accidens in the conclusion: and this on the part of a proper attribute. But either way something will be per accidens in the premises, inasmuch as the subject is subsumed under the middle, for example, triangle under brazen, or vice versa. However, it has been previously established that in a demonstration both the conclusion and the premises are per se and not per accidens. It is required, therefore, in demonstrations that the middle and the extremes be of the same genus.
Deinde cum dicit: propter hoc geometriae etc., infert duas conclusiones ex praemissis. Quarum prima est quod nulla scientia demonstrat aliquid de subiecto alterius scientiae, sive sit scientiae communioris sive alterius scientiae disparatae; sicut geometria non demonstrat quod contrariorum eadem est scientia: contraria enim pertinent ad scientiam communem, scilicet ad philosophiam primam vel dialecticam. Et similiter geometria non demonstrat quod duo cubi sint unus cubus, idest quod ex ductu unius numeri cubici in alium numerum cubicum surgat numerus cubicus. Dicitur autem numerus cubicus, qui consurgit ex ductu unius numeri in seipsum bis; sicut octonarius est numerus cubicus, surgit enim ex ductu binarii in seipsum bis, quia bis duo bis sunt octo. Et eadem ratione vigintiseptem est numerus cubicus, et radix eius est tria, quia ter tria ter faciunt vigintiseptem. Si ergo ducantur octo in vigintiseptem consurgit numerus cubicus, idest ducenta sexdecim, cuius radix est sex: quia sexies sex sexies sunt ducenta sexdecim. Hoc ergo habet probare arithmeticus, non geometra. Et similiter, quod est unius scientiae non habet probare alia scientia, nisi forte una scientia sit sub altera; sicut se habet perspectiva ad geometriam, et consonantia vel harmonica, idest musica, ad arithmeticam. Then (75b13) he infers two conclusions from the foregoing. The first conclusion is that no science demonstrates anything about the subject of another science, when this other science is more common than o entirely disparate from the first. Thus, geometry does not demonstrate that one and the same science deals with the both of two contraries: for contraries pertain to the common science, namely, to first philosophy or to dialectics. In like manner, geometry does not demonstrate that “two cubes are one cube,” i.e., that the product of one cube and another cube is a cube. “Cube” here means the number which results from multiplying a number by itself twice: for example, 8 is a cube, for it is the result of multiplying 2 by itself twice, for 2 times 2 times 2 are 8. Similarly,, 27 is a cube, whose root is 3, because 3 times 3 times 3 make 27. Now if 8 be multiplied by 27 the product is 216, which is a cube whose root is 6, because 6 times 6 times 6 make 216. Therefore, arithmetic, not, geometry has: the power to prove this. Similarly, that which pertains to one science cannot be proved by another science unless the one happens~, to be under the other, as optics is under geometry, and consonance or harmony, i.e., music, is under arithmetic.
Secunda conclusio ponitur; ibi: neque si aliquid et cetera. Et est quod scientia etiam de proprio subiecto non probat quodlibet accidens, sed accidens quod est sui generis. Sicut si aliquid inest lineis, non secundum quod sunt lineae, neque secundum propria principia linearum, hoc non demonstrat geometra de lineis; sicut quod linea recta sit pulcherrima linearum, aut recta linea si est contraria circulari vel non. Haec enim non sunt secundum proprium genus lineae, sed secundum aliquid communius. Pulchrum enim et contrarium genus lineae transcendunt. The second conclusion (75b17) states that a science cannot prove just any random accident of its subject, but the accidents proper to its genus Thus, if something belongs to lines not as lines or not according to the proper principles of lines, geometry does not demonstrate it of lines: for example, that a straight line is the most beautiful of lines, or whether straight line is contrary or not to the curved. For these matters are outside the proper genus of line and belong to something more general. For beauty and contrary transcend the genus of line.

Lectio 16
Caput 8
Φανερὸν δὲ καὶ ἐὰν ὦσιν αἱ προτάσεις καθόλου ἐξ ὧν ὁ συλλογισμός, ὅτι ἀνάγκη καὶ τὸ συμπέρασμα ἀΐδιον εἶναι τῆς τοιαύτης ἀποδείξεως καὶ τῆς ἁπλῶς εἰπεῖν ἀποδείξεως. οὐκ ἔστιν ἄρα ἀπόδειξις τῶν φθαρτῶν οὐδ' ἐπιστήμη ἁπλῶς, b21. It is also clear that if the premisses from which the syllogism proceeds are commensurately universal, the conclusion of such i.e. in the unqualified sense — must also be eternal. Therefore no attribute can be demonstrated nor known by strictly scientific knowledge to inhere in perishable things.
ἀλλ' οὕτως ὥσπερ κατὰ συμβεβηκός, ὅτι οὐ καθ' ὅλου αὐτοῦ ἐστιν ἀλλὰ ποτὲ καὶ πώς. ὅταν δ' ᾖ, ἀνάγκη τὴν ἑτέραν μὴ καθόλου εἶναι πρότασιν καὶ φθαρτήν—φθαρτὴν μὲν ὅτι ἔσται καὶ τὸ συμπέρασμα οὔσης, μὴ καθόλου δὲ ὅτι τῷ μὲν ἔσται τῷ δ' οὐκ ἔσται ἐφ' ὧν—ὥστ' οὐκ ἔστι συλλογίσασθαι καθόλου, ἀλλ' ὅτι νῦν. b24. The proof can only be accidental, because the attribute's connexion with its perishable subject is not commensurately universal but temporary and special. If such a demonstration is made, one premiss must be perishable and not commensurately universal (perishable because only if it is perishable will the conclusion be perishable; not commensurately universal, because the predicate will be predicable of some instances of the subject and not of others); so that the conclusion can only be that a fact is true at the moment — not commensurately and universally.
ὁμοίως δ' ἔχει καὶ περὶ ὁρισμούς, ἐπείπερ ἐστὶν ὁ ὁρισμὸς ἢ ἀρχὴ ἀποδείξεως ἢ ἀπόδειξις θέσει διαφέρουσα ἢ συμπέρασμά τι ἀποδείξεως. b30. The same is true of definitions, since a definition is either a primary premiss or a conclusion of a demonstration, or else only differs from a demonstration in the order of its terms.
αἱ δὲ τῶν πολλάκις γινομένων ἀποδείξεις καὶ ἐπιστῆμαι, οἷον σελήνης ἐκλείψεως, δῆλον ὅτι ᾗ μὲν τοιοῦδ' εἰσίν, ἀεὶ εἰσίν, ᾗ δ' οὐκ ἀεί, κατὰ μέρος εἰσίν. ὥσπερ δ' ἡ ἔκλειψις, ὡσαύτως τοῖς ἄλλοις. b32. Demonstration and science of merely frequent occurrences — e.g. of eclipse as happening to the moon — are, as such, clearly eternal: whereas so far as they are not eternal they are not fully commensurate. Other subjects too have properties attaching to them in the same way as eclipse attaches to the moon.
Postquam ex superioribus philosophus concluserat quod demonstratio non concludit ex extraneis principiis, nunc iterum ex superioribus intendit concludere quod demonstratio non est de corruptibilibus. Et circa hoc duo facit: primo, ostendit quod sempiternorum et non corruptibilium est demonstratio; secundo, ostendit qualiter sit eorum quae sunt ut frequenter; ibi: eorum autem et cetera. Circa primum duo facit: primo, ostendit quod demonstratio non sit corruptibilium, sed sempiternorum; secundo, ostendit idem de definitione; ibi: similiter se habet et cetera. Circa primum duo facit: primo, proponit conclusionem intentam; secundo, ponit rationem probantem ipsam; ibi: quod autem universaliter et cetera. After concluding from the above that demonstration does not conclude from extraneous principles, the Philosopher intends to conclude something else from the above, namely, that demonstration is not of destructible things. Concerning this he does two things. First, he shows that demonstration is of eternal and not of destructible things. Secondly, he shows how it is of things that occur now and then (75b32). Concerning the first he does two things. First, he shows that demonstration is not of perishable but of eternal things. Secondly, he shows that the same is true of definition (75b30). In regard to the first he does two things. First, he proposes the intended conclusion. Secondly, he sets down the reason proving it (75b24).
Primo ergo ponit duas conclusiones, quarum una sequitur ex altera. Prima est quod necesse est conclusionem demonstrationis huius de qua nunc agitur, et quam possumus dicere simpliciter demonstrationem, esse perpetuam; quod quidem sequitur ex hoc, quod supra habitum est, scilicet quod propositiones, ex quibus fit syllogismus, debent esse universales: quod significavit per dici de omni. Secunda conclusio est quod neque demonstratio, neque scientia est corruptibilium, loquendo simpliciter, sed solum secundum accidens. Therefore first (75b21), he sets down two conclusions, one of which follows from the other. The first is that the conclusion of this demonstration which we are now discussing and which we can call demonstration in the full sense must be eternal: which, of course, follows from what has been stated so far, namely, that the propositions of a syllogism should be universal: which he signified by “said of all.” The second conclusion is that neither demonstration nor science is of destructible things, i.e., absolutely speaking, but only accidentally.
Deinde cum dicit: quod autem universaliter etc., inducit rationem ad probandum propositas conclusiones: quae talis est. Conclusionis corruptibilis, et non sempiternae, non est in se continere quod est universaliter, sed aliquando et sic. Dictum est enim supra quod dici de omni duo continet, scilicet quod non in quodam sic et in quodam non, et iterum, quod non aliquando sic et aliquando non. In omnibus autem corruptibilibus invenitur aliquando sic et aliquando non. Unde patet quod in corruptibilibus non invenitur dici de omni, sive quod est universaliter. Sed ubi conclusio est non universalis, oportet aliquam praemissarum esse non universalem. Conclusio ergo corruptibilis oportet quod sequatur ex praemissis, quarum altera non sit universalis. Cum ergo huic coniunxerimus quod demonstratio simpliciter semper debet esse ex universalibus, sequitur quod demonstratio non possit habere conclusionem corruptibilem, sed sempiternam. Then (75b24) he presents an argument to prove the proposed conclusions. It is this: It is not the character of a destructible non-eternal conclusion to contain what is universally so, but what is so for a time and in certain instances. For it has been established above that “said of all” contains two things, namely, that it is not such as to be so in one case and not in another, or so at one time and not at another. But in destructible things we find that at one time something is so and at another not so. Hence it is clear that “said of all” or “said universally” are not found in destructible things. But where a conclusion is not universal, at least one of the premises is not universal. Therefore, a destructible conclusion must have followed from premises, one of which is not universal. Accordingly, when we add to this the fact that demonstration absolutely must be from what is universal, it follows that a demonstration cannot have a destructible conclusion, but must have an eternal one.
Deinde cum dicit: similiter se habet etc., ostendit quod etiam definitio est non corruptibilium, sed sempiternorum, tali ratione. Demonstratio quantum ad principia et conclusiones est sempiternorum et non corruptibilium; sed definitio vel est principium, vel conclusio demonstrationis, vel demonstratio positione differens; ergo definitio non est corruptibilium, sed sempiternorum. Then (75b30) he shows that definition, too, is not of destructible but of eternal things. The reason for this is that demonstration, both as to its principles and conclusions, is of eternal and not of destructible things. But a definition is either a principle of a demonstration, a conclusion of a demonstration, or a demonstration with a different ordering of its terms. Therefore, a definition is not of destructible, but of eternal things.
Ad intellectum autem huius literae sciendum est quod contingit definitiones diversas dari eiusdem rei, sumptas ex diversis causis. Causae autem ad invicem ordinem habent: nam ex una sumitur ratio alterius. Ex forma enim sumitur ratio materiae: talem enim oportet esse materiam, qualem forma requirit. Efficiens autem est ratio formae: quia enim agens agit sibi simile, oportet quod secundum modum agentis sit etiam modus formae, quae ex actione consequitur. Ex fine autem sumitur ratio efficientis: nam omne agens agit propter finem. Oportet ergo quod definitio, quae sumitur a fine, sit ratio et causa probativa aliarum definitionum, quae sumuntur ex aliis causis. For a better understanding of this passage it should be noted that it is possible to give different definitions of the same thing, depending on the different causes mentioned. But causes are arranged in a definite order to one another: for the reason of one is derived from another. Thus the reason of matter is derived from the form, for the matter must be such as the form requires. Again, the agent is the reason for the form: for since an agent produces something like unto itself, the mode of the form which results from the action must be according to the mode of the agent. Finally, it is from the end that the reason of the agent is derived, for every agent acts because of an end. Consequently, a definition which is formulated from the end is the reason and cause proving the other definitions which are formulated from the other causes.
Ponamus ergo duas definitiones domus, quarum una sumatur a causa materiali, quae sit talis: domus est cooperimentum constitutum ex lapidibus, cemento et lignis. Alia sumatur ex causa finali, quae sit talis: domus est cooperimentum prohibens nos a pluviis, frigore et calore. Potest ergo prima definitio demonstrari ex secunda, hoc modo: omne cooperimentum prohibens nos a pluviis, frigore et calore oportet quod sit constitutum ex lapidibus, cemento et lignis; domus est huiusmodi; ergo et cetera. Therefore, let us lay down two definitions, one of which is formulated from the material cause, for example, that a house is a shelter composed of stones, cement and wood; the other being formulated from the final cause, namely, that a house is a shelter protecting us from the rain and heat and cold. Now the first definition can be demonstrated from the second in the following way: Every shelter protecting us from rain, heat and cold should be composed of wood, cement and stones; but a house is such a thing: therefore...
Patet ergo quod definitio, quae sumitur a fine, est principium demonstrationis; illa autem, quae sumitur a materia, est demonstrationis conclusio. Potest tamen utraque coniungi, ut sit una definitio, hoc modo: domus est cooperimentum constitutum ex dictis, defendens a pluvia, frigore et calore. Talis autem definitio continet totum quod est in demonstratione, scilicet medium et conclusionem. Et ideo talis definitio est demonstratio positione differens; quia in hoc solo differt a demonstratione, quia non est ordinata in modo et figura. Thus it is clear that the definition formulated from the end is the principle of the demonstration, whereas the one formulated from the matter is the conclusion of the demonstration. However, the two can be combined in the following way to form one definition: A house is a shelter composed of the aforesaid, protecting us from rain, cold and heat. But such a definition contains all the elements of a demonstration, namely, a middle and a conclusion. Accordingly, such a definition is a demonstration differing merely in the arrangement of its terms, because the only way it differs from the demonstration is that it is not arranged in a syllogistic mode and figure.
Sciendum est autem quod quia demonstratio non est corruptibilium, sed sempiternorum, neque definitio, Plato coactus fuit ponere ideas. Cum enim ista sensibilia sint corruptibilia, videbatur quod eorum non posset esse neque demonstratio, neque definitio. Et ideo videbatur quod oporteret ponere quasdam substantias incorruptibiles, de quibus et demonstrationes et definitiones darentur. Et has substantias sempiternas vocabat species vel ideas. Here we might remark that because demonstration, as well as definition, is not of destructible but of eternal things, Plato was led to posit “Ideas.” For since sensible things are destructible, it appeared that there could be neither demonstration nor definition of them. As a consequence it seemed necessary to postulate certain indestructible substances concerning which demonstrations and definitions could be given. These eternal substances are what he calls “Forms” or “Ideas.”
Sed huic opinioni occurrit Aristoteles superius dicens quod demonstratio non est corruptibilium nisi per accidens. Etsi enim ista sensibilia corruptibilia sint in particulari, in universali tamen quamdam sempiternitatem habent. Cum ergo demonstratio detur de istis sensibilibus in universali, non autem in particulari, sequitur quod demonstratio non sit corruptibilium, nisi per accidens; sempiternorum autem est per se. However, Aristotle opposed this opinion above, when he said that demonstration is not of destructible things except per accidens. For although those sensible things are destructible as individuals, nevertheless in the universal they have a certain everlasting status. Therefore, since demonstration is made about those sensible things universally and not individually, it follows that demonstration is not of destructible things except per accidens, but of eternal things per se.
Deinde cum dicit: eorum autem quae etc., ostendit quomodo eorum, quae sunt ut frequenter, possit esse demonstratio, dicens: quod eorum quae saepe fiunt, sunt etiam demonstrationes et scientiae: sicut de defectu lunae, qui tamen non semper est. Non enim luna semper deficit, sed aliquando. Haec autem quae sunt frequenter, secundum quod huiusmodi sunt, idest secundum quod de eis demonstrationes dantur, sunt semper: sed secundum quod non sunt semper, sunt particularia. De particularibus autem non potest esse demonstratio, ut ostensum est, sed solum de universalibus. Unde patet quod huiusmodi, secundum quod de eis est demonstratio, sunt semper. Et sicut est de defectu lunae, ita est de omnibus aliis similibus. Then (75b32) he shows how there can be demonstration of things that occur now and then, saying “that there are science and demonstrations of things that occur frequently, as the eclipse of the moon,” which is not always. For the moon is not always being eclipsed, but only now and then. Now things that occur frequently, so far as they are such, i.e., so far as demonstrations are given concerning them, are always; but they are particular, so far as they are not always. But demonstration cannot be of particulars, as we have shown, but only of universals. Hence it is clear that these things, insofar as there is demonstration of them, are always. And as in the case of the eclipse of the moon, so in all kindred matters.
Consideranda tamen est differentia inter ea. Quaedam enim non sunt semper secundum tempus, sunt autem semper per comparationem ad causam: quia nunquam deficit, quin posita tali causa, sequatur effectus; sicut est de defectu lunae. Nunquam enim deficit, quin semper sit lunae eclypsis, quandocunque terra diametraliter interponitur inter solem et lunam. In quibusdam vero contingit quod non semper sunt, etiam per comparationem ad causam: quia videlicet causae impediri possunt. Non enim semper ex semine hominis generatur homo habens duas manus; sed quandoque fit defectus vel propter impedimentum causae agentis vel materiae. In utrisque autem sic ordinandae sunt demonstrationes, ut ex universalibus propositionibus inferatur universalis conclusio, removendo illa, in quibus potest esse defectus vel ex parte temporis tantum vel etiam ex parte causae. However, there are certain differences to be noted among them. For some are not always with respect to time, but they are always in respect to their cause, because it never fails that under given conditions the effect follows, as in the eclipse of the moon. For the moon never fails to be eclipsed when the earth is diametrically interposed between sun and moon. But others happen not to be always even in respect to their causes, i.e., in those cases where the causes can be impeded. For it is not always that from a human seed a man with two hands is generated, but now and then a failure occurs, owing to a defect in the efficient cause or material cause. However, in both cases the demonstration must be so set up that a universal conclusion may be inferred from universal propositions by ruling out whatever can be an exception either on the part of time alone, or also of some cause.

Lectio 17
Caput 9
Ἐπεὶ δὲ φανερὸν ὅτι ἕκαστον ἀποδεῖξαι οὐκ ἔστιν ἀλλ' ἢ ἐκ τῶν ἑκάστου ἀρχῶν, ἂν τὸ δεικνύμενον ὑπάρχῃ ᾗ ἐκεῖνο, οὐκ ἔστι τὸ ἐπίστασθαι τοῦτο, ἂν ἐξ ἀληθῶν καὶ ἀναποδείκτων δειχθῇ καὶ ἀμέσων. b37. It is clear that if the conclusion is to show an attribute inhering as such, nothing can be demonstrated except from its 'appropriate' basic truths. Consequently a proof even from true, indemonstrable, and immediate premisses does not constitute knowledge.
ἔστι γὰρ οὕτω δεῖξαι, ὥσπερ Βρύσων τὸν τετραγωνισμόν. κατὰ κοινόν τε γὰρ δεικνύουσιν οἱ τοιοῦτοι λόγοι, ὃ καὶ ἑτέρῳ ὑπάρξει· διὸ καὶ ἐπ' ἄλλων ἐφαρμόττουσιν (76a.) οἱ λόγοι οὐ συγγενῶν. οὐκοῦν οὐχ ᾗ ἐκεῖνο ἐπίσταται, ἀλλὰ κατὰ συμβεβηκός· οὐ γὰρ ἂν ἐφήρμοττεν ἡ ἀπόδειξις καὶ ἐπ' ἄλλο γένος. Ἕκαστον δ' ἐπιστάμεθα μὴ κατὰ συμβεβηκός, ὅταν κατ' ἐκεῖνο γινώσκωμεν καθ' ὃ ὑπάρχει, ἐκ τῶν ἀρχῶν τῶν ἐκείνου ᾗ ἐκεῖνο, οἷον τὸ δυσὶν ὀρθαῖς ἴσας ἔχειν, ᾧ ὑπάρχει καθ' αὑτὸ τὸ εἰρημένον, ἐκ τῶν ἀρχῶν τῶν τούτου. ὥστ' εἰ καθ' αὑτὸ κἀκεῖνο ὑπάρχει ᾧ ὑπάρχει, ἀνάγκη τὸ μέσον ἐν τῇ αὐτῇ συγγενείᾳ εἶναι. b40. Such proofs are like Bryson's method of squaring the circle; for they operate by taking as their middle a common character — a character, therefore, which the subject may share with another — and consequently they apply equally to subjects different in kind. They therefore afford knowledge of an attribute only as inhering accidentally, not as belonging to its subject as such: otherwise they would not have been applicable to another genus. Our knowledge of any attribute's connexion with a subject is accidental unless we know that connexion through the middle term in virtue of which it inheres, and as an inference from basic premisses essential and 'appropriate' to the subject — unless we know, e.g. the property of possessing angles equal to two right angles as belonging to that subject in which it inheres essentially, and as inferred from basic premisses essential and 'appropriate' to that subject: so that if that middle term also belongs essentially to the minor, the middle must belong to the same kind as the major and minor terms.
εἰ δὲ μή, ἀλλ' ὡς τὰ ἁρμονικὰ δι' ἀριθμητικῆς. τὰ δὲ τοιαῦτα δείκνυται μὲν ὡσαύτως, διαφέρει δέ· τὸ μὲν γὰρ ὅτι ἑτέρας ἐπιστήμης (τὸ γὰρ ὑποκείμενον γένος ἕτερον), τὸ δὲ διότι τῆς ἄνω, ἧς καθ' αὑτὰ τὰ πάθη ἐστίν. ὥστε καὶ ἐκ τούτων φανερὸν ὅτι οὐκ ἔστιν ἀποδεῖξαι ἕκαστον ἁπλῶς ἀλλ' ἢ ἐκ τῶν ἑκάστου ἀρχῶν. ἀλλὰ τούτων αἱ ἀρχαὶ ἔχουσι τὸ κοινόν. a8. The only exceptions to this rule are such cases as theorems in harmonics which are demonstrable by arithmetic. Such theorems are proved by the same middle terms as arithmetical properties, but with a qualification — the fact falls under a separate science (for the subject genus is separate), but the reasoned fact concerns the superior science, to which the attributes essentially belong. Thus, even these apparent exceptions show that no attribute is strictly demonstrable except from its 'appropriate' basic truths, which, however, in the case of these sciences have the requisite identity of character.
Εἰ δὲ φανερὸν τοῦτο, φανερὸν καὶ ὅτι οὐκ ἔστι τὰς ἑκάστου ἰδίας ἀρχὰς ἀποδεῖξαι· ἔσονται γὰρ ἐκεῖναι ἁπάντων ἀρχαί, καὶ ἐπιστήμη ἡ ἐκείνων κυρία πάντων. a17. It is no less evident that the peculiar basic truths of each inhering attribute are indemonstrable; for basic truths from which they might be deduced would be basic truths of all that is, and the science to which they belonged would possess universal sovereignty.
καὶ γὰρ ἐπίσταται μᾶλλον ὁ ἐκ τῶν ἀνώτερον αἰτίων εἰδώς· ἐκ τῶν προτέρων γὰρ οἶδεν, ὅταν ἐκ μὴ αἰτιατῶν εἰδῇ αἰτίων. ὥστ' εἰ μᾶλλον οἶδε καὶ μάλιστα, κἂν ἐπιστήμη ἐκείνη εἴη καὶ μᾶλλον καὶ μάλιστα. a19. This is so because he knows better whose knowledge is deduced from higher causes, for his knowledge is from prior premisses when it derives from causes themselves uncaused: hence, if he knows better than others or best of all, his knowledge would be science in a higher or the highest degree.
ἡ δ' ἀπόδειξις οὐκ ἐφαρμόττει ἐπ' ἄλλο γένος, ἀλλ' ἢ ὡς εἴρηται αἱ γεωμετρικαὶ ἐπὶ τὰς μηχανικὰς ἢ ὀπτικὰς καὶ αἱ ἀριθμητικαὶ ἐπὶ τὰς ἁρμονικάς. a23. But, as things are, demonstration is not transferable to another genus, with such exceptions as we have mentioned of the application of geometrical demonstrations to theorems in mechanics or optics, or of arithmetical demonstrations to those of harmonics.
Ostenderat supra philosophus quod demonstratio non procedit ex principiis extraneis; hic autem ostendit quod non procedit ex communibus. Et circa hoc duo facit: primo, ostendit propositum; secundo, inducit quandam conclusionem ex dictis; ibi: si autem hoc est et cetera. Above the Philosopher showed that demonstration does not proceed from extraneous principles; here he shows that it does not proceed from common principles. Concerning this he does two things. First, he states his proposition. Secondly, he draws a conclusion from what he has said (76a17).
Circa primum tria facit. Primo, proponit intentum dicens quod, quia manifestum est quod non contingit unumquodque per unumquodque demonstrare, sed oportet quod demonstratio fiat ex unoquoque principiorum, hoc modo, quod id quod demonstratur sit secundum quod est illud, idest, oportet quod principia demonstrationis insint per se ei, quod demonstratur; si, inquam, ita est, non sufficit, ad hoc quod aliquid sciatur, quod demonstretur ex veris et immediatis, sed oportet ulterius quod demonstretur ex principiis propriis. Concerning the first he does two things. First (75b37), he states his intention, saying that since it is clear that it is not just through anything at random that something is demonstrated, but it is required that demonstration be from one or another of a thing’s principles in such a way that what is demonstrated be in line with what the thing as such is, i.e., it is required that the principles of the demonstration belong per se to that which is demonstrated; if, I say, this is so, then in order that something be scientifically known, it is not enough that it be demonstrated from true and immediate principles, but it is further required that it be demonstrated from proper principles.
Secundo; ibi: est enim sic demonstrare etc., probat propositum, scilicet quod non sufficiat ex veris et immediatis aliquid demonstrare, quia sic contingeret aliquid demonstrare, sicut Bryso demonstravit tetragonismum, idest quadraturam circuli, ostendens aliquod quadratum esse circulo aequale per aliqua principia communia, hoc modo: in quocunque genere est invenire aliquid maius et minus alicui, in eodem est invenire et illi aequale; in genere autem quadratorum est invenire aliquod quadratum minus circulo, quod scilicet scribitur intra circulum, et aliquod maius circulo, intra quod circulus describitur; ergo est invenire aliquod quadratum circulo aequale. Secondly (75b40), he proves his proposition, namely, that it is not enough to demonstrate something from true and immediate principles, because then something could be demonstrated in the way that Bryson proved squaring, i.e., the squaring of a circle. For he showed that some square is equal to a circle, using common principles in the following way: In any matter in which it is possible to have something greater and something less than something else, one can find something equal to it. But in the genus of squares it is possible to find one which is less than a given circle, namely, one inscribed in the circle, and another which is greater, namely, one circumscribed about the circle. Therefore, one can be found which is equal to the circle.
Haec quidem probatio est secundum commune: aequale enim, et maius, et minus, excedunt genus quadranguli et circuli. Unde patet quod huiusmodi rationes demonstrant secundum aliquod commune, quia medium alteri inest, quam ei de quo fit demonstratio; et ideo huiusmodi rationes conveniunt aliis, et non conveniunt istis, de quibus dantur, tanquam proximis. Unde patet quod qui scit per huiusmodi rationes, non scit secundum quod illud est, idest per se, sed per accidens tantum. Si enim esset secundum se, non conveniret demonstratio in aliud genus. Unumquodque enim scimus secundum accidens, cum non cognoscimus illud secundum quod est ex principiis illius, idest secundum quod est ex principiis per se. Sicut habere tres angulos aequales duobus rectis inest per se triangulo, idest secundum quod est ex principiis illius. Quare si per se inesset medium acceptum conclusioni, necesse esset in eadem proximitate esse, idest proximum esse secundum genus conclusioni. But this proof is according to something common: for equal and greater and less transcend the genus of square and circle. Hence such characteristics demonstrate according to something common, because the middle is present in other things besides the one with which the demonstration is concerned. Therefore, such characteristics, since they belong to other things as well, do not belong to the things of which they are said as to proximate subjects. Consequently, one who knows through such characteristics does not know according to what the thing is as such, i.e., per se, but only per accidens. For if it were based on what the thing is according to itself, the demonstration would not apply to something of another genus. For we know a thing according to what it accidentally is, when we do not know it according to what it is in virtue of its own principles, i.e., according to what comes per se from its principles, as “having three angles equal to two right angles” belongs per se to triangle, i.e., according to what it is in virtue of its principles. Consequently, if the middle which one used belonged per se to the conclusion, it would necessarily have to be in the same proximity, i.e., proximate according to genus, to the conclusion.
Tertio; ibi: si vero non etc., excludit quandam dubitationem. Contingit enim aliquando medium demonstrationis non esse in eodem genere cum conclusione. Quod qualiter contingat ostendit dicens: si vero non sit medium in eadem proximitate conclusioni, sed hoc modo sicut demonstratur aliquid in harmonica, idest in musica, per arithmeticam; verum quidem est quod huiusmodi etiam similiter demonstratur. Fit enim demonstratio in inferiori scientia per principia superioris scientiae, ut ostensum est; sicut et in scientia superiori per principia superioris. Thirdly (760), he excludes a doubt. For it sometimes happens that the middle of a demonstration is not in the same genus as the conclusion. How this can happen he shows when he suggests that if the middle is not in the same proximate genus as the conclusion, but is present in the way that theorems in harmonics, (i.e., in music), are demonstrable by arithmetic, it is nevertheless true that such things are also demonstrated. For in a lower science there is demonstration through the principles of a higher science, as we have established, just as in the higher science there is demonstration through the principles of that higher science.
Sed in hoc differt, quod alterius scientiae, scilicet inferioris, est scire ipsum quia tantum: genus enim subiectum inferioris scientiae est alterum a genere subiecto superioris scientiae, ex qua sumuntur principia. Sed scire propter quid est superioris scientiae, cuius sunt per se illae passiones. Cum enim passio insit subiecto propter medium, illa scientia considerabit propter quid, ad quam pertinet medium, cuius per se est passio, quae demonstratur. Si vero subiectum sit ad aliam scientiam pertinens, illius scientiae non erit propter quid, sed quia tantum; nec tali subiecto per se conveniet passio demonstrata de ipso, sed per medium extraneum. Si vero medium et subiectum pertineant ad eamdem scientiam, tunc illius scientiae erit scire quia et propter quid. But there is this difference: on the part of the other science, namely, the lower one, there is only knowledge quia [i.e., knowledge of the fact], for the generic subject of the lower science is different from the generic subject of the higher science, from which the principles are borrowed. But knowledge propter quid [i.e., of the cause why] is proper to the higher science to which those proper attributes belong per se. For since it is in virtue of the middle that a proper attribute is in a subject, that science will consider the propter quid to which pertains the middle in which the proper attribute being demonstrated inheres per se. But if the subject belongs to another science, it will not pertain to that science to know propter quid but only quia. Nor will the proper attribute demonstrated of such a subject belong per se to it, but it will be through an alien middle. However, if the middle and the subject pertain to the same science, then it will be characteristic of that science to know quia and propter quid.
Remota autem dubitatione, ulterius conclusionem intentam principaliter inducit, dicens quod ex praedictis patet quod non est demonstrare unumquodque simpliciter, idest quocunque modo, sed secundum hoc quod demonstratur ex propriis principiis uniuscuiusque. Sed et principia propria singularum scientiarum habent aliquod commune prius eis. Having settled this doubt, he draws the conclusion chiefly intended, saying that in the light of the foregoing it is clear that one should not demonstrate haphazardly, i.e., in any random way, but in such a way that a demonstration is made from the principles proper to each thing. But even the principles proper to the particular sciences have something common prior to them.
Deinde cum dicit: si autem hoc etc., inducit quandam conclusionem sequentem ex dictis. Et circa hoc tria facit. Primo, inducit conclusionem dicens quod, si hoc verum est, scilicet quod demonstrationes in singulis scientiis non fiunt ex communibus principiis, et iterum quod principia scientiarum habent aliquid prius se, quod est commune; manifestum est quod non est uniuscuiusque scientiae demonstrare principia sua propria. Illa enim priora principia, per quae possent probari singularum scientiarum propria principia, sunt communia principia omnium, et illa scientia, quae considerat huiusmodi principia communia, est propria omnibus, idest ita se habet ad ea, quae sunt communia omnibus, sicut se habent aliae scientiae particulares ad ea, quae sunt propria. Sicut cum subiectum arithmeticae sit numerus, ideo arithmetica considerat ea, quae sunt propria numeri: similiter prima philosophia, quae considerat omnia principia, habet pro subiecto ens, quod est commune ad omnia; et ideo considerat ea, quae sunt propria entis, quae sunt omnibus communia, tanquam propria sibi. Then (76a17) he draws a conclusion that follows from the aforesaid. And in regard to this he does three things. First, he draws the conclusion that if demonstrations in the particular sciences do not proceed from common principles, and if the principles of those sciences have something prior to them which is common, then it is clear that it is not the business of each science to prove its proper principles. For those prior principles through which the proper principles of the particular sciences can be proved are principles common to all; and the science which considers such common principles is proper to all, i.e., is related to things which are common to all, as those other particular sciences are related to things respectively proper to each. For example, since the subject of arithmetic is number, arithmetic considers things proper to number. In like manner, first philosophy, which considers all principles, has for its subject “being,” which is common to all. Therefore, it considers the things proper to being (which are common to all) as proper to itself.
Secundo, cum dicit: et namque scivit etc., ostendit praeeminentiam huiusmodi scientiae, quae considerat principia communia, scilicet primae philosophiae, ad alias. Semper enim oportet illud, per quod aliquid probatur, esse magis scitum vel notum. Qui enim scit aliquid ex superioribus causis, oportet quod sit magis intelligens illas causas, quia scivit ex prioribus simpliciter, cum non sciat ex causatis causas: quando enim aliquis scit ex causatis causas, tunc non intelligit ex prioribus et ex magis notis simpliciter, sed ex magis notis et prioribus quoad nos. Cum autem principia inferioris scientiae probantur ex principiis superioris, non proceditur ex causatis in causas, sed e converso. Unde oportet quod talis processus sit ex prioribus et ex magis notis simpliciter. Oportet ergo magis esse scitum quod est superioris scientiae, ex quo probatur id quod est inferioris, et maxime esse scitum id, quo omnia alia probantur, et ipsum non probatur ex alio priori. Et per consequens scientia superior erit magis scientia, quam inferior; et scientia suprema, scilicet philosophia prima, erit maxime scientia. Secondly (76a19), he shows the pre-eminence of this science which considers common principles, namely, first philosophy, over the others. For that through which something is proved must always be scientifically more known or at least more known. For one whose science of something proceeds from higher causes must understand those causes even better, because his science proceeded from the absolutely prior, since it is not through something caused that he knows these causes. For when one’s knowledge of causes proceeds from caused things, he does not derive his knowledge from the absolutely prior and better known, but from what is prior and better known in reference to us. But when the principles of a lower science are proved by the principles of a higher science, the process is not from caused things to causes, but conversely. Therefore, such a process must go from the absolutely prior and from the absolutely more known. Thus, those items of a higher science that are used in proving matters of a lower science must be more known. Furthermore, that by which all other things are proved and which is itself not proved by anything prior must be the most known. Consequently, a higher science will be science in a fuller sense than a lower; and the highest science, namely, first philosophy, will be science in the fullest sense.
Tertio, ibi: sed demonstratio etc., redit ad principalem conclusionem: et dicit quod demonstratio non procedit in aliud genus, nisi sicut dictum est quod demonstratio geometriae procedit ad scientias inferiores; sicut sunt artes mechanicae, quae utuntur mensuris; aut speculativae, sicut scientiae quae sunt de visu, ut perspectivae, quae sunt de visuali; et similiter est de arithmetica in comparatione ad harmonicam, idest musicam. Thirdly (7643), he returns to the principal conclusion and says that demonstration does not cross over into another genus except, as already mentioned, when a demonstration from geometry is applied to certain subordinate sciences, as the mechanical arts, which employ measurements, or the perspective arts, such as the sciences which deal with vision, as optics which deals with the visual. The same applies to arithmetic in relation to harmonics, i.e., music.

Lectio 18
Caput 9 cont.

Χαλεπὸν δ' ἐστὶ τὸ γνῶναι εἰ οἶδεν ἢ μή. χαλεπὸν γὰρ τὸ γνῶναι εἰ ἐκ τῶν ἑκάστου ἀρχῶν ἴσμεν ἢ μή· ὅπερ ἐστὶ τὸ εἰδέναι. οἰόμεθα δ', ἂν ἔχωμεν ἐξ ἀληθινῶν τινῶν συλλογισμὸν καὶ πρώτων, ἐπίστασθαι. τὸ δ' οὐκ ἔστιν, ἀλλὰ συγγενῆ δεῖ εἶναι τοῖς πρώτοις.

a26. It is hard to be sure whether one knows or not; for it is hard to be sure whether one's knowledge is based on the basic truths appropriate to each attribute — the differentia of true knowledge. We think we have scientific knowledge if we have reasoned from true and primary premisses. But that is not so: the conclusion must be homogeneous with the basic facts of the science.
Chapter 10
Λέγω δ' ἀρχὰς ἐν ἑκάστῳ γένει ταύτας ἃς ὅτι ἔστι μὴ ἐνδέχεται δεῖξαι. a31. I call the basic truths of every genus those elements in it the existence of which cannot be proved.
τί μὲν οὖν σημαίνει καὶ τὰ πρῶτα καὶ τὰ ἐκ τούτων, λαμβάνεται, ὅτι δ' ἔστι, τὰς μὲν ἀρχὰς ἀνάγκη λαμβάνειν, τὰ δ' ἄλλα δεικνύναι· οἷον τί μονὰς ἢ τί τὸ εὐθὺ καὶ τρίγωνον, εἶναι δὲ τὴν μονάδα λαβεῖν καὶ μέγεθος, τὰ δ' ἕτερα δεικνύναι. a32. As regards both these primary truths and the attributes dependent on them the meaning of the name is assumed. The fact of their existence as regards the primary truths must be assumed; but it has to be proved of the remainder, the attributes. Thus we assume the meaning alike of unity, straight, and triangular; but while as regards unity and magnitude we assume also the fact of their existence, in the case of the remainder proof is required.
Ἔστι δ' ὧν χρῶνται ἐν ταῖς ἀποδεικτικαῖς ἐπιστήμαις τὰ μὲν ἴδια ἑκάστης ἐπιστήμης τὰ δὲ κοινά, κοινὰ δὲ κατ' ἀναλογίαν, ἐπεὶ χρήσιμόν γε ὅσον ἐν τῷ ὑπὸ τὴν ἐπιστήμην γένει· a37. Of the basic truths used in the demonstrative sciences some are peculiar to each science, and some are common, but common only in the sense of analogous, being of use only in so far as they fall within the genus constituting the province of the science in question.
ἴδια μὲν οἷον γραμμὴν εἶναι τοιανδὶ καὶ τὸ εὐθύ, κοινὰ δὲ οἷον τὸ ἴσα ἀπὸ ἴσων ἂν ἀφέλῃ, ὅτι ἴσα τὰ λοιπά. a40. Peculiar truths are, e.g. the definitions of line and straight; common truths are such as 'take equals from equals and equals remain'.
ἱκανὸν δ' ἕκαστον τούτων ὅσον ἐν τῷ γένει· ταὐτὸ γὰρ ποιήσει, (76b.) κἂν μὴ κατὰ πάντων λάβῃ ἀλλ' ἐπὶ μεγεθῶν μόνον, τῷ δ' ἀριθμητικῷ ἐπ' ἀριθμῶν. a42. Only so much of these common truths is required as falls within the genus in question: for a truth of this kind will have the same force even if not used generally but applied by the geometer only to magnitudes, or by the arithmetician only to numbers.
Ἔστι δ' ἴδια μὲν καὶ ἃ λαμβάνεται εἶναι, περὶ ἃ ἡ ἐπιστήμη θεωρεῖ τὰ ὑπάρχοντα καθ' αὑτά, οἷον μονάδας ἡ ἀριθμητική, ἡ δὲ γεωμετρία σημεῖα καὶ γραμμάς. ταῦτα γὰρ λαμβάνουσι τὸ εἶναι καὶ τοδὶ εἶναι. τὰ δὲ τούτων πάθη καθ' αὑτά, τί μὲν σημαίνει ἕκαστον, λαμβάνουσιν, οἷον ἡ μὲν ἀριθμητικὴ τί περιττὸν ἢ ἄρτιον ἢ τετράγωνον ἢ κύβος, ἡ δὲ γεωμετρία τί τὸ ἄλογον ἢ τὸ κεκλάσθαι ἢ νεύειν, ὅτι δ' ἔστι, δεικνύουσι διά τε τῶν κοινῶν καὶ ἐκ τῶν ἀποδεδειγμένων. καὶ ἡ ἀστρολογία ὡσαύτως. πᾶσα γὰρ ἀποδεικτικὴ ἐπιστήμη περὶ τρία ἐστίν, ὅσα τε εἶναι τίθεται (ταῦτα δ' ἐστὶ τὸ γένος, οὗ τῶν καθ' αὑτὰ παθημάτων ἐστὶ θεωρητική), καὶ τὰ κοινὰ λεγόμενα ἀξιώματα, ἐξ ὧν πρώτων ἀποδείκνυσι, καὶ τρίτον τὰ πάθη, ὧν τί σημαίνει ἕκαστον λαμβάνει. b2. Also peculiar to a science are the subjects the existence as well as the meaning of which it assumes, and the essential attributes of which it investigates, e.g. in arithmetic units, in geometry points and lines. Both the existence and the meaning of the subjects are assumed by these sciences; but of their essential attributes only the meaning is assumed. For example arithmetic assumes the meaning of odd and even, square and cube, geometry that of incommensurable, or of deflection or verging of lines, whereas the existence of these attributes is demonstrated by means of the axioms and from previous conclusions as premisses. Astronomy too proceeds in the same way. For indeed every demonstrative science has three elements: (1) that which it posits, the subject genus whose essential attributes it examines; (2) the so-called axioms, which are primary premisses of its demonstration; (3) the attributes, the meaning of which it assumes.
ἐνίας μέντοι ἐπιστήμας οὐδὲν κωλύει ἔνια τούτων παρορᾶν, οἷον τὸ γένος μὴ ὑποτίθεσθαι εἶναι, ἂν ᾖ φανερὸν ὅτι ἔστιν (οὐ γὰρ ὁμοίως δῆλον ὅτι ἀριθμὸς ἔστι καὶ ὅτι ψυχρὸν καὶ θερμόν), καὶ τὰ πάθη μὴ λαμβάνειν τί σημαίνει, ἂν ᾖ δῆλα· ὥσπερ οὐδὲ τὰ κοινὰ οὐ λαμβάνει τί σημαίνει τὸ ἴσα ἀπὸ ἴσων ἀφελεῖν, ὅτι γνώριμον. ἀλλ' οὐδὲν ἧττον τῇ γε φύσει τρία ταῦτά ἐστι, περὶ ὅ τε δείκνυσι καὶ ἃ δείκνυσι καὶ ἐξ ὧν. b16. Yet some sciences may very well pass over some of these elements; e.g. we might not expressly posit the existence of the genus if its existence were obvious (for instance, the existence of hot and cold is more evident than that of number); or we might omit to assume expressly the meaning of the attributes if it were well understood. In the way the meaning of axioms, such as 'Take equals from equals and equals remain', is well known and so not expressly assumed. Nevertheless in the nature of the case the essential elements of demonstration are three: the subject, the attributes, and the basic premisses.
Postquam ostendit quod demonstratio non procedit ex principiis communibus sed ex propriis, hic ad evidentiam praemissorum determinat de principiis propriis et communibus. Et circa hoc duo facit. After showing that demonstration does not proceed from common but from proper principles, the Philosopher, to elucidate this point, decides questions concerning proper and common principles. With respect to this he does two things.
Primo, ostendit necessitatem huiusmodi determinationis, dicens quod difficile est cognoscere utrum sciamus ex principiis propriis (quod solum est vere scire) aut non ex propriis. Opinantur enim multi se scire, si habeant syllogismum ex aliquibus veris et primis. Sed hoc non est verum: immo oportet, ad hoc quod sciamus, quod principia sint proxima illis quae debent demonstrari (quae hic dicuntur prima, sicut et supra dicebantur extrema); vel oportet proxima esse primis principiis indemonstrabilibus. First (76a26), he shows the need for such a determination, saying that it is difficult to discern whether we know from proper principles, which alone is truly scientific knowing, or do not know from proper principles. For many believe that they know scientifically if they possess a syllogism composed of things true and first. But this is not so; indeed, to know in a scientific manner it is required that the principles be proximate to the things to be demonstrated—here they are called “first,” just as above they were called “extremes”—or proximate to the first indemonstrable principles.
Secundo, ibi: dico autem principia etc., determinat de principiis propriis et communibus. Et circa hoc duo facit: primo enim determinat de principiis propriis et communibus; secundo, ostendit qualiter ad huiusmodi principia se habeant demonstrativae scientiae; ibi: non contingere autem et cetera. Circa primum duo facit: primo, distinguit principia a non principiis; secundo, principia ad invicem; ibi: sunt autem quibus et cetera. Secondly (76a3l), he determines concerning proper and common principles. And in regard to this he does two things. First he determines concerning common and proper principles. Secondly, he shows how the demonstrative sciences are related to such principles (77a10) [L. 20]. Concerning the first he does two things. First, he distinguishes principles from non-principles. Secondly, he distinguishes the principles from each other(76a37).
Circa primum duo facit. Primo, ostendit quae sint principia, dicens quod principia in unoquoque genere sunt illa quae, cum sint vera, tamen non contingit ea demonstrare vel simpliciter si sint principia prima, vel ad minus non est demonstrare in illa scientia in qua sumuntur ut principia. Dicit autem, cum sint vera, ad differentiam falsorum, quae non demonstrantur in aliqua scientia. In regard to the first he does two things. First (76a3l), he shows what the principles are and says that the principles in any genus are those which, since they are true, happen not to be demonstrated, either not at all, if they are first principles, or at least not in that science in which they are accepted as principles. And he says, “the existence of which,” (i.e., whose truth), “cannot be proved,” to distinguish from the false which are not demonstrated in any science.
Secundo, ibi: quid quidem igitur etc., ostendit convenientiam et differentiam inter principia et non principia. Conveniunt enim principia cum non principiis in hoc, quod de utrisque oportet accipere, quasi supponendo quid significent, et prima, idest principia, et quae sunt ex his, idest quae ex principiis sumuntur: quia quod quid est proprie pertinet ad scientiam quae est de substantia, scilicet ad philosophiam primam, a qua omnes aliae hoc accipiunt. Secondly (76a32), he shows the points of agreement and of difference between principles and non-principles. For principles agree with non-principles in the fact that with respect to each, i.e., with respect both to the first, namely, the principles, and to the things that proceed from them, i.e., things assumed from the principles, one must accept the supposition signify, because the quod quid est [that which the definition of what they signifies] of a thing pertains properly to the science concerned with substance, i.e., to first philosophy, from which all other sciences accept this.
Sed in hoc differunt principia, et quae sunt ex principiis, quia de principiis oportet accipere supponendo quod sunt; de aliis autem, quae sunt ex principiis, oportet demonstrare quia sunt. Sicut in mathematicis accipitur supponendo et quid est unitas, quae est principium, et quid est rectum, et quid est triangulus, quae non sunt principia, sed passiones: sed quod unitas sit, aut quod magnitudo sit, accipit mathematicus quasi principia; alia vero demonstrat, scilicet quae sunt ex principiis. Demonstrat enim triangulum aequilaterum et angulum rectum, et etiam hanc lineam rectam esse. But they are unlike in the fact that in regard to the principles one must accept the supposition that they are, whereas in regard to things from principles one is required to demonstrate that they are. Thus in mathematics one accepts by supposing both what unity is (which is a principle) and what straight and triangle are (which are not principles but proper attributes): but the fact that unity is or that magnitude is, the mathematician accepts as principles; but the other things, namely, things that are from principles, he demonstrates. For he demonstrates that a triangle is equilateral, or that an angle is right, or even that this line is straight.
Deinde cum dicit: sunt autem quibus etc., distinguit principia ad invicem: et primo, principia propria a communibus; secundo, communia ad invicem; ibi: non est autem suppositio et cetera. Prima dividitur in duas; in prima, dividit principia propria et communia; in secunda, manifestat quoddam quod poterat esse dubium; ibi: quasdam tamen scientias et cetera. Then (76a37) he distinguishes principles from one another. First, the proper from the common. Secondly, the common, one from the other (76b23) [L. 19]. The first is divided into two parts. First, he divides proper and common principles. Secondly, he settles something which might be doubtful (76b16).
Circa primum tria facit. Primo, ponit divisionem, dicens quod principiorum, quibus utimur in demonstrativis scientiis, alia sunt propria uniuscuiusque scientiae, alia vero communia. Et quia hoc posset videri contrarium ei, quod supra ostensum est, quia scientiae demonstrativae non procedunt ex communibus, ideo subiungit quod communia principia accipiuntur in unaquaque scientia demonstrativa secundum analogiam, idest secundum quod sunt proportionata illi scientiae. Et hoc est quod subdit exponens, quod utile est accipere huiusmodi principia in scientiis, quantum pertinet ad genus subiectum, quod continetur sub illa scientia. In regard to the first he does three things. First (76a37), he lays down a division, saying that “of the principles which we use in demonstrative sciences, some are proper to each single science but others common.” Then, because this statement might seem contrary to what he established above, namely, that demonstrative sciences do not proceed from common principles, he hastens to add that “the common principles are taken in each demonstrative science according to an analogy,” i.e., as proportionate to that science. And this is what he means when by way of explanation he further states that “it is useful” to accept such principles in the sciences insofar as they pertain to the genus of the subject which is investigated in that science.
Secundo, ibi: propria principia etc., exemplificat de utrisque, dicens quod propria principia sunt, ut lineam esse huiusmodi, vel rectum. Tam enim subiecti quam passionis definitio in scientiis pro principio habetur. Communia vero principia sunt, ut, si ab aequalibus aequalia demas, quae remanent sunt aequalia, et aliae communes animi conceptiones. Secondly (76a40), he gives examples of each, saying that proper principles are, for example, that a line or a right angle is such and such. For in the sciences the definitions, both of the subject and of the proper attribute, are held as principles. But common principles are, for example, that if equals be subtracted from equals, the remainders are equal, and other common conceptions in the mind.
Tertio, ibi: sufficiens autem est etc., ostendit quomodo praemissis principiis scientiae demonstrativae utantur. Et primo quidem de communibus dicit quod sufficiens est accipere unumquodque istorum communium, quantum pertinet ad genus subiectum, de quo est scientia. Idem enim faciet geometria, si non accipiat praemissum principium commune in sua communitate, sed solum in magnitudinibus, et arithmetica in solis numeris. Ita enim poterit concludere geometria, si dicat: si ab aequalibus magnitudinibus aequales auferas magnitudines, quae remanent sunt aequales; sicut si diceret: si ab aequalibus aequalia demas, quae remanent sunt aequalia. Et similiter dicendum est de numeris. Thirdly (76a42), he shows how the demonstrative sciences use the aforesaid principles. First, in regard to the common principles, he says that “it suffices to accept each of those common ones,” so far as it pertains to the generic subject with which the science is concerned. For geometry does this if it takes the above-mentioned common principle not in its generality but only in regard to magnitudes, and arithmetic in regard to numbers. For the geometer will then be able to reach his conclusion by saying that if equal magnitudes be taken from equal magnitudes, the remaining magnitudes are equal, just as if he were to say that if equals are taken from equals, the remainders are equal. The same must also be said for numbers.
Secundo, ibi: sunt autem propria etc., ostendit qualiter demonstrativae scientiae utantur propriis principiis, dicens quod propria principia sunt quae supponuntur esse in scientiis, scilicet subiecta, circa quae scientia speculatur ea quae per se insunt eis. Sicut arithmetica considerat unitates, et geometria considerat signa, idest puncta et lineas. Praedictae enim supponunt esse et hoc esse, idest supponunt de eis, et quia sunt et quid sunt. Sed de passionibus supponunt praedictae scientiae quid significet unaquaeque; sicut arithmetica supponit quid est par, et quid est impar, aut quid est numerus quadratus aut cubicus; et geometria supponit quid est rationale in lineis. Dicitur enim linea rationalis, de qua possumus ratiocinari per lineam datam: huiusmodi autem est omnis linea commensurabilis lineae datae; quae vero est ei non commensurabilis, vocatur irrationalis vel surda. Similiter et geometria supponit quid est reflexum aut curvum. Sed praedictae scientiae demonstrant de omnibus praedictis passionibus quod sint per principia communia, et ex illis principiis, quae demonstrantur ex communibus. Et quod dictum est de geometria et arithmetica, intelligendum est etiam de astrologia. Secondly (76b2), he shows how the demonstrative sciences employ proper principles. And he says that proper principles are things supposed in the sciences as existing, namely, the subjects, whose proper attributes are investigated in the sciences. In this way arithmetic considers unities, and geometry considers “signs,” i.e., points, and lines. For they suppose these things to be and to be this, i.e., they suppose of them that they are and what they are. But in regard to the proper attributes they suppose what each signifies. Thus arithmetic supposes what “even” is and what “odd” is and what a “square or cubic number” is. Similarly, geometry supposes what is rational in lines. (For a rational line is one about which we can reason, the line being given. For example, a rational line is any line commensurable with the given lines; but one which is incommensurable with it is called irrational and surd). In like manner, geometry supposes what a reflex or what a curved line are. However, these sciences demonstrate concerning all the above-mentioned proper attributes that they are, and they do so through common principles and principles demonstrated from the common principles. And what has been said 6f geometry and arithmetic should also be understood of astronomy.
Omnis enim scientia demonstrativa est circa tria: quorum unum est genus subiectum, cuius per se passiones scrutantur; et aliud est communes dignitates, ex quibus sicut ex primis demonstrat; tertium autem passiones, de quibus unaquaeque scientia accipit quid significent. For every demonstrative science is concerned with three things: one is the generic subject whose per se attributes are investigated; another is the common (axioms) dignities from which, as from basic truths, it demonstrates; the third are the proper attributes concerning which each science supposes what their names signify.
Deinde cum dicit: quasdam tamen scientias etc., manifestat quoddam, quod poterat esse dubium. Quia enim dixerat quod scientiae supponunt de principiis quia sunt, de passionibus quid sunt, de subiectis autem utrumque, posset aliquis credere quod oporteret specialem fieri mentionem de omnibus istis. Unde hoc removet, dicens quod nihil prohibet quasdam scientias despicere quaedam praedictorum, idest non facere mentionem expressam de praemissis, sicut quandoque non facit mentionem de hoc quod supponat genus subiectum esse, si sit manifestum quod sit, quia non est similiter manifestum de omnibus quod sint, sicut quod sit numerus, et quod sit calidum vel frigidum: quorum unum est propinquum rationi, alterum sensui. Similiter et quaedam scientiae non supponunt de passionibus quid significent, expressam mentionem de eis faciendo. Sicut etiam non oportet quod de communibus principiis semper scientiae faciant mentionem, quia nota sunt. Nihilominus tamen, tria praedicta naturaliter sunt in qualibet scientia supponenda. Then (76b16) he clarifies something about which there might be doubt. For since he had said that the sciences suppose concerning the principles that they are and concerning the proper attributes what they are, but concerning the subject both that it is and what it is, someone might believe that he should have made special mention of all these. Hence he removes this by saying that nothing hinders certain sciences from neglecting some of the aforesaid, i.e., from making express mention of them, as for example, not mentioning that it takes the existence of its generic subject for granted, if it is already obvious that it does exist. For we not have the same evidence in all cases that they do exist, as we do in the case of number and in the case of hot and cold, the one being close to reason and the other to sense. Again, certain sciences do not suppose what the proper attributes signify in the sense of making express mention of them, just as they do not think it necessary always to make express mention of the common principles, because they are known. Be that as it may, the three above-mentioned items are naturally to be supposed in each science.

Lectio 19
Caput 10 cont.
Οὐκ ἔστι δ' ὑπόθεσις οὐδ' αἴτημα, ὃ ἀνάγκη εἶναι δι' αὑτὸ καὶ δοκεῖν ἀνάγκη. οὐ γὰρ πρὸς τὸν ἔξω λόγον ἡ ἀπόδειξις, ἀλλὰ πρὸς τὸν ἐν τῇ ψυχῇ, ἐπεὶ οὐδὲ συλλογισμός. ἀεὶ γὰρ ἔστιν ἐνστῆναι πρὸς τὸν ἔξω λόγον, ἀλλὰ πρὸς τὸν ἔσω λόγον οὐκ ἀεί. b23. That which expresses necessary self-grounded fact, and which we must necessarily believe, is distinct both from the hypotheses of a science and from illegitimate postulate — I say 'must believe', because all syllogism, and therefore a fortiori demonstration, is addressed not to the spoken word, but to the discourse within the soul, and though we can always raise objections to the spoken word, to the inward discourse we cannot always object.
ὅσα μὲν οὖν δεικτὰ ὄντα λαμβάνει αὐτὸς μὴ δείξας, ταῦτ', ἐὰν μὲν δοκοῦντα λαμβάνῃ τῷ μανθάνοντι, ὑποτίθεται, καὶ ἔστιν οὐχ ἁπλῶς ὑπόθεσις ἀλλὰ πρὸς ἐκεῖνον μόνον, ἂν δὲ ἢ μηδεμιᾶς ἐνούσης δόξης ἢ καὶ ἐναντίας ἐνούσης λαμβάνῃ τὸ αὐτό, αἰτεῖται. καὶ τούτῳ διαφέρει ὑπόθεσις καὶ αἴτημα· ἔστι γὰρ αἴτημα τὸ ὑπεναντίον τοῦ μανθάνοντος τῇ δόξῃ, ἢ ὃ ἄν τις ἀποδεικτὸν ὂν λαμβάνῃ καὶ χρῆται μὴ δείξας. b27. That which is capable of proof but assumed by the teacher without proof is, if the pupil believes and accepts it, hypothesis, though only in a limited sense hypothesis — that is, relatively to the pupil; if the pupil has no opinion or a contrary opinion on the matter, the same assumption is an illegitimate postulate. Therein lies the distinction between hypothesis and illegitimate postulate: the latter is the contrary of the pupil's opinion, demonstrable, but assumed and used without demonstration.
Οἱ μὲν οὖν ὅροι οὐκ εἰσὶν ὑποθέσεις (οὐδὲν γὰρ εἶναι ἢ μὴ λέγεται), ἀλλ' ἐν ταῖς προτάσεσιν αἱ ὑποθέσεις, τοὺς δ' ὅρους μόνον ξυνίεσθαι δεῖ· τοῦτο δ' οὐχ ὑπόθεσις (εἰ μὴ καὶ τὸ ἀκούειν ὑπόθεσίν τις εἶναι φήσει), ἀλλ' ὅσων ὄντων τῷ ἐκεῖνα εἶναι γίνεται τὸ συμπέρασμα. M. The definition — viz. those which are not expressed as statements that anything is or is not — are not hypotheses: but it is in the premisses of a science that its hypotheses are contained. Definitions require only to be understood, and this is not hypothesis — unless it be contended that the pupil's hearing is also an hypothesis required by the teacher. Hypotheses, on the contrary, postulate facts on the being of which depends the being of the fact inferred.
(οὐδ' ὁ γεωμέτρης ψευδῆ ὑποτίθεται, ὥσπερ τινὲς ἔφασαν, λέγοντες ὡς οὐ δεῖ τῷ ψεύδει χρῆσθαι, τὸν δὲ γεωμέτρην ψεύδεσθαι λέγοντα ποδιαίαν τὴν οὐ ποδιαίαν ἢ εὐθεῖαν τὴν γεγραμμένην οὐκ εὐθεῖαν (77a.) οὖσαν. ὁ δὲ γεωμέτρης οὐδὲν συμπεραίνεται τῷ τήνδε εἶναι γραμμὴν ἣν αὐτὸς ἔφθεγκται, ἀλλὰ τὰ διὰ τούτων δηλούμενα.) b39. Nor are the geometer's hypotheses false, as some have held, urging that one must not employ falsehood and that the geometer is uttering falsehood in stating that the line which he draws is a foot long or straight, when it is actually neither. The truth is that the geometer does not draw any conclusion from the being of the particular line of which he speaks, but from what his diagrams symbolize.
ἔτι τὸ αἴτημα καὶ ὑπόθεσις πᾶσα ἢ ὡς ὅλον ἢ ὡς ἐν μέρει, οἱ δ' ὅροι οὐδέτερον τούτων. a3. A further distinction is that all hypotheses and illegitimate postulates are either universal or particular, whereas a definition is neither.
Chapter 11
Εἴδη μὲν οὖν εἶναι ἢ ἕν τι παρὰ τὰ πολλὰ οὐκ ἀνάγκη, εἰ ἀπόδειξις ἔσται, εἶναι μέντοι ἓν κατὰ πολλῶν ἀληθὲς εἰπεῖν ἀνάγκη - οὐ γὰρ ἔσται τὸ καθόλου, ἂν μὴ τοῦτο ᾖ — ἐὰν δὲ τὸ καθόλου μὴ ᾖ, τὸ μέσον οὐκ ἔσται, ὥστ' οὐδ' ἀπόδειξις. δεῖ ἄρα τι ἓν καὶ τὸ αὐτὸ ἐπὶ πλειόνων εἶναι μὴ ὁμώνυμον. a5. So demonstration does not necessarily imply the being of Forms nor a One beside a Many, but it does necessarily imply the possibility of truly predicating one of many; since without this possibility we cannot save the universal, and if the universal goes, the middle term goes witb. it, and so demonstration becomes impossible. We conclude, then, that there must be a single identical term unequivocally predicable of a number of individuals.
Postquam divisit Aristoteles principia communia a propriis, hic distinguit communia principia ad invicem. After dividing common principles from proper principles, Aristotle now distinguishes among the common principles.
Et dividitur in partes tres: in prima, ponit distinctionem communium principiorum ad invicem; in secunda, ostendit differentiam definitionis a quodam genere principiorum communium, ibi: termini igitur non etc.; in tertia, excludit quemdam errorem, ibi: species quidem igitur et cetera. Circa primum duo facit: primo, distinguit communes animi conceptiones a petitionibus, sive suppositionibus; secundo, petitiones et suppositiones ad invicem, ibi: quaecunque quidem igitur et cetera. His treatment is divided into three parts. In the first he lays down a distinction among the common principles. In the second he shows the difference between a definition and a certain genus of common principles (76b35). In the third he excludes an error (77a5). Concerning the first he does two things. First, he distinguishes the common conceptions in the mind from postulates or suppositions. Secondly, he distinguishes among the latter (76b27).
Circa primum considerandum est quod communes animi conceptiones habent aliquid commune cum aliis principiis demonstrationis, et aliquid proprium. Commune quidem habent, quia necesse est tam ista, quam alia principia per se esse vera. Proprium autem est horum principiorum quod non solum necesse est ea per se vera esse, sed etiam necesse est videri quod per se sint vera. Nullus enim potest opinari contraria eorum. With respect to the first (76b23) it should be noted that the common. conceptions in the mind have something in common with the other principles of demonstration and something proper: something common, because both they and the others must be true in virtue of themselves. But what is proper to the former is that it is necessary not only that they be true of themselves but that they be seen to be such. For no one can think their contraries. He says, therefore, that that principle of which it is not only required that it be in virtue of itself but further required that it be seen, namely, a common conception in the mind or a dignity, is neither a postulate nor a supposition.
Dicit ergo quod illud principium, quod necesse est non solum per seipsum esse, sed etiam ulterius necesse est, ipsum videri, scilicet communis animi conceptio vel dignitas, non est neque petitio neque suppositio. Quod sic probat. Petitio et suppositio exteriori ratione confirmari possunt, idest argumentatione aliqua. Sed communis animi conceptio non est ad exterius rationem, quia non potest probari per aliquam argumentationem, sed est ad eam, quae est in anima, quia lumine naturalis rationis statim fit nota. Et quod non sit ad exterius rationem patet, quia non fit syllogismus ad probandas huiusmodi communes animi conceptiones. Et quod huiusmodi non sunt notae per exteriorem rationem, sed per interiorem, probat per hoc, quod exteriori rationi potest instari vel vere vel apparenter: interiori autem rationi non est possibile semper instari. Et hoc ideo, quia nihil est adeo verum, quin voce possit negari. Nam et hoc principium notissimum, quod non contingat idem esse et non esse, quidam ore negaverunt. Quaedam autem adeo vera sunt, quod eorum opposita intellectu capi non possunt; et ideo interiori ratione eis obviari non potest, sed solum exteriori, quae est per vocem. Et huiusmodi sunt communes animi conceptiones. He proves this in the following way: A postulate and a supposition can be confirmed by a reason from without, i.e., by some argumentation; but a common conception in the mind does not bear on a reason from without (because it cannot be proved by any argument), but bears on that reason which is in the soul, because it is made known at once by the natural light of reason. That it does not bear on any reason from without is shown by the fact that a syllogism is not formed to prove such common conceptions of the mind. Furthermore, that these are not made known by an outward reason but by the inward he proves by the fact that it is possible to contest an outward reason, either truly or apparently, but it is not always possible to do so with the inward reason. This is so because nothing is so true that it cannot be denied orally. (For even this most evident principle that the same thing cannot be and not be has been orally denied by some). On the other hand, some things are so true that their opposites cannot be conceived by the intellect. Therefore, they cannot be challenged in the inward reason but only by an outward reason which is by the voice. Such are the common conceptions in the mind.
Deinde, cum dicit: quaecunque igitur etc., distinguit suppositiones et petitiones ad invicem. Sciendum tamen est, quod aliquid commune habent, et in aliquo differunt. Hoc quidem commune est eis, quod cum sint demonstrabilia, tamen demonstrator accipit ea non demonstrans, et praecipue, quia non sunt demonstrabilia per suam scientiam, sed per aliam, ut supra dictum est. Unde et inter immediata principia computantur, quia demonstrator utitur eis absque medio, eo quod non habeant medium in illa scientia. Then (76b27) he distinguishes suppositions and postulates from one another. Here, too, it should be noted that they have something in common and something in which they differ. What is common to them is that although they can be demonstrated, the demonstrator assumes them without demonstrating, chiefly because they are not demonstrable by his own science but by another, as explained above. Hence they are reckoned among the immediate principles because the demonstrator uses them without a middle, since they do not have a middle in that science.
Differunt autem ad invicem: quia si quidem talis propositio sit probabilis addiscenti, cui fit demonstratio, dicitur suppositio. Et sic suppositio dicitur non simpliciter, sed ad aliquem. Si vero ille nec sit eiusdem opinionis, neque contrariae, oportet quod demonstrator hoc ab eo petat, et tunc dicitur petitio. Si autem sit contrariae opinionis, tunc erit quaestio, de qua oportet disputari inter eos. Hoc tamen omnibus commune est, quod unoquoque eorum utitur demonstrator non demonstrans, cum sit demonstrabile. Yet they do differ, because if such a proposition is accepted as reasonable by the pupil to whom a demonstration is being made, it is called a “Supposition.” In that case it is not called a supposition absolutely, but relative to him. But if the pupil has no reason for or against the proposition, the demonstrator must request him to admit it as reasonable, and then it is called a “postulate.” However, if the pupil has a contrary opinion, then it will be a “question,” which must be settled between them. At any rate, what is common to all of them is that the demonstrator uses each of thein without demonstrating, although they are demonstrable.
Deinde, cum dicit: termini igitur non sunt etc., distinguit definitiones a suppositionibus per duas rationes; quarum secunda incipit ibi: amplius petitio et cetera. Then (76b35) he distinguishes definitions from suppositions with two reasons, the second of which begins at (77a3). In regard to the first he does two things.
Circa primum duo facit: primo, ponit rationem, quae talis est: omnis petitio, vel suppositio dicit aliquid esse vel non esse; termini autem, idest definitiones, non dicunt aliquid esse vel non esse; termini ergo non sunt suppositiones neque petitiones, per se sumpti. Sed in propositionibus assumpti sunt suppositiones; ut cum dicitur, homo est animal rationale mortale. Sed terminos per se sumptos, oportet solum intelligere; intelligere autem non est supponere, sicut nec audire. Sed illa supponuntur quorumcunque existentium, idest ex quibuscumque existentibus fit conclusio, in eo quod illa sunt, idest propter praemissa. First, he presents this reason: Every postulate or supposition declares something to be or not to be; but terms, i.e., definitions, do not declare that something is or is not. Terms, therefore, taken by themselves, are neither postulates nor suppositions. But when they are employed in propositions they are suppositions, as in the statement, “Man is a rational mortal animal.” But terms taken by themselves are only understood; and understanding is not supposing, any more than hearing is. Rather supposition bears on things that exist such that a conclusion Js made from them in regard to what they are, i.e., in virtue of the premises.
Secundo, ibi: neque geometra etc., excludit quamdam dubitationem. Dicebant enim quidam quod geometra falsa suppositione utebatur, cum diceret lineam esse unius pedis, quae non est unius pedis; aut lineam descriptam in pulvere esse rectam, quae non est recta. Sed ipse dicit quod geometra non supponit falsum propter hoc. Cum enim geometra nihil demonstret de particularibus, sed de universalibus, ut supra dictum est; hae autem lineae sunt quaedam particularia; manifestum est quod de his lineis nihil demonstrat, neque etiam ex eis, sed utitur eis ut exemplis universalium, quae per haec exempla intelliguntur, de quibus et ex quibus demonstrat. Secondly (76b39), he excludes a doubt. For there were some who said that a geometer uses a false supposition when he says that a line not one foot long is one foot long, or that a line traced in the sand is straight when it really is not. But he says that a geometer does not on this account suppose something false. For since the geometer demonstrates nothing of singulars but of universals, as explained above, and these lines are singulars, it is obvious that he is not demonstrating anything about these lines or from these lines; rather he is using them as examples of the universals (which are understood by examples) from which and about which he demonstrates.
Deinde, cum dicit: amplius petitio etc., ponit secundam rationem, quae talis est: omnis suppositio vel petitio est in toto vel in parte, idest est propositio universalis vel particularis; sed definitiones neutrum horum sunt, quia in eis nihil ponitur sive praedicatur, neque universaliter, neque particulariter; ergo et cetera. Then (77a3) he presents the second reason which is this: Every supposition or postulate is in whole or in part, i.e., is either a universal proposition or a particular. But definitions are neither of these, because nothing is placed or predicated in them universally or particularly. Therefore...
Deinde, cum dicit: species quidem esse etc., ostendit ex praemissis quod non est necessarium ponere ideas, ut Plato posuit. Ostensum est enim supra quod demonstrationes de universalibus sunt, et hoc modo sunt de sempiternis. Non igitur necesse est ad hoc quod demonstratio sit, species esse, idest ideas, aut quodcunque unum extra multa, sicut ponebant Platonici mathematica separata cum ideis, ut sic demonstrationes possint esse de sempiternis. Sed necessarium est esse unum in multis et de multis, si demonstratio debet esse, quia non erit universale, nisi sit unum de multis; et si non sit universale, non erit medium demonstrationis; ergo nec demonstratio. Then (77a5) he shows from the foregoing that it is not necessary to posit “Ideas,” as Plato did. For it was shown above that demonstrations are concerned with universals and in that sense with eternal things. Therefore, it is not necessary for the validity of demonstration that there be “Forms,” i.e., “Ideas,” or a “one outside the many,” as Plato posited separated mathematical beings along with Ideas in order that thereby demonstrations might bear on eternal things. What is required is that there be “one in many and about many,” if there is to be demonstration, because it will not be universal unless it is “one about many.” And if it is not universal, there will not be a middle of demonstration and, consequently, no demonstration.
Et quod oporteat medium demonstrationis esse universale, patet per hoc quod oportet medium demonstrationis esse unum et idem de pluribus praedicatum non aequivoce, sed secundum rationem eamdem: quod est ratio universalis. Si autem aequivocum esset, posset accidere vitium in arguendo. That the middle of a demonstration must be universal is plain from the fact that the middle of a demonstration must be some one same thing predicated of many not equivocally but according to the same aspect, which is a universal aspect. But if it should happen to be equivocal, a defect in reasoning would occur.

Lectio 20
Caput 11 cont.
τὸ δὲ μὴ ἐνδέχεσθαι ἅμα φάναι καὶ ἀποφάναι οὐδεμία λαμβάνει ἀπόδειξις, ἀλλ' ἢ ἐὰν δέῃ δεῖξαι καὶ τὸ συμπέρασμα οὕτως. δείκνυται δὲ λαβοῦσι τὸ πρῶτον κατὰ τοῦ μέσου, ὅτι ἀληθές, ἀποφάναι δ' οὐκ ἀληθές. τὸ δὲ μέσον οὐδὲν διαφέρει εἶναι καὶ μὴ εἶναι λαβεῖν, ὡς δ' αὔτως καὶ τὸ τρίτον. εἰ γὰρ ἐδόθη, καθ' οὗ ἄνθρωπον ἀληθὲς εἰπεῖν, εἰ καὶ μὴ ἄνθρωπον ἀληθές, ἀλλ' εἰ μόνον ἄνθρωπον ζῷον εἶναι, μὴ ζῷον δὲ μή, ἔσται [γὰρ] ἀληθὲς εἰπεῖν Καλλίαν, εἰ καὶ μὴ Καλλίαν, ὅμως ζῷον, μὴ ζῷον δ' οὔ. αἴτιον δ' ὅτι τὸ πρῶτον οὐ μόνον κατὰ τοῦ μέσου λέγεται ἀλλὰ καὶ κατ' ἄλλου διὰ τὸ εἶναι ἐπὶ πλειόνων, ὥστ' οὐδ' εἰ τὸ μέσον καὶ αὐτό ἐστι καὶ μὴ αὐτό, πρὸς τὸ συμπέρασμα οὐδὲν διαφέρει. a10. The law that it is impossible to affirm and deny simultaneously the same predicate of the same subject is not expressly posited by any demonstration except when the conclusion also has to be expressed in that form; in which case the proof lays down as its major premiss that the major is truly affirmed of the middle but falsely denied. It makes no difference, however, if we add to the middle, or again to the minor term, the corresponding negative. For grant a minor term of which it is true to predicate man — even if it be also true to predicate not-man of it — still grant simply that man is animal and not not-animal, and the conclusion follows: for it will still be true to say that Callias — even if it be also true to say that not-Callias — is animal and not not-animal. The reason is that the major term is predicable not only of the middle, but of something other than the middle as well, being of wider application; so that the conclusion is not affected even if the middle is extended to cover the original middle term and also what is not the original middle term.
τὸ δ' ἅπαν φάναι ἢ ἀποφάναι ἡ εἰς τὸ ἀδύνατον ἀπόδειξις λαμβάνει, καὶ ταῦτα οὐδ' ἀεὶ καθόλου, ἀλλ' ὅσον ἱκανόν, ἱκανὸν δ' ἐπὶ τοῦ γένους. λέγω δ' ἐπὶ τοῦ γένους οἷον περὶ ὃ γένος τὰς ἀποδείξεις φέρει, ὥσπερ εἴρηται καὶ πρότερον. a22. The law that every predicate can be either truly affirmed or truly denied of every subject is posited by such demonstration as uses reductio ad impossibile, and then not always universally, but so far as it is requisite; within the limits, that is, of the genus — the genus, I mean (as I have already explained), to which the man of science applies his demonstrations.
Ἐπικοινωνοῦσι δὲ πᾶσαι αἱ ἐπιστῆμαι ἀλλήλαις κατὰ τὰ κοινά (κοινὰ δὲ λέγω οἷς χρῶνται ὡς ἐκ τούτων ἀποδεικνύντες, ἀλλ' οὐ περὶ ὧν δεικνύουσιν οὐδ' ὃ δεικνύουσιν), a26. In virtue of the common elements of demonstration — I mean the common axioms which are used as premisses of demonstration, not the subjects nor the attributes demonstrated as belonging to them — all the sciences have communion with one another,
καὶ ἡ διαλεκτικὴ πάσαις, καὶ εἴ τις καθόλου πειρῷτο δεικνύναι τὰ κοινά, οἷον ὅτι ἅπαν φάναι ἢ ἀποφάναι, ἢ ὅτι ἴσα ἀπὸ ἴσων, ἢ τῶν τοιούτων ἄττα. ἡ δὲ διαλεκτικὴ οὐκ ἔστιν οὕτως ὡρισμένων τινῶν, οὐδὲ γένους τινὸς ἑνός. οὐ γὰρ ἂν ἠρώταο ἀποδεικνύντα γὰρ οὐκ ἔστιν ἐρωτᾶν διὰ τὸ τῶν ἀντικειμένων ὄντων μὴ δείκνυσθαι τὸ αὐτό. δέδεικται δὲ τοῦτο ἐν τοῖς περὶ συλλογισμοῦ. a29. and in communion with them all is dialectic and any science which might attempt a universal proof of axioms such as the law of excluded middle, the law that the subtraction of equals from equals leaves equal remainders, or other axioms of the same kind. Dialectic has no definite sphere of this kind, not being confined to a single genus. Otherwise its method would not be interrogative; for the interrogative method is barred to the demonstrator, who cannot use the opposite facts to prove the same nexus. This was shown in my work on the syllogism.
Postquam determinavit philosophus de principiis propriis et communibus, hic ostendit qualiter demonstrativae scientiae se habeant ad principia communia et propria. Et dividitur in duas partes: in prima, ostendit qualiter se habeant demonstrativae scientiae circa communia: in secunda, qualiter se habeant circa propria; ibi: si autem idem est interrogatio et cetera. Circa primum duo facit: primo, ostendit qualiter se habeant demonstrativae scientiae circa prima principia inter communia; secundo, quomodo se habeant communiter circa omnia principia communia; ibi: communicant autem omnes scientiae et cetera. Circa primum duo facit: primo, ostendit quomodo se habeant demonstrativae scientiae circa hoc principium, quod, non contingit simul affirmare et negare; secundo, quomodo se habeant circa istud principium, de quolibet est affirmatio vel negatio vera; ibi: omne autem affirmare et cetera. Haec enim duo principia sunt omnium prima, ut probatur in IV metaphysicae. After determining about proper and common principles, the Philosopher now shows how the demonstrative sciences behave in regard to the common and to the proper principles. His treatment falls into two parts. In the first he shows how the demonstrative sciences are related to common principles. In the second he shows how they are related to proper principles (77a36) [L. 21]. Concerning the first he does two things. First, he shows how the demonstrative sciences are related to the first of the common principles. Secondly, how they are related generally to all the common principles (77a29). Concerning the first he does two things. First, he shows how the demonstrative sciences behave relative to the principle that one should not affirm and deny the same thing. Secondly, how they behave relative to the principle that there is either true affirmation or true negation of each thing, for, as is proved in Metaphysics IV, these two principles are the first of all principles (77a22).
Dicit ergo primo quod nulla demonstratio accipit hoc principium, quod, non contingit simul affirmare et negare. Si enim aliqua demonstratio eo uteretur ad ostendendum aliquam conclusionem, oporteret quod sic eo uteretur, quod acciperet primum, idest maiorem extremitatem, affirmari de medio et non negari. Quia si acciperet affirmationem et negationem ex parte medii, nihil differret utrum sic vel sic esset; et eadem ratio est de tertio, idest de minori extremitate per comparationem ad medium. Verbi gratia: sit animal primum, homo medium, et Callias tertium. Si quis vellet uti praedicto principio in demonstratione, oporteret sic arguere: He says therefore first (77a10) that no demonstration makes use of the principle that one does not affirm and deny at one and the same time. For if a demonstration were to use it to show some conclusion, it would have to use it in such a way as to assert that the first, i.e., the major, extreme is affirmed of the middle and not denied; because if it were to admit affirmation and negation on the part of the middle, it would make no difference whether it was this way or that. And the same reasoning holds for the third, i.e., minor, extreme in relation to the middle. For example, take “animal” as the first, “man” as the middle’ and “Callias” as the third. If someone wished to use this principle in a demonstration, he would have to argue in the following way:
omnis homo est animal et non est non animal;
Callias est homo;
ergo Callias est animal et non est non animal.
Every man is an animal and is not a non-animal;
But Callias is a man:
Therefore, Callias is an animal and is not a non-animal.
Cum vero dicat, omnis homo est animal, nihil differt, utrum etiam haec sit vera, non homo est animal, vel non sit vera. Et similiter in conclusione non differt, ex quo Callias est animal, utrum non Callias sit animal, vel non animal. For when we say that every man is an animal, it matters not whether it is also true that a non-man is an animal, or whether it is not true. Similarly, in the conclusion it matters not, Callias being an animal, whether nonCallias is an animal or not.
Et huius causa est, quia primum non oportet dici de solo medio, sed potest etiam dici de quodam alio, quod est diversum a medio, quod significatur per negationem medii; propter hoc quod primum dicitur de pluribus quandoque quam medium, sicut animal de pluribus quam homo; unde dicitur de equo, qui est non homo. Unde si accipiatur medium idem et non idem, idest, si accipiatur medium affirmativum et negativum, ut cum dico, homo et non homo est animal, nihil facit ad conclusionem. Cum autem accipitur affirmatio et negatio ex parte maioris extremitatis, differt quidem quantum ad conclusionem et etiam quantum ad veritatem praemissarum. Si enim homo esset non animal, non esset verum quod homo est animal, neque sequeretur quod Callias esset animal. Tamen nihil plus certificatur cum dicitur, homo est animal et non est non animal, quam cum dicitur solum, homo est animal. Idem enim intelligitur per utrumque. Et sic manifestum est quod demonstrationes non utuntur hoc principio, scilicet quod affirmatio et negatio non sint simul vera, neque ex parte praedicati, neque ex parte subiecti. Now the reason for this is that the first is not limited to being said only *of the middle but can also be said of something else diverse from the middle and described by a negation of the riliddle (since the first is sometimes said of things other than the middle, as “animal” is said of things other than man; for it is said of horse, which is not man). Hence if the middle is taken the same and not the same, i.e., if an affirmative and a negative middle be taken, as when I say, “man” and “non-man” are animals, it contributes nothing to the conclusion. However, if the affirmation and negation are taken on the part of the major extreme, it does make a difference both as to the conclusion and as to the truth of the premises. For if man were not an animal, it would not be true that man is an animal, nor would it follow that Callias is an animal. Yet nothing more is verified by stating that man is an animal and is not a non-animal than by merely stating that man is an animal, for the same thing is conveyed by each. And thus it is clear that demonstrations do not use the principle that affirmation and negation are not simultaneously true, either on the part of the predicate or on the part of the subject.
Deinde, cum dicit: omne autem affirmare etc., ostendit quomodo demonstrativae scientiae utantur hoc principio, de quolibet est affirmatio vel negatio vera. Et dicit quod hoc principium accipit demonstratio, quae est ad impossibile. In hac enim demonstratione probatur aliquid esse verum per hoc quod eius oppositum est falsum. Quod nequaquam contingeret, si possibile esset utrumque oppositorum esse falsum. Then (77a22) he shows how demonstrative sciences use the principle that “of anything there is either true affirmation or negation.” And he says that this principle is utilized in a demonstration leading to the impossible. For in this demonstration something is proved to be true by the fact that its opposite is false. (This of course would never happen, if it were possible for the two opposites to be false).
Non tamen semper utitur praedicta demonstratio hoc principio, quia quandoque illud oppositum quod ostenditur esse falsum, non est negatio, sed contrarium immediatum. Sicut si ostenderetur aliquem numerum esse parem per hoc quod falsum est ipsum esse imparem ducendo ad impossibile. Neque etiam utitur hoc principio universaliter, idest in sua universalitate, sub his terminis, ens et non ens, sed quantum sufficiens est in genere aliquo. Et dico de illo genere, circa quod sunt demonstrationes. Sicut si in geometria accipiatur rectum et non rectum; ut cum ostenditur aliqua linea esse recta per hoc quod est falsum eam esse non rectam ducendo ad impossibile. Nevertheless such a demonstration does not always employ this principle, for sometimes that opposite which is shown to be false is not a negation but an immediate contrary. For example, if a number were shown to be even on the ground that its opposite, namely, “it is odd,” is false, and this were done by leading to the impossible. Neither does it use this principle universally, i.e., in its universality, namely, under the terms “being” and “non-being,” but only so far as is sufficient for a genus or so far as it is narrowed to a generic subject. And I mean the first genus involved in the demonstration. For example, in leading to the impossible in geometry the terms would be “straight” and “non-straight” in the case where it is shown that some line is straight on the ground that it is false to state that it is not straight.
Deinde, cum dicit: communicant autem etc., ostendit qualiter demonstrativae scientiae se habeant communiter ad omnia principia communia. Et circa hoc duo facit. Primo, dicit quod omnes scientiae in communibus principiis communicant hoc modo, quod omnes utuntur eis, sicut ex quibus demonstrant, quod est uti eis ut principiis: sed non utuntur eis, ut de quibus aliquid demonstrant, ut de subiectis, neque sicut quod demonstrant, quasi conclusionibus. Then (77a26) he shows how the sciences as a community function relative to all common principles. In regard to this he does two things. First, he says that all the sciences share alike in the common principles in the sense that they all use them as items from which they demonstrate-which is to use them as principles. But they do not use them as things about which they demonstrate something, i.e., as subjects, or as things which they demonstrate, i.e., as conclusions.
Secundo, ibi: et dialectica de omnibus etc., ostendit quod quaedam scientiae utuntur principiis communibus alio modo quam dictum est. Dialectica enim est de communibus; et aliqua alia scientia est etiam de communibus, scilicet philosophia prima, cuius subiectum est ens et considerat ea quae consequuntur ens, ut proprias passiones entis. Sciendum tamen est quod alia ratione dialectica est de communibus et logica et philosophia prima. Philosophia enim prima est de communibus, quia eius consideratio est circa ipsas res communes, scilicet circa ens et partes et passiones entis. Et quia circa omnia quae in rebus sunt habet negotiari ratio, logica autem est de operationibus rationis; logica etiam erit de his, quae communia sunt omnibus, idest de intentionibus rationis, quae ad omnes res se habent. Non autem ita, quod logica sit de ipsis rebus communibus, sicut de subiectis. Considerat enim logica, sicut subiecta, syllogismum, enunciationem, praedicatum, aut aliquid huiusmodi. Secondly (77a29), he shows that certain sciences employ the common principles in a manner other than has been described. For dialectics is concerned with common things, and a certain other science is concerned with common things, namely, first philosophy, whose subject is being and which considers the things which follow upon being as the proper attributes of being. Yet it should be noted that dialectics is concerned with common things under an aspect different from logic and first philosophy. For first philosophy is concerned with common things because its consideration is focused on those common things, namely, on being and on the parts and attributes of being. But because reason occupies itself with all things that are, and logic studies the operations of reason, logic will also be concerned with matters common to all things, i.e., with reason’s intentionalities which bear on all things, but not in such a way that logic has these common things as its subject-for logic considers as its subject the syllogism, enunciation, predication, and things of that type.
Pars autem logicae, quae demonstrativa est, etsi circa communes intentiones versetur docendo, tamen usu demonstrativae scientiae non est in procedendo ex his communibus intentionibus ad aliquid ostendendum de rebus, quae sunt subiecta aliarum scientiarum. Sed hoc dialectica facit, quia ex communibus intentionibus procedit arguendo dialecticus ad ea quae sunt aliarum scientiarum, sive sint propria sive communia, maxime tamen ad communia. Sicut argumentatur quod odium est in concupiscibili, in qua est amor, ex hoc quod contraria sunt circa idem. Est ergo dialectica de communibus non solum quia pertractat intentiones communes rationis, quod est commune toti logicae, sed etiam quia circa communia rerum argumentatur. Quaecunque autem scientia argumentatur circa communia rerum, oportet quod argumentetur circa principia communia, quia veritas principiorum communium est manifesta ex cognitione terminorum communium, ut entis et non entis, totius et partis, et similium. But although the part of logic which is demonstrative is engaged in teaching about common intentionalities, the use of a demonstrative science does not consist in proceeding from common intentionalities to show anything about the things which are the subjects of the other sciences. But dialectics does this, for it goes from common intentions and argues to things that pertain to other sciences, whether they be proper or common things, but mainly common things. Thus, it argues that hatred is in the concupiscible appetite, because love is, on the ground that contraries are concerned with a same thing. Consequently, dialectics is concerned with common things not only because it treats concerning the common intentionalities of reason, which is common to all logic, but also because it argues about common characteristics of things. But any science that argues about the common characteristics of things must argue about the common principles, because the truth of the common principles is made manifest from the knowledge of common terms, as “being” and “non-being,” “whole” and “part” and the like.
Dicit autem signanter: et si aliqua scientia tentet monstrare communia, quia philosophia prima non demonstrat principia communia, sunt enim indemonstrabilia simpliciter; sed aliqui errantes tentaverunt ea demonstrare, ut patet in IV metaphysicae. Vel etiam quia, etsi non possunt demonstrari simpliciter, tamen philosophus primus tentat ea monstrare eo modo, quo est possibile, scilicet contradicendo negantibus ea, per ea quae oportet ab eis concedi, non per ea, quae sunt magis nota. It is significant that he says, “and any science which might attempt,” because first philosophy does not demonstrate the common principles, since they are absolutely indemonstrable, although some have unwittingly attempted to demonstrate them, as is stated in Metaphysics IV. Or else, because even though they cannot, strictly speaking, be demonstrated, the first philosopher attempts to uphold them in a way that is possible, namely, by contradicting those who deny them, appealing to things that must be conceded by them, though not to things which are more known.
Sciendum est etiam quod primus philosophus non solum hoc modo demonstrat ea, sed etiam monstrat aliquid de eis, sicut de subiectis; sicut quod, impossibile est mente concipere opposita eorum, ut patet in IV metaphysicae. Cum ergo disputet circa haec principia et philosophus primus et dialecticus, tamen aliter et aliter. Dialecticus enim non procedit ex aliquibus principiis demonstrativis, neque assumit alteram partem contradictionis tantum, sed se habet ad utramque (contingit enim utramque quandoque vel probabilem esse, vel ex probabilibus ostendi, quae accipit dialecticus). Et propter hoc interrogat. Demonstrator autem non interrogat, quia non se habet ad opposita. Et haec differentia utriusque posita est in his, quae de syllogismo sunt, idest in libro priorum. Philosophia ergo prima procedit circa communia per modum demonstrationis, et non per modum dialecticae disputationis. It should also be noted that the first philosopher demonstrates them not only in this way, but also shows something about them as about subjects: for example, that “it is impossible for the mind to think their opposites,” as is clear from Metaphysics IV. Therefore’ although both the first philosopher and the dialectician debate about these first principles, yet one does so in one way and the other in another way. For the dialectician neither proceeds from demonstrative principles nor takes only one side of a contradiction but is open to both. For each side might happen to be probable or be upheld by probable statements, which the dialectician utilizes: and that is why he asks [his questions in terms of two alternatives]. But the demonstrator does not ask [in that way], because he is not open to opposites. And this is the difference between the two, as was laid down in the treatment of the syllogism, namely, in the beginning of the Prior Analytics. Therefore, first philosophy in treating common principles proceeds after the manner of a demonstration and not after the manner of a dialectical disputation.

Lectio 21
Caput 12
Εἰ δὲ τὸ αὐτό ἐστιν ἐρώτημα συλλογιστικὸν καὶ πρότασις ἀντιφάσεως, προτάσεις δὲ καθ' ἑκάστην ἐπιστήμην ἐξ ὧν ὁ συλλογισμὸς ὁ καθ' ἑκάστην, εἴη ἄν τι ἐρώτημα ἐπιστημονικόν, ἐξ ὧν ὁ καθ' ἑκάστην οἰκεῖος γίνεται συλλογισμός. δῆλον ἄρα ὅτι οὐ πᾶν ἐρώτημα γεωμετρικὸν ἂν εἴη οὐδ' ἰατρικόν, ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων· a36. If a syllogistic question is equivalent to a proposition embodying one of the two sides of a contradiction, and if each science has its peculiar propositions from which its peculiar conclusion is developed, then there is such a thing as a distinctively scientific question, and it is the interrogative form of the premisses from which the 'appropriate' conclusion of each science is developed. Hence it is clear that not every question will be relevant to geometry, nor to medicine, nor to any other science:
ἀλλ' ἐξ (77b.) ὧν δείκνυταί τι περὶ ὧν ἡ γεωμετρία ἐστίν, ἢ ἃ ἐκ τῶν αὐτῶν δείκνυται τῇ γεωμετρίᾳ, ὥσπερ τὰ ὀπτικά. ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων. a42. only those questions will be geometrical which form premisses for the proof of the theorems of geometry or of any other science, such as optics, which uses the same basic truths as geometry. Of the other sciences the like is true.
καὶ περὶ μὲν τούτων καὶ λόγον ὑφεκτέον ἐκ τῶν γεωμετρικῶν ἀρχῶν καὶ συμπερασμάτων, περὶ δὲ τῶν ἀρχῶν λόγον οὐχ ὑφεκτέον τῷ γεωμέτρῃ ᾗ γεωμέτρης· ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων ἐπιστημῶν. b2. Of these questions the geometer is bound to give his account, using the basic truths of geometry in conjunction with his previous conclusions; of the basic truths the geometer, as such, is not bound to give any account. The like is true of the other sciences.
οὔτε πᾶν ἄρα ἕκαστον ἐπιστήμονα ἐρώτημα ἐρωτητέον, οὔθ' ἅπαν τὸ ἐρωτώμενον ἀποκριτέον περὶ ἑκάστου, ἀλλὰ τὰ κατὰ τὴν ἐπιστήμην διορισθέντα. b6. There is a limit, then, to the questions which we may put to each man of science; nor is each man of science bound to answer all inquiries on each several subject, but only such as fall within the defined field of his own science.
εἰ δὲ διαλέξεται γεωμέτρῃ ᾗ γεωμέτρης οὕτως, φανερὸν ὅτι καὶ καλῶς, ἐὰν ἐκ τούτων τι δεικνύῃ· εἰ δὲ μή, οὐ καλῶς. δῆλον δ' ὅτι οὐδ' ἐλέγχει γεωμέτρην ἀλλ' ἢ κατὰ συμβεβηκός· ὥστ' οὐκ ἂν εἴη ἐν ἀγεωμετρήτοις περὶ γεωμετρίας διαλεκτέον· λήσει γὰρ ὁ φαύλως διαλεγόμενος. ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων ἔχει ἐπιστημῶν. b8. If, then, in controversy with a geometer qua geometer the disputant confines himself to geometry and proves anything from geometrical premisses, he is clearly to be applauded; if he goes outside these he will be at fault, and obviously cannot even refute the geometer except accidentally. One should therefore not discuss geometry among those who are not geometers, for in such a company an unsound argument will pass unnoticed. This is correspondingly true in the other sciences.
Postquam philosophus ostendit quomodo scientiae demonstrativae se habeant circa principia communia, hic ostendit quomodo se habeant circa propria. Et dividitur in duas partes: in prima, ostendit quod in qualibet scientia sunt propriae interrogationes, responsiones et disputationes; in secunda, ostendit quomodo in qualibet scientia sunt propriae deceptiones; ibi: quoniam autem sunt geometricae et cetera. Circa primum duo facit: primo, ostendit quod in qualibet scientia sunt propriae interrogationes; secundo, ostendit quod in qualibet scientia sunt propriae responsiones et disputationes; ibi: neque omnem interrogationem utique et cetera. Circa primum duo facit: primo, ostendit quod in qualibet scientia sunt interrogationes propriae; secundo, quae sunt illae; ibi: sed ex quibus aut demonstratur et cetera. After showing how demonstrative sciences function in relation to common principles, the Philosopher now shows how they employ proper principles. And his treatment falls into two parts. In the first he shows that in each science there are questions, responses and disputations peculiar to each. In the second he shows how in each science there are deceptions, peculiar to each (77b16) [L. 22]. Concerning the first he does two things. First, he, shows that in each science there are its own questions. Secondly, that each science employs the responses and disputations proper to it (77b6). Concerning the first he does two things. First, he shows that in each science there are questions peculiar to each. Secondly, what these are (77a42).
Primum sic ostendit. Idem est secundum substantiam interrogatio syllogistica et propositio, quae accipit alteram partem contradictionis, licet in modo proferendi differant (hoc enim, quod ad interrogationem respondetur, assumitur ut propositio in aliquo syllogismo); in unaquaque autem scientia sunt propriae propositiones, ex quibus fit syllogismus: ostensum est enim quod quaelibet scientia ex propriis procedit; ergo in qualibet scientia est propria interrogatio. Non ergo quaelibet interrogatio est geometrica, vel medicinalis; et sic de aliis scientiis. He shows the first (77a36) in the following way: A syllogistic question and a proposition which takes one definite side of a contradiction are the same as to content, but they have not the same function. For the answer which one gives to the question is made to serve as a proposition in some syllogism. But in each of the sciences, the propositions from which a syllogism is formed, are peculiar to it, for it has been shown that every science proceeds from things proper to it. Therefore, in each science there are questions peculiar to it. Hence not any random question is pertinent to geometry or to medicine or to some one of the other sciences.
Sciendum tamen est quod interrogatio aliter est in scientiis demonstrativis et aliter est in dialectica. In dialectica enim non solum interrogatur de conclusione, sed etiam de praemissis: de quibus demonstrator non interrogat, sed ea sumit quasi per se nota, vel per talia principia probata; sed interrogat tantum de conclusione. Sed cum eam demonstraverit, utitur ea, ut propositione, ad aliam conclusionem demonstrandam. However, it should be noted that questioning occurs one way in demonstrative sciences and another way in dialectics. For in dialectics not only the conclusion but also the premises are open to question; but in demonstrative sciences the demonstrator takes premises as per se known or proved by such principles. Hence, he asks only about the conclusion. And when he has demonstrated it, he uses it as a proposition to demonstrate some other conclusion.
Deinde cum dicit: sed ex quibus etc., ostendit quae interrogationes sunt propriae unicuique scientiae. Then (77a42) he explains which questions are peculiar to each science. First, insofar as they are taken as propositions from which the demonstrator proceeds. Secondly, when taken as conclusions (77b2).
Et primo, in quantum assumuntur ut propositiones, ex quibus demonstrator procedit; secundo, in quantum sumuntur ut conclusiones; ibi: et de his quidem et cetera. Dicit ergo primo quod interrogationes geometricae sunt ex quibus demonstratur aliquid circa illa, de quibus est geometria, aut circa illa, quae demonstrantur ex principiis eiusdem geometriae; sicut illa, ex quibus demonstratur aliquid in speculativa scientia, idest in perspectiva, quae procedit ex principiis geometriae. Et quod dictum est de geometria, intelligendum est de aliis scientiis: quia scilicet propositio, vel interrogatio dicitur proprie alicuius scientiae, ex qua demonstratur vel in ipsa scientia, vel in scientia ei subalternata. He says therefore first (77a42) that geometric questions are ones from which something is demonstrated pertaining to matters of geometry or pertaining to matters demonstrated from the principles of geometry—for example, the ones from which something is demonstrated in optical science, i.e., perspective, which proceeds from the principles of geometry. And what is said of geometry applies to other sciences, namely, that a proposition or question is peculiar to a science, if it is one from which a demonstration in that science or in a subalternate science proceeds.
Deinde cum dicit: et de his quidem rationem etc., notificat geometricam interrogationem, prout est conclusio, dicens quod de interrogationibus geometricis ponenda est ratio, demonstrando scilicet veritatem ipsarum ex principiis geometricis et conclusionibus, quae per illa principia demonstrantur. Non enim cuiuslibet demonstrationis geometricae ratio redditur ex primis geometriae principiis, sed interdum ex his quae per prima principia sunt conclusa. Interrogationum autem, quae semper sunt conclusiones in demonstrativis scientiis, ratio reddi potest in eisdem, sed principiorum ratio non potest reddi a geometra, secundum quod geometra est. Et similiter est in aliis scientiis. Nulla enim scientia probat sua principia, secundum quod ostensum est supra. Dicit autem, secundum quod geometra est, quia contingit in aliqua scientia probari principia illius scientiae, in quantum illa scientia assumit principia alterius scientiae; sicut geometra probat sua principia secundum quod assumit formam philosophi primi, idest metaphysici. Then (77b2) he analyzes geometric questions insofar as they are conclusions, saying that solutions to geometric questions must be proved by demonstrating their truth from geometric principles and from conclusions already demonstrated by those principles. (For sometimes the proofs of certain geometric demonstrations are not drawn from the first principles of geometry, but from matters concluded from those first principles). Hence when it is a matter of questions which are always conclusions in demonstrative sciences, the proof can be obtained in the science itself, but the proof of the principles cannot be drawn from geometry qua geometry. The same also applies to other sciences. For no science proves its own principles, as we have explained above. And he says, “from geometry qua geometry,” because it may happen that a science proves its own principles, insofar as that science assumes the principles of another science, as a geometer proves his own principles insofar as he assumes the role of first philosopher, i.e., of metaphysician.
Deinde cum dicit: neque omnem interrogationem etc., ostendit quod in qualibet scientia sunt propriae responsiones et disputationes. Et primo quod sint propriae responsiones; secundo, quod sint propriae disputationes; ibi: si autem disputat et cetera. Then (77b6) he shows that each has its own responses and disputations. First, that each has its own responses. Secondly, its own disputations (77b8).
Dicit ergo primo, quod ex praedictis patet quod non contingit unumquemque scientem de qualibet quaestione interrogare. Unde etiam patet quod non contingit de quolibet interrogato respondere: sed solum de his quae sunt secundum propriam scientiam: eo quod ad eamdem scientiam pertinet interrogatio et responsio. He says therefore (77b6) that from the foregoing it is clear that each scientific knower does not ask just any question whatever. Hence it is also clear that he will not answer just any random question but only those which conform to his own science on the ground that a question and its answer pertain to the same science.
Et quia ex interrogatione et responsione fit disputatio, consequenter ostendit quod in qualibet scientia est propria disputatio, dicens quod si disputet geometra cum geometra, secundum quod geometra, idest de his quae ad geometriam pertinent, manifestum est quod bene procedit disputatio, si tamen non solum fiat disputatio de eo quod est geometriae, sed etiam ex principiis geometricis procedatur. Si vero non sic fiat disputatio in geometria, non bene disputatur. Si enim aliquis disputet cum geometra non de geometricis, manifestum est quod non arguit, idest non convincit, nisi per accidens: puta si sit disputatio de musica et contingat geometram per accidens esse musicum. Unde manifestum est quod non est in non geometricis de geometria disputandum, quia non poterit iudicari per principia illius scientiae, utrum bene disputetur vel male. Et similiter se habet in aliis scientiis.  Then (77b8) because disputations are concerned with a question and its answer, he shows that there are disputations - peculiar to each science, saying that if geometer disputes with geometer precisely as geometer, i.e., in matters pertaining to geometry, then obviously the disputation goes well so long as the disputation not only concerns a point of geometry but proceeds from the principles of geometry. But it does not go well, if the disputation in geometry does not proceed along these lines. For if someone disputes with a geometer in matters alien to geometry, he does not argue, i.e., does not convince, except accidentally: for example, if the dispute concerns music, and the geometer accidentally happens to be a musician. Hence it is clear that one should not dispute in geometry about matters not geometric, because it will not be possible to judge by the principles of that science whether the dispute went favorably or unfavorably: and the same holds for other sciences.

Lectio 22
Caput 12 cont.
Ἐπεὶ δ' ἔστι γεωμετρικὰ ἐρωτήματα, ἆρ' ἔστι καὶ ἀγεωμέτρητα; b16. Since there are 'geometrical' questions, does it follow that there are also distinctively 'ungeometrical' questions?
καὶ παρ' ἑκάστην ἐπιστήμην τὰ κατὰ τὴν ἄγνοιαν τὴν ποίαν γεωμετρικά ἐστιν; b17. Further, in each special science — geometry for instance — what kind of error is it that may vitiate questions, and yet not exclude them from that science?
καὶ πότερον ὁ κατὰ τὴν ἄγνοιαν συλλογισμὸς ὁ ἐκ τῶν ἀντικειμένων συλλογισμός, b19. Again, is the erroneous conclusion one constructed from premisses opposite to the true premisses, or is it formal fallacy though drawn from geometrical premisses?
ἢ ὁ παραλογισμός, κατὰ γεωμετρίαν δέ, ἢ <ὁ> ἐξ ἄλλης τέχνης, οἷον τὸ μουσικόν ἐστιν ἐρώτημα ἀγεωμέτρητον περὶ γεωμετρίας, b21. Or, perhaps, the erroneous conclusion is due to the drawing of premisses from another science; e.g. in a geometrical controversy a musical question is distinctively ungeometrical,
τὸ δὲ τὰς παραλλήλους συμπίπτειν οἴεσθαι γεωμετρικόν πως καὶ ἀγεωμέτρητον ἄλλον τρόπον; διττὸν γὰρ τοῦτο, ὥσπερ τὸ ἄρρυθμον, καὶ τὸ μὲν ἕτερον ἀγεωμέτρητον τῷ μὴ ἔχειν [ὥσπερ τὸ ἄρρυθμον], τὸ δ' ἕτερον τῷ φαύλως ἔχειν· καὶ ἡ ἄγνοια αὕτη καὶ ἡ ἐκ τῶν τοιούτων ἀρχῶν ἐναντία. b23. whereas the notion that parallels meet is in one sense geometrical, being ungeometrical in a different fashion: the reason being that 'ungeometrical', like 'unrhythmical', is equivocal, meaning in the one case not geometry at all, in the other bad geometry? It is this error, i.e. error based on premisses of this kind — 'of' the science but false — that is the contrary of science.
ἐν δὲ τοῖς μαθήμασιν οὐκ ἔστιν ὁμοίως ὁ παραλογισμός, ὅτι τὸ μέσον ἐστὶν ἀεὶ τὸ διττόν· κατά τε γὰρ τούτου παντός, καὶ τοῦτο πάλιν κατ' ἄλλου λέγεται παντός (τὸ δὲ κατηγορούμενον οὐ λέγεται πᾶν), ταῦτα δ' ἔστιν οἷον ὁρᾶν τῇ νοήσει, ἐν δὲ τοῖς λόγοις λανθάνει. ἆρα πᾶς κύκλος σχῆμα; ἂν δὲ γράψῃ, δῆλον. τί δέ; τὰ ἔπη κύκλος; φανερὸν ὅτι οὐκ ἔστιν. b27. In mathematics the formal fallacy is not so common, because it is the middle term in which the ambiguity lies, since the major is predicated of the whole of the middle and the middle of the whole of the minor (the predicate of course never has the prefix 'all'); and in mathematics one can, so to speak, see these middle terms with an intellectual vision, while in dialectic the ambiguity may escape detection. E.g. 'Is every circle a figure?' A diagram shows that this is so, but the minor premiss 'Are epics circles?' is shown by the diagram to be false.
Οὐ δεῖ δ' ἔνστασιν εἰς αὐτὸ φέρειν, ἂν ᾖ ἡ πρότασις ἐπακτική. ὥσπερ γὰρ οὐδὲ πρότασίς ἐστιν ἣ μὴ ἔστιν ἐπὶ πλειόνων (οὐ γὰρ ἔσται ἐπὶ πάντων, ἐκ τῶν καθόλου δ' ὁ συλλογισμός), δῆλον ὅτι οὐδ' ἔνστασις. αἱ αὐταὶ γὰρ προτάσεις καὶ ἐνστάσεις· ἣν γὰρ φέρει ἔνστασιν, αὕτη γένοιτ' ἂν πρότασις ἢ ἀποδεικτικὴ ἢ διαλεκτική. b34. If a proof has an inductive minor premiss, one should not bring an 'objection' against it. For since every premiss must be applicable to a number of cases (otherwise it will not be true in every instance, which, since the syllogism proceeds from universals, it must be), then assuredly the same is true of an 'objection'; since premisses and 'objections' are so far the same that anything which can be validly advanced as an 'objection' must be such that it could take the form of a premiss, either demonstrative or dialectical.
Συμβαίνει δ' ἐνίους ἀσυλλογίστως λέγειν διὰ τὸ λαμβάνειν ἀμφοτέροις τὰ ἑπόμενα, οἷον καὶ ὁ Καινεὺς ποιεῖ, (78a.) ὅτι τὸ πῦρ ἐν τῇ πολλαπλασίᾳ ἀναλογίᾳ· καὶ γὰρ τὸ πῦρ ταχὺ γεννᾶται, ὥς φησι, καὶ αὕτη ἡ ἀναλογία. οὕτω δ' οὐκ ἔστι συλλογισμός· ἀλλ' εἰ τῇ ταχίστῃ ἀναλογίᾳ ἕπεται ἡ πολλαπλάσιος καὶ τῷ πυρὶ ἡ ταχίστη ἐν τῇ κινήσει ἀναλογία. b40. On the other hand, arguments formally illogical do sometimes occur through taking as middles mere attributes of the major and minor terms. An instance of this is Caeneus' proof that fire increases in geometrical proportion: 'Fire', he argues, 'increases rapidly, and so does geometrical proportion'. There is no syllogism so, but there is a syllogism if the most rapidly increasing proportion is geometrical and the most rapidly increasing proportion is attributable to fire in its motion.
ἐνίοτε μὲν οὖν οὐκ ἐνδέχεται συλλογίσασθαι ἐκ τῶν εἰλημμένων, ὁτὲ δ' ἐνδέχεται, ἀλλ' οὐχ ὁρᾶται. a5. Sometimes, no doubt, it is impossible to reason from premisses predicating mere attributes: but sometimes it is possible, though the possibility is overlooked.
Εἰ δ' ἦν ἀδύνατον ἐκ ψεύδους ἀληθὲς δεῖξαι, ῥᾴδιον ἂν ἦν τὸ ἀναλύειν· ἀντέστρεφε γὰρ ἂν ἐξ ἀνάγκης. ἔστω γὰρ τὸ Α ὄν· τούτου δ' ὄντος ταδὶ ἔστιν, ἃ οἶδα ὅτι ἔστιν, οἷον τὸ Β. ἐκ τούτων ἄρα δείξω ὅτι ἔστιν ἐκεῖνο. a8. If false premisses could never give true conclusions 'resolution' would be easy, for premisses and conclusion would in that case inevitably reciprocate. I might then argue thus: let A be an existing fact; let the existence of A imply such and such facts actually known to me to exist, which we may call B. I can now, since they reciprocate, infer A from B.
ἀντιστρέφει δὲ μᾶλλον τὰ ἐν τοῖς μαθήμασιν, ὅτι οὐδὲν συμβεβηκὸς λαμβάνουσιν (ἀλλὰ καὶ τούτῳ διαφέρουσι τῶν ἐν τοῖς διαλόγοις) ἀλλ' ὁρισμούς. a10. Reciprocation of premisses and conclusion is more frequent in mathematics, because mathematics takes definitions, but never an accident, for its premisses — a second characteristic distinguishing mathematical reasoning from dialectical disputations.
Αὔξεται δ' οὐ διὰ τῶν μέσων, ἀλλὰ τῷ προσλαμβάνειν, οἷον τὸ Α τοῦ Β, τοῦτο δὲ τοῦ Γ, πάλιν τοῦτο τοῦ Δ, καὶ τοῦτ' εἰς ἄπειρον· καὶ εἰς τὸ πλάγιον, οἷον τὸ Α καὶ κατὰ τοῦ Γ καὶ κατὰ τοῦ Ε, οἷον ἔστιν ἀριθμὸς ποσὸς ἢ καὶ ἄπειρος τοῦτο ἐφ' ᾧ Α, ὁ περιττὸς ἀριθμὸς ποσὸς ἐφ' οὗ Β, ἀριθμὸς περιττὸς ἐφ' οὗ Γ· ἔστιν ἄρα τὸ Α κατὰ τοῦ Γ. καὶ ἔστιν ὁ ἄρτιος ποσὸς ἀριθμὸς ἐφ' οὗ Δ, ὁ ἄρτιος ἀριθμὸς ἐφ' οὗ Ε· ἔστιν ἄρα τὸ Α κατὰ τοῦ Ε. a13. A science expands not by the interposition of fresh middle terms, but by the apposition of fresh extreme terms. E.g. A is predicated of B, B of C, C of D, and so indefinitely. Or the expansion may be lateral: e.g. one major A, may be proved of two minors, C and E. Thus let A represent number — a number or number taken indeterminately; B determinate odd number; C any particular odd number. We can then predicate A of C. Next let D represent determinate even number, and E even number. Then A is predicable of E.
Postquam ostendit philosophus quod in qualibet scientia sunt propriae interrogationes, responsiones et disputationes; hic ostendit quod in qualibet scientia sunt propriae deceptiones et ignorantiae. Et dividitur in partes duas: in prima, movet quasdam quaestiones; in secunda, solvit; ibi: secundum geometriam vero et cetera. After showing that each science has its own questions, responses and disputations, the Philosopher shows that each science has its own deceptions and errors. And his treatment is divided into two parts. In the first he raises certain questions. In the second he solves them (77b21).
Ponit ergo primo tres quaestiones, quarum prima est. Cum sint quaedam interrogationes geometricae, ut ostensum est, nonne sunt etiam quaedam non geometricae? Et quod quaeritur de geometria, potest de qualibet alia scientia quaeri. Accordingly, he poses three questions, the first of which (77b16) is this: Since there are geometric questions, as we have shown, are there not also non-geometric ones? And what is asked of geometry can be asked of every other science.
Secundam quaestionem ponit; ibi: et secundum unamquamque etc., quae talis est. Utrum interrogationes quae sunt secundum ignorantiam, quae est in unaquaque scientia, possint dici geometricae, et similiter alicui alii scientiae propriae? Dicuntur autem interrogationes secundum ignorantiam alicuius scientiae, quando interrogatur de his, quae sunt contra veritatem scientiae illius. Then (77b17) he poses the second question, namely: May questions which arise from ignorance bearing on some particular science be called geometric; and likewise for questions proper to any other science? (Questions arising from ignorance bearing on some science are those which ask about matters contrary to the truths of that science).
Tertiam quaestionem ponit; ibi: et utrum secundum ignorantiam etc., quae talis est. In unaquaque quidem scientia accidit decipi per aliquem syllogismum, quem vocat secundum ignorantiam. Contingit autem per aliquem syllogismum deceptionem accidere dupliciter: uno modo, quia peccat in materia, procedens ex falsis; alio modo, quia peccat in forma, non servando debitam figuram et modum. Et est differentia inter hos modos duos: quia ille qui peccat in materia, syllogismus est, cum observentur omnia, quae ad formam syllogismi pertinent. Ille autem qui peccat in forma non est syllogismus, sed paralogismus, idest apparens syllogismus. In dialecticis quidem utroque modo contingit deceptionem fieri. Unde et in I topicorum Aristoteles facit mentionem de litigioso, qui est syllogismus, et de peccante in forma, qui non est syllogismus, sed apparens. Est ergo quaestio, utrum syllogismus ignorantiae, qui fit in scientiis demonstrativis, sit syllogismus ex oppositis scientiae, idest ex falsis procedens, aut paralogismus, scilicet peccans in forma: qui non est syllogismus, sed apparens. Then (77b19) he poses the third question, namely: In each science it is possible to be deceived by a syllogism which he calls “according to ignorance.” But deception through a syllogism can occur in two ways: in one way when it fails as to form, not observing the correct form and mode of a syllogism. In another way, when it fails in matter, proceeding from the false. Now there is a difference between these two ways, because one that fails in matter is still a syllogism, since everything is observed that pertains to the form of a syllogism. But one that fails in form is not even a syllogism, but a paralogism, i.e., an apparent syllogism. In dialectics, deception can occur in both these ways. Hence in Topics I Aristotle speaks of the contentious, which is a syllogism, and of the one defective in form which is not a syllogism but an apparent one. Hence the question is this: Whether or not a syllogism of ignorance which is used in the demonstrative sciences is a syllogism based on matters opposed to the science, i.e., one that proceeds from false premises, or a paralogism, namely, one that fails in form, which is not a syllogism, but an apparent one?
Deinde cum dicit: secundum geometriam etc., solvit praedictas quaestiones: et primo, solvit primam; secundo, secundam; ibi: de geometria autem etc.; tertio, tertiam; ibi: in doctrinis autem et cetera. Then (77b21) he solves these questions. First, he solves the first Secondly, the second (77b23). Thirdly, the third (77b27).
Dicit ergo primo quod interrogatio omnino non geometrica est illa, quae omnino fit ex alia arte, sicut ex musica. Ut si quaeratur in geometria, utrum tonus possit dividi in duo semitonia aequalia; talis interrogatio est omnino non geometrica: quia est ex his, quae nullo modo ad geometriam pertinent. He says therefore first (77b21) that a completely non-geometric question is one which is formed entirely from another art, say, music. For example, if one asks in geometry whether a tone could be divided into two equal semi-tones, such a question would be entirely non-geometric, because it concerns matters which do not pertain at all to geometry.
Deinde cum dicit: de geometria autem etc., solvit secundam quaestionem dicens quod interrogatio de geometria, idest de his quae pertinent ad geometriam, cum interrogatur de aliquo quod est contra veritatem geometriae (sicut si fiat quaestio de hoc quod est parallelas subire, idest lineas aeque distantes concurrere), est quodammodo geometrica et quodammodo non geometrica. Sicut enim arrhythmon, idest quod est sine rhythmo vel sono, dupliciter dicitur, uno modo, quod nullo modo habet sonum, ut lana, alio modo, quod habet pravum sonum, sicut Campana non bene sonans; ita et interrogatio non geometrica dicitur dupliciter. Uno modo, quia est omnino non geometrica, quasi nihil habens geometriae, sicut quaestio de musica proposita. Alio modo, quia prave habet id quod geometriae est; quia videlicet habet contrarium geometricae veritati. Ista ergo interrogatio, quae est de concursu linearum aeque distantium, non est non geometrica primo modo, cum sit de rebus geometricis, sed secundo modo, quia prave habet id quod geometriae est. Et ignorantia haec, scilicet quae est in prave utendo principiis geometriae, contraria est veritati geometriae. Then (77b23) he solves the second question, saying that a question about geometry, i.e., about matters pertinent to geometry, when a person is asked about something which is against the truth of geometry (for example, if the questions concerned parallels meeting, i.e., equidistant lines coming together), would be geometric in one sense and non-geometric in another. For just as arrythmon, i.e., without rhythm or sound, can be taken in two senses: in one sense for that which has no sound at all, as wool, and in another sense for that which does not give a good sound, as a poorly-sounding bell, so a question is called non-geometric in two ways. In one way, because it is completely alien to geometry, as a question about music. In another way, because it mistakenly holds something in the field of geometry, namely, because it holds something contrary to geometric truth. Therefore, the question about the convergence of parallel lines is non-geometric not in the first way, since it touches on a point of geometry, but in the second way, because it is mistaken about some point of geometry. Such ignorance, namely, which consists in wrongly using the principles of geometry is contrary to the truth of geometry.
Deinde cum dicit: in doctrinis autem etc., solvit tertiam quaestionem. Et circa hoc duo facit: primo, ostendit quod in demonstrativis scientiis non sit paralogismus in dictione; secundo, quod non sit paralogismus extra dictionem; ibi: non oportet autem et cetera. Then (77b27) he solves the third question. And he does two things. First, he shows that in demonstrative sciences there is no paralogism in language. Secondly, nor apart from language (77b34).
Cum autem secundum sex locos sophisticos fiat paralogismus in dictione, ex his accipit unum, scilicet paralogismum qui fit secundum aequivocationem, ostendens quod talis paralogismus in scientiis demonstrativis esse non potest: de quo tamen magis videtur. Dicit ergo quod in doctrinis non sit paralogismus, scilicet syllogismus peccans in forma, sicut in dialecticis. In demonstrativis enim oportet medium idem semper esse dupliciter, idest ad duo extrema comparari: quia et de medio maior extremitas universaliter praedicatur, et medium iterum universaliter praedicatur de minori extremitate. Sed quod praedicatur, non dicitur omne, idest signum universale non apponitur ad praedicatum. Now although paralogism. in language occurs in any of six sophistical ways, he takes one of them, namely, the paralogism which proceeds by way of equivocation, and shows that such,a paralogism cannot occur in demonstrative sciences, being easier to detect. He says, therefore, that “formal paralogism,” i.e., a syllogism defective in form, as in dialectics, “does not occur in the disciplines.” For in a demonstrative syllogism the middle must always be the same in two ways, i.e., the same middle must be compared to the two extremes: for the major extreme is predicated universally of the middle, and the middle is predicated universally of the minor extreme, even though when it is predicated, we do not say “every,” i.e., the sign of universality is not applied to the predicate.
In fallacia vero aequivocationis est quidem idem medium secundum vocem, non autem secundum rem. Et ideo quando in voce proponitur, latet, sed si ad sensum demonstretur, non potest ibi esse aliqua deceptio. Sicut hoc nomen circulus aequivoce dicitur de figura et de poemate. In rationibus ergo, idest in argumentationibus, latet, idest deceptio potest accidere; ut si dicatur: omnis circulus est figura; poema Homeri est circulus; ergo poema Homeri est figura. Si vero describatur ad sensum circulus, nulla potest esse deceptio: manifestum enim erit quod carmina non sunt circulus. But in the fallacy of equivocation the middle is the same according to vocal sound but not according to reality. Consequently, it escapes notice when it is proposed orally; but if it is demonstrated to the senses, the deception cannot succeed. For example, the name “circle” is said equivocally of a figure and a poem. Therefore, in reasons, i.e., in argumentations, the ambiguity may go unnoticed, i.e., deception is possible, as for example, if one were to say: “Every circle is a figure; Homer’s poem is a circle: therefore, Homer’s poem is a figure.” But if a circle is drawn for someone to look at, there cannot be deception. For it will be obvious that songs are not circles.
Sicut autem haec deceptio excluditur per hoc quod medium demonstratur ad sensum, ita et in demonstrativis excluditur per hoc quod medium demonstratur ad intellectum. Cum enim aliquid definitur, ita se habet ad intellectum, sicut id quod sensibiliter describitur se habet ad visum. Et ideo dicit quod haec, scilicet definita, in demonstrativis scientiis sunt quae videntur in intellectu. In demonstrationibus autem semper proceditur ex definitionibus. Unde non potest ibi esse deceptio secundum fallaciam aequivocationis: et multo minus secundum alias fallacias in dictione. Now just as in this case the deception is prevented by presenting the middle to the senses, so in demonstrative syllogisms, deception is prevented by the fact that the middle is shown to the intellect. For when something is defined, it is as plain to the intellect as something sensibly drawn is plain to the sight. Hence he says that in demonstrative syllogisms “these,” i.e., things defined, “are seen by the intellect.” But in demonstrations one always proceeds from definitions. Hence, deception through equivocation has no place there, much less through any of the other fallacies of language.
Deinde cum dicit: non oportet autem etc., ostendit quod non potest fieri paralogismus in demonstrativis secundum fallaciam extra dictionem. Et quia huiusmodi paralogismis frequenter obviatur ferendo instantiam, per quam ostenditur defectus in forma syllogizandi; ideo primo ostendit qualiter ferenda esset instantia in demonstrativis; secundo, ostendit quod in eis non potest esse paralogismus secundum fallaciam extra dictionem; ibi: contingit autem quosdam et cetera. Then (77b34) he shows that in demonstrative syllogisms paralogism according to fallacy outside of language cannot occur. And because a paralogism of this kind is frequently challenged by citing an objection, through which the defect in the form of syllogizing is shown: therefore: First, he shows how an objection should be presented in demonstrative matters. Secondly, he shows that in demonstrative matters there cannot be paralogism according to fallacy outside the language (77b40).
Dicit ergo primo, quod non oportet in demonstrativis ferre instantiam in ipsum, idest in aliquem paralogismum, sumendo aliquam propositionem inductivam, idest particularem: nam inductio ex particularibus procedit, sicut syllogismus ex universalibus. Et hoc ideo est, quia in demonstrativis non sumitur propositio, nisi quae est in pluribus: nisi enim sit in pluribus, non erit in omnibus; oportet autem syllogismum demonstrativum ex universalibus procedere. Unde manifestum est quod neque instantia potest esse in demonstrativis, nisi universalis, quia eaedem sunt propositiones et instantiae. Tam enim in dialecticis quam in demonstrativis, illud quod sumitur ut instantia, postea sumitur ut propositio ad syllogizandum contra illum qui proponebat. He says therefore first (77b34) that “one should not bring an objection against it,” i.e., against a paralogism. by citing an inductive, i.e., particular, proposition. (For an induction proceeds from particulars as a syllogism proceeds from universals). The reason for this is that in demonstrative matters no proposition is admitted unless it is verified in the greater number of cases, for if it is not verified in the greater number it will not be in all. But a demonstrative syllogism must proceed from universals. Therefore, it is obvious that in demonstrative matters an objection must be universal, because the propositions and the objections must be the same. For in dialectical and in demonstrative matters that which is taken as an objection is later used as a proposition to syllogize against the one who proposed.
Deinde cum dicit: contingit autem quosdam etc., ostendit quod in demonstrativis non accidit deceptio per paralogismum extra dictionem. Et sicut supra ostenderat quod non est paralogismus in dictione in demonstrativis, ostendendo de uno, scilicet de paralogismo secundum fallaciam aequivocationis; ita hic ostendit quod in demonstrativis non est paralogismus extra dictionem, ostendendo de uno, qui fit secundum fallaciam consequentis. Patet enim quod secundum alias fallacias extra dictionem non potest esse paralogismus in demonstrativis. Neque enim secundum accidens, cum demonstrationes procedant ex his quae sunt per se; neque secundum quid et simpliciter, cum ea quae in demonstrationibus sumuntur, sint universaliter, et semper, et non secundum quid. Then (77b40) he shows that in demonstrative matters deception through paralogisms outside of language does not occur. And just as above he showed that there was no paralogism in language in demonstrative matters by showing it for one, namely, for the paralogism which employs the fallacy of equivocation, so now he shows that in demonstrative matters there is not paralogism outside of language by showing it of the one which relies on fallacy of consequent. For it is obvious that paralogism. in demonstrative matters cannot occur according to the other fallacies outside of language: not the fallacy according to accident, because demonstration proceeds from things that are per se; nor according to qualified and absolute, because the statements used in demonstrations are taken universally and always, and without qualification.
Circa hoc ergo duo facit: primo, ostendit qualiter fiat paralogismus secundum fallaciam consequentis; secundo, quod ex hoc modo non accidit deceptio in demonstrativis; ibi: aliquando quidem et cetera. Therefore, he does two things in regard to his thesis. First, he shows how paralogism according to fallacy of consequent works. Secondly, that deception does not take place in this manner in demonstrative matters (78a5).
Dicit ergo primo quod quosdam contingit non syllogistice dicere, idest non servare formam syllogismi, propter hoc, quod accipiunt utrisque inhaerentia, idest quia accipiunt medium affirmative praedicatum de utroque extremorum; quod est syllogizare in secunda figura ex duabus propositionibus affirmativis; quod facit fallaciam consequentis. Sicut fecit quidam philosophus nomine Caeneus ad ostendendum quod ignis sit in multiplicata analogia, idest quod in maiori quantitate generatur ignis, quam fuerit corpus ex quo generatur: eo quod ignis, cum sit rarissimum corpus, per rarefactionem ex aliis corporibus generatur. Unde oportet quod materia prioris corporis sub maioribus dimensionibus extendatur, formam ignis assumens. Ad hoc autem probandum utebatur tali syllogismo: quod generatur in multiplicata analogia, cito generatur; sed ignis cito generatur; ergo ignis generatur in multiplicata analogia. He says therefore first (77b40) that “illogical arguments do sometimes occur,” i.e., do not observe syllogistic form, “because they admit both inherences,” i.e., they accept a middle predicated affirmatively of each extreme, which is the same as syllogizing in the second figure but employing two affirmative propositions, thus committing the fallacy of consequent. This is what the philosopher Caeneus did in order to show that “fire is in multiple proportion,” i.e., that fire is generated in greater quantity than was the body from which it is generated, on the ground that fire, being the most rarified of bodies, is generated from other bodies through rarefaction. Hence it is required that the matter of the previous body be spread out under larger dimensions when assuming the form of fire. To prove this he used the following syllogism: “Whatever is generated in multiplied proportion is generated quickly; but fire is generated quickly: therefore, fire is generated in multiplied proportion.”
Deinde cum dicit: aliquando quidem igitur etc., ostendit quod per hunc modum syllogizandi non accidit deceptio in demonstrativis scientiis. Et circa hoc duo facit: primo, manifestat quod ex hoc modo syllogizandi non semper accidit deceptio, dicens quod aliquando, secundum praedictum modum arguendi, non contingit syllogizare ex acceptis, quando scilicet termini non sunt convertibiles. Non enim sequitur, si omnis homo est animal, quod quidquid est animal sit homo. Aliquando vero contingit syllogizare, scilicet in terminis convertibilibus. Sicut enim sequitur: si est homo, est animal rationale mortale; ita etiam sequitur e converso quod, si est animal rationale mortale, est homo. Sed tamen non videtur quod sequatur syllogistice, quia non servatur debita forma syllogismi. Then (78a5) he shows that deception arising from this form of reasoning does not occur in demonstrative sciences. In regard to this he does two things. First, he makes it clear that this form of syllogizing does not always end in deception, saying that according to this form of arguing there are cases when one cannot syllogize from the premises, namely, when the terms are not convertible-for it does not follow, if every man is an animal, that every animal is a man. But now and then there are cases when one can syllogize, namely, when the terms are convertible. For just as it follows that if a thing is a man, it is a rational mortal animal, so conversely, if a thing is a rational mortal animal, it is a man. However, it does not seem to follow syllogistically, because the due form of a syllogism is not observed.
Secundo cum dicit: si autem esset impossibile etc., ostendit quod in demonstrativis scientiis contingit praedicto modo syllogizari absque deceptione. Et hoc ostendit tripliciter. Primo sic. Secundum praedictum modum syllogizandi accidit deceptio ex eo, quod non convertitur consequentia, quae putatur converti. In quo non accideret deceptio, si quemadmodum conclusio est vera, ita et praemissae sint verae: tunc enim in convertendo non accidet deceptio. Sicut si dicam de Socrate: Socrates est homo; ergo Socrates est animal; nulla deceptio falsitatis sequitur, sicut si e converso arguatur sic: est animal; ergo est homo. Secondly (78a7), he shows that it is possible to syllogize in the abovementioned manner without deception. And he shows this in three ways, the first of which is this: According to the above-mentioned manner of syllogizing, deception occurs because the consequence which was assumed convertible is not converted. But in this situation deception would not occur, if to the extent that the conclusion is true, so are the premises true: for there will then be no deception in converting. For if it is stated of Socrates that he is a man, therefore he is an animal; no deception or falsity follows: but it does if it is argued conversely that he is an animal, therefore he is a man.
Sed si praemissa est falsa, conclusione existente vera, tunc in convertendo accidit deceptio. Sicut si dicam: si asinus est homo, est animal; ergo si est animal, est homo. Si ergo impossibile esset ex falsis ostendere verum, et semper oporteret verum ostendi ex veris, tunc facile esset resolvere conclusionem in principia absque deceptione; quia nulla falsitas esset, si ex conclusione inferretur aliqua praemissarum. Tali enim suppositione facta, converterentur de necessitate conclusio et praemissa, quantum ad veritatem. Sicut enim praemissa existente vera, conclusio est vera, ita et e converso. Sit enim, quod a sit; et hoc posito, sequatur ea esse de quibus certum est mihi quod sunt vera, sicut b. Unde cum utrumque sit verum, ex hoc etiam, scilicet ex b, potero iterum inferre a. Sic ergo una ratio est, quare deceptio non accidit in demonstrativis scientiis per fallaciam consequentis, quia in demonstrativis scientiis impossibile est syllogizari verum ex falsis, sicut ostensum est supra. But if the premise is true and the conclusion false, then deception occurs in converting. For example, if I were to say: “If an ass is a man, it is an animal; therefore, if it is an animal, it is a man.” Consequently, if it were impossible to deduce the true from the false, and it were always necessary to deduce the true from the true, then it would be easy to analyze a conclusion into its principles without deception, because there would be no falsity if either of the premises were to be inferred from the conclusion. On this supposition the conclusion and premises would be converted of necessity as to truth. For just as the conclusion is true if the premise is true, so vice versa. For suppose that A is, and granting this, suppose that the same things follow as are known by me to be true, say B. Hence since both are true, I can then also infer A from B. And so, one reason why deception through fallacy of consequent does not occur in demonstrative sciences is that it is impossible in demonstrative sciences for the true to be syllogized from the false, as we have explained above.
Secundam rationem ponit; ibi: convertuntur autem magis et cetera. In terminis enim convertibilibus non accidit deceptio secundum fallaciam consequentis, eo quod in his consequentia convertitur. Illa vero, quae sunt in mathematicis, idest in demonstrativis scientiis, ut plurimum sunt convertibilia, quia non recipiunt pro medio aliquod praedicatum per accidens, sed solum definitiones, quae sunt demonstrationis principia, ut supra dictum est. Et in hoc differunt ab his, quae sunt in dialogis, idest in dialecticis syllogismis, in quibus frequenter recipiuntur accidentia. Then (78a10) he gives the second reason. For in cases of convertible terms, deception through fallacy of consequent does not occur, because in those cases the consequent is converted. Now the things which are used in mathematical, i.e., in demonstrative sciences, are for the most part convertible, because these sciences do not admit as a middle anything predicated per accidens, but only definitions which are principles of demonstration, as we have explained. And this is their point of difference from the statements in dialogues, i.e., in dialectical syllogisms, in which accidens are frequently admitted.
Tertiam rationem ponit ibi: augentur autem etc., quae talis est. In demonstrativis scientiis sunt determinata principia, ex quibus proceditur ad conclusiones. Unde ex conclusionibus potest rediri in principia, sicut ex determinato in determinatum. Quod autem demonstrationes ex determinatis principiis procedant, ex hoc ostendit, quia demonstrationes non augentur per media, idest in demonstrationibus non assumuntur plura media ad unam conclusionem demonstrandam. Quod intelligendum est in demonstrationibus propter quid, de quibus loquitur. Unius enim effectus non potest esse nisi una propria causa, propter quam est. Then (78a13) he gives the third reason and it is this: In demonstrative sciences there are determinate principles from which one proceeds to the conclusions. Hence it is possible to return from the conclusions to the principles as from something determinate to something determinate. That demonstrations do proceed from determinate principles he shows by the fact that “demonstrations are not increased by middles,” i.e., in demonstrations one does not use a series of different middle terms to demonstrate one conclusion. (This, of course, is to be understood of demonstrations propter quid, of which he is speaking). For of one effect there can be but one sole cause why it is.
Sed licet non multiplicentur per media demonstrationes, multiplicantur tamen duobus modis. Uno modo, in post assumendo, idest in assumendo medium sub medio. Sicut si sub a sumatur b, et sub bc, et sub cd; et sic in infinitum. Sicut cum habere tres angulos probatur de triangulo per hoc, quod est figura habens angulum extrinsecum aequalem duobus intrinsecis sibi oppositis, et de isoscele per hoc quod est triangulus. Alio modo multiplicantur demonstrationes in latus; sicut cum a probatur de c et de e. Verbi gratia: omnis numerus quantus aut est finitus aut infinitus. Et hoc ponatur in quo sit a, scilicet esse finitum vel infinitum. Sed impar numerus est numerus quantus. Et hoc, scilicet numerus quantus, ponatur in quo est b; sed numerus impar ponatur in quo est c. Sequitur ergo quod a praedicetur de c, idest quod numerus impar sit finitus vel infinitus. Et similiter potest idem concludi de numero pari, et per idem medium. But although demonstrations are not increased by using several middles of demonstration, nevertheless they are increased in two ways: in one way, “by assuming one after another,” i.e., subsuming one middle under another, say B under A, C under B, D under C, and so on. For example, when “having three angles equal to two right angles” is proved of triangle on the ground that it is a figure having an external angle equal to the two opposite interior angles, and is then proved of isosceles on the ground that it is a triangle. Another way they are increased is laterally, as when A is proved of C and of E. For example, “Every quantified number is either finite or infinite.” Let A be this predicate, i.e., “to be finite or infinite.” But an odd number is a quantified number. Let B represent “quantified number,” and C, “odd number.” It follows therefore, that A (to be finite or infinite) is predicated of C (odd number), i.e., that an odd number is either a finite or an infinite number. And he says that it is possible along the same lines to conclude concerning even number, and this through the same middle.
Potest autem et haec pars, quae incipit ibi: augentur autem etc., introduci aliter. Ut quia dixerat quod in demonstrativis assumuntur definitiones pro mediis; unius autem rei una est definitio; ex hoc sequitur quod demonstrationes non augeantur per media. The passage (78aI3), which begins, “A science expands,” can be introduced in another way. Since he had just said that in demonstrative matters, definitions are used for middles, and since there is one sole definition of one thing, it follows that it is not through middles that, demonstrations are increased.

Lectio 23
Caput 13
Τὸ δ' ὅτι διαφέρει καὶ τὸ διότι ἐπίστασθαι, πρῶτον μὲν ἐν τῇ αὐτῇ ἐπιστήμῃ, καὶ ἐν ταύτῃ διχῶς, ἕνα μὲν τρόπον ἐὰν μὴ δι' ἀμέσων γίνηται ὁ συλλογισμός (οὐ γὰρ λαμβάνεται τὸ πρῶτον αἴτιον, ἡ δὲ τοῦ διότι ἐπιστήμη κατὰ τὸ πρῶτον αἴτιον), ἄλλον δὲ εἰ δι' ἀμέσων μέν, ἀλλὰ μὴ διὰ τοῦ αἰτίου ἀλλὰ τῶν ἀντιστρεφόντων διὰ τοῦ γνωριμωτέρου. κωλύει γὰρ οὐδὲν τῶν ἀντικατηγορουμένων γνωριμώτερον εἶναι ἐνίοτε τὸ μὴ αἴτιον, ὥστ' ἔσται διὰ τούτου ἡ ἀπόδειξις, a22. Knowledge of the fact differs from knowledge of the reasoned fact. To begin with, they differ within the same science and in two ways: (1) when the premisses of the syllogism are not immediate (for then the proximate cause is not contained in them — a necessary condition of knowledge of the reasoned fact): (2) when the premisses are immediate, but instead of the cause the better known of the two reciprocals is taken as the middle; for of two reciprocally predicable terms the one which is not the cause may quite easily be the better known and so become the middle term of the demonstration.
οἷον ὅτι ἐγγὺς οἱ πλάνητες διὰ τοῦ μὴ στίλβειν. ἔστω ἐφ' ᾧ Γ πλάνητες, ἐφ' ᾧ Β τὸ μὴ στίλβειν, ἐφ' ᾧ Α τὸ ἐγγὺς εἶναι. ἀληθὲς δὴ τὸ Β κατὰ τοῦ Γ εἰπεῖν· οἱ γὰρ πλάνητες οὐ στίλβουσιν. ἀλλὰ καὶ τὸ Α κατὰ τοῦ Β· τὸ γὰρ μὴ στίλβον ἐγγύς ἐστι· τοῦτο δ' εἰλήφθω δι' ἐπαγωγῆς ἢ δι' αἰσθήσεως. ἀνάγκη οὖν τὸ Α τῷ Γ ὑπάρχειν, ὥστ' ἀποδέδεικται ὅτι οἱ πλάνητες ἐγγύς εἰσιν. οὗτος οὖν ὁ συλλογισμὸς οὐ τοῦ διότι ἀλλὰ τοῦ ὅτι ἐστίν· οὐ γὰρ διὰ τὸ μὴ στίλβειν ἐγγύς εἰσιν, ἀλλὰ διὰ τὸ ἐγγὺς εἶναι οὐ στίλβουσιν. a30. Thus (2) (a) you might prove as follows that the planets are near because they do not twinkle: let C be the planets, B not twinkling, A proximity. Then B is predicable of C; for the planets do not twinkle. But A is also predicable of B, since that which does not twinkle is near — we must take this truth as having been reached by induction or sense-perception. Therefore A is a necessary predicate of C; so that we have demonstrated that the planets are near. This syllogism, then, proves not the reasoned fact but only the fact; since they are not near because they do not twinkle, but, because they are near, do not twinkle.
ἐγχωρεῖ δὲ καὶ διὰ θατέρου θάτερον δειχθῆναι, καὶ ἔσται τοῦ διότι ἡ ἀπόδειξις, οἷον ἔστω τὸ Γ πλάνητες, ἐφ' ᾧ Β (78b.) τὸ ἐγγὺς εἶναι, τὸ Α τὸ μὴ στίλβειν· ὑπάρχει δὴ καὶ τὸ Β τῷ Γ καὶ τὸ Α τῷ Β, ὥστε καὶ τῷ Γ τὸ Α [τὸ μὴ στίλβειν]. καὶ ἔστι τοῦ διότι ὁ συλλογισμός· εἴληπται γὰρ τὸ πρῶτον αἴτιον. a39. The major and middle of the proof, however, may be reversed, and then the demonstration will be of the reasoned fact. Thus: let C be the planets, B proximity, A not twinkling. Then B is an attribute of C, and A — not twinkling — of B. Consequently A is predicable of C, and the syllogism proves the reasoned fact, since its middle term is the proximate cause.
πάλιν ὡς τὴν σελήνην δεικνύουσιν ὅτι σφαιροειδής, διὰ τῶν αὐξήσεων—εἰ γὰρ τὸ αὐξανόμενον οὕτω σφαιροειδές, αὐξάνει δ' ἡ σελήνη, φανερὸν ὅτι σφαιροειδής—οὕτω μὲν οὖν τοῦ ὅτι γέγονεν ὁ συλλογισμός, ἀνάπαλιν δὲ τεθέντος τοῦ μέσου τοῦ διότι· οὐ γὰρ διὰ τὰς αὐξήσεις σφαιροειδής ἐστιν, ἀλλὰ διὰ τὸ σφαιροειδὴς εἶναι λαμβάνει τὰς αὐξήσεις τοιαύτας. b3. Another example is the inference that the moon is spherical from its manner of waxing. Thus: since that which so waxes is spherical, and since the moon so waxes, clearly the moon is spherical. Put in this form, the syllogism turns out to be proof of the fact, but if the middle and major be reversed it is proof of the reasoned fact; since the moon is not spherical because it waxes in a certain manner, but waxes in such a manner because it is spherical. (Let C be the moon, B spherical, and A waxing.)
σελήνη ἐφ' ᾧ Γ, σφαιροειδὴς ἐφ' ᾧ Β, αὔξησις ἐφ' ᾧ Α. ἐφ' ὧν δὲ τὰ μέσα μὴ ἀντιστρέφει καὶ ἔστι γνωριμώτερον τὸ ἀναίτιον, τὸ ὅτι μὲν δείκνυται, τὸ διότι δ' οὔ. b10. Again (b), in cases where the cause and the effect are not reciprocal and the effect is the better known, the fact is demonstrated but not the reasoned fact.
Postquam philosophus determinavit de demonstratione propter quid, hic ostendit differentiam inter demonstrationem quia, et demonstrationem propter quid. Et circa hoc duo facit: primo, ostendit differentiam utriusque in eadem scientia; secundo, in diversis; ibi: alio autem modo et cetera. Circa primum duo facit: primo, ponit duplicem differentiam utriusque demonstrationis in eadem scientia; secundo, manifestat per exempla; ibi: ut quod prope sint planetae et cetera. After determining about demonstration propter quid, the Philosopher here shows the difference between demonstration quia and demonstration propter quid. And he does two things about this. First, he shows how they differ in the same science. Secondly, in diverse sciences (78b33) [L. 25]. Concerning the first he does two things. First, he states the twofold difference between the two in the same science. Secondly, he clarifies this with examples (78a30).
Dicit ergo primo: superius dictum est quod demonstratio est syllogismus faciens scire, et quod demonstratio ex causis rei procedit et primis et immediatis. Quod intelligendum est de demonstratione propter quid. Sed tamen differt scire quia ita est, et propter quid ita est. Et cum demonstratio sit syllogismus faciens scire, ut dictum est, oportet etiam quod demonstratio quae facit scire quia, differat a demonstratione quae facit scire propter quid. Et horum quidem differentia primo consideranda est in eadem scientia; postea consideranda est in diversis. He says therefore first (78a22) that, as said above, demonstration is a syllogism causing scientific knowledge and proceeds from the causes both first and immediate of a thing. Now this is to be understood as referring to demonstration propter quid. But there is a difference between knowing that a thing is so and why it is so. Therefore, since demonstration is a syllogism causing scientific knowledge, as has been said, it is necessary that a demonstration quia which makes one know that a thing is so should differ from the demonstration propter quid which makes one know why. Consequently, this difference must be considered first in the same science and later in sciences that are diverse.
In una autem scientia dupliciter differt utrumque praedictorum, secundum duo quae requirebantur ad demonstrationem simpliciter, quae facit scire propter quid; scilicet quod sit ex causis, et quod sit ex immediatis. Uno igitur modo differt scire quia ab hoc quod est scire propter quid; quia scire quia est si non fiat syllogismus demonstrativus per non medium, idest per immediatum, sed fiat per mediata. Sic enim non accipietur prima causa, cum tamen scientia, quae est propter quid, sit secundum primam causam. Et ita non erit scientia propter quid. In one and the same science each of the above is said to differ in regard to the two things required for demonstration in the strict sense—which causes knowledge of the why—namely, that it be from causes and from immediate causes. Hence one way that scientific knowledge quia differn from propter quid is that it is the former if the syllogism is not through, immediate principles but through mediate ones. For in that case the first cause will not be employed, whereas science propter quid is according to the first cause; consequently, the former will not be science propter quid.
Alio modo differunt, quia scire quia est quando fit syllogismus non quidem per media, idest per mediata, sed per immediata, sed non fit per causam: sed fit per convertentiam, idest per effectus convertibiles et immediatos. Et tamen talis demonstratio fit per notius, scilicet nobis: alias non faceret scire. Non enim pervenimus ad cognitionem ignoti, nisi per aliquid magis notum. Nihil enim prohibet duorum aeque praedicantium, idest convertibilium, quorum unum sit causa, et aliud effectus, notius esse aliquando non causam, sed magis effectum. Nam effectus aliquando est notior causa quoad nos et secundum sensum, licet causa sit semper notior simpliciter, et secundum naturam. Et ita per effectum notiorem causa potest fieri demonstratio non faciens scire propter quid, sed tantum quia. It differs in another way, because it is science quia when the syllogism, although not through middles, i.e., mediate, but through immediate things, is not through the cause but through “convertence,” i.e., through effects convertible and immediate. Hence a demonstration of this kind is through the better known, namely, to us; otherwise it would not effect scientific knowledge. For we do not reach a knowledge of the unknown except through something better known. However in the case of two things equally predicable, i.e., convertible, one of which is the cause and the other the effect, there is nothing to preclude that now and then the better known will not be the cause but the effect. For sometimes the effect is better known than the cause both in respect to us and according to sense-perception, although absolutely and according to nature the cause is the better known. Consequently, through an effect better known than the cause there can be demonstration which does not engender propter quid knowledge but only quia.
Deinde cum dicit: ut quod prope etc., manifestat praedictam differentiam per exempla. Et dividitur in duas partes: in prima, ponit exempla de demonstratione quia, quae est per effectum; in secunda, de demonstratione quia, quae est per causam mediatam; ibi: amplius in quibus medium et cetera. Prima in duas: in prima, ponit exempla de syllogismo qui fit per effectum convertibilem; in secunda, de syllogismo qui fit per effectum non convertibilem; ibi: in quibus autem media et cetera. Prima dividitur in duas partes secundum duo exempla quae ponit; secunda pars incipit ibi: item sic lunam et cetera. Circa primum duo facit: primo, ponit exemplum de demonstratione quia, quae est per effectum; secundo, docet quomodo posset converti in demonstrationem propter quid; ibi: contingit autem et cetera. Then (78a30) he clarifies these differences by examples. And this is divided into two parts. In the first he develops an example of the demonstration quia which is through an effect. In the second of demonstration quia which is through a mediate cause (78b13) [L. 24]. The first is divided into two parts. In the first he gives an example of a syllogism which is through a convertible effect. In the second of a syllogism through a nonconvertible effect (78b10). The first is divided into two parts according to the two examples he gives, the second of which begins at (78b3). Concerning the first he does two things. First, he gives an example of demonstration quia which is through an effect. Secondly, he states when it can be converted into a demonstration propter quid (78a39).
Dicit ergo primo quod demonstratio quia per effectum est, si quis concludat quod planetae sunt prope propter hoc quod non scintillant. Non enim non scintillare est causa quod planetae sint prope, sed e converso. Propter hoc enim non scintillant planetae, quia sunt prope. Stellae enim fixae scintillant, quia visus in comprehensione earum caligat propter earum distantiam. Formetur ergo syllogismus sic: omne non scintillans est prope; planetae sunt non scintillantes; ergo sunt prope. Sit in quo c planetae, idest accipiatur planetae quasi minor extremitas. In quo autem b sit non scintillare, idest non scintillare accipiatur medius terminus. In quo autem a sit prope esse, idest prope esse accipiatur ut maior extremitas. Vera igitur est haec propositio: omne c est b, quia planetae non scintillant. Et iterum verum est quod omne b est a, quia omnis stella non scintillans prope est. Huiusmodi autem propositionis veritas oportet quod accipiatur per inductionem, aut per sensum, quia effectus hic est notior causa quantum ad sensum. Et sic sequitur conclusio quod omne c sit a. Et sic demonstratum est quod planetae sive stellae erraticae sunt prope. Hic igitur syllogismus non est propter quid, sed est quia. Non enim propter hoc quod non scintillant, planetae sunt prope, sed propter id quod prope sunt, non scintillant. He says therefore first (78a30) that demonstration quia is through an effect if one concludes for example that the planets are near because they do not twinkle. For non-twinkling is not the cause why the planets are near, but vice versa: for the planets do not twinkle because they are near. For the fixed stars twinkle because in gazing at them the sight is beclouded on account of the distance. Therefore, the syllogism might be formed in the following way: “Whatever does not twinkle is near; but the planets do not twinkle: therefore, they are near.” Here we let C be the planets, i.e., let “planets” be the minor extreme, and let B consist in not twinkling, and A “to be near” be the major extreme. Then the proposition, “Every C is B,” is true, namely, the planets do not twinkle. Also it is true that “Every B is A,” i.e., every star that does not twinkle is near. Rowever, the truth of such a proposition must be obtained through induction or through sense perception, because the effect here is better known than the cause. And so, the conclusion, “Every C is A,” follows. In this way, then, it has been demonstrated that the planets, i.e., the wandering stars, are near. Consequently, this syllogism is not propter quid but quia. For it is not because they do not twinkle that planets are iiear but rather, because they are near, they do not twinkle.
Deinde cum dicit: contingit autem et per alterum etc., docet quomodo demonstratio quia convertatur in demonstrationem propter quid, dicens quod contingit et per alterum demonstrare alterum, idest per hoc quod est prope esse, demonstrare quod non scintillant; et sic erit demonstratio propter quid. Ut sit c erraticae, idest accipiatur stella erratica minor extremitas; in quo b sit prope esse, idest prope esse accipiatur ut medius terminus, quod supra erat maior extremitas; a sit non scintillare, idest accipiatur non scintillare maior extremitas, quod supra erat medius terminus. Est igitur et b in c, quia omnis planeta prope est; et a est in b, quia omnis planeta, qui prope est, non scintillat; quare sequitur quod et a sit in c, scilicet, quod omnis planeta non scintillet. Et sic erit syllogismus propter quid, cum accepta sit prima et immediata causa. Then (78a39) he teaches how a demonstration quia is changed to a demonstration propter quid. And he says that “it is possible to demonstrate the one through the other,” i.e., to demonstrate that they do not twinkle, because they are near. Then the demonstration will be propter quid. Thus let C be the wanderers, i.e., let “wandering star” be the minor extreme; let B consist in being near, i.e., let “to be near,” which was the major extreme above, be the middle term; and let A consist in not twinkling, i.e., let “not to twinkle,” which above was the middle term, now be the major term. Therefore, B is in C, i.e., “Every planet is near”; and A is in B, i.e., “Any planet which is near does not twinkle.” Wherefore, it follows that A is in C, i.e., “A planet does not twinkle.” In this way we have a syllogism propter quid, since it rests on the first and immediate cause.
Deinde cum dicit: item sic lunam demonstrant etc., ponit aliud exemplum ad idem, dicens quod sic (idest demonstratione faciente scire quia), demonstrant quod luna sit circularis per incrementa, quibus scilicet omni mense augetur et minuitur, sic argumentantes: omne quod sic augetur quasi circulariter, circulare est; augetur autem sic luna; ergo est circularis. Sic igitur factus est syllogismus demonstrans quia. Sed e converso, posito medio ipsius, fit syllogismus propter quid, scilicet si ponatur circulare ut medius terminus, et augmentum ut maior extremitas. Non enim ideo circularis est luna, quia sic augetur, sed quia circularis est, ideo talia augmenta recipit. Sit ergo luna in quo c, idest minor extremitas; augmentum in quo b, idest medius terminus; circularis autem in quo a, idest maior extremitas. Et hoc intelligendum est in syllogismo quia. E converso autem in syllogismo propter quid. Then (78b3) he presents another example of this, saying that “in this way” (i.e., by means of a demonstration quia), “one demonstrates that the moon is round because of its phases,” according to which it waxes and wanes every month. They argue thus: “Everything which waxes thus circularly is circular; but the moon waxes thus: therefore, it is circular.” “Put in this form it is a syllogism demonstrating quia. But if the middle be interchanged,” i.e., if “circular” be made the middle term and “waxes” the major term, “it becomes a demonstration propter quid.” For the moon is not circular because it waxes in that way, but because it is circular it undergoes such phases. Therefore, let C be “the moon,” i.e., the minor extreme; let “waxing” be A, i.e., the major extreme, and let “circular” be B, the middle term. This will be the situation in the syllogism propter quid.
Deinde cum dicit: in quibus autem media etc., ostendit quod sit demonstratio quia per effectum non convertibilem, dicens quod in illis etiam syllogismis in quibus media non convertuntur cum extremis, et accipitur ut notius quoad nos, scilicet loco medii, quod non est causa, sed magis effectus, demonstratur quidem quia, sed non propter quid. Et quidem si tale medium convertatur cum maiori extremitate, et excedat minorem, manifestum est quod conveniens fit syllogismus. Sicut si probetur de Venere quod sit prope, quia non scintillat. Si autem e converso minor terminus esset in plus quam medium assumptum; non esset conveniens syllogismus. Non enim potest de stella universaliter concludi quod sit prope, propter hoc quod non scintillat. In comparatione autem ad maiorem terminum est e converso. Nam si medium sit in minus quam maior terminus conveniens fit syllogismus. Sicut si per hoc, quod est moveri motu progressivo, probetur de aliquo quod habeat animam sensibilem. Si autem sit in plus, non fit conveniens syllogismus. Nam ab effectu, qui a pluribus causis procedere potest, non potest una illarum concludi. Sicut non potest concludi, quod aliquis habeat febrem, ex excitatione pulsus. Then (78b10) he shows that a demonstration through a non-convertible effect is quid. He says, therefore, that even in those syllogisms in which the middles are not converted with the extremes, and in which an effect rather than a cause is taken as the middle better known in reference to us, even in those cases the demonstration is quia and not propter quid. If the middle be such that it can be converted with the major extreme and it exceeds the minor, then obviously it is a fitting syllogism; for example, if one proves that Venus is near because it does not twinkle. On the other hand, if the minor exceeded the middle, it would not be a fitting syllogism: for one cannot conclude universally of stars that they are near because they do not twinkle. Quite the contrary is true in comparison to the major term: for if the middle is in less things than is the major term, the syllogism is fitting, as when it is proved that someone has a sensible soul on the ground that he is capable of progressive local motion. But if it is in more, than the syllogism is not fitting, for from an effect which can proceed from several causes, one of them cannot be concluded. Thus, one cannot conclude from a rapid pulse that he has a fever.

Lectio 24
Caput 13 cont.
Ἔτι ἐφ' ὧν τὸ μέσον ἔξω τίθεται. καὶ γὰρ ἐν τούτοις τοῦ ὅτι καὶ οὐ τοῦ διότι ἡ ἀπόδειξις· οὐ γὰρ λέγεται τὸ αἴτιον. b13. This also occurs (1) when the middle falls outside the major and minor, for here too the strict cause is not given, and so the demonstration is of the fact, not of the reasoned fact.
οἷον διὰ τί οὐκ ἀναπνεῖ ὁ τοῖχος; ὅτι οὐ ζῷον. εἰ γὰρ τοῦτο τοῦ μὴ ἀναπνεῖν αἴτιον, ἔδει τὸ ζῷον εἶναι αἴτιον τοῦ ἀναπνεῖν, οἷον εἰ ἡ ἀπόφασις αἰτία τοῦ μὴ ὑπάρχειν, ἡ κατάφασις τοῦ ὑπάρχειν, ὥσπερ εἰ τὸ ἀσύμμετρα εἶναι τὰ θερμὰ καὶ τὰ ψυχρὰ τοῦ μὴ ὑγιαίνειν, τὸ σύμμετρα εἶναι τοῦ ὑγιαίνειν, —ὁμοίως δὲ καὶ εἰ ἡ κατάφασις τοῦ ὑπάρχειν, ἡ ἀπόφασις τοῦ μὴ ὑπάρχειν. ἐπὶ δὲ τῶν οὕτως ἀποδεδομένων οὐ συμβαίνει τὸ λεχθέν· οὐ γὰρ ἅπαν ἀναπνεῖ ζῷον. b14. For example, the question 'Why does not a wall breathe?' might be answered, 'Because it is not an animal'; but that answer would not give the strict cause, because if not being an animal causes the absence of respiration, then being an animal should be the cause of respiration, according to the rule that if the negation of causes the non-inherence of y, the affirmation of x causes the inherence of y; e.g. if the disproportion of the hot and cold elements is the cause of ill health, their proportion is the cause of health; and conversely, if the assertion of x causes the inherence of y, the negation of x must cause y's non-inherence. But in the case given this consequence does not result; for not every animal breathes.
ὁ δὲ συλλογισμὸς γίνεται τῆς τοιαύτης αἰτίας ἐν τῷ μέσῳ σχήματι. οἷον ἔστω τὸ Α ζῷον, ἐφ' ᾧ Β τὸ ἀναπνεῖν, ἐφ' ᾧ Γ τοῖχος. τῷ μὲν οὖν Β παντὶ ὑπάρχει τὸ Α (πᾶν γὰρ τὸ ἀναπνέον ζῷον), τῷ δὲ Γ οὐθενί, ὥστε οὐδὲ τὸ Β τῷ Γ οὐθενί· οὐκ ἄρα ἀναπνεῖ ὁ τοῖχος. b24. A syllogism with this kind of cause takes place in the second figure. Thus: let A be animal, B respiration, C wall. Then A is predicable of all B (for all that breathes is animal), but of no C; and consequently B is predicable of no C; that is, the wall does not breathe.
ἐοίκασι δ' αἱ τοιαῦται τῶν αἰτιῶν τοῖς καθ' ὑπερβολὴν εἰρημένοις· τοῦτο δ' ἔστι τὸ πλέον ἀποστήσαντα τὸ μέσον εἰπεῖν, οἷον τὸ τοῦ Ἀναχάρσιος, ὅτι ἐν Σκύθαις οὐκ εἰσὶν αὐλητρίδες, οὐδὲ γὰρ ἄμπελοι. b27. Such causes are like far-fetched explanations, which precisely consist in making the cause too remote, as in Anacharsis' account of why the Scythians have no flute-players; namely because they have no vines.
Κατὰ μὲν δὴ τὴν αὐτὴν ἐπιστήμην καὶ κατὰ τὴν τῶν μέσων θέσιν αὗται διαφοραί εἰσι τοῦ ὅτι πρὸς τὸν τοῦ διότι συλλογισμόν· b32. Thus, then, do the syllogism of the fact and the syllogism of the reasoned fact differ within one science and according to the position of the middle terms.
Postquam manifestavit philosophus per exempla, qualiter demonstratur quia per effectum; hic ostendit qualiter demonstratur quia per non immediata. Et circa hoc duo facit: primo, manifestat propositum; secundo, ostendit qualiter in huiusmodi demonstrationibus media se habeant ad conclusiones; ibi: comparantur autem huiusmodi et cetera. Circa primum tria facit: primo, proponit intentum; secundo, manifestat per exemplum; ibi: ut quare non respirat etc.; tertio, ordinat in forma syllogistica; ibi: syllogismus autem et cetera. After clarifying with examples how there is demonstration quia through effect, the Philosopher here shows how there is demonstration quia through things not immediately connected. First, he amplifies his proposition. Secondly, he shows how the middles relate themselves to the conclusions in this type of demonstration (78b27). Concerning the first he does three things. First, he states his intention. Secondly, he clarifies it with examples (78b14). Thirdly, he puts it in syllogistic form (78b24).
Dicit ergo primo quod non solum in his quae probantur per effectum demonstratur quia et non propter quid, sed etiam in quibus medium extra ponitur. Dicitur autem medium extra poni quando est diversum a maiori termino, ut accidit in syllogismis negativis. Vel medium extra poni dicitur, quando est extra genus, quasi communius, et non convertitur cum maiori termino. Quod autem per tale medium non possit demonstrari propter quid probat ex hoc, quod demonstratio propter quid est per causam. Tale autem medium non est causa proprie loquendo. He says therefore first (78b13) that there is demonstration quia and not propter quid not only in those matters that are proved through an effect but also “in matters in which the middle is set outside.” Now a middle is said to be set outside when it is diverse from the major term, as in negative syllogisms; or a middle is said to be set outside when it is outside the genus as being something more common and not convertible with the major term. That something cannot be demonstrated propter quid through such a middle he proves on the ground that demonstration propter quid is through a cause. But the middle in question is not, properly speaking, a cause.
Deinde cum dicit: ut quare non respirat etc., manifestat quod dixerat per exemplum, dicens: ut si quis velit probare, quod non respirat paries, quia non est animal, non demonstrat propter quid, nec accipit causam. Quia si non esse animal esset causa non respirandi, oporteret quod esse animal esset causa respirandi: quod falsum est. Multa enim sunt animalia quae non respirant. Oportet enim, si negatio est causa negationis, quod affirmatio sit causa affirmationis; sicut non esse calidum et frigidum in mensura est causa quod aliquis non sanetur, et esse calidum et frigidum in mensura est causa quod aliquis sanetur. Similiter autem est e converso, quod si affirmatio est causa affirmationis, et negatio est causa negationis. In praemissis autem hoc non contingit, quia affirmatio non est causa affirmationis; quia non omne quod est animal respirat. Then (78b14) he clarifies what he had said with an example, saying that if someone attempted to prove that a wall does not breathe because it is not an animal, he would not be demonstrating propter quid or giving the cause. Because if the fact of not being an animal were the cause of not breathing, it would be required that being an animal would be the cause of breathing-which is false. For there are many animals which do not breathe. For it is required, if a negation is the cause of a negation, that the affirmation be the cause of the affirmation, as the fact of not being warm and cold in due measure is the cause of someone’s not getting well, and the fact of being warm and cold in due measure is the cause of someone’s getting well. The converse is also true, namely, that if an affirmation is the cause of an affirmation, the negation is the cause of the negation. But this does not occur in the case at hand, because the affirmation is not the cause of an affirmation, since not every animal breathes.
Deinde cum dicit: syllogismus autem etc., ordinat praedictum exemplum in forma syllogistica, dicens quod syllogismum praedictum oportet fieri in media figura. Et hoc ideo est, quia in prima figura non potest esse, quando conclusio est negativa, quod maior sit affirmativa; quod oportet in praedicto exemplo esse. Nam respirare, quod est maior terminus, oportet quod coniungatur cum animali, quod est medius terminus, secundum affirmationem. Sed paries, quod est minor terminus, oportet quod coniungatur cum animali, quod est medium, secundum negationem. Et sic oportet quod maior sit affirmativa et minor negativa. Quod quidem nunquam fit in prima figura; sed solum in secunda. Then (78b24) he arranges the aforesaid example in syllogistic form and says that it should be arranged in the second figure. And this is so because in the first figure there cannot be a negative conclusion such that the major would be affirmative, as our example requires. For “to breathe,” which is the major extreme, must be joined with “animal,” which is the middle term, according to affirmation. But “wall,” which is the minor extreme, must be joined with “animal,” which is the middle, according to negation. Consequently, the major will be affirmative and the minor negative. But such a thing never occurs in the first figure, but only in the second.
Accipiatur ergo animal a, idest medius terminus; b respirare, idest maior extremitas; et paries c, idest minor extremitas. Sic igitur a est in omni b, quia omne respirans est animal; in nullo autem c est a, quia nullus paries est animal: quare sequitur quod etiam b in nullo c sit, scilicet quod nullus paries respiret. Si autem acciperetur medium propinquum, esset demonstratio propter quid. Ut si ostenderetur quod paries non respiret, quia non habet pulmonem. Omne enim habens pulmonem respirat, et e converso. Let “animal,” then, be A, i.e., the middle term, “breathe” be B, i.e., the major extreme, and “wall” be C, i.e., the minor extreme. it therefore follows that A is in every B, because everything that breathes is an animal; but A is in no C, because no wall is an animal. Hence it follows that B, too, is in no C, i.e., that no wall breathes. Furthermore, if the most proximate middle were used, it would be demonstration propter quid: for example, if it were shown that a wall does not breathe because it does not have lungs. For whatever has lungs breathes, and conversely.
Deinde cum dicit: comparantur huiusmodi etc., ostendit quomodo media se habeant ad conclusiones, dicens quod huiusmodi causae remotae comparantur dictis secundum excellentiam, quia scilicet excedunt communitatem conclusionis probandae. Et huiusmodi medium contingit dicere quod est multum distans. Ut patet in probatione Anacharsidis, qui probat quod apud Scythas non sunt sibilatores, propter hoc, quod non sunt ibi vites. Hoc enim est medium valde remotum. Propinquum enim esset non habere vinum; et adhuc propinquius non bibere vinum; ex quo sequitur laetitia cordis quae movet ad cantandum, ut sic sibilatio pro cantu intelligatur. Vel melius potest dici quod sibilus hic accipitur non pro quolibet cantu; sed pro cantu vindemiantium, qui vocatur celeuma. Then (78b27) he shows how these middles are related to the conclusion, saying that such far-away causes are compared to what they explain as being too remote, because they go beyond the pale of the conclusion to be proved. Furthermore, a middle of this sort happens to assign what is far-fetched, as in Anacharsis’ proof that there are no flutists among the Scythians because there are no vines there. For this is quite a far-fetched middle. A nearer one would be that they have no wine, and nearer still, that they do not drink wine, from which follows a merry heart which moves one to sing-if fluting is taken to mean singing. Or it might be better to say that fluting is not taken for just any singing but for the song of the grape harvesters, called “celeuma.”
Deinde cum dicit: secundum quidem etc., epilogat quod dixerat, dicens: quod hae sunt differentiae syllogismi quia ad syllogismum qui est propter quid, in eadem scientia, et secundum eorumdem positionem, idest eorum qui habent eumdem ordinem. Quod dicitur ad removendum illud, quod post dicet, quod una scientia est sub altera. Then (78b32) he summarizes what he had said and declares that these are the differences between the syllogism quia and the syllogism propter quid in the same science, “and according to the position of the same,” i.e., of those that have the same order. This he says to exclude what he will discuss later, namely, that one science is under another.

Lectio 25
Caput 13 cont.
ἄλλον δὲ τρόπον διαφέρει τὸ διότι τοῦ ὅτι τῷ δι' ἄλλης ἐπιστήμης ἑκάτερον θεωρεῖν. b34. But there is another way too in which the fact and the reasoned fact differ, and that is when they are investigated respectively by different sciences.
τοιαῦτα δ' ἐστὶν ὅσα οὕτως ἔχει πρὸς ἄλληλα ὥστ' εἶναι θάτερον ὑπὸ θάτερον, οἷον τὰ ὀπτικὰ πρὸς γεωμετρίαν καὶ τὰ μηχανικὰ πρὸς στερεομετρίαν καὶ τὰ ἁρμονικὰ πρὸς ἀριθμητικὴν καὶ τὰ φαινόμενα πρὸς ἀστρολογικήν. b35. This occurs in the case of problems related to one another as subordinate and superior, as when optical problems are subordinated to geometry, mechanical problems to stereometry, harmonic problems to arithmetic, the data of observation to astronomy.
σχεδὸν δὲ συνώνυμοί εἰσιν ἔνιαι τούτων τῶν ἐπιστημῶν, οἷον ἀστρολογία ἥ τε μαθηματικὴ (79a.) καὶ ἡ ναυτική, καὶ ἁρμονικὴ ἥ τε μαθηματικὴ καὶ ἡ κατὰ τὴν ἀκοήν. a1. (Some of these sciences bear almost the same name; e.g. mathematical and nautical astronomy, mathematical and acoustical harmonics.)
ἐνταῦθα γὰρ τὸ μὲν ὅτι τῶν αἰσθητικῶν εἰδέναι, τὸ δὲ διότι τῶν μαθηματικῶν· οὗτοι γὰρ ἔχουσι τῶν αἰτίων τὰς ἀποδείξεις, καὶ πολλάκις οὐκ ἴσασι τὸ ὅτι, καθάπερ οἱ τὸ καθόλου θεωροῦντες πολλάκις ἔνια τῶν καθ' ἕκαστον οὐκ ἴσασι δι' ἀνεπισκεψίαν. ἔστι δὲ ταῦτα ὅσα ἕτερόν τι ὄντα τὴν οὐσίαν κέχρηται τοῖς εἴδεσιν. τὰ γὰρ μαθήματα περὶ εἴδη ἐστίν· οὐ γὰρ καθ' ὑποκειμένου τινός· εἰ γὰρ καὶ καθ' ὑποκειμένου τινὸς τὰ γεωμετρικά ἐστιν, ἀλλ' οὐχ ᾗ γε καθ' ὑποκειμένου. a3. Here it is the business of the empirical observers to know the fact, of the mathematicians to know the reasoned fact; for the latter are in possession of the demonstrations giving the causes, and are often ignorant of the fact: just as we have often a clear insight into a universal, but through lack of observation are ignorant of some of its particular instances. These connexions have a perceptible existence though they are manifestations of forms. For the mathematical sciences concern forms: they do not demonstrate properties of a substratum, since, even though the geometrical subjects are predicable as properties of a perceptible substratum, it is not as thus predicable that the mathematician demonstrates properties of them.
ἔχει δὲ καὶ πρὸς τὴν ὀπτικήν, ὡς αὕτη πρὸς τὴν γεωμετρίαν, ἄλλη πρὸς ταύτην, οἷον τὸ περὶ τῆς ἴριδος· τὸ μὲν γὰρ ὅτι φυσικοῦ εἰδέναι, τὸ δὲ διότι ὀπτικοῦ, ἢ ἁπλῶς ἢ τοῦ κατὰ τὸ μάθημα. a10. As optics is related to geometry, so another science is related to optics, namely the theory of the rainbow. Here knowledge of the fact is within the province of the natural philosopher, knowledge of the reasoned fact within that of the optician, either qua optician or qua mathematical optician.
πολλαὶ δὲ καὶ τῶν μὴ ὑπ' ἀλλήλας ἐπιστημῶν ἔχουσιν οὕτως, οἷον ἰατρικὴ πρὸς γεωμετρίαν· ὅτι μὲν γὰρ τὰ ἕλκη τὰ περιφερῆ βραδύτερον ὑγιάζεται, τοῦ ἰατροῦ εἰδέναι, διότι δὲ τοῦ γεωμέτρου. a13. Many sciences not standing in this mutual relation enter into it at points; e.g. medicine and geometry: it is the physician's business to know that circular wounds heal more slowly, the geometer's to know the reason why.
Postquam ostendit philosophus qualiter demonstratio quia differt a demonstratione propter quid in eadem scientia; hic ostendit quomodo differt in diversis scientiis. Et circa hoc duo facit. Primo, proponit intentum, dicens quod alio modo a praedictis differt propter quid ab ipso quia, propter hoc quod in diversis scientiis considerantur, idest quod ad unam scientiam pertinet scire propter quid, et ad aliam scientiam pertinet scire quia. After showing how demonstration quia differs from demonstration propter quid in the same science, the Philosopher shows how they differ in sciences that are diverse. And he does two things: First, he states his proposition, saying (78b34) that in a way other than the above the propter quid differs from the quia due to the fact that they are considered in diverse sciences, i.e., that the propter quid pertains to one science and the quia to another.
Secundo cum dicit: huiusmodi autem sunt etc., manifestat propositum. Et circa hoc duo facit: primo, manifestat propositum in scientiis, quarum una est sub altera; secundo, in scientiis, quarum una non est sub altera; ibi: multae autem non sibi et cetera. Circa primum duo facit: primo, ostendit qualiter se habeant scientiae ad invicem, quarum una est sub altera, ad quarum unam pertinet propter quid, ad alteram autem quia; secundo, ostendit quomodo in praedictis scientiis ad unam earum pertinet quia, et ad aliam propter quid; ibi: hoc enim ipsum et cetera. Circa primum duo facit: primo, ostendit quomodo praedictae scientiae se habeant ad invicem secundum ordinem; secundo, ostendit qualiter se habeant ad invicem secundum convenientiam; ibi: fere autem univocae et cetera. Secondly (78b34) he elucidates his proposition. First, he elucidates it in sciences one of which is under the other. Secondly, in sciences one of which is not under the other (79a13). Concerning the first he does two things. First, he shows how those sciences are related, one of which is under the other and to one of which pertains the quia and to the other the propter quid. Secondly, he shows how in these sciences the quia pertains to one and the propter quid to the other (790). Concerning the first he does two things. First, he shows how such sciences relate to one another as to order. Secondly, how they relate to one another as to agreement (79a1).
Dicit ergo primo quod huiusmodi scientiae sunt (scilicet ad quarum unam pertinet quia, ad aliam autem propter quid) quaecunque sic se habent ad invicem, quod altera est sub altera. Sed intelligendum est unam scientiam esse sub altera dupliciter. Uno modo, quando subiectum unius scientiae est species subiecti superioris scientiae; sicut animal est species corporis naturalis, et ideo scientia de animalibus est sub scientia naturali. Alio modo, quando subiectum inferioris scientiae, non est species subiecti superioris scientiae; sed subiectum inferioris scientiae comparatur ad subiectum superioris, sicut materiale ad formale. Et hoc modo accipit hic unam scientiam esse sub altera, sicut speculativa, idest perspectiva, se habet ad geometriam. Geometria enim est de linea et aliis magnitudinibus: perspectiva autem est circa lineam determinatam ad materiam, idest circa lineam visualem. Linea autem visualis non est species lineae simpliciter, sicut nec triangulus ligneus est species trianguli: non enim ligneum est differentia trianguli. Et similiter machinativa, idest scientia de faciendis machinis, se habet ad stereometriam, idest ad scientiam quae est de mensurationibus corporum. Et haec scientia dicitur esse sub scientia per applicationem formalis ad materiale. Nam mensurae corporum simpliciter comparantur ad mensuras lignorum et aliarum materierum, quae requiruntur ad machinas, per applicationem formalis ad materiale. Et similiter se habet harmonica, idest musica, ad arithmeticam. Nam musica applicat numerum formalem (quem considerat arithmeticus) ad materiam, idest ad sonos. Et similiter se habet apparentia, idest scientia navalis, quae considerat signa apparentia serenitatis vel tempestatis, ad astrologiam, quae considerat motus et situs astrorum. He says therefore first (78b35) that these sciences (i.e., to one of which pertains quia and to the other propter quid) are the ones so related that one is under the other. But this occurs in two ways: in one way, when the subject of one science is a species of the subject of the higher science, as animal is a species of natural body—consequently, the science of animals is under natural science; in another way, when the subject of the lower science is not a species of the subject of the higher science but is compared to the latter as material to formal. An example of this latter way of one science being under another is the way “specular,” i.e., optics, is under geometry. For geometry is concerned with lines and other magnitudes, whereas optics is concerned with a line determined to matter, i.e., the visual line. Now the visual line is not, strictly speaking, a species of line any more than wooden triangle is a species of triangle: for wooden is not a difference in respect to triangle. In like manner mechanical engineering, i.e., the science of making machines, is related to stereometry, i.e., the science of measuring bodies. This science is said to be under the other as applying the formal to the material. For the measures of bodies’ absolutely are compared to the measures of wood and other material required for machines as the application of the formal to the material. In like manner, harmonics, i.e., music, is related to arithmetic: for music applies formal number, which arithmetic considers, to matter, i.e., to sounds. And the same is true of “appearance,” i.e., nautical science (which considers the signs indicative of calm or storm), as compared to astronomy, which considers the motions and positions of the stars.
Deinde cum dicit: fere autem univocae etc., ostendit qualiter se habent praedictae scientiae ad invicem secundum convenientiam, dicens quod fere huiusmodi scientiae sunt univocae ad invicem. Dicit autem fere, quia communicant in nomine generis, et non in nomine speciei. Dicuntur enim omnes praedictae scientiae mathematicae; quaedam quidem quia sunt de subiecto abstracto a materia, ut geometria et arithmetica, quae simpliciter mathematicae sunt; quaedam autem per applicationem principiorum mathematicorum ad res materiales, sicut astrologia dicitur mathematica et etiam navalis scientia, et similiter harmonica, idest musica, dicitur mathematica, et quae est secundum auditum, idest practica musicae, quae cognoscit ex experientia auditus sonos. Vel potest dici quod sunt univocae, quia etiam in nomine speciei conveniunt. Nam et navalis dicitur astrologia, et practica musicae dicitur musica. Dicit autem fere, quia hoc non contingit in omnibus, sed in pluribus. Then (79a1) he shows how the aforesaid sciences relate to one another in point of agreement. And he says that these sciences are almost mutually univocal. And he says, “almost,” because they agree in the name of their genus and not in the name of their species. Thus, all the sciences mentioned above are called mathematical: some, indeed, because they are concerned with a subject which is abstracted from matter, as geometry and arithmetic, which are absolutely mathematical; but others are so through applying mathematical principles to material things, as astronomy is called mathematical and as nautical science is. Similarly, harmonics, i.e., music, is called mathematical and so is acoustics, i.e., practical music, which knows sounds through experience based on hearing. Or it might be said that they are univocal because they also agree in the name of their species, for we speak of nautical astronomy as astronomy and of practical music as music. But because this is not so in all but only in some, he says “almost.”
Deinde cum dicit: hoc enim ipsum etc., manifestat quomodo in praedictis scientiis ad unam scientiam pertinet quia, et ad aliam propter quid. Et circa hoc duo facit: primo, ostendit quomodo scientiae, quae sub se continent alias, habent dicere propter quid; secundo, quomodo scientiae, quae sub eis continentur, habent dicere propter quid respectu aliarum scientiarum; ibi: habet autem se et cetera. Then (79a3) he shows how in these sciences the quia pertains to the one science and the propter quid to the other. In regard to this there are two points. First, he shows how those sciences which contain others under them state the propter quid. Secondly, how sciences which are contained under another one state the propter quid of some others (79a10).
Sciendum ergo est circa primum quod in omnibus praenominatis scientiis, illae quae continentur sub aliis, applicant principia mathematicae ad sensibilia. Quae autem sub se continent alias sunt magis mathematicae. Et ideo dicit primo philosophus quod scire quia est sensibilium, idest scientiarum inferiorum, quae applicant ad sensibilia: sed scire propter quid est mathematicorum, idest scientiarum, quarum principia applicantur ad sensibilia. Huiusmodi enim habent demonstrare ea, quae assumuntur ut causae in inferioribus scientiis. One should note, therefore, with respect to the first point that in all the above-mentioned sciences, the ones which are contained under others apply mathematical principles to sensible things, while the ones which contain others under them are more mathematical. Accordingly, the Philosopher first of all says (79a3) that to know the quia pertains to the sensible, i.e., to the lower sciences, which make application to sensible things; but to know the propter quid pertains to the mathematical, i.e., to those sciences whose principles are not applied to sensible things. For these latter sciences are concerned with demonstrating matters which are accepted as causes in the lower sciences.
Et quia posset aliquis credere quod qui sciret propter quid, sciret etiam de necessitate quia, consequenter hoc removet dicens quod multoties illi, qui sciunt propter quid, nesciunt quia. Et hoc manifestat per exemplum: sicut considerantes universale, multoties nesciunt quaedam singularia, propter hoc, quod non intendunt per considerationem; sicut qui scit omnem mulam esse sterilem, nescit de ista, quam non considerat. Et similiter mathematicus qui demonstrat propter quid, nescit quandoque quia, quia non applicat principia superioris scientiae ad ea, quae demonstrantur in inferiori scientia. But because someone might suppose that whoever knows the propter quid also necessarily knows the quia, he dismisses this, saying that very frequently those who know the propter quid do not know the quia. And he gives as an example of this those who, when considering the universal often take no account of the singulars precisely because their speculation does not consider them: thus, one who knows that every mule is sterile, does not know this of the one he does not consider. In like manner, a mathematician who demonstrates propter quid, now and then does not know the quia, because he does not apply the principles of the higher sciences to matters demonstrated in the lower science.
Et quia dixerat quod scire propter quid est mathematicorum, vult ostendere cuiusmodi genus causae a mathematicis sumatur. Unde dicit quod istae scientiae, quae accipiunt propter quid a mathematicis, sunt alterum quiddam, idest differunt ab eis secundum subiectum, scilicet in quantum applicant ad materiam. Unde huiusmodi scientiae utuntur speciebus, idest formalibus principiis, quae accipiunt a mathematicis. Mathematicae enim scientiae sunt circa species. Non enim earum consideratio est de subiecto, idest de materia; quia quamvis ea, de quibus geometria considerat, sint in materia, sicut linea, superficies et huiusmodi; non tamen considerat de eis geometria, secundum quod sunt in materia, sed secundum quod sunt abstracta. Nam geometria ea, quae sunt in materia secundum esse, abstrahit a materia secundum considerationem. Scientiae autem ei subalternatae e converso accipiunt ea, quae sunt considerata in abstractione a geometra, et applicant ad materiam. Unde patet quod geometra dicit propter quid in istis scientiis secundum causam formalem. And because he had said that it belongs to mathematics to know the propter quid, he proposes to indicate which genus of cause is used by mathematics. Hence he says that those sciences which receive the propter quid from mathematics “are something else,” i.e., they differ from the mathematical according to subject, i.e., insofar as they make application to matter. Hence these latter “use forms’ “ i.e., formal principles, which they receive from mathematics: “for the mathematical sciences concern forms.” For their considerations do not bear on the subject, i.e., on matter, because although the items which geometry considers exist in matter, for example, the line, plane and so on, nevertheless geometry does not consider them precisely as they are in matter, but as abstracted. For those things that are in matter according to existence, geometry abstracts from matter according to consideration. Conversely, the sciences subalternated to it accept those things which were considered in the abstract by geometry and apply them to matter. Hence it is plain that it is according to the formal cause that geometry states the propter quid in those sciences.
Deinde cum dicit: habet autem se etc., ostendit quod etiam scientia subalternata dicit propter quid, non respectu subalternantis, sed respectu cuiusdam alterius. Perspectiva enim subalternatur geometriae. Et si comparemus perspectivam ad geometriam, perspectiva dicit quia et geometria propter quid. Sed sicut perspectiva subalternatur geometriae, ita scientia de iride subalternatur perspectivae. Applicat enim principia, quae perspectiva tradit simpliciter, ad determinatam materiam. Unde ipsius physici, qui tractat de iride, est scire quia; sed perspectivi est scire propter quid. Dicit enim physicus conversionem visus ad nubem, aliquo modo dispositam ad solem, esse causam iridis. Propter quid autem sumit a perspectivo. Then (79a10) he shows that even the subalternated sciences state the propter quid, not of its subalternating science but of some other science. Thus, optics is subalternated to geometry, so that if we compare the one with the other, optics states the quia and geometry the propter quid. But just as optics is subalternated to geometry, so the science of the rainbow is subalternated to optics, for it applies to a determinate matter the principles which optics hands down absolutely. Hence it belongs to the naturalist who treats of the rainbow to know the quia, but to the expert in optics to know the propter quid. For the naturalist says that the cause of the rainbow is the convergence of a visual line at a cloud arranged in some relation to the sun; but the propter quid he takes from optics.
Deinde cum dicit: multae autem et non sub etc., ostendit quomodo quia et propter quid differunt in diversis scientiis non subalternatis, dicens quod multae scientiarum, quae non sunt sub invicem, sic se habent ad invicem, scilicet quod ad unam pertinet quia, et ad alteram pertinet propter quid. Sicut patet de medicina et geometria. Non enim subiectum medicinae sumitur sub subiecto geometriae, sicut subiectum perspectivae; sed tamen ad aliquam conclusionem, in medicina consideratam, applicabilia sunt principia geometriae. Sicut quod vulnera circularia tardius sanentur, medici est scire quia, qui hoc experitur, sed propter quid scire est geometrae, ad quem pertinet cognoscere quod circulus est figura sine angulo. Unde partes circularis vulneris non appropinquant sibi, ut possint de facili coniungi. Sciendum autem est quod illa differentia quia et propter quid, quae est secundum diversas scientias, continetur sub altero praedictorum modorum, scilicet quando fit demonstratio per causam remotam. Then (79a13) he shows how quia and propter quid differ among sciences that are diverse but not subalternate. And he says that many sciences which are not subalternate are nevertheless related, i.e., in such a way that one states the quia and the other the propter quid. This is true of medicine and geometry. For the subject of medicine is not subsumed under the subject of geometry as the subject of optics is. Nevertheless, the principles of geometry are applicable to certain conclusions reached in medicine: for example, it belongs to the man of medicine who observes it to know quia that circular wounds heal rather slowly; but to know the propter quid belongs to the geometer, whose business it is to know that a circle is a figure without corners. Hence the edges of a circular wound are not close enough to each other to allow them to be easily joined. It should also be noted that this difference of quia and propter quid between sciences that are diverse is contained under one of the modes previously discussed, namely, when the demonstration is made through a remote cause.

Lectio 26
Caput 14
Τῶν δὲ σχημάτων ἐπιστημονικὸν μάλιστα τὸ πρῶτόν ἐστιν. αἵ τε γὰρ μαθηματικαὶ τῶν ἐπιστημῶν διὰ τούτου φέρουσι τὰς ἀποδείξεις, οἷον ἀριθμητικὴ καὶ γεωμετρία καὶ ὀπτική, καὶ σχεδὸν ὡς εἰπεῖν ὅσαι τοῦ διότι ποιοῦνται τὴν σκέψιν· ἢ γὰρ ὅλως ἢ ὡς ἐπὶ τὸ πολὺ καὶ ἐν τοῖς πλείστοις διὰ τούτου τοῦ σχήματος ὁ τοῦ διότι συλλογισμός. a17. Of all the figures the most scientific is the first. Thus, it is the vehicle of the demonstrations of all the mathematical sciences, such as arithmetic, geometry, and optics, and practically all of all sciences that investigate causes: for the syllogism of the reasoned fact is either exclusively or generally speaking and in most cases in this figure — a second proof that this figure is the most scientific; for grasp of a reasoned conclusion is the primary condition of knowledge.
ὥστε κἂν διὰ τοῦτ' εἴη μάλιστα ἐπιστημονικόν· κυριώτατον γὰρ τοῦ εἰδέναι τὸ διότι θεωρεῖν. εἶτα τὴν τοῦ τί ἐστιν ἐπιστήμην διὰ μόνου τούτου θηρεῦσαι δυνατόν. ἐν μὲν γὰρ τῷ μέσῳ σχήματι οὐ γίνεται κατηγορικὸς συλλογισμός, ἡ δὲ τοῦ τί ἐστιν ἐπιστήμη καταφάσεως· ἐν δὲ τῷ ἐσχάτῳ γίνεται μὲν ἀλλ' οὐ καθόλου, τὸ δὲ τί ἐστι τῶν καθόλου ἐστίν· οὐ γὰρ πῇ ἐστι ζῷον δίπουν ὁ ἄνθρωπος. a23. Thirdly, the first is—Thirdly, the first is the only figure which enables us to pursue knowledge of the essence of a thing. In the second figure no affirmative conclusion is possible, and knowledge of a thing's essence must be affirmative; while in the third figure the conclusion can be affirmative, but cannot be universal, and essence must have a universal character: e.g. man is not two-footed animal in any qualified sense, but universally.
ἔτι τοῦτο μὲν ἐκείνων οὐδὲν προσδεῖται, ἐκεῖνα δὲ διὰ τούτου καταπυκνοῦται καὶ αὔξεται, ἕως ἂν εἰς τὰ ἄμεσα ἔλθῃ. φανερὸν οὖν ὅτι κυριώτατον τοῦ ἐπίστασθαι τὸ πρῶτον σχῆμα. a30. Finally, the first figure has no need of the others, while it is by means of the first that the other two figures are developed, and have their intervals closepacked until immediate premisses are reached. Clearly, therefore, the first figure is the primary condition of knowledge.
Chapter 15
Ὥσπερ δὲ ὑπάρχειν τὸ Α τῷ Β ἐνεδέχετο ἀτόμως, οὕτω καὶ μὴ ὑπάρχειν ἐγχωρεῖ. λέγω δὲ τὸ ἀτόμως ὑπάρχειν ἢ μὴ ὑπάρχειν τὸ μὴ εἶναι αὐτῶν μέσον· οὕτω γὰρ οὐκέτι ἔσται κατ' ἄλλο τὸ ὑπάρχειν ἢ μὴ ὑπάρχειν. a33. Just as an attribute A may (as we saw) be atomically connected with a subject B, so its disconnexion may be atomic. I call 'atomic' connexions or disconnexions which involve no intermediate term; since in that case the connexion or disconnexion will not be mediated by something other than the terms themselves.
ὅταν μὲν οὖν ἢ τὸ Α ἢ τὸ Β ἐν ὅλῳ τινὶ ᾖ, ἢ καὶ ἄμφω, οὐκ ἐνδέχεται τὸ Α τῷ Β πρώτως μὴ ὑπάρχειν. ἔστω γὰρ τὸ Α ἐν ὅλῳ τῷ Γ. οὐκοῦν εἰ τὸ Β μὴ ἔστιν ἐν ὅλῳ τῷ Γ (ἐγχωρεῖ γὰρ τὸ μὲν Α εἶναι ἔν τινι ὅλῳ, τὸ δὲ Β μὴ εἶναι ἐν τούτῳ), συλλογισμὸς ἔσται τοῦ μὴ ὑπάρχειν τὸ Α τῷ Β· εἰ γὰρ τῷ μὲν (79b.) Α παντὶ τὸ Γ, τῷ δὲ Β μηδενί, οὐδενὶ τῷ Β τὸ Α. a36. It follows that if either A or B, or both A and B, have a genus, their disconnexion cannot be primary. Thus: let C be the genus of A. Then, if C is not the genus of B — for A may well have a genus which is not the genus of B — there will be a syllogism proving A's disconnexion from B thus:
all A is C,
no B is C,
therefore no B is A.
ὁμοίως δὲ καὶ εἰ τὸ Β ἐν ὅλῳ τινί ἐστιν, οἷον ἐν τῷ Δ· τὸ μὲν γὰρ Δ παντὶ τῷ Β ὑπάρχει, τὸ δὲ Α οὐδενὶ τῷ Δ, ὥστε τὸ Α οὐδενὶ τῷ Β ὑπάρξει διὰ συλλογισμοῦ. τὸν αὐτὸν δὲ τρόπον δειχθήσεται καὶ εἰ ἄμφω ἐν ὅλῳ τινί ἐστιν. Or if it is B which has a genus D, we have
all B is D,
no D is A,
therefore no B is A, by syllogism;
and the proof will be similar if both A and B have a genus.
ὅτι δ' ἐνδέχεται τὸ Β μὴ εἶναι ἐν ᾧ ὅλῳ ἐστὶ τὸ Α, ἢ πάλιν τὸ Α ἐν ᾧ τὸ Β, φανερὸν ἐκ τῶν συστοιχιῶν, ὅσαι μὴ ἐπαλλάττουσιν ἀλλήλαις. εἰ γὰρ μηδὲν τῶν ἐν τῇ Α Γ Δ συστοιχίᾳ κατὰ μηδενὸς κατηγορεῖται τῶν ἐν τῇ Β Ε Ζ, τὸ δ' Α ἐν ὅλῳ ἐστὶ τῷ Θ συστοίχῳ ὄντι, φανερὸν ὅτι τὸ Β οὐκ ἔσται ἐν τῷ Θ· ἐπαλλάξουσι γὰρ αἱ συστοιχίαι. ὁμοίως δὲ καὶ εἰ τὸ Β ἐν ὅλῳ τινί ἐστιν. b5. That the genus of A need not be the genus of B and vice versa, is shown by the existence of mutually exclusive coordinate series of predication. If no term in the series ACD... is predicable of any term in the series BEF..., and if G — a term in the former series — is the genus of A, clearly G will not be the genus of B; since, if it were, the series would not be mutually exclusive. So also if B has a genus, it will not be the genus of A.
ἐὰν δὲ μηδέτερον ᾖ ἐν ὅλῳ μηδενί, μὴ ὑπάρχῃ δὲ τὸ Α τῷ Β, ἀνάγκη ἀτόμως μὴ ὑπάρχειν. εἰ γὰρ ἔσται τι μέσον, ἀνάγκη θάτερον αὐτῶν ἐν ὅλῳ τινὶ εἶναι. ἢ γὰρ ἐν τῷ πρώτῳ σχήματι ἢ ἐν τῷ μέσῳ ἔσται ὁ συλλογισμός. εἰ μὲν οὖν ἐν τῷ πρώτῳ, τὸ Β ἔσται ἐν ὅλῳ τινί (καταφατικὴν γὰρ δεῖ τὴν πρὸς τοῦτο γενέσθαι πρότασιν), εἰ δ' ἐν τῷ μέσῳ, ὁπότερον ἔτυχεν (πρὸς ἀμφοτέροις γὰρ ληφθέντος τοῦ στερητικοῦ γίνεται συλλογισμός· ἀμφοτέρων δ' ἀποφατικῶν οὐσῶν οὐκ ἔσται). b12. If, on the other hand, neither A nor B has a genus and A does not inhere in B, this disconnexion must be atomic. If there be a middle term, one or other of them is bound to have a genus, for the syllogism will be either in the first or the second figure. If it is in the first, B will have a genus — for the premiss containing it must be affirmative: if in the second, either A or B indifferently, since syllogism is possible if either is contained in a negative premiss, but not if both premisses are negative.
Φανερὸν οὖν ὅτι ἐνδέχεταί τε ἄλλο ἄλλῳ μὴ ὑπάρχειν ἀτόμως, καὶ πότ' ἐνδέχεται καὶ πῶς, εἰρήκαμεν. b21. Hence it is clear that one thing may be atomically disconnected from another, and we have stated when and how this is possible.
Postquam philosophus determinavit de materia syllogismi demonstrativi, hic determinat de forma ipsius, ostendens in qua figura praecipue fiat syllogismus demonstrativus. Et dividitur in duas partes. In prima ostendit quod syllogismus demonstrativus maxime fit in prima figura. Et quia in prima figura proceditur etiam ex negativis, et oportet demonstrationem ex immediatis procedere, ostendit in secunda parte quomodo contingit propositionem negativam esse immediatam; ibi: sicut autem esse a in b et cetera. After determining about the material of the syllogism, the Philosopher here determines about its form, showing in which figure chiefly the, demonstrative syllogism is formed. His treatment is divided into two parts. In the first he shows that the demonstrative syllogism is formed first and foremost in the first figure. In the second, because it is possible in the first figure to proceed from negatives and because a demonstration must proceed from things immediate, he shows how a negative proposition happens to be immediate (79a33).
Primum ostendit tribus rationibus, quarum prima talis est. In quacunque figura maxime fit syllogismus propter quid, illa figura maxime est faciens scire, et propter hoc est magis accommoda demonstrationibus; cum demonstratio sit syllogismus faciens scire. Sed in prima figura maxime fit syllogismus propter quid (quod patet ex hoc quod mathematicae scientiae, ut arithmetica et geometria, et quaecunque aliae propter quid demonstrant, ut plurimum prima figura utuntur); ergo prima figura est maxime faciens scire et maxime accommoda demonstrationibus. He shows the first with three reasons, the first of which (79a17) is this: In whatever figure the propter quid syllogism is best made, that figure is the best for causing scientific knowledge and for that reason is most suitable for demonstrations, since demonstration is a syllogism which causes scientific knowledge. But a syllogism propter quid is best made in the first figure, and this is evidenced by the fact that the mathematical sciences of arithmetic and geometry and all other sciences which demonstrate propter quid employ the first figure in most cases. Therefore, the first figure is the one which first and foremost causes scientific knowledge and is most suitable for demonstrations.
Causa autem, quare demonstratio propter quid maxime fit in prima figura, haec est. Nam in prima figura medius terminus subiicitur maiori extremitati, quae est praedicatum conclusionis, et praedicatur de minori termino, qui est subiectum conclusionis. Oportet autem in demonstratione propter quid medium esse causam passionis, quae praedicatur in conclusione de subiecto. Et unus modus dicendi per se est quando subiectum est causa praedicati, ut interfectum interiit, sicut supra dictum est; et hoc competit primae figurae, in qua medium subiicitur maiori extremitati, ut dictum est. But the reason why demonstration propter quid is best made in the first figure is this: in the first figure the middle term is both subjected, to the major extreme, which is the predicate of the conclusion, an(I predicated of the minor extreme, which is the subject of the conclusion. Now in demonstrations propter quid the middle must be the cause of the proper attribute which is predicated of the subject in the conclusion. Furthermore, one of the modes of “saying per se” is when the subject is the cause of the predicate, as “being butchered, he died,” as has been explained above; and this is verified in the first figure in which the middle is subject to the major extreme, as has been said.
Secundam rationem ponit; ibi: postea ipsius quod quid est etc., quae talis est. Quod quid est potissimum locum in demonstrativis scientiis habet, quia, sicut dictum est, definitio aut est principium demonstrationis, aut conclusio, aut demonstratio positione differens. Ad investigandum autem definitionem sola prima figura convenit. Nam in sola prima figura concluditur universalis affirmativa, quae sola competit ad scientiam quod quid est. Nam quod quid est per affirmationem cognoscitur: praedicatur enim definitio de definito affirmative et universaliter; non enim quidam homo est animal bipes, sed omnis homo. Ergo prima figura maxime est faciens scire et accommoda demonstrationibus. The second reason (79a23) is this: the quod quid est [i.e., the definition as signifying the essence] plays a most important role in demonstrative sciences because, as has been said, a definition is either the principle of a demonstration or the conclusion or a demonstration differing in the position of its terms. Now when it is a question of formulating a definition the first figure alone is suitable. For that is the only figure in which a universal affirmative is concluded, which alone produces science of the quod quid est. For the quod quid est is known through an affirmation. Furthermore, the definition is predicated affirmatively and universally of the defined: for it is not some man that is a two-legged animal, but every man. Therefore, the first figure is foremost in causing scientific knowledge and is best accommodated to demonstrations.
Tertiam rationem ponit; ibi: amplius haec quidem etc., quae talis est. Aliae figurae in demonstrationibus indigent prima; prima autem non indiget aliis; ergo prima figura efficacius facit scire quam aliae. Quod autem aliae figurae indigeant prima ex hoc manifestum est, quod oportet ad perfectam scientiam habendam, quod propositiones mediatae, quae sumuntur in demonstrationibus, ad immediata reducantur. Quod quidem fit dupliciter, scilicet densando media et augmentando. Then (79a30) he gives the third reason which is this: For purposes of demonstration the other figures need the first, but the first does not need them. Therefore, the first figure is better equipped for causing scientific knowledge than the others. That the others need the first is evidenced by the fact that if complete scientific knowledge is to be had, the mediate propositions present in demonstrations must be reduced to immediate ones. But this reduction is made in two ways, namely, by condensing middles [i.e., by inserting fresh middles, the movement proceeding from the extremes toward the middle first employed] and by expanding outwards [i.e., by going from the middle toward the remote extreme].
Densando quidem, quando medium acceptum mediate coniungitur utrique extremorum, vel alteri. Unde, quando accipiuntur media alia inter medium primum et extrema, fit quasi quaedam condensatio mediorum. Sicut si acciperetur primo sic: omne e est c; omne c est a: et deinde inter c et e, sumatur medium d; et inter c et a medium b. Augmentando autem, quando medium est immediatum minori extremitati, et mediatum maiori. Tunc enim oportet accipere plura media alia supra medium primo acceptum. Ut si dicatur: omne e est d; omne d est a; et postea supra d accipiantur alia media. It is done by condensing when the middle actually employed is joined mediately to each of the extremes or only to the minor extreme. Hence when other middles are introduced between the first middle and the extremes, the middles, as it were, close in. Thus, if someone first says, “Every E is C,” [minor], “Every C is A,” [major], and then a middle, say D, is introduced between C and E, and a middle B between C and A. It is done by expanding when the middle is immediate to the minor extreme and mediate to the major: for then it is necessary to introduce several other middles more general than the middle first taken. Thus, if one says, “Every E is D,” “Every D is A,” and the later other middles more general than D are introduced [between D and A].
Haec autem condensatio et augmentatio mediorum fit solum per primam figuram: tum, quia solum in prima figura concluditur universalis affirmativa; tum, quia solum in prima figura medium sumitur inter extrema. In secunda autem figura medium accipitur extra extrema, quasi praedicatum de eis. In tertia vero figura, infra extrema, quasi subiectum de eis. Now these processes of condensing and expanding can be performed only in the first figure, both because the first figure is the only one which yields a universal affirmative conclusion, and because it is only in the first figure that the middle lies between both extremes. For in the second figure the middle lies outside the extremes, as being predicated of them; in the third figure [the middle] is below each of the extremes, as being subjected to them.
Deinde cum dicit: sicut autem esse etc., docet quomodo propositio negativa possit esse immediata. Et circa hoc duo facit. Then (79a33) he teaches how a negative proposition can be immediate. In regard to this he does two things.
Primo, proponit intentum, dicens quod sicut contingit a esse in b individualiter, idest immediate, sic et conceditur non esse, idest ita potest concedi quod propositio significans a non esse in b sit immediata. Unde exponit, quid est individualiter esse vel non esse, scilicet quando affirmativa vel negativa non habet medium per quod probetur. First (79a33), he states his intention, saying that “just as it is possible for every A to be in B atomically,” i.e., immediately, “so too not to be,” i.e., so too it is possible for a proposition signifying that A is not in B to be immediate. Then he goes on to explain what it is to be or not to be atomically, namely, when the affirmation or negation does not have middle through which it might be proved.
Secundo; ibi: cum igitur aut a quidem etc., manifestat propositum. Et circa hoc duo facit: primo, ostendit quomodo propositio negativa sit mediata; secundo, quomodo sit immediata; ibi: si vero neutrum et cetera. Circa primum duo facit: primo, manifestat propositum; secundo, ostendit quoddam quod supposuerat; ibi: quod autem contingit b non esse et cetera. Secondly (79a36), he elucidates his proposition. In regard to this he does two things. First, he shows when a negative proposition is mediate. Secondly, when immediate (79b12). Concerning the first he does two things. First, he clarifies his proposition. Secondly, he proves something he had presupposed (79b5).
Dicit ergo primo quod cum a, idest maior terminus, aut b, idest minor terminus, sunt in quodam toto, sicut species in genere, aut etiam ambo sunt sub aliquo genere, non contingit a non esse in b primo, idest non contingit quod haec propositio, nullum b est a, sit immediata. Et primo manifestat hoc quando a est in quodam toto, scilicet c; b autem in nullo; ut puta, si a sit homo, c substantia, b quantitas: potest enim syllogismus fieri ad probandum quod a nulli b insit per hoc, quod c omni a inest, b autem nulli; ut si fiat syllogismus in secunda figura, talis: He says therefore first (79a36), that when A, i.e., the major term, or B, i.e., the minor term, is in some whole as a species in its genus, or both are under diverse genera or predicaments, it does not occur that A is not in B first, i.e., it does not occur that this proposition, “No B is A,” is immediate. First, then, he manifests this when A is in some whole, say in Q and B is in no whole. For example, if A is “man,” C is “substance,” and B is “quantity,” it is possible to form a syllogism to prove that A is in no B on the ground that C is in every A and in no B. Thus we get a syllogism in the second figure:
omnis homo est substantia;
nulla quantitas est substantia;
ergo nulla quantitas est homo.
Every man is a substance;
No quantity is a substance:
Therefore, no quantity is a man.
Et similiter est, si b, idest minor terminus, sit in quodam toto, ut in d, a autem non sit in aliquo toto; syllogizari poterit quod a sit in nullo b. Ut sit a substantia, b linea, d quantitas; et fiat syllogismus in prima figura sic: In like manner, if B, i.e., the minor term, is in some whole, say D, but A is not in that whole, it is possible to syllogize that A is in no B. For example if A is “substance,” B “line,” and D “quantity,” we get a syllogism in the first figure:
nulla quantitas est substantia;
omnis linea est quantitas;
ergo nulla linea est substantia.
No quantity is a substance;
Every line is a quantity;
Therefore, no line is a substance.
Eodem autem modo poterit demonstrari conclusio negativa, si utrumque sit in quodam toto; ut si sit a linea, c quantitas, b albedo, et d qualitas; potest syllogizari in secunda figura, et in prima. In secunda figura sic: In the same way, a negative conclusion could be demonstrated if either is in some whole. For example, if A is “line,” C “quantity,” B “whitenem” and D “quality,” it is possible to form a syllogism in the first and in the second figure. In the second figure thus:
omnis linea est quantitas;
nulla albedo est quantitas;
ergo nulla albedo est linea.
Every line is a quantity;
No whiteness is a quantity:
Therefore, no whiteness is a line.
In prima figura sic: And in the first figure thus:
nulla qualitas est linea;
omnis albedo est qualitas;
ergo nulla albedo est linea.
No quality is a line;
Every whiteness is a quality:
Therefore, no whiteness is a line.
Est autem intelligendum, propositionem negativam esse mediatam, utroque terminorum existente in quodam toto, non quidem in eodem, sed in diversis. Si enim sint in eodem toto, erit propositio immediata, sicut, nullum rationale est irrationale, vel nullum bipes est quadrupes. We should understand, however, that a negative proposition is mediate, when both terms exist in some whole which is not the same but different for each. For if they are in the same whole, the proposition will be immediate, as “No rational being is irrational,” or “No biped is a quadruped.”
Deinde cum dicit: quod autem contingit etc., manifestat quod supposuerat, scilicet quod, altero extremorum existente in aliquo toto, alterum non sit in eodem, dicens quod manifestum est ex coordinationibus, scilicet praedicamentorum diversorum, quae non commutantur ad invicem. Scilicet quia id quod est in uno praedicamento, non est in altero, manifestum est quod contingat b non esse in toto, in quo est a, aut e converso, quia videlicet contingit unum terminorum accipi in uno praedicamento, in quo non est alius. Sit enim una coordinatio praedicamenti acd, puta praedicamentum substantiae; et alia coordinatio sit bef, puta praedicamentum quantitatis. Si ergo nihil eorum, quae sunt in coordinatione acd, de nullo praedicatur eorum, quae sunt in coordinatione bef; a autem sit in p, quasi in quodam generalissimo, quod sit principium totius primae coordinationis; manifestum est quod b non erit in p, quia sic coordinationes, idest praedicamenta, commutarentur. Similiter autem est si b sit in quodam toto, ut puta in e; manifestum est quod a non erit in e. Then (79b5) he explains something he had presupposed, namely, “on condition that one of the extremes exist in some whole and that the other be not in the same,” saying that it is “clear from the ‘orderings’ of the various predicaments, which are” not mutually interchangeable. In other words, because that which is in one predicament is not in another, it is plain that B happens not to be in the whole in which A is, or vice versa, because one of the terms happens to be taken from one predicament in which the other is not found. Thus, let one ordering of the predicaments be ACD, say the predicament of substance, and another ordering be BEF, say the predicament of quantity. Then if none of those in the ordering ACD be predicated of none in the ordering BEF, while A is in P as in that most general item which is the principle of the whole first ering, it is plain that B is not in P, because then the orderings, i.e., the predicaments, would be interchanged. Similarly, if B is in some whole, say in E, it is plain that A is not in E.
Deinde cum dicit: si vero neutrum etc., ostendit quomodo propositio negativa sit immediata dicens quod, si neutrum sit in toto aliquo, scilicet, neque a neque b; et tamen a non sit in b, necesse est quod haec sit immediata, nullum b est a. Quia si acciperetur aliquod medium ad syllogizandum eam, oporteret quod alterum ipsorum esset in aliquo toto; oporteret enim syllogismum fieri, aut in prima figura, aut in secunda. In tertia enim figura non potest concludi universalis negativa, qualem oportet esse propositionem immediatam. Si quidem syllogismus fit in prima figura, oportet quod b sit in quodam toto, quia b est minor extremitas, et in prima figura oportet semper minorem propositionem esse affirmativam. Non enim fit syllogismus in prima figura ex maiori affirmativa et minori negativa. Then (79b12) he indicates how a negative proposition may be immediate, saying that “if neither is in some whole,” i.e., neither A nor B, and A is not in B, it is necessary that this proposition, “No B is A,” be immediate. Because if a middle were taken to syllogize it, then one of them would have to be in some whole, for the syllogism would have to be made either in the first figure or in the second, since in the third figure a universal negative cannot be concluded, as is required for an immediate proposition. However, if it is made in the first, B would have to be in some whole, because B is the minor extreme, and in the first figure the minor proposition must always be affirmative. For a syllogism with the major affirmative and the minor negative cannot be formed in the first figure.
Sed si syllogismus erit in media figura, contingit quodcunque, idest vel a vel b, esse in toto quodam; quia in media figura potest esse negativa tam prima quam secunda propositio. Nunquam tamen potest esse, neque in prima neque in secunda, utraque propositio negativa. Et ideo oportet quod, altera existente affirmativa, alterum extremorum sit in quodam toto. Sic igitur patet quod propositio negativa est immediata, quando neutrum terminorum est in quodam toto. Non autem potest dici quod quamvis neutrum sit in quodam toto, potest tamen accipi medium ad ipsam concludendam, scilicet si accipiatur medium convertibile: quia oportet tale medium esse quod sit prius et notius; et hoc est vel genus vel definitio, quae non est sine genere. But if it be in the second figure, either may, i.e., A or B, may be in a whole, because in the second figure the first proposition may be negative in some moods and the minor in other moods. Of course it is never permitted, neither in the first nor the second, to have both propositions negative. And so it is required that when either proposition is affirmative, one of the extremes must be in some whole. Thus it is clear that a negative proposition is immediate, when neither of its terms is in some whole. This does not mean, however, that although neither is in some whole, a middle could be found to conclude it, namely, if one were to take a convertible middle, because it is necessary that such a middle be prior and better known. And this is either the genus itself or the definition, which is not without a genus.
Deinde cum dicit: manifestum igitur est etc., concludendo epilogat quod dictum est. Et litera plana est ex dictis. Then (79b21) he concludes and summarizes what has been said. Here the text is sufficiently clear.

Lectio 27
Ἄγνοια δ' ἡ μὴ κατ' ἀπόφασιν ἀλλὰ κατὰ διάθεσιν λεγομένη ἔστι μὲν ἡ διὰ συλλογισμοῦ γινομένη ἀπάτη, b23. Ignorance — defined not as the negation of knowledge but as a positive state of mind — is error produced by inference.
αὕτη δ' ἐν μὲν τοῖς πρώτως ὑπάρχουσιν ἢ μὴ ὑπάρχουσι συμβαίνει διχῶς· ἢ γὰρ ὅταν ἁπλῶς ὑπολάβῃ ὑπάρχειν ἢ μὴ ὑπάρχειν, ἢ ὅταν διὰ συλλογισμοῦ λάβῃ τὴν ὑπόληψιν. τῆς μὲν οὖν ἁπλῆς ὑπολήψεως ἁπλῆ ἡ ἀπάτη, τῆς δὲ διὰ συλλογισμοῦ πλείους. (1) Let us first consider propositions asserting a predicate's immediate connexion with or disconnexion from a subject. Here, it is true, positive error may befall one in alternative ways; for it may arise where one directly believes a connexion or disconnexion as well as where one's belief is acquired by inference. The error, however, that consists in a direct belief is without complication; but the error resulting from inference — which here concerns us — takes many forms.
μὴ ὑπαρχέτω γὰρ τὸ Α μηδενὶ τῷ Β ἀτόμως· οὐκοῦν ἐὰν συλλογίζηται ὑπάρχειν τὸ Α τῷ Β, μέσον λαβὼν τὸ Γ, ἠπατημένος ἔσται διὰ συλλογισμοῦ. ἐνδέχεται μὲν οὖν ἀμφοτέρας τὰς προτάσεις εἶναι ψευδεῖς, ἐνδέχεται δὲ τὴν ἑτέραν μόνον. εἰ γὰρ μήτε τὸ Α μηδενὶ τῶν Γ ὑπάρχει μήτε τὸ Γ μηδενὶ τῶν Β, εἴληπται δ' ἑκατέρα ἀνάπαλιν, ἄμφω ψευδεῖς ἔσονται. ἐγχωρεῖ δ' οὕτως ἔχειν τὸ Γ πρὸς τὸ Α καὶ Β ὥστε μήτε ὑπὸ τὸ Α εἶναι μήτε καθόλου τῷ Β. τὸ μὲν γὰρ Β ἀδύνατον εἶναι ἐν ὅλῳ τινί (πρώτως γὰρ ἐλέγετο αὐτῷ τὸ Α μὴ ὑπάρχειν), τὸ δὲ Α οὐκ ἀνάγκη πᾶσι τοῖς οὖσιν εἶναι καθόλου, ὥστ' ἀμφότεραι ψευδεῖς. ἀλλὰ καὶ τὴν ἑτέραν ἐνδέχεται ἀληθῆ λαμβάνειν, οὐ μέντοι ὁποτέραν ἔτυχεν, ἀλλὰ τὴν (80a.) Α Γ· ἡ γὰρ Γ Β πρότασις ἀεὶ ψευδὴς ἔσται διὰ τὸ ἐν μηδενὶ εἶναι τὸ Β, τὴν δὲ Α Γ ἐγχωρεῖ, οἷον εἰ τὸ Α καὶ τῷ Γ καὶ τῷ Β ὑπάρχει ἀτόμως (ὅταν γὰρ πρώτως κατηγορῆται ταὐτὸ πλειόνων, οὐδέτερον ἐν οὐδετέρῳ ἔσται). διαφέρει δ' οὐδέν, οὐδ' εἰ μὴ ἀτόμως ὑπάρχει. Thus, let A be atomically disconnected from all B: then the conclusion inferred through a middle term C, that all B is A, will be a case of error produced by syllogism. Now, two cases are possible. Either (a) both premisses, or (b) one premiss only, may be false. (a) If neither A is an attribute of any C nor C of any B, whereas the contrary was posited in both cases, both premisses will be false. (C may quite well be so related to A and B that C is neither subordinate to A nor a universal attribute of B: for B, since A was said to be primarily disconnected from B, cannot have a genus, and A need not necessarily be a universal attribute of all things. Consequently both premisses may be false.) On the other hand, (b) one of the premisses may be true, though not either indifferently but only the major A-C since, B having no genus, the premiss C-B will always be false, while A-C may be true. This is the case if, for example, A is related atomically to both C and B; because when the same term is related atomically to more terms than one, neither of those terms will belong to the other. It is, of course, equally the case if A-C is not atomic.
Ἡ μὲν οὖν τοῦ ὑπάρχειν ἀπάτη διὰ τούτων τε καὶ οὕτω γίνεται μόνως (οὐ γὰρ ἦν ἐν ἄλλῳ σχήματι τοῦ ὑπάρχειν συλλογισμός), Error of attribution, then, occurs through these causes and in this form only — for we found that no syllogism of universal attribution was possible in any figure but the first.
Postquam philosophus determinavit de syllogismo demonstrativo, per quem acquiritur scientia, hic determinat de syllogismo, per quem inducitur in nobis ignorantia sive deceptio. Et circa hoc duo facit: primo enim ostendit qualis ignorantia per syllogismum induci possit; secundo, ostendit modum, quo talis syllogismus procedit; ibi: simplicis quidem igitur opinionis et cetera. After determining about the demonstrative syllogism through which science is acquired, the Philosopher here determines concerning the syllogism through which ignorance or deception is produced in us. In regard to this he does two things. First, he shows what sort of ignorance can be induced by a syllogism. Secondly, he shows the mode in which such a syllogism proceeds (79b28).
Distinguit ergo primo duplicem ignorantiam; quarum una est secundum negationem; alia est secundum dispositionem. Ignorantia quidem secundum negationem est quando homo omnino nihil scit de re. Et haec est ignorantia in non attingendo, ut philosophus dicit in IX Metaph.; sicut patet de rustico, qui omnino nihil scit de triangulo, an habeat tres angulos aequales duobus rectis. Ignorantia autem secundum dispositionem est quando aliquis habet quidem aliquam dispositionem in cognoscendo, sed corruptam: dum scilicet existimat aliquid circa rem sed falso; dum vel existimat esse, quod non est, vel non esse quod est. Et haec ignorantia idem est, quod error. He distinguishes therefore first (79b23) a twofold ignorance, one of which is negative and the other a positive state. There is ignorance in a negative way, when a man has no scientific knowledge at all about a thing. This ignorance consists in not attaining, as the Philosopher says in Metaphysics IX, and is exemplified in a peasant who knows absolutely nothing about whether a triangle has three angles equal to two right angles. But ignorance is present as a positive state, when one does have a definite opinion but it is unsound. For example, when he falsely thinks something about a thing, either because he thinks something to be which is not, or something not to be which is. And this ignorance is the same as error.
Prima ergo ignorantia non fit per syllogismum: sed secunda per syllogismum fieri potest. Et tunc vocatur deceptio. Haec autem ignorantia sive deceptio potest contingere circa duo. Uno quidem modo circa ea quae sunt prima principia et immediata, dum scilicet opinatur quis opposita principiis: quae quidem etsi non possit opinari interius in mente, ut supra dictum est, quia non cadunt sub apprehensione; potest tamen eis contradicere secundum vocem, et secundum quamdam falsam imaginationem, ut dicitur de quibusdam negantibus principia in IV Metaphys. Alio modo circa conclusiones, quae non sunt prima et immediata. Et prima quidem ignorantia sive deceptio opponitur cognitioni intellectus. Secunda autem cognitioni scientiae. Now the first ignorance is not produced by a syllogism, but the second can be, and then it is called deception. Such ignorance or deception can concern two things: first, those things which are first and immediate principles, namely, when a person opines things opposed to the principles. And although he cannot so opine inwardly in the mind, as has been stated above, because these things do not fall under apprehension, nevertheless he can contradict them orally and according to a false imagination, as is said in Metaphysics IV of some who deny the principles. Secondly, it might be concerned with conclusions which are not first and immediate. Furthermore, the first of these ignorances or deceptions is opposed to the knowledge which is understanding, and the second to the knowledge which is science.
Utraque autem ignorantia dispositionis, sive sit de his quae sunt prima, sive sit de his quae non sunt prima, potest homini provenire dupliciter. Uno modo simpliciter, quando scilicet absolute absque aliquo ductu rationis existimat falsum, sive affirmando, sive negando. Alio modo, quando inducitur ad falsum existimandum, per aliquam rationem syllogisticam, sicut philosophus dicit in IV Metaphys., quod quidam principiis contradicunt, velut rationibus persuasi; alii vero, non quasi ratione persuasi, sed propter ineruditionem vel pertinaciam, volentes in omnibus quaerere demonstrationem. Each of these states of ignorance, whether concerned with things that are first or things not first, can befall a man in two ways: first, straightway, when independently of any process of reasoning he thinks a falsehood by affirming or denying. In another way, when he is brought to his opinion through some syllogized reason. Hence the Philosopher says in Metaphysics IV that some contradict the principles as though persuaded by reasons; but others, not as though persuaded by reasons, but through lack of erudition or through wilfullness which demands to have a demonstration in all matters.
Deinde cum dicit: simplicis quidem igitur etc., ostendit quomodo praedictae ignorantiae causantur. Et primo quomodo causatur ignorantia, quae est per syllogismum; secundo, quomodo proveniat ignorantia homini sine syllogismo; ibi: manifestum est autem et cetera. Circa primum duo facit; primo, ostendit quomodo causetur ignorantia per syllogismum in primis et immediatis; secundo, quomodo causetur in his, quae non sunt prima et immediata; ibi: in his autem quae non individua et cetera. Circa primum duo facit: primo, ostendit quomodo causetur ignorantia, qua existimatur esse quod non est; secundo, quomodo causetur ignorantia, qua existimatur non esse quod est; ibi: quae vero ipsius non esse et cetera. Circa primum tria facit: primo, proponit modum, quo praedicta ignorantia communiter causatur; secundo, assignat diversitates circa hoc contingentes; ibi: contingit quidem etc.; tertio, respondet tacitae quaestioni; ibi: ipsius quidem igitur esse et cetera. Then (79b28) he shows how these ignorances are caused. First, how the ignorance produced by a syllogism is caused. Secondly, how ignorance befalls a man without syllogizing (81a38) [L. 30]. Concerning the first he does two things. First, he states how ignorance in regard to first and immediate principles is engendered by a syllogism. Secondly, how it is caused in regard to things not first and immediate (80b17) [L. 29]. Concerning the first he does two things. First, he states how the ignorance is caused whereby that which is not is believed to be. Secondly, that whereby that which is is believed not to be (80a8) [L. 28]. Regarding the first he does three things. First, he states how the aforesaid ignorance is generally caused. Secondly, he assigns its various possible forms (79b31). Thirdly, he answers a tacit question (80a6).
Dicit ergo primo quod falsae opinionis, quam supra vocavit simplicem, est simplex deceptio, idest uno solo modo ad hoc pervenitur. Non enim causatur ex ratione, quae diversificari potest, sed magis ex defectu rationis, qui non diversificatur per diversos modos, sicut nec aliae negationes secundum propriam rationem. He says therefore first (79b28) that the deception according to false opinion which above was said to occur straightway is simple, i.e., is engendered in only one way. For it is not caused by a reason, which might vary, but by the lack of a reason which [lack] is not diversified into various modes each having its own characteristics any more than other negations are.
Sed quia ratio falsa multipliciter variari potest, inde est quod huiusmodi ignorantia, quae fit per syllogismum, multipliciter accidere potest, secundum quod multipliciter potest esse falsus syllogismus. Ponit autem communem modum dicens: non sit enim a in nullo b individualiter, idest sit haec propositio vera immediata, nullum b est a: ponuntur enim duae negationes, loco unius; puta, si dicamus, nulla quantitas est substantia, secundum doctrinam supra positam de negativis immediatis. Si quis ergo concludat oppositum huius, per aliquem syllogismum, ostendens scilicet omne b esse a, accipiens pro medio c, erit deceptio per syllogismum. Yet because false reasons can be various and many, this ignorance, when it is engendered by a syllogism can occur in many ways according to the several ways in which a syllogism can be false. The common way is set forth when he says, “Let A be in no B individually” (79b29), i.e., let this proposition be immediately true, “No B is A.” For two negations are stated in the place of one— for instance if we should say, “No quality is a substance”— according to the doctrine on immediate negative propositions presented above. Therefore, if someone were to conclude the opposite of this with a syllogism, taking C as a middle to show that every B is A, there will be a deception through syllogism.
Deinde cum dicit: contingit quidem igitur etc., ostendit quot modis potest hoc variari. Est autem sciendum quod falsa conclusio non concluditur nisi falso syllogismo. Syllogismus autem potest esse falsus dupliciter. Uno modo, quia deficit in forma syllogistica. Et hic non est syllogismus, sed apparens. Alio modo, quia utitur falsis propositionibus. Et hic quidem est syllogismus propter syllogisticam formam, est autem falsus propter falsas propositiones assumptas. In disputatione ergo dialectica, quae fit circa probabilia, usus est utriusque falsi syllogismi, quia talis disputatio procedit ex communibus. Et ita in ea error attendi potest et circa materiam quam assumit, quae est communis, et etiam circa formam, quae est communis. Then (79b31) he shows in how many ways this can vary. And first of all it should be noted that a false conclusion is not concluded except when the syllogism is false. But a syllogism can be false in two ways: first, because it lacks syllogistic form, in which case it is not a syllogism but appears to be one. In a second way, because it employs false propositions, in which case it is a syllogism as to form, but it is a false one because of the false propositions used. Therefore, in a dialectical disputation, which bears on probables, use could be made of both types of false syllogism, because such a disputation proceeds from common premises. Consequently, error can arise in it both as regards the matter employed, which is common, and also as regards the form, which is common.
Sed in disputatione demonstrativa, quae est circa necessaria, non est usus, nisi illius syllogismi qui est falsus propter materiam; quia, ut dicitur in I topicorum, paralogismus disciplinae procedit ex propriis disciplinae, sed non ex veris. Unde, cum forma syllogistica sit inter communia computanda, paralogismus disciplinae, de quo nunc agitur, non peccat in forma, sed solum in materia, et circa propria, non circa communia. Et ideo primo, ostendit quomodo huiusmodi syllogismus procedat ex duabus falsis; secundo, quomodo procedat ex altera falsa; ibi: sed alteram contingit et cetera. Primum autem contingit dupliciter, quia falsa propositio, aut est contraria verae, aut contradictoria. Primo ergo ostendit quomodo huiusmodi syllogismus procedat ex duabus falsis contrariis veris; secundo, quomodo accipitur contradictio; ibi: potest autem sic se habere et cetera. But in a demonstrative disputation, which bears on necessary things, the type of syllogism used is the one which could be false only on account of the matter, because as it is stated in Topics I, a paralogism in a given discipline proceeds from things proper to the discipline, but not from true things. Hence, since syllogistic form must be counted among the common things, paralogism within a discipline-which is the matter under discussion-is not defective in form but only in matter, and furthermore in regard to proper and not in regard to common things. And so: First, he tells how such a syllogism might proceed from two false premises. Secondly, how it might proceed from one or the other premise being false (79b40). But since the first can occur in two ways, namely, the false proposition might be contrary to the true or contradictory to it, therefore: First, he shows how such a syllogism might proceed from two false statements that are contrary to true statements. Secondly, how a contradiction is accepted (79b34).
Dicit ergo primo quod in praedicto syllogismo deceptionem causante, contingit quandoque utrasque propositiones esse falsas, et quandoque alteram tantum. Utrasque autem falsas, uno modo si accipiamus contrarias verarum. Habeat enim ita se c ad a et ad b, quod nullum c sit a, et quod nullum b sit c. Si ergo accipiantur contraria horum, scilicet, omne c est a; omne b est c; utraeque propositiones erunt falsae totaliter. Puta si dicam: omnis qualitas est substantia; omnis quantitas est qualitas; ergo omnis quantitas est substantia. First, then (79b30) he says that in a syllogism causing deception it sometimes happens that both premises are false, and sometimes only one. Let us take”the case of two false premises, each contrary to what is true. For example, suppose that C is so related to A and to B. that no C is A and no B is C. Now if we take the contraries of these, namely, that every C is A and every B is C, these two propositions will be completely false. For example, if I should say: “Every quality is a substance; every quantity is a quality: therefore, every quantity is a substance.”
Deinde cum dicit: potest autem sic se habere etc., ostendit quomodo possunt esse ambae falsae, et non sunt contrariae veris, sed contradictoriae. Puta, si sic se habeat c ad a et ad b, quod nec contineatur totaliter sub a, neque universaliter insit b. Puta si accipiamus, perfectum vel ens in actu, et procedamus sic: omne perfectum est substantia; omnis quantitas est perfecta; ergo et cetera. Manifestum est quod utraque est falsa, sed non totaliter. Sunt enim contradictoriae earum verae, scilicet, quoddam perfectum non est substantia, et, quaedam quantitas non est perfecta. Contrariae autem sunt falsae, scilicet, nullum perfectum est substantia, et nulla quantitas est perfecta. Then (79b34) he shows how both premises can be false and not contrary, but contradictory, to what is true. For example, if C is so related to A and to B, that it is neither contained totally under A nor is universally in B. Thus one might take C as “perfect” or “actual being,” and proceed in the following way: “Every perfect thing is a substance; every quantity is perfect: therefore, every quantity is a substance. Now although both are false, they are not entirely so, for their contradictories are true, namely, “Some perfect things are not substances,” and “Some quantity is not perfect”; but their contraries are false, namely, “No perfect thing is a substance,” and “No quantity is perfect.”
Quod autem c non universaliter insit b (idest, quod ista non sit vera: omnis quantitas est perfecta, quae erat minor; ut, omne b est c) probat per hoc, quod b non potest contineri sub aliquo toto, quod de eo universaliter praedicetur. Et hoc ideo, quia haec propositio: nullum b est a, dicebatur esse immediata, quod est a non inesse b primo. Dictum est autem supra illas negativas esse immediatas, quarum neuter terminorum est sub aliquo toto. But that C is not in B universally (i.e., that the statement, “Every quantity is perfect,” which was the minor, is not true), i.e., that “Every B is C,” is not true he proves by the fact that B cannot be contained under any whole that might be predicated universally of it. And this is so because the proposition, “No B is A,” was said to be immediate, which means that A is universally not in B. But, as has been stated above, those negative propositions are immediate, neither of whose terms is under a whole.
Sed videtur haec probatio non esse sufficiens, quia de eo etiam quod non est sub aliquo toto, sicut species sub genere, potest aliquid universaliter praedicari. Non enim solum genus aut differentia universaliter praedicatur, sed etiam proprium. Sed dicendum est quod licet praedicta probatio non sit efficax, communiter loquendo, est tamen efficax in proposito. Quia, sicut in I topicorum dicitur, paralogismus disciplinae, de quo hic loquitur, procedit ex convenientibus disciplinae. Unde intendit uti talibus mediis, qualibus utitur demonstrator. Demonstrationis autem medium est definitio, ut supra dictum est. Unde et syllogismus, de quo hic loquitur, intendit uti definitione pro medio. Definitio autem continet genus et differentiam. Unde oportet id quod universaliter praedicatur in hoc syllogismo, continere id, in quo est subiectum, sicut in toto. Nevertheless, this proof does not seem sufficient, because something can be predicated universally even of that which is not under some whole as a species under a genus. For the genus and the difference are not the only things predicated universally, but a property is too. But it must be said that although the proof in question is not efficacious generally speaking, it is in this case. Because, as it is stated in Topics I, a paralogism within a discipline— with which we are concerned here— proceeds from items that accord with the discipline. Hence he intends to use such middles as a demonstrator uses. But the middle of a demonstration is the definition, as has been stated above. Therefore, even in the syllogism of which he is speaking, he intends to use a definition as the middle. Now a definition contains the genus and difference. Hence that which is predicated universally in this syllogism should contain that in which the subject is, as in a whole.
Quod autem a non universaliter insit ipsi c (idest, quod ista non sit universaliter vera: omne perfectum est substantia, quae erat maior, ut, omne c est a), probat per hoc, quod non est necesse de quocunque universali, quod insit universaliter omnibus quae sunt: quia nullum praedicamentum praedicatur de his, quae continentur sub alio praedicamento; neque etiam universaliter praedicatur de his quae communiter consequuntur ens, quae sunt actus et potentia, perfectum et imperfectum, prius et posterius, et alia huiusmodi Again, that A is not in C universally, i.e., that the statement, “Every perfect thing is a substance,” is not universally true is proved by the fact that it is not required of every universal that it be universally in all things that are: because no predicament is predicated of things contained under another predicament nor universally predicated of those items that commonly follow upon being, namely, act and potency, perfect and imperfect, prior and posterior, and the like.
Deinde cum dicit: sed alteram contingit etc., ostendit quomodo praedictus syllogismus procedat ex altera vera, et altera falsa. Et dicit quod in praedicto syllogismo contingit accipere alteram veram, scilicet maiorem, quae est a.c, altera existente falsa, scilicet minore, quae est b.c. Et quod propositio minor, quae est b.c, semper sit falsa probat, sicut et supra, per hoc quod b in nullo est, sicut in toto. Sed quod haec propositio a.c possit esse vera, altera existente falsa, probat in terminis. Sit enim ita, quod a insit b et c individualiter, idest immediate, sicut genus proximis speciebus, ut color albedini et nigredini. Manifestum est enim secundum hoc, quod maior erit vera, scilicet, omne c est a, puta: omnis albedo est color; minor autem est falsa, scilicet, omnis nigredo est albedo; quia quando aliquid primo praedicatur de pluribus, neutrum istorum plurium de neutro praedicatur. Prima enim praedicatio generis est de oppositis speciebus. Then (79b40) he shows how the aforesaid syllogism might proceed from one true and one false. And he says that in the aforesaid syllogism we might take one true, i.e., the major, which is AC, and the other false, namely, the minor, which is BC. That the minor proposition, which is BC, is always the false one he proves, as he did above, on the ground that B is in no C as in a whole. But that this proposition AC could be true, while the other is false he proves in terms. Let A be in B and in C individually, i.e., immediately, as a genus in its species; for example, as color is to whiteness and blackness. Now under these conditions it is obvious that the major will be true, namely, “Every C is A,” for example, “Every whiteness is a color,” and the minor false, namely, “Every blackness is whiteness”: for when something is predicated “first” of several, none of the several is predicated of any other of them. For the first predication of a genus is of species which are opposite.
Est autem circa hoc dubitatio: quia his terminis positis, non sequitur conclusio falsa, sed vera. Erit enim conclusio quod a insit b, cui suppositum est inesse individualiter. Sed dicendum quod hoc exemplum ponitur solum ad manifestandum quomodo possit esse maior vera et minor falsa. Sed hoc exemplum non habet locum in proposito, ubi quaeritur conclusio falsa. Et ideo philosophus subiungit: differt autem nihil, nec si non individualiter insit. Possumus enim accipere tales terminos, quod a non insit b individualiter, neque aliquo modo; immo potius ab eo individualiter removeatur. Nec est etiam necesse quod insit c individualiter, quia non est necessarium quod demonstrator utatur solum propositionibus immediatis; sed etiam his, quae per immediata fidem acceperunt. Accipere ergo possumus alios terminos, ad propositum pertinentes, ut si accipiamus pro medio substantiam intellectualem: omnis enim intelligentia est substantia; minor autem est falsa: omnis quantitas est intelligentia. Unde sequitur conclusio falsa. Here, too, a doubt arises, because from the terms used here, not a false but a true conclusion follows. For the conclusion will be that A is in B, it having also been assumed that it is in B individually. But it must be answered that this example was used merely to illustrate how the major could be true and the minor false; although it is of no use in a case where a false conclusion is sought. Hence the Philosopher at once adds, “It is equally the case of AC if not atomic.” However, we could take terms such that A is not in B either individually or in any way, but is rather immediately removed from B. Neither is it necessary that it be in C individually, because it is not necessary that a demonstrator employ only immediate propositions: for he may use ones which are supported by immediate propositions. We can, therefore, take other terms pertaining to the present case, for example, “intellectual substance,” as the middle, for “Every intelligence is a substance,” and take as the false minor, “Every quantity is an intelligence.” Hence a false conclusion follows.
Deinde cum dicit: ipsius quidem igitur etc., respondet tacitae quaestioni. Posset enim aliquis ab eo requirere quod ostenderet diversitatem huius syllogismi in aliis figuris. Sed ipse respondet quod deceptio, quae est ipsius esse, idest per quam aliquis existimat propositionem affirmativam falsam, potest fieri solum per primam figuram: quia in alia figura, scilicet in secunda, non potest fieri syllogismus affirmativus. Tertia autem figura non pertinet ad propositum, quia in ea non potest concludi universalis, quae principaliter intenditur in demonstratione, et in hoc syllogismo. Then (80a6) he answers a tacit question. For someone might request that he exemplify the diversity of such a syllogism in the other figures. But he answers that a deception, which bears on being, i.e., through which someone opines a false affirmative proposition, can be derived only by the first figure, because in the next figure, i.e., in the second, an affirmative syllogism cannot be formed. As for the third figure, it has no bearing on the case, because it cannot conclude a universal, which is principally intended in demonstration and in this syllogism.

Lectio 28
Caput 16 cont.
ἡ δὲ τοῦ μὴ ὑπάρχειν ἔν τε τῷ πρώτῳ καὶ ἐν τῷ μέσῳ σχήματι. πρῶτον οὖν εἴπωμεν ποσαχῶς ἐν τῷ πρώτῳ γίνεται, καὶ πῶς ἐχουσῶν τῶν προτάσεων. a8. On the other hand, an error of non-attribution may occur either in the first or in the second figure. Let us therefore first explain the various forms it takes in the first figure and the character of the premisses in each case.
ἐνδέχεται μὲν οὖν ἀμφοτέρων ψευδῶν οὐσῶν, οἷον εἰ τὸ Α καὶ τῷ Γ καὶ τῷ Β ὑπάρχει ἀτόμως· ἐὰν γὰρ ληφθῇ τὸ μὲν Α τῷ Γ μηδενί, τὸ δὲ Γ παντὶ τῷ Β, ψευδεῖς αἱ προτάσεις. a11. (c) It may occur when both premisses are false; e.g. supposing A atomically connected with both C and B, if it be then assumed that no C is and all B is C, both premisses are false.
ἐνδέχεται δὲ καὶ τῆς ἑτέρας ψευδοῦς οὔσης, καὶ ταύτης ὁποτέρας ἔτυχεν. ἐγχωρεῖ γὰρ τὴν μὲν Α Γ ἀληθῆ εἶναι, τὴν δὲ Γ Β ψευδῆ, τὴν μὲν Α Γ ἀληθῆ ὅτι οὐ πᾶσι τοῖς οὖσιν ὑπάρχει τὸ Α, τὴν δὲ Γ Β ψευδῆ ὅτι ἀδύνατον ὑπάρχειν τῷ Β τὸ Γ, ᾧ μηδενὶ ὑπάρχει τὸ Α· οὐ γὰρ ἔτι ἀληθὴς ἔσται ἡ Α Γ πρότασις· ἅμα δέ, εἰ καὶ εἰσὶν ἀμφότεραι ἀληθεῖς, καὶ τὸ συμπέρασμα ἔσται ἀληθές. a14. (d) It is also possible when one is false. This may be either premiss indifferently. A-C may be true, C-B false — A-C true because A is not an attribute of all things, C-B false because C, which never has the attribute A, cannot be an attribute of B; for if C-B were true, the premiss A-C would no longer be true, and besides if both premisses were true, the conclusion would be true.
ἀλλὰ καὶ τὴν Γ Β ἐνδέχεται ἀληθῆ εἶναι τῆς ἑτέρας οὔσης ψευδοῦς, οἷον εἰ τὸ Β καὶ ἐν τῷ Γ καὶ ἐν τῷ Α ἐστίν· ἀνάγκη γὰρ θάτερον ὑπὸ θάτερον εἶναι, ὥστ' ἂν λάβῃ τὸ Α μηδενὶ τῷ Γ ὑπάρχειν, ψευδὴς ἔσται ἡ πρότασις. φανερὸν οὖν ὅτι καὶ τῆς ἑτέρας ψευδοῦς οὔσης καὶ ἀμφοῖν ἔσται ψευδὴς ὁ συλλογισμός. a20. Or again, C-B may be true and A-C false; e.g. if both C and A contain B as genera, one of them must be subordinate to the other, so that if the premiss takes the form No C is A, it will be false. This makes it clear that whether either or both premisses are false, the conclusion will equally be false.
Ἐν δὲ τῷ μέσῳ σχήματι ὅλας μὲν εἶναι τὰς προτάσεις ἀμφοτέρας ψευδεῖς οὐκ ἐνδέχεται· ὅταν γὰρ τὸ Α παντὶ τῷ Β ὑπάρχῃ, οὐδὲν ἔσται λαβεῖν ὃ τῷ μὲν ἑτέρῳ παντὶ θατέρῳ δ' οὐδενὶ ὑπάρξει· δεῖ δ' οὕτω λαμβάνειν τὰς προτάσεις ὥστε τῷ μὲν ὑπάρχειν τῷ δὲ μὴ ὑπάρχειν, εἴπερ ἔσται συλλογισμός. εἰ οὖν οὕτω λαμβανόμεναι ψευδεῖς, δῆλον ὡς ἐναντίως ἀνάπαλιν ἕξουσι· τοῦτο δ' ἀδύνατον. ἐπί τι δ' ἑκατέραν οὐδὲν κωλύει ψευδῆ εἶναι, οἷον εἰ τὸ Γ καὶ τῷ Α καὶ τῷ Β τινὶ ὑπάρχοι· ἂν γὰρ τῷ μὲν Α παντὶ ληφθῇ ὑπάρχον, τῷ δὲ Β μηδενί, ψευδεῖς μὲν ἀμφότεραι αἱ προτάσεις, οὐ μέντοι ὅλαι ἀλλ' ἐπί τι. καὶ ἀνάπαλιν δὲ τεθέντος τοῦ στερητικοῦ ὡσαύτως. a27. In the second figure the premisses cannot both be wholly false; for if all B is A, no middle term can be with truth universally affirmed of one extreme and universally denied of the other: but premisses in which the middle is affirmed of one extreme and denied of the other are the necessary condition if one is to get a valid inference at all. Therefore if, taken in this way, they are wholly false, their contraries conversely should be wholly true. But this is impossible. On the other hand, there is nothing to prevent both premisses being partially false; e.g. if actually some A is C and some B is C, then if it is premised that all A is C and no B is C, both premisses are false, yet partially, not wholly, false. The same is true if the major is made negative instead of the minor.
τὴν δ' ἑτέραν εἶναι ψευδῆ καὶ ὁποτερανοῦν ἐνδέχεται. ὃ γὰρ ὑπάρχει τῷ Α παντί, καὶ τῷ Β ὑπάρχει· ἐὰν οὖν ληφθῇ τῷ μὲν Α ὅλῳ ὑπάρχειν (80b.) τὸ Γ, τῷ δὲ Β ὅλῳ μὴ ὑπάρχειν, ἡ μὲν Γ Α ἀληθὴς ἔσται, ἡ δὲ Γ Β ψευδής. πάλιν ὃ τῷ Β μηδενὶ ὑπάρχει, οὐδὲ τῷ Α παντὶ ὑπάρξει· εἰ γὰρ τῷ Α, καὶ τῷ Β· ἀλλ' οὐχ ὑπῆρχεν. ἐὰν οὖν ληφθῇ τὸ Γ τῷ μὲν Α ὅλῳ ὑπάρχειν, τῷ δὲ Β μηδενί, ἡ μὲν Γ Β πρότασις ἀληθής, ἡ δ' ἑτέρα ψευδής. a38. Or one premiss may be wholly false, and it may be either of them. Thus, supposing that actually an attribute of all A must also be an attribute of all B, then if C is yet taken to be a universal attribute of all but universally non-attributable to B, C-A will be true but C-B false. Again, actually that which is an attribute of no B will not be an attribute of all A either; for if it be an attribute of all A, it will also be an attribute of all B, which is contrary to supposition; but if C be nevertheless assumed to be a universal attribute of A, but an attribute of no B, then the premiss C-B is true but the major is false.
ὁμοίως δὲ καὶ μετατεθέντος τοῦ στερητικοῦ. ὃ γὰρ μηδενὶ ὑπάρχει τῷ Α, οὐδὲ τῷ Β οὐδενὶ ὑπάρξει· ἐὰν οὖν ληφθῇ τὸ Γ τῷ μὲν Α ὅλῳ μὴ ὑπάρχειν, τῷ δὲ Β ὅλῳ ὑπάρχειν, ἡ μὲν Γ Α πρότασις ἀληθὴς ἔσται, ἡ ἑτέρα δὲ ψευδής. καὶ πάλιν, ὃ παντὶ τῷ Β ὑπάρχει, μηδενὶ λαβεῖν τῷ Α ὑπάρχον ψεῦδος. ἀνάγκη γάρ, εἰ τῷ Β παντί, καὶ τῷ Α τινὶ ὑπάρχειν· ἐὰν οὖν ληφθῇ τῷ μὲν Β παντὶ ὑπάρχειν τὸ Γ, τῷ δὲ Α μηδενί, ἡ μὲν Γ Β ἀληθὴς ἔσται, ἡ δὲ Γ Α ψευδής. b6. The case is similar if the major is made the negative premiss. For in fact what is an attribute of no A will not be an attribute of any B either; and if it be yet assumed that C is universally non-attributable to A, but a universal attribute of B, the premiss C-A is true but the minor wholly false. Again, in fact it is false to assume that that which is an attribute of all B is an attribute of no A, for if it be an attribute of all B, it must be an attribute of some A. If then C is nevertheless assumed to be an attribute of all B but of no A, C-B will be true but C-A false.
φανερὸν οὖν ὅτι καὶ ἀμφοτέρων οὐσῶν ψευδῶν καὶ τῆς ἑτέρας μόνον ἔσται συλλογισμὸς ἀπατητικὸς ἐν τοῖς ἀτόμοις. b14. It is thus clear that in the case of atomic propositions erroneous inference will be possible not only when both premisses are false but also when only one is false.
Postquam philosophus ostendit quomodo concludatur per syllogismum affirmativa falsa, contraria negativae immediatae, hic ostendit quomodo per syllogismum concludatur negativa falsa, contraria affirmativae immediatae. Et primo, in prima figura; secundo, in secunda; ibi: sed in media figura et cetera. Circa primum duo facit. Primo, ostendit de quo est intentio. Et dicit quod cum negativa universalis concludi possit in prima et in secunda figura, primo dicendum est quot modis syllogismus ignorantiae fiat in prima figura, et qualiter se habentibus propositionibus in veritate et falsitate. Secundo; ibi: contingit quidem etc., prosequitur propositum. Et primo, ostendit quomodo fiat talis syllogismus in prima figura ex duabus falsis; secundo, quomodo fiat ex altera vera et altera falsa; ibi: contingit autem et altera et cetera. After showing how a false affirmative conclusion contrary to an immediate negative is obtained by syllogizing, the Philosopher here shows how by syllogizing a false negative is concluded contrary to an immediate affirmative. First, in the first figure. Secondly, in the second (8048). Concerning the first he does two things. First (80a8), he states his intention and says that since a universal negative may be concluded in the first as well as in the second figure, we must first show in which moods a syllogism of ignorance is formed in the first figure and under which conditions of truth and falsity in the propositions. Secondly (80a11), he establishes his proposition. First, he shows how such a proposition is formed from two false premises in the first figure. Secondly, how it is formed from one false and one true premise (80a14).
Dicit ergo primo quod praedictus syllogismus fieri potest ex utrisque falsis. Quod patet si a sit et in c et in b individualiter, idest immediate. Est autem immediate genus in proximis speciebus, in quas primo dividitur, sicut color in albedine et nigredine. Genus enim per se praedicatur de specie, quia primo ponitur in eius definitione; et immediate praedicatur de specie proxima, quia immediate in eius definitione ponitur, non autem ex hoc, quod ponatur in definitione alicuius partis definientis, sicut se habet genus remotum ad ultimam speciem. Sint ergo termini, color, albedo, nigredo. Si ergo accipiatur a quidem in nullo c esse, utpote si dicamus: nulla albedo est color; c autem in omni b, ut puta si dicamus: omnis nigredo est albedo; falsae sunt ambae propositiones, et falsa est conclusio, scilicet: nulla nigredo est color. He says therefore first (80a11) that the aforesaid syllogism can be formed from premises, both of which are false. This is clear if A is both in B and in C individually, i.e., immediately. It is thus that a genus is immediately in the proximate species into which it is first divided, as color into blackness and whiteness. For the genus is predicated per se of the species, because the former is placed first in the definition of the latter; and it is predicated immediately of a proximate species, because it is put in its definition immediately and not in the way that a remote genus-which is put in the definition of a defining part-is related to an ultimate species. Therefore, let the terms be “color,” “whiteness” and “blackness.” If, then, we assume that A is in no C, for example, if we say, “No whiteness is a color,” but C is in every B, say “All blackness is whiteness,” both propositions are false, as is the conclusion, “No blackness is a color.”
Deinde cum dicit: contingit autem etc., ostendit quomodo possit esse in praedicto syllogismo altera vera et altera falsa. Et primo ostendit quomodo possit esse maior vera et minor falsa; secundo, quomodo contingit e converso; ibi: sed et eam quae est et cetera. Then (80a14) he shows how there can be one false and one true premise in the syllogism under discussion. First, he shows how the major can be true and the minor false. Secondly, how it might be the reverse (8040).
Dicit ergo primo quod contingit syllogismum ignorantiae negativum fieri in prima figura, falsa existente altera propositionum indifferenter, quaecunque sit illa. Potest enim contingere quod haec propositio a.c, quae est maior, sit vera, et propositio, quae est b.c, sit falsa, quae est minor. Et quod propositio maior possit esse vera, probat per hoc, quod iste terminus a, quicunque sit ille, non est necesse quod insit omnibus, sicut color non praedicatur de omnibus entibus. Quod autem minor sit falsa, probat per hoc, quia non potest accipi aliquis terminus, a quo universaliter negetur a, qui quidem terminus praedicetur de b: supponimus enim quod haec sit vera et immediata: omne b est a. Si ergo aliquid universaliter praedicetur de b, ita quod haec sit vera, omne b est c, non potest esse quod de illo universaliter negetur a. Et ita haec propositio: nullum c est a, non erit vera; quae erat maior. Si enim omne b est a, ut supponitur, et omne b est c, ut assumitur, sequitur in tertia figura: quoddam c est a, quae est contradictoria maioris. Falsa ergo erit ista: nullum c est a. Si ergo haec sit vera, quae est maior, necesse est quod haec sit falsa, quae est minor: omne b est c. He says therefore first (80a14) that a negative syllogism of ignorance can be formed in the first figure no matter which one of the propositions happens to be false. For it might happen that the proposition AC, which is the major, is true, and the proposition BC, which is the minor, is false. That the major proposition could be true he proves by the fact that the term A, whatever it be, need not be in all things, as color is not predicated of all beings. That the minor would be false he proves on the ground that it is not possible to assume a term of which A would be universally denied and which would also be predicated of B: for we are supposing that the proposition, “Every B is A,” is true and immediate. Therefore, if something were universally predicated of B, so that “Every B is C” would be true, then A cannot be universally denied of C. Consequently, this proposition, “No C is A,” which was the major, will not be true. For if every B is A, as we supposed, and every B is C, as we are now assuming, it follows in the third figure that some C is A, which contradicts the major. Therefore the proposition, “No C is A,” will be false. Hence if this is true, which is the major, it is required that this be false, which is the minor, i.e., “Every B is C.”
Secundo, probat per hoc quod ex duabus veris non potest concludi falsa, ut supra probatum est. Datur autem haec esse vera: nullum c est a. Si ergo etiam haec sit vera: omne b est c; sequitur quod conclusio sit vera: nullum b est a; quae tamen supponitur esse falsa, utpote contraria huic immediatae propositioni: omne b est a. Then he proves the same thing on the ground that if both premises are true, then as has been proved above, a false conclusion cannot follow—which is out of place in a syllogism of ignorance, which ought to conclude to a false conclusion. But it was given that this is true, namely, “No C is A”’: if then it is also true that “Every B is C,” it follows that the conclusion, “No B is A,” is true, whereas it is supposed to be false, being contrary to this immediate proposition, “Every B is A.”
Deinde cum dicit: sed et eam etc., ostendit quomodo minor sit vera, maiori existente falsa. Et dicit quod propositio c.b, scilicet minor, potest esse vera, cum maior sit falsa. Quia enim haec propositio: omne b est a, cuius contraria debet concludi, est immediata, necesse est quod b sit in a sicut pars in toto, sicut albedo in colore. Potest autem accipi aliquid aliud, in quo etiam sit b sicut in toto, non tamen immediate, et sit illud qualitas quae sit c. Necesse est ergo, secundum praedicta, quod horum duorum, scilicet a et c, alterum sit sub altero, idest color sub qualitate. Si ergo aliquis accipiat a in nullo c esse, ut puta, si dicat: nulla qualitas est color, falsa erit propositio. Minor autem erit vera, scilicet: omnis albedo est qualitas. Conclusio autem erit falsa, et immediatae contraria, scilicet: nulla albedo est color. Sic ergo manifestum est quod potest fieri syllogismus ignorantiae negativus in prima figura, et altera propositione falsa et utrisque. Then (80a20) he shows how the minor can be true, the major being false. And he says that the proposition CB, namely, the minor, can be true, while the major is false. For since this proposition, “Every B is A,” whose contrary is to be concluded, is immediate, it is necessary that B exist in A as a part in a whole, as “whiteness” in “color.” But it is possible to take something else in which B also exists as in a whole, though not immediately—let this other thing be “quality,” i.e., C. It is necessary, therefore, according to the aforesaid, that as between these two, namely, A and C, one should be under the other, i.e., color under quality. Now if someone assumes that A is in no C and says, “No quality is a color,” the proposition will be false. But the minor will be true, namely, “Every whiteness is a quality.” The conclusion, however, “No whiteness is a color,” will be false and contrary to an immediate proposition. And so it is clear that a negative syllogism of ignorance can be formed in the first figure when either one or both of the premises are false.
Deinde cum dicit: sed in media figura etc., ostendit quomodo syllogismus ignorantiae negativus fiat in secunda figura. Et primo, quando utraque est falsa; secundo, quando altera tantum; ibi: similiter autem et alteram esse falsam et cetera. Then (8047) he shows how a negative syllogism of ignorance is formed in the second figure. First, when both are false. Secondly, when one or the other is false (80a38).
Dicit ergo primo quod in media figura non contingit utrasque propositiones esse totas falsas. Et dicit totas falsas illas, quae sunt contrariae propositionibus veris. Et hoc probat. Quia cum debeamus concludere negativam falsam contrariam affirmativae immediatae, necesse est accipere quod haec sit vera et immediata, omne b est a, puta, omnis albedo est color. Sic autem se habentibus terminis, non potest inveniri aliquis medius terminus, qui universaliter praedicetur de uno termino, et universaliter removeatur ab altero. Detur enim quod ille terminus c universaliter removeatur ab a, et universaliter praedicetur de b; erit ergo haec vera: nullum a est c; quare et conversa erit vera: nullum c est a; sed omne b est c, ergo nullum b est a; cuius contrarium fuit suppositum. He says therefore first (8047) that in the second figure it does not happen that both propositions are entirely false. And he calls those propositions entirely false which are contrary to true propositions. He proves this: For since we are trying to conclude a false negative contrary to an immediate affirmative, we must assume that this proposition, “Every B is A,” is true and immediate, say, “Every whiteness is a color.” But with terms so related it is impossible to find a middle term which would be predicated universally of one and universally removed from the other. For suppose that the term C could be universally removed from A and universally predicated of B. Then the proposition, “No A is C,” will be true; consequently, its converse, “No C is A,” will also be true. But every B is C. Therefore, no B is A, the contrary of which was supposed.
Similiter etiam non potest esse quod universaliter removeatur a b, et universaliter praedicetur de a; quia si haec est vera: omne a est c, et conversa erit vera: quoddam c est a. Si autem haec est vera: nullum b est c, et conversa erit vera: nullum c est b. Sic ergo ex his duabus propositionibus: quoddam c est a; nullum c est b; sequitur, quoddam b non est a, quae est contradictoria eius, quae supponebatur, omne b est a. Relinquitur ergo quod impossibile est inveniri aliquod medium, quod, praedicto modo se habentibus a et b, de uno praedicetur, et ab alio removeatur. Et tamen oportet, si debeat fieri syllogismus in secunda figura, ut medium de uno extremorum praedicetur, et de alio negetur. Et ideo si ambae sunt falsae totaliter, oportet quod earum contrariae sint verae; quod est impossibile, ut probatum est. Similarly, it cannot be universally removed from B and universally predicated of A. For if it is true that every A is C, the converse, “Some C is A,” will be true. But if it is true that no B is C, its converse, “No C is B,” will be true. So, then, from these two propositions, “Some C is A” and “No C is B,” there follows, “Some B is not A,” which is the contradictory of what was supposed, namely, that “Every B is A.” What remains, therefore, is that it is impossible to find any middle which, A and B being related in the way we have supposed, can be predicated of one and removed from the other. Yet if a syllogism is to be formed in the second figure, the middle must be predicated of one of the extremes and denied of the other. Therefore, if both are totally false, the contraries would have to be true, which is impossible as has been proved.
Nihil tamen prohibet utramque propositionem esse falsam particulariter. Puta, si accipiamus quoddam medium, quod particulariter praedicetur de a et de b, puta masculus, quod particulariter praedicatur de animali et de homine. Si ergo accipiatur c esse in omni a, puta, si accipiamus: omne animal esse masculum; et accipiamus c in nullo b esse, puta si dicamus: nullus homo est masculus; utraque propositio est falsa, non tamen totaliter, sed particulariter. Et eadem ratio est, si e converso maior sit negativa, et minor affirmativa. Ut si dicamus: nullum animal est masculum; omnis homo est masculus. However, nothing prevents both from being false particularly: thus we may take some middle which is predicated particularly of A and of B, say “male,” which is predicated particularly of animal and of man. Now if C is taken in every A, say “Every animal is male,” and in no B, say “No man is male,” each proposition will be false, not entirely but particularly. And the same holds if, conversely, the major is negative and the minor affirmative, i.e., if we should say, “No animal is male” and “Every man is male.”
Deinde cum dicit: similiter autem alteram etc., ostendit quomodo contingit alteram esse falsam. Et primo in secundo modo secundae figurae; secundo in primo; ibi: similiter autem fit transposito et cetera. Then (80a38) he shows how it happens when one is false. First, in the second mode of the second figure. Secondly, in the first mode (80b6).
Dicit ergo primo quod contingit in hac figura alteram propositionem esse falsam indifferenter, quaecunque sit illa. Quod patet ex hoc, quia cum supponatur a per se et immediate praedicari de b, quidquid est in omni a est in omni b; sicut omne quod universaliter praedicatur de animali, praedicatur universaliter de homine. Si ergo accipiatur aliquod medium c, quod universaliter praedicetur de a, ut si dicamus: omne animal est vivum; et universaliter removeatur a b, ut si dicamus: nullus homo est vivus: patet quod a.c, quae est maior propositio, erit vera; sed b.c quae est minor, erit falsa. He says therefore first (80a38) that in this figure it occurs that either proposition may be false. This is clear from the fact that if A is supposed to be predicated per se and immediately of B, whatever is in every A is in every B, as whatever is predicated universally of animal is predicated universally of man. Therefore, if some middle, C, be taken which is universally predicated of A, say “Every animal is living,” and universally removed from B, say “No man is living,” it is evident that AC, which is the major proposition, will be true, but BC, which is the minor, will be false.
Et similiter probat quod e converso contingit maiorem esse falsam. Non enim potest esse quod aliquid universaliter removeatur a b, et universaliter praedicetur de a, terminis sic se habentibus. Dictum est enim quod si aliquid est in a universaliter, sequitur quod sit in b. Si ergo aliquid removeatur a b universaliter, non potest esse quod universaliter praedicetur de a. Sicut quod universaliter removetur ab homine, non potest universaliter praedicari de animali. Si ergo accipiatur aliquid, quod universaliter removeatur ab homine, puta, irrationale, et dicatur sic: omne animal est irrationale; nullus homo est irrationalis; sequitur quod minor propositio sit vera, et maior falsa. Sed in his terminis, maior propositio non est totaliter falsa. Potest autem accipi terminus in quo sit totaliter falsa, puta si accipiamus inanimatum pro medio. Similarly, he proves that the converse occurs when the major is false. For it cannot be that something be universally removed from B and universally predicated of A, when the terms have that position. For it has been stated that if something is in A universally, it follows that it is also in B. Consequently, if something be removed universally from B, it cannot be that it is predicated universally of A. For example, anything universally removed from “man” cannot be universally predicated of “animal.” Therefore, if something is taken which is universally removed from man, say “irrational,” and you state that “Every animal is irrational” and “No man is irrational,” it follows that the minor proposition is true and the major false. But in these terms the major premise is not totally false. However, one can take a term in which it is totally false, for example, if we should take “inanimate” as the middle.
Deinde cum dicit: similiter autem fit etc., ostendit idem in primo modo secundae figurae, in quo maior est negativa. Manifestum est enim quod, praedictis terminis, scilicet a et b, sic se habentibus ut dictum est, quod universaliter removetur ab a, non poterit esse in nullo b. Si ergo accipiatur c medium, quod universaliter removetur ab a, et universaliter praedicetur de b; erit maior propositio vera et minor falsa. Puta si sint isti termini, inanimatum, animal, homo. Then (80b6) he shows the same in the first mode of the second figure, where the major is negative. For it is clear that with the terms A and B so related, as was said, something universally removed from A cannot be in any B. Therefore, if a middle, C, be taken which is universally removed from A and universally predicated of B, the major will be true and the minor false. For example, if the terms are “inanimate,” “animal” and “man.”
Et similiter ostendit quod potest esse minor vera, et maior falsa. Manifestum est enim, secundum praedicta, quod id quod universaliter praedicatur de b, non potest removeri universaliter ab omni a: quia quod universaliter praedicatur de b, ad minus oportet in quodam a esse. Si ergo accipiatur c medium, quod universaliter praedicetur de b, puta, rationale, vel vivum, et universaliter negetur de a; minor propositio erit vera, scilicet: omnis homo est rationale, vel vivum. Maior autem: nullum animal est rationale, est falsa in parte; nullum animal est vivum, est falsa in toto. Similarly, he shows that the minor can be true and the major false. For it is clear, according to the aforesaid, that that which is universally predicated of B cannot be universally removed from every A, because what is universally predicated of B must be in some A at least. Therefore, if C be taken as a middle which is universally predicated of B, say, “rational” or “living,” and universally denied of A, there will be a true minor proposition, namely, “Every man is rational” or “living.” But the major, namely, “No animal is rational,” is false in part, while “No animal is living,” is false entirely.
Deinde epilogando concludit quod syllogismus deceptivus potest fieri in immediatis, utrisque propositionibus existentibus falsis, vel altera tantum. Then summarizing (80b14) he concludes that a deceptive syllogism can be formed in immediates, when both propositions are false or only one is false.

Lectio 29
Caput 17
Ἐν δὲ τοῖς μὴ ἀτόμως ὑπάρχουσιν [ἢ μὴ ὑπάρχουσιν], ὅταν μὲν διὰ τοῦ οἰκείου μέσου γίνηται τοῦ ψεύδους ὁ συλλογισμός, οὐχ οἷόν τε ἀμφοτέρας ψευδεῖς εἶναι τὰς προτάσεις, ἀλλὰ μόνον τὴν πρὸς τῷ μείζονι ἄκρῳ. (λέγω δ' οἰκεῖον μέσον δι' οὗ γίνεται τῆς ἀντιφάσεως ὁ συλλογισμός.) ὑπαρχέτω γὰρ τὸ Α τῷ Β διὰ μέσου τοῦ Γ. ἐπεὶ οὖν ἀνάγκη τὴν Γ Β καταφατικὴν λαμβάνεσθαι συλλογισμοῦ γινομένου, δῆλον ὅτι ἀεὶ αὕτη ἔσται ἀληθής· οὐ γὰρ ἀντιστρέφεται. ἡ δὲ Α Γ ψευδής· ταύτης γὰρ ἀντιστρεφομένης ἐναντίος γίνεται ὁ συλλογισμός. b17. In the case of attributes not atomically connected with or disconnected from their subjects, (a) (i) as long as the false conclusion is inferred through the 'appropriate' middle, only the major and not both premisses can be false. By 'appropriate middle' I mean the middle term through which the contradictory — i.e. the true-conclusion is inferrible. Thus, let A be attributable to B through a middle term C: then, since to produce a conclusion the premiss C-B must be taken affirmatively, it is clear that this premiss must always be true, for its quality is not changed. But the major A-C is false, for it is by a change in the quality of A-C that the conclusion becomes its contradictory — i.e. true.
ὁμοίως δὲ καὶ εἰ ἐξ ἄλλης συστοιχίας ληφθείη τὸ μέσον, οἷον τὸ Δ εἰ καὶ ἐν τῷ Α ὅλῳ ἐστι καὶ κατὰ τοῦ Β κατηγορεῖται παντός· ἀνάγκη γὰρ τὴν μὲν Δ Β πρότασιν μένειν, τὴν δ' ἑτέραν ἀντιστρέφεσθαι, ὥσθ' ἡ μὲν ἀεὶ ἀληθής, ἡ δ' ἀεὶ ψευδής. καὶ σχεδὸν ἥ γε τοιαύτη ἀπάτη ἡ αὐτή ἐστι τῇ διὰ τοῦ οἰκείου μέσου. b27. Similarly (ii) if the middle is taken from another series of predication; e.g. suppose D to be not only contained within A as a part within its whole but also predicable of all B. Then the premiss D-B must remain unchanged, but the quality of A-D must be changed; so that D-B is always true, A-D always false. Such error is practically identical with that which is inferred through the 'appropriate' middle.
ἐὰν δὲ μὴ διὰ τοῦ οἰκείου μέσου γίνηται ὁ συλλογισμός, ὅταν μὲν ὑπὸ τὸ Α ᾖ τὸ μέσον, τῷ δὲ Β μηδενὶ ὑπάρχῃ, ἀνάγκη ψευδεῖς εἶναι ἀμφοτέρας. ληπτέαι γὰρ ἐναντίως ἢ ὡς ἔχουσιν αἱ προτάσεις, εἰ μέλλει συλλογισμὸς ἔσεσθαι· οὕτω δὲ λαμβανομένων ἀμφότεραι γίνονται ψευδεῖς. οἷον εἰ τὸ μὲν Α ὅλῳ τῷ Δ ὑπάρχει, τὸ δὲ Δ μηδενὶ τῶν Β· ἀντιστραφέντων γὰρ τούτων συλλογισμός τ' ἔσται καὶ αἱ προτάσεις ἀμφότεραι ψευδεῖς. ὅταν δὲ μὴ ᾖ ὑπὸ τὸ Α τὸ μέσον, οἷον τὸ Δ, ἡ (81a.) μὲν Α Δ ἀληθὴς ἔσται, ἡ δὲ Δ Β ψευδής. ἡ μὲν γὰρ Α Δ ἀληθής, ὅτι οὐκ ἦν ἐν τῷ Α τὸ Δ, ἡ δὲ Δ Β ψευδής, ὅτι εἰ ἦν ἀληθής, κἂν τὸ συμπέρασμα ἦν ἀληθές· ἀλλ' ἦν ψεῦδος. b32. On the other hand, (b) if the conclusion is not inferred through the 'appropriate' middle — (i) when the middle is subordinate to A but is predicable of no B, both premisses must be false, because if there is to be a conclusion both must be posited as asserting the contrary of what is actually the fact, and so posited both become false: e.g. suppose that actually all D is A but no B is D; then if these premisses are changed in quality, a conclusion will follow and both of the new premisses will be false. When, however, (ii) the middle D is not subordinate to A, A-D will be true, D-B false — A-D true because A was not subordinate to D, D-B false because if it had been true, the conclusion too would have been true; but it is ex hypothesi false.
Διὰ δὲ τοῦ μέσου σχήματος γινομένης τῆς ἀπάτης, ἀμφοτέρας μὲν οὐκ ἐνδέχεται ψευδεῖς εἶναι τὰς προτάσεις ὅλας (ὅταν γὰρ ᾖ τὸ Β ὑπὸ τὸ Α, οὐδὲν ἐνδέχεται τῷ μὲν παντὶ τῷ δὲ μηδενὶ ὑπάρχειν, καθάπερ ἐλέχθη καὶ πρότερον), τὴν ἑτέραν δ' ἐγχωρεῖ, καὶ ὁποτέραν ἔτυχεν. εἰ γὰρ τὸ Γ καὶ τῷ Α καὶ τῷ Β ὑπάρχει, ἐὰν ληφθῇ τῷ μὲν Α ὑπάρχειν τῷ δὲ Β μὴ ὑπάρχειν, ἡ μὲν Γ Α ἀληθὴς ἔσται, ἡ δ' ἑτέρα ψευδής. πάλιν δ' εἰ τῷ μὲν Β ληφθείη τὸ Γ ὑπάρχον, τῷ δὲ Α μηδενί, ἡ μὲν Γ Β ἀληθὴς ἔσται, ἡ δ' ἑτέρα ψευδής.

Ἐὰν μὲν οὖν στερητικὸς ᾖ τῆς ἀπάτης ὁ συλλογισμός, εἴρηται πότε καὶ διὰ τίνων ἔσται ἡ ἀπάτη·

a5. When the erroneous inference is in the second figure, both premisses cannot be entirely false; since if B is subordinate to A, there can be no middle predicable of all of one extreme and of none of the other, as was stated before. One premiss, however, may be false, and it may be either of them. Thus, if C is actually an attribute of both A and B, but is assumed to be an attribute of A only and not of B, C-A will be true, C-B false: or again if C be assumed to be attributable to B but to no A, C-B will be true, C-A false.

We have stated when and through what kinds of premisses error will result in cases where the erroneous conclusion is negative.

ἐὰν δὲ καταφατικός, ὅταν μὲν διὰ τοῦ οἰκείου μέσου, ἀδύνατον ἀμφοτέρας εἶναι ψευδεῖς· ἀνάγκη γὰρ τὴν Γ Β μένειν, εἴπερ ἔσται συλλογισμός, καθάπερ ἐλέχθη καὶ πρότερον. ὥστε ἡ Α Γ ἀεὶ ἔσται ψευδής· αὕτη γάρ ἐστιν ἡ ἀντιστρεφομένη. a16. If the conclusion is affirmative, (a) (i) it may be inferred through the 'appropriate' middle term. In this case both premisses cannot be false since, as we said before, C-B must remain unchanged if there is to be a conclusion, and consequently A-C, the quality of which is changed, will always be false.
ὁμοίως δὲ καὶ εἰ ἐξ ἄλλης συστοιχίας λαμβάνοιτο τὸ μέσον, ὥσπερ ἐλέχθη καὶ ἐπὶ τῆς στερητικῆς ἀπάτης· ἀνάγκη γὰρ τὴν μὲν Δ Β μένειν, τὴν δ' Α Δ ἀντιστρέφεσθαι, καὶ ἡ ἀπάτη ἡ αὐτὴ τῇ πρότερον. a20. This is equally true if (ii) the middle is taken from another series of predication, as was stated to be the case also with regard to negative error; for D-B must remain unchanged, while the quality of A-D must be converted, and the type of error is the same as before.
ὅταν δὲ μὴ διὰ τοῦ οἰκείου, ἐὰν μὲν ᾖ τὸ Δ ὑπὸ τὸ Α, αὕτη μὲν ἔσται ἀληθής, ἡ ἑτέρα δὲ ψευδής· ἐγχωρεῖ γὰρ τὸ Α πλείοσιν ὑπάρχειν ἃ οὐκ ἔστιν ὑπ' ἄλληλα. ἐὰν δὲ μὴ ᾖ τὸ Δ ὑπὸ τὸ Α, αὕτη μὲν ἀεὶ δῆλον ὅτι ἔσται ψευδής (καταφατικὴ γὰρ λαμβάνεται), τὴν δὲ Δ Β ἐνδέχεται καὶ ἀληθῆ εἶναι καὶ ψευδῆ· οὐδὲν γὰρ κωλύει τὸ μὲν Α τῷ Δ μηδενὶ ὑπάρχειν, τὸ δὲ Δ τῷ Β παντί, οἷον ζῷον ἐπιστήμῃ, ἐπιστήμη δὲ μουσικῇ. οὐδ' αὖ μήτε τὸ Α μηδενὶ τῶν Δ μήτε τὸ Δ μηδενὶ τῶν Β. [φανερὸν οὖν ὅτι μὴ ὄντος τοῦ μέσου ὑπὸ τὸ Α καὶ ἀμφοτέρας ἐγχωρεῖ ψευδεῖς εἶναι καὶ ὁποτέραν ἔτυχεν.] a25. (b) The middle may be inappropriate. Then (i) if D is subordinate to A, A-D will be true, but D-B false; since A may quite well be predicable of several terms no one of which can be subordinated to another. If, however, (ii) D is not subordinate to A, obviously A-D, since it is affirmed, will always be false, while D-B may be either true or false; for A may very well be an attribute of no D, whereas all B is D, e.g. no science is animal, all music is science. Equally well A may be an attribute of no D, and D of no B. It emerges, then, that if the middle term is not subordinate to the major, not only both premisses but either singly may be false.
Ποσαχῶς μὲν οὖν καὶ διὰ τίνων ἐγχωρεῖ γίνεσθαι τὰς κατὰ συλλογισμὸν ἀπάτας ἔν τε τοῖς ἀμέσοις καὶ ἐν τοῖς δι' ἀποδείξεως, φανερόν. a35. Thus we have made it clear how many varieties of erroneous inference are liable to happen and through what kinds of premisses they occur, in the case both of immediate and of demonstrable truths.
Postquam philosophus ostendit quomodo syllogismus ignorantiae fit in propositionibus immediatis, hic ostendit quomodo fit in propositionibus mediatis. Et primo, quomodo concludatur propositio negativa falsa, quae opponitur affirmativae verae; secundo, quomodo concludatur affirmativa falsa, quae opponitur negativae verae; ibi: si vero sit affirmativus et cetera. Circa primum duo facit: primo, ostendit quomodo hoc fiat in prima figura; secundo, in secunda: ibi: sed per mediam figuram et cetera. Circa primum tria facit: primo, ostendit quomodo fit syllogismus ignorantiae in propositionibus mediatis, per medium proprium; secundo, quomodo fit per medium quidem non proprium, sed tamen similem habitudinem habens ad terminos, sicut medium proprium; ibi: similiter autem est, et si ex alia ordinatione etc.; tertio, ostendit quomodo fit praedictus syllogismus per medium extraneum; ibi: si vero non per proprium medium et cetera. After showing how a syllogism of ignorance in regard to immediate propositions is made, the Philosopher here shows how it is made in regard to mediate propositions. First, how a false negative proposition is concluded which is opposed to a true affirmative. Secondly, how a false affirmative is concluded which is opposed to a true negative (81a16). Concerning the first he does two things. First, he shows this in the first figure. Secondly, in the second (81a4). In regard to the first he does three things. First, he shows how a syllogism of ignorance is constructed in mediate propositions through a proper middle. Secondly, how one is constructed through a middle which although not proper has a relationship to the terms, a relationship akin to that of a proper middle (80b27). Thirdly, he shows how such a syllogism is constructed through an extraneous middle (80b32).
Dicit ergo primo quod, quando syllogismus concludens falsum, fit in propositionibus, quae non sunt individuae, idest immediatae, si accipiatur proprium medium, unde fit syllogismus, non potest esse utrasque propositiones esse falsas, sed solum maiorem. Et exponit quid nominet proprium medium. Ex quo enim propositio, cuius contraria syllogizatur, est mediata, oportet quod praedicatum syllogizetur de subiecto per aliquod medium. Potest ergo illud idem medium accipi ad concludendum oppositum. Puta, haec est propositio mediata: omnis triangulus habet tres angulos aequales duobus rectis; medium autem per quod syllogizatur praedicatum de subiecto, est figura habens angulum extrinsecum aequalem duobus intrinsecis sibi oppositis. Si ergo velimus probare quod nullus triangulus habet tres angulos aequales duobus rectis per hoc idem medium, erit syllogismus falsitatis per proprium medium. Et ideo dicit quod medium proprium est, per quod fit syllogismus contradictionis, idest ad oppositum. Puta in praedicto exemplo: sit a triangulus, b habere tres, medium c figura talis. In prima autem figura necesse est minorem esse affirmativam, et ideo oportet quod illa, quae erat minor in syllogismo vero, maneat non conversa nec transmutata in suam oppositam in syllogismo falsitatis. Unde oportet quod semper sit vera. Sed maior propositio veri syllogismi convertitur in negativam contrariam; et ideo oportet quod maior sit falsa. Puta si dicamus: nulla figura habens etc. habet tres etc.; omnis triangulus est figura talis; ergo et cetera. He says therefore first (80b17) that when a syllogism is constructed which concludes something false in propositions which are not individual, i.e., not immediate, if the “proper middle” through which the syllogism is formed be taken, then both propositions cannot be false, but only the major. He explains what he means by “proper middle.” For since the proposition whose contrary is to be syllogized is mediate, it is required that the predicate be syllogized of the subject through some middle. Therefore, the same middle can be employed to conclude the opposite. Say that the mediate proposition is “Every triangle has three angles equal to two right angles.” The middle through which the predicate is syllogized of the subject is, “A figure having an exterior angle equal to the two opposite interior angles.” Now if we would prove through the same middle that no triangle has three angles equal to two right angles, it will be a syllogism of falsity through a proper middle. Hence he says that a proper middle is one through which the syllogism of contradiction is made, i.e., leading to the opposite conclusion. In the above example, A would be “triangle,” B, “having three...” and C, the middle, “such a figure.” Now in the first figure the minor must be affirmative; therefore, that which was the minor in the true syllogism must remain unconverted and not changed into its opposite in the syllogism of falsity. Hence it must always be true. But the major proposition of the true syllogism is changed into its contrary negative; hence the major must be false. For example, we might say: “No figure having an exterior angle equal to the two opposite interior angles has three angles equal to two right angles; but the triangle is such a figure: therefore, no triangle has three angles equal to two right angles.”
Deinde cum dicit: similiter autem est etc., ostendit quomodo fit praedictus syllogismus per medium extraneum, sed simile proprio. Et dicit quod similiter syllogizabitur, si medium accipiatur ex alia ordinatione. Puta si a demonstretur de b per c, et accipiamus in syllogismo falsitatis medium non c, sed d, ita tamen quod d etiam contineatur universaliter sub a et praedicetur universaliter de b, puta si accipiamus pro medio figuram contentam tribus lineis rectis; quia hic etiam necesse est minorem propositionem, scilicet db, manere sicut erat in syllogismo concludente verum, quamvis per proprium medium; maiorem autem propositionem necesse est transmutari in contrariam: et ideo semper minor erit vera, et maior semper erit falsa. Et quantum ad modum arguendi ista deceptio est similis ei, quae fit per proprium medium. Then (80b27) he shows how the aforesaid syllogism is constructed through a middle which is extraneous but like the proper. And he says that it will be syllogized in like fashion if the middle is taken from another ordering. For example, if A had been demonstrated of B through C, and we were to take in the syllogism of falsity not C but D as the middle, in such a way, however, that D is also contained universally under A and predicated universally of B: say if we took for the middle, “a closed figure of three lines,” because here too the minor proposition DB must remain as it was in the syllogism which concluded the true, although through a proper middle. But the major proposition will have to be changed into its contrary. And so the minor will always be true and the major always false. But as to the mode of the argument, this deception is similar to that which is formed through the proper middle.
Deinde cum dicit: si vero non per proprium etc., ostendit quomodo fit syllogismus falsitatis per medium extraneum, et dissimile proprio. Potest autem hoc medium hoc modo accipi, ut contineatur universaliter sub a, et de nullo b praedicetur. Et in hoc casu oportebit utrasque propositiones esse falsas, quia oportebit, ad hoc quod fiat syllogismus in prima figura, accipere propositiones e contrario, ut scilicet accipiamus maiorem negativam et dicamus, nullum d est a, et minorem affirmativam, et dicamus: omne b est d: et sic patet utrasque esse falsas. Et haec quidem terminorum habitudo inveniri non potest in convertibilibus, sicut in subiecto et passione, quae per aliquod medium de subiecto concluditur. Manifestum est enim quod non potest accipi aliquid, de quo passio universaliter praedicetur, quod a subiecto universaliter removeatur. Sed haec habitudo potest inveniri, quando propositio est mediata, ex hoc quod superius genus vel passio superioris generis praedicatur de ultima specie; puta si dicamus: omnis homo est vivus. Vivum enim potest concludi de homine per medium, quod est animal. Si ergo accipiamus aliquid, de quo vivum universaliter praedicetur, sicut est oliva, quae vere removetur ab homine universaliter, erit habitudo terminorum, quam quaerimus. Haec enim erit falsa: nulla oliva est viva; et minor erit similiter falsa: omnis homo est oliva; et similiter conclusio erit falsa: nullus homo est vivus, quod est contrarium propositioni verae mediatae. Then (80b32) he shows how a syllogism of falsity is made through a middle which is extraneous and unlike the proper. For a middle can be taken such that it is contained universally under A but is predicated of no B. In this case both propositions will have to be false, because in order that the syllogism be formed in the first figure, it will be necessary to take propositions to the contrary, namely, a major which is negative, for example, “No D is A,” and the minor affirmative, for example, “Every B is D.” Clearly then both are false. Now this relationship of terms cannot be found in things convertible, say in a subject and its proper attribute which is concluded of the subject through some middle. For it is obvious that no middle can be taken such that the proper attribute would be universally predicated of it, and that middle be removed universally from the subject. But this relationship can be found when the proposition is mediate, for example, when a higher genus or the proper attribute of a higher genus is predicated of an ultimate species, as when we say, “Every man is living.” For “living” can be concluded of man through the middle, “animal.” Therefore, if we should take something of which “living” would be universally predicated, say “olive,” but would be universally removed from man, the relationship of terms that we are seeking will result. For this will be false, “No olive is living”; and the minor, too, will be false, “Every man is an olive.” Similarly, the conclusion will be false, “No man is living,” which is contrary to the true mediate proposition.
Contingit etiam maiorem esse veram et minorem esse falsam; puta si accipiamus pro medio aliquid, quod non contineatur sub a, puta lapidem. Tunc enim maior, quae est a.d, erit vera, scilicet, nullus lapis est vivens; quia lapis non continetur sub vivo: sed minor erit falsa, scilicet, omnis homo est lapis. Si enim esset haec vera, prima existente vera, sequeretur quod conclusio esset vera, cum tamen dictum sit quod sit falsa. Non autem potest esse e converso quod minor sit vera si sit medium extraneum, quia medium extraneum non poterit universaliter praedicari de b. Oportet autem semper minorem affirmativam accipere in prima figura. It also happens that the major may be true and the minor false. For example, if we take as middle something which is not contained under A, say “stone”; then the major, AB, will be true, namely, “No stone is living,” because “stone” is not contained under “living,” but the minor will be false, namely, “Every man is a stone.” For if the minor remained true, while the first Was true, then the conclusion would be true, whereas it has been said that it is to be false. However, the converse does not occur, i.e., that the minor be true when the middle is extraneous, because such a middle cannot be predicated universally of B. But in the first figure the minor taken must always be an affirmative statement.
Deinde cum dicit: sed per mediam figuram etc., ostendit quomodo fit syllogismus ignorantiae negativus in secunda figura. Et dicit quod non potest contingere in secunda figura, quod utraque propositio sit falsa totaliter. Si enim debeat concludi haec falsa, nullum b est a, contraria verae; oportet quod a universaliter praedicetur de b. Unde non poterit aliquid inveniri, quod universaliter praedicetur de uno, et universaliter negetur de altero; sicut supra dictum est, cum agebatur de syllogismo ignorantiae in immediatis. Then (81a5) he shows how a negative syllogism of ignorance is made in the second figure. And he says that in the second figure it cannot occur that both propositions be totally false. For if we are to conclude the false proposition, “No B is A,” contrary to the true, it would have been required that A be predicated universally of B. Hence nothing will be able to be found which would be universally predicated of one and universally denied of the other, as has been established above when we treated concerning the syllogism of ignorance in immediates.
Potest tamen altera tantum esse totaliter falsa, quaecunque sit illa. Et hoc manifestat primo in secundo modo secundae figurae, in quo maior est affirmativa et minor negativa. Sit ergo medium sic se habens ad extrema, ut universaliter de utroque praedicetur sicut vivum praedicatur universaliter et de homine et de animali. Si ergo accipiatur maior affirmativa, ut dicamus, omne animal est vivum; et accipiatur minor negativa, ut dicatur, nullus homo est vivus; maior erit vera et minor falsa, et conclusio falsa. Similiter etiam si accipiamus in primo modo secundae figurae maiorem negativam, ut dicamus, nullum animal est vivum; et minorem affirmativam, ut dicamus, omnis homo est vivus; erit maior falsa et minor vera, et conclusio falsa. But one or the other of them can be totally false. And he manifests this first in the second mood of the second figure where the major is affirmative and the minor negative. Thus, let the middle be related to the extremes so that it is predicated universally of both, as “living” is predicated universally of man and of animal. Then if we take the major affirmative, say “Every animal is living,” and the minor negative, . say “No man is living,” the major will be true, the minor false, and the conclusion false. In like manner also, if in the first mood of the second figure we take the major negative, say “No animal is living,” and the minor affirmative, say “Every man is living,” the major will be false, the minor true, and the conclusion false.
Ex his dictis epilogando concludit dictum esse quando et per quae possit fieri deceptio, si syllogismus deceptivus sit privativus. Having said these things he sums up and concludes that he has stated when and through which kinds of premises deception can occur, if the deceptive syllogism is privative [negative].
Deinde cum dicit: si vero sit affirmativus etc., ostendit quomodo fiat affirmativus syllogismus deceptionis in propositionibus mediatis. Et primo, quando fit per proprium medium; secundo quando fit per medium simile proprio; ibi: similiter autem et si ex alia etc.; tertio, quando fit per medium extraneum; ibi: cum vero fit per non proprium et cetera. Then (81a16) he shows how an affirmative syllogism of deception is formulated in mediate propositions. First, when it is formulated through a proper middle. Secondly, when it is formulated through a middle similar to a proper middle (8140). Thirdly, when it is formulated through an extraneous middle (81a25).
Dicit ergo primo quod si fiat syllogismus deceptionis affirmativus in propositionibus mediatis, si accipiatur proprium medium, ut supra expositum est, impossibile est quod utraque sit falsa. Quia cum talis syllogismus non possit fieri nisi in prima figura, utraque existente affirmativa, necesse est quod minor propositio maneat hoc modo, sicut erat in vero syllogismo. Unde oportebit maiorem propositionem esse mutatam, scilicet de negativa in affirmativam; unde oportebit quod sit falsa. Puta si velimus concludere quod omnis homo sit quantitas, quod est contrarium huic, nullus homo est quantitas, cuius proprium medium est substantia; accipiemus istam falsam, omnis substantia est quantitas, et hanc veram, omnis homo est substantia. He says therefore first (8106), that if an affirmative syllogism of deception is to be formulated in mediate propositions, if a proper middle such as explained above be taken, it is impossible that both propositions be false. For since a syllogism of this kind can be formed only in the first figure, both propositions being affirmative, it is required that the minor proposition remain as it was in the true syllogism. Hence the major will have to be changed, namely, from negative to affirmative, so that it will have to be false. For example, if we desire to conclude that “Every man is a quantity,” which is the contrary of the statement, “No man is a quantity,” whose proper middle is “substance,” we will take the false proposition, “Every substance is a quantity,” and the true proposition, “Every man is a substance.”
Deinde cum dicit: similiter autem et si etc., ostendit quomodo fit syllogismus ignorantiae, quando accipitur medium non proprium, quod sit eiusdem ordinis, sed ex alia coordinatione. Puta si dicerem: omne agens est quantitas; omnis homo est agens; ergo omnis homo est quantitas. Oportet enim hic minorem manere, maiorem vero mutari de negativa in affirmativam. Unde et haec deceptio similis est priori deceptioni, sicut dicebatur in syllogismo privativo. Then (81a20) he shows how a syllogism of ignorance is formed when a non-proper middle is taken which is not of the same order but from some other ordering. For example, if I say, “Every agent is a quantity; every man is an agent: therefore, every man is a quantity.” For it is necessary in this case for the minor to remain, but the major will have to be changed from negative to affirmative. Hence this deception is similar to the previous deception, as was stated in the privative syllogism.
Deinde cum dicit: cum vero sit non etc., ostendit quomodo fiat syllogismus deceptionis affirmativus per medium extraneum; et dicit quod si accipiatur tale medium extraneum, quod contineatur sub maiori extremitate, tunc maior propositio erit vera, et minor falsa. Potest enim a, quae est maior extremitas, de pluribus universaliter praedicari, quae non sunt sub invicem; puta habitus de grammatica et virtute. Haec enim est mediata, nulla grammatica est virtus. Possumus ergo concludere contrarium huius, scilicet: omnis grammatica est virtus, per aliquod medium, quod contineatur sub virtute; et tunc maior erit vera, et minor falsa. Puta si dicamus: omnis temperantia est virtus; omnis grammatica est temperantia; ergo omnis grammatica est virtus. Then (81a25) he shows how an affirmative syllogism of deception is formulated through an extraneous middle. And he says that if an extraneous middle be taken such that it is contained under the major extreme, then the major will be true and the minor false. For A, the major extreme, can be predicated universally of many things that are not under one another, say “habit” of grammar and virtue. For this is mediate, “No grammar is a virtue.” There we can conclude the contrary of this, namely, “All grammar is virtue,” through a middle which is contained under virtue. Then the major will be true and the minor false. For example, we might say: “All temperance is a virtue; grammar is temperance: therefore, all grammar is a virtue.”
Si vero accipiatur aliquod medium, quod non sit sub a, maior semper erit falsa, quia accipitur affirmativa. Sed minorem contingit esse cum hac quandoque quidem falsam, et tunc ambae erunt falsae, puta si dicamus: omnis albedo est virtus; omnis grammatica est albedo; ergo etc.; quandoque autem potest esse vera: nihil enim prohibet, sic se habentibus terminis, quod a removeatur ab omni d, et d sit in omni b, sicut est in his terminis, animal, scientia, musica. Animal enim, quod est maior extremitas, removetur universaliter ab omni scientia; unde haec, quae sumitur ut maior in syllogismo ignorantiae, omnis scientia est animal, est falsa. Minor vero, scilicet, omnis musica est scientia, est vera; sed conclusio falsa contraria negativae verae mediatae. Contingit etiam quod et a sit in nullo d, et d in nullo b, ut dictum est. But if a middle is taken which is not under the major extreme, the major will always be false, because it will be affirmative. But the minor may sometimes be false with such a major; then both will be false. For example, if we should say: “Every whiteness is a virtue; all grammar is whiteness: therefore....” But sometimes it can be true. For when the terms are so related, there is nothing to hinder A from being removed from every D, and D from being in every B, as happens in these terms, namely, “animal,” “science,” “music.” For the major extreme, “animal,” is removed universally from all science; hence this proposition which is taken as the major in the syllogism of ignorance is false, namely, “Every science is an animal.” But the minor, namely, “All music is science,” is true. And the conclusion will be false, being contrary to the true mediate negative. It can also happen that A is in no D, and D in no B, as has been said.
Sic igitur patet quod quando medium non continetur sub maiori extremitate, possunt esse utraeque falsae et altera earum, quaecunque contingit, quia et maior et minor potest esse falsa: maior autem non potest esse vera, sic se habentibus terminis, ut supra dictum est. Thus it is evident that when the middle is not contained under the major extreme, they may both be false or just one of them, because the major and the minor may be false. However, with the three terms so related, as we have said above, the major cannot be true.
Ultimo autem epilogando concludit manifestum esse ex praedictis, quot modis et per quas propositiones veras vel falsas possunt fieri deceptiones per syllogismum, tam in propositionibus immediatis, quam in propositionibus mediatis, quae demonstratione probantur. Finally (81a35), he summarizes and concludes that it is plain from the foregoing how many ways and through which alignment of true and false propositions it is possible to construct deceptions through syllogisms, both in immediate propositions and in mediate propositions, which are proved by demonstration.

Lectio 30
Caput 18
Φανερὸν δὲ καὶ ὅτι, εἴ τις αἴσθησις ἐκλέλοιπεν, ἀνάγκη καὶ ἐπιστήμην τινὰ ἐκλελοιπέναι, ἣν ἀδύνατον λαβεῖν, a38. It is also clear that the loss of any one of the senses entails the loss of a corresponding portion of knowledge, and that,
εἴπερ μανθάνομεν ἢ ἐπαγωγῇ ἢ ἀποδείξει, ἔστι δ' ἡ μὲν ἀπόδειξις (81b.) ἐκ τῶν καθόλου, ἡ δ' ἐπαγωγὴ ἐκ τῶν κατὰ μέρος, ἀδύνατον δὲ τὰ καθόλου θεωρῆσαι μὴ δι' ἐπαγωγῆς (ἐπεὶ καὶ τὰ ἐξ ἀφαιρέσεως λεγόμενα ἔσται δι' ἐπαγωγῆς γνώριμα ποιεῖν, ὅτι ὑπάρχει ἑκάστῳ γένει ἔνια, καὶ εἰ μὴ χωριστά ἐστιν, ᾗ τοιονδὶ ἕκαστον), ἐπαχθῆναι δὲ μὴ ἔχοντας αἴσθησιν ἀδύνατον. τῶν γὰρ καθ' ἕκαστον ἡ αἴσθησις· οὐ γὰρ ἐνδέχεται λαβεῖν αὐτῶν τὴν ἐπιστήμην· οὔτε γὰρ ἐκ τῶν καθόλου ἄνευ ἐπαγωγῆς, οὔτε δι' ἐπαγωγῆς ἄνευ τῆς αἰσθήσεως. a39. since we learn either by induction or by demonstration, this knowledge cannot be acquired. Thus demonstration develops from universals, induction from particulars; but since it is possible to familiarize the pupil with even the so-called mathematical abstractions only through induction — i.e. only because each subject genus possesses, in virtue of a determinate mathematical character, certain properties which can be treated as separate even though they do not exist in isolation — it is consequently impossible to come to grasp universals except through induction. But induction is impossible for those who have not sense-perception. For it is sense-perception alone which is adequate for grasping the particulars: they cannot be objects of scientific knowledge, because neither can universals give us knowledge of them without induction, nor can we get it through induction without sense-perception.
Postquam philosophus determinavit de ignorantia dispositionis, quae fit per syllogismum; hic determinat de ignorantia simplicis negationis, quae fit absque syllogismo. Et primo, ostendit in quibus habeatur talis ignorantia ex necessitate; secundo, probat propositum; ibi: siquidem addiscimus et cetera. After the Philosopher has determined concerning ignorance through deception which is caused by a syllogism, he now determines concerning the ignorance of simple negation which is produced without a syllogism. First, he shows in which type of person this ignorance is had of necessity. Secondly, he proves his proposition (81a39).
Dicit ergo primo quod si alicui deficiat aliquis sensus, puta visus aut auditus, necesse est quod deficiat ei scientia propriorum sensibilium illius sensus. Puta, si cui deficit sensus visus, necesse est quod deficiat ei scientia de coloribus. Et sic habebit de coloribus ignorantiam negationis, omnino ignorans colorem. Sed hoc intelligendum est quando nunquam habuit sensum visus, sicut patet in caeco nato. Si quis enim amittat visum prius habitum, non propter hoc oportet quod careat scientia colorum; quia ex his, quae prius sensit, remanet in eo memoria colorum. He says therefore first (81a38) that if a person lacks any of the senses, say, sight or hearing, then necessarily the science of the sensible objects proper to those senses will be lacking. Thus, if a person lacks the sense of sight, then of necessity the science of colors will be lacking in him. And so he will have ignorance of negation in regard to colors, being entirely ignorant of colors. However, this must be understood of persons who never had the sense of sight, as a person born blind. But if someone loses the sight he once had, he does not on that account necessarily lack a science of colors, because the memory of colors previously sensed remains in him.
Contingit autem de aliquibus rebus haberi ignorantiam negationis, quae tamen cognosci possunt per sensum quem habemus: sicut si aliquis habens visum semper fuisset in tenebris, careret quidem scientia colorum, sed non ex necessitate, quia posset huiusmodi scientiam accipere sentiendo colores: quod non contingit in eo, qui caret sensu visus. Et ideo addit quod impossibile est accipere; quia videlicet ille, qui caret potentia visiva, non potest percipere cognitionem colorum. But it is possible that ignorance of negation be had of certain things which can nevertheless be known through a sense we possess.. For example, if someone with sight were always in the dark, he would de facto lack a science of colors, but not of necessity, because he could acquire it by sensing colors-which does not occur in one who lacks the sense of sight. Hence he adds that it is impossible to receive it because one who lacks the power of sight cannot even acquire a knowledge of colors.
Deinde cum dicit: siquidem addiscimus etc., probat propositum per hoc quod duplex est modus acquirendi scientiam. Unus quidem per demonstrationem, alius autem per inductionem; quod etiam in principio huius libri positum est. Differunt autem hi duo modi, quia demonstratio procedit ex universalibus; inductio autem procedit ex particularibus. Si ergo universalia, ex quibus procedit demonstratio, cognosci possent absque inductione, sequeretur quod homo posset accipere scientiam eorum, quorum non habet sensum. Sed impossibile est universalia speculari absque inductione. Et hoc quidem in rebus sensibilibus est magis manifestum, quia in eis per experientiam, quam habemus circa singularia sensibilia, accipimus universalem notitiam, sicut manifestatur in principio metaphysicae. Then (81a39) he proves his proposition on the ground that there are two ways of acquiring science: one is through demonstration, and the other is through induction, as we have stated in the beginning of this book. But these two ways differ, because demonstration proceeds from universals, but induction from particulars. Therefore, if any universals from which demonstration proceeds could be known without induction, it would follow that a person could acquire science of things of which he does not have sense experience. But it is impossible that universals be known scientifically without induction. This is quite obvious in sensible things, because we receive the universal aspect in them through the experience which we have in regard to sensible things, as is explained in Metaphysics I. But this might be doubted in things which are abstract, as in mathematics. For although experience begins from sense, as it is stated in Metaphysics 1, it seems that this plays no role in studying things already isolated or abstracted from sensible matter.
Sed maxime hoc videtur dubium in his, quae dicuntur secundum abstractionem, sicut in mathematicis. Cum enim experientia a sensu ortum habeat, ut dicitur in principio metaphysicae, videtur quod hoc locum non habeat in his, quae sunt abstracta a materia sensibili. Et ideo ad hoc excludendum dicit quod etiam ea, quae dicuntur secundum abstractionem, contingit nota facere per inductionem; quia in unoquoque genere abstractorum sunt quaedam particularia, quae non sunt separabilia a materia sensibili, secundum quod unumquodque eorum est hoc. Quamvis enim linea secundum abstractionem dicatur, tamen haec linea, quae est in materia sensibili, in quantum est individuata abstrahi non potest, quia individuatio eius est ex hac materia. Non autem manifestantur nobis principia abstractorum, ex quibus demonstrationes in eis procedunt, nisi ex particularibus aliquibus, quae sensu percipimus. Puta ex hoc, quod videmus aliquod totum singulare sensibile, perducimur ad cognoscendum quid est totum et pars, et cognoscimus quod omne totum est maius sua parte, considerando hoc in pluribus. Sic igitur universalia, ex quibus demonstratio procedit, non fiunt nobis nota, nisi per inductionem. Therefore, to exclude this he says that even those things that are abstract happen to be made known through induction, because in each genus of abstract things are certain particulars which are not isolable from sensible matter, so far as each of them is a “this something.” For although “line” is studied in isolation from its sensible matter, nevertheless “this line”, which is in sensible matter, so far forth as it is individualized, cannot be so isolated, because its individuation is from this matter. Furthermore, the principles of abstracted [isolated] things, from which demonstrations regarding them are formed, are not made manifest to us except from certain particulars which we perceive by sense. Thus, from the fact that we see some single sensible whole we are led to know what a whole is and what a part, and we know that every whole is greater than its part by considering this in many. Thus the universals from which demonstration proceeds are made known to us only through induction.
Homines autem carentes sensu aliquo non possunt inductionem facere de singularibus pertinentibus ad sensum illum, quia singularium, ex quibus procedit inductio, est solum cognitio sensus. Unde oportet quod omnino sint huiusmodi singularia ignota, quia non contingit quod aliquis carens sensu accipiat talium singularium scientiam; quia neque ex universalibus potest demonstrare sine inductione, per quam universalia cognoscuntur, ut dictum est; neque per inductionem potest aliquid cognosci sine sensu, qui est singularium, ex quibus procedit inductio. Now men who lack any of the senses cannot make an induction from singulars pertaining to that sense, because sense is the sole knower of the singulars from which induction proceeds. Hence such singulars are utterly unknown, because it does not occur that anyone lacking a sense receives science of such singulars: first, because he cannot demonstrate from universals without induction through which universals are known, as has been said; secondly, because nothing can be known through induction without the sense which is concerned with the singulars from which induction proceeds.
Est autem considerandum quod per verba philosophi, quae hic inducuntur, excluditur duplex positio. Prima quidem est positio Platonis; qui ponebat quod nos habebamus scientiam de rebus per species participatas ab ideis. Quod si esset verum, universalia fierent nobis nota absque inductione; et ita possemus acquirere scientiam eorum, quorum sensum non habemus. Unde et hoc argumento utitur Aristoteles contra Platonem in fine I metaphysicae. Secunda est positio dicentium quod possumus in hac vita cognoscere substantias separatas, intelligendo quidditates earum; quae tamen per sensibilia quae cognoscimus, quae ab eis omnimode transcenduntur, cognosci non possunt. Unde, si ipsae cognoscerentur secundum suas essentias, sequeretur quod aliqua cognoscerentur absque inductione et sensu: quod philosophus hic negat, etiam de abstractis. It should be noted that by these words of the Philosopher two positions are excluded: the first is Plato’s, who stated that we do not have science of things except through Forms participated from ideas. If this were so, universals could be made known to us without induction, and we would be able to acquire a science of things of which we have no sense. Hence Aristotle also uses this argument against Plato at the end of Metaphysics I. The second is the position of those who claim that in this life we can know separated substances by understanding their quiddities, which however cannot be known through sensible objects which we know and which are entirely transcended by them. Hence if they were known according to their essences, it would follow that some things would be known without induction and sense perception, which the Philosopher here denies even in regard to abstracted things.

Lectio 31
Caput 19
Ἔστι δὲ πᾶς συλλογισμὸς διὰ τριῶν ὅρων, καὶ ὁ μὲν δεικνύναι δυνάμενος ὅτι ὑπάρχει τὸ Α τῷ Γ διὰ τὸ ὑπάρχειν τῷ Β καὶ τοῦτο τῷ Γ, ὁ δὲ στερητικός, τὴν μὲν ἑτέραν πρότασιν ἔχων ὅτι ὑπάρχει τι ἄλλο ἄλλῳ, τὴν δ' ἑτέραν ὅτι οὐχ ὑπάρχει. b10. Every syllogism is effected by means of three terms. One kind of syllogism serves to prove that A inheres in C by showing that A inheres in B and B in C; the other is negative and one of its premisses asserts one term of another, while the other denies one term of another.
φανερὸν οὖν ὅτι αἱ μὲν ἀρχαὶ καὶ αἱ λεγόμεναι ὑποθέσεις αὗταί εἰσι· λαβόντα γὰρ ταῦτα οὕτως ἀνάγκη δεικνύναι, οἷον ὅτι τὸ Α τῷ Γ ὑπάρχει διὰ τοῦ Β, πάλιν δ' ὅτι τὸ Α τῷ Β δι' ἄλλου μέσου, καὶ ὅτι τὸ Β τῷ Γ ὡσαύτως. b14. It is clear, then, that these are the fundamentals and so-called hypotheses of syllogism. Assume them as they have been stated, and proof is bound to follow — proof that A inheres in C through B, and again that A inheres in B through some other middle term, and similarly that B inheres in C.
κατὰ μὲν οὖν δόξαν συλλογιζομένοις καὶ μόνον διαλεκτικῶς δῆλον ὅτι τοῦτο μόνον σκεπτέον, εἰ ἐξ ὧν ἐνδέχεται ἐνδοξοτάτων γίνεται ὁ συλλογισμός, ὥστ' εἰ καὶ μὴ ἔστι τι τῇ ἀληθείᾳ τῶν Α Β μέσον, δοκεῖ δὲ εἶναι, ὁ διὰ τούτου συλλογιζόμενος συλλελόγισται διαλεκτικῶς· πρὸς δ' ἀλήθειαν ἐκ τῶν ὑπαρχόντων δεῖ σκοπεῖν. b18. If our reasoning aims at gaining credence and so is merely dialectical, it is obvious that we have only to see that our inference is based on premisses as credible as possible: so that if a middle term between A and B is credible though not real, one can reason through it and complete a dialectical syllogism. If, however, one is aiming at truth, one must be guided by the real connexions of subjects and attributes.
ἔχει δ' οὕτως· ἐπειδὴ ἔστιν ὃ αὐτὸ μὲν κατ' ἄλλου κατηγορεῖται μὴ κατὰ συμβεβηκός—λέγω δὲ τὸ κατὰ συμβεβηκός, οἷον τὸ λευκόν ποτ' ἐκεῖνό φαμεν εἶναι ἄνθρωπον, οὐχ ὁμοίως λέγοντες καὶ τὸν ἄνθρωπον λευκόν· ὁ μὲν γὰρ οὐχ ἕτερόν τι ὢν λευκός ἐστι, τὸ δὲ λευκόν, ὅτι συμβέβηκε τῷ ἀνθρώπῳ εἶναι λευκῷ—ἔστιν οὖν ἔνια τοιαῦτα ὥστε καθ' αὑτὰ κατηγορεῖσθαι. b24. Thus: since there are attributes which are predicated of a subject essentially or naturally and not coincidentally — not, that is, in the sense in which we say 'That white (thing) is a man', which is not the same mode of predication as when we say 'The man is white': the man is white not because he is something else but because he is man, but the white is man because 'being white' coincides with 'humanity' within one substratum — therefore there are terms such as are naturally subjects of predicates.
Ἔστω δὴ τὸ Γ τοιοῦτον ὃ αὐτὸ μὲν μηκέτι ὑπάρχει ἄλλῳ, τούτῳ δὲ τὸ Β πρώτῳ, καὶ οὐκ ἔστιν ἄλλο μεταξύ. καὶ πάλιν τὸ Ε τῷ Ζ ὡσαύτως, καὶ τοῦτο τῷ Β. ἆρ' οὖν τοῦτο ἀνάγκη στῆναι, ἢ ἐνδέχεται εἰς ἄπειρον ἰέναι; b30. Suppose, then, C such a term not itself attributable to anything else as to a subject, but the proximate subject of the attribute B — i.e. so that B-C is immediate; suppose further E related immediately to F, and F to B. The first question is, must this series terminate, or can it proceed to infinity?
καὶ πάλιν εἰ τοῦ μὲν Α μηδὲν κατηγορεῖται καθ' αὑτό, τὸ δὲ Α τῷ Θ ὑπάρχει πρώτῳ, μεταξὺ δὲ μηδενὶ προτέρῳ, καὶ τὸ Θ τῷ Η, καὶ τοῦτο τῷ Β, ἆρα καὶ τοῦτο ἵστασθαι ἀνάγκη, ἢ καὶ τοῦτ' ἐνδέχεται εἰς ἄπειρον ἰέναι; διαφέρει δὲ τοῦτο τοῦ πρότερον τοσοῦτον, ὅτι τὸ μέν ἐστιν, ἆρα ἐνδέχεται ἀρξαμένῳ ἀπὸ τοιούτου ὃ μηδενὶ ὑπάρχει ἑτέρῳ ἀλλ' ἄλλο ἐκείνῳ, ἐπὶ τὸ ἄνω εἰς ἄπειρον ἰέναι, θάτερον δὲ ἀρξάμενον ἀπὸ τοιούτου (82a.) ὃ αὐτὸ μὲν ἄλλου, ἐκείνου δὲ μηδὲν κατηγορεῖται, ἐπὶ τὸ κάτω σκοπεῖν εἰ ἐνδέχεται εἰς ἄπειρον ἰέναι. b34. The second question is as follows: Suppose nothing is essentially predicated of A, but A is predicated primarily of H and of no intermediate prior term, and suppose H similarly related to G and G to B; then must this series also terminate, or can it too proceed to infinity? There is this much difference between the questions: the first is, is it possible to start from that which is not itself attributable to anything else but is the subject of attributes, and ascend to infinity? The second is the problem whether one can start from that which is a predicate but not itself a subject of predicates, and descend to infinity?
Ἔτι τὰ μεταξὺ ἆρ' ἐνδέχεται ἄπειρα εἶναι ὡρισμένων τῶν ἄκρων; λέγω δ' οἷον εἰ τὸ Α τῷ Γ ὑπάρχει, μέσον δ' αὐτῶν τὸ Β, τοῦ δὲ Β καὶ τοῦ Α ἕτερα, τούτων δ' ἄλλα, ἆρα καὶ ταῦτα εἰς ἄπειρον ἐνδέχεται ἰέναι, ἢ ἀδύνατον; a2. A third question is, if the extreme terms are fixed, can there be an infinity of middles? I mean this: suppose for example that A inheres in C and B is intermediate between them, but between B and A there are other middles, and between these again fresh middles; can these proceed to infinity or can they not?
ἔστι δὲ τοῦτο σκοπεῖν ταὐτὸ καὶ εἰ αἱ ἀποδείξεις εἰς ἄπειρον ἔρχονται, καὶ εἰ ἔστιν ἀπόδειξις ἅπαντος, ἢ πρὸς ἄλληλα περαίνεται. a7. This is the equivalent of inquiring, do demonstrations proceed to infinity, i.e. is everything demonstrable? Or do ultimate subject and primary attribute limit one another?
Ὁμοίως δὲ λέγω καὶ ἐπὶ τῶν στερητικῶν συλλογισμῶν καὶ προτάσεων, οἷον εἰ τὸ Α μὴ ὑπάρχει τῷ Β μηδενί, ἤτοι πρώτῳ, ἢ ἔσται τι μεταξὺ ᾧ προτέρῳ οὐχ ὑπάρχει (οἷον εἰ τῷ Η, ὃ τῷ Β ὑπάρχει παντί), καὶ πάλιν τούτου ἔτι ἄλλῳ προτέρῳ, οἷον εἰ τῷ Θ, ὃ τῷ Η παντὶ ὑπάρχει. καὶ γὰρ ἐπὶ τούτων ἢ ἄπειρα οἷς ὑπάρχει προτέροις, ἢ ἵσταται. a9. I hold that the same questions arise with regard to negative conclusions and premisses: viz. if A is attributable to no B, then either this predication will be primary, or there will be an intermediate term prior to B to which a is not attributable — G, let us say, which is attributable to all B — and there may still be another term H prior to G, which is attributable to all G. The same questions arise, I say, because in these cases too either the series of prior terms to which a is not attributable is infinite or it terminates.
Ἐπὶ δὲ τῶν ἀντιστρεφόντων οὐχ ὁμοίως ἔχει. οὐ γὰρ ἔστιν ἐν τοῖς ἀντικατηγορουμένοις οὗ πρώτου κατηγορεῖται ἢ τελευταίου πάντα γὰρ πρὸς πάντα ταύτῃ γε ὁμοίως ἔχει, εἴτ' ἐστὶν ἄπειρα τὰ κατ' αὐτοῦ κατηγορούμενα, εἴτ' ἀμφότερά ἐστι τὰ ἀπορηθέντα ἄπειρα· πλὴν εἰ μὴ ὁμοίως ἐνδέχεται ἀντιστρέφειν, ἀλλὰ τὸ μὲν ὡς συμβεβηκός, τὸ δ' ὡς κατηγορίαν. a15. One cannot ask the same questions in the case of reciprocating terms, since when subject and predicate are convertible there is neither primary nor ultimate subject, seeing that all the reciprocals qua subjects stand in the same relation to one another, whether we say that the subject has an infinity of attributes or that both subjects and attributes — and we raised the question in both cases — are infinite in number. These questions then cannot be asked — unless, indeed, the terms can reciprocate by two different modes, by accidental predication in one relation and natural predication in the other.
Postquam philosophus determinavit de syllogismo demonstrativo, ostendens ex quibus et qualibus procedat, et in qua figura demonstrationes fieri possunt; hic inquirit utrum demonstrationes possint in infinitum procedere. Et primo, movet quaestionem; secundo, determinat eam; ibi: quod quidem igitur non contingit media et cetera. Circa primum duo facit: primo, praemittit quaedam, quae sunt necessaria ad intellectum quaestionis; secundo, movet quaestionem; ibi: sit igitur c huiusmodi et cetera. Circa primum duo facit: primo, praemittit de forma syllogistica, quam oportet in demonstrationibus observare; secundo, resumit qualis debeat esse demonstrationis materia; ibi: manifestum igitur est quod principia et cetera. After the Philosopher has determined concerning the demonstrative syllogism by showing from what and what sort of things it proceeds and in which figure demonstrations can be formed, he now inquires whether demonstrations can proceed to infinity. First, he raises the question. Secondly, he settles it (82a21) [L. 32]. Concerning the first he does two things. First, he sets down certain prefatory remarks needed for understanding the question. Secondly, he raises the question (81b30). Concerning the first he does two things. First, he prefaces something about the syllogistic form one must observe in demonstrations. Secondly, he re views what the matter of demonstration should be (81b14).
Circa primum tria tangit. Quorum primum est commune omni syllogismo, scilicet quod omnis syllogismus est per tres terminos; ut manifestum est in libro priorum. Secundum autem pertinet ad syllogismum affirmativum; cuius forma est talis quod concludit a esse in c propter id, quod a est in b, et b est in c; et haec est forma syllogistica in prima figura, in qua sola potest concludi affirmativa universalis, quae maxime quaeritur in demonstrationibus. Tertium est quod pertinet ad syllogismum negativum, qui de necessitate unam propositionem habet affirmativam, aliam autem negativam; differenter tamen in prima figura et in secunda, ut patet per ea, quae in libro priorum ostensa sunt. With respect to the first (81b10) he touches on three things, the first of which is common to every syllogism, namely, that every syllogism is formed in three terms, as is indicated in Prior Analytics I. The second however, pertains to an affirmative syllogism whose form is such that it concludes A to be in C, because A is in B and B in C. And this is the form of a syllogism in the first figure in which alone can be concluded a universal affirmative, the chief quest in demonstration. The third pertains to a negative syllogism which of necessity has one affirmative proposition and one negative, but differently in the first figure and in the second, as is clear from what has been shown in the Prior Analytics.
Deinde cum dicit: manifestum igitur est etc., resumit quae sit materia demonstrationum. Et circa hoc tria facit: primo enim proponit demonstrationis materiam; secundo, ostendit differentiam huius materiae ad materiam syllogismi dialectici; ibi: secundum quidem igitur opinationem etc.; tertio, differentiam positam manifestat; ibi: habet autem sic se et cetera. Then (81b14) he reviews what the matter of demonstration should be. In regard to this he does three things. First, he states what this matter is. Secondly, he shows the difference between this matter and the matter of a dialectical syllogism (81b18). Thirdly, he clarifies this difference (81b24).
Dicit ergo primo quod, cum syllogismus habeat tres terminos, ex quibus formantur duae propositiones concludentes tertiam, manifestum est quod hae propositiones, ex quibus proceditur in syllogismo demonstrativo secundum formam praedictam, sunt principia et suppositiones, de quibus in praecedentibus dictum est. Qui enim accipit huiusmodi principia, sic demonstrat per ea, sicut expositum est in forma syllogistica, ut scilicet quia a sit in c probatur per b; et si propositio a.b sit iterum mediata, quod a sit in b demonstratur per aliud medium. Et simile est si propositio minor, scilicet b.c, sit mediata. He says therefore first (81b14) that since a syllogism has three terms from which are formed the two propositions which conclude the third, it is clear that these propositions, from which one proceeds in a demonstrative syllogism according to the aforesaid form, are the principles and suppositions we discussed earlier. For one who accepts such principles demonstrates through them in the syllogistic form we have mentioned, namely, that A is in C is proved through B; and if the proposition AB is mediate, another middle is used to demonstrate that A is in B. The like is done if the minor proposition, BC, is mediate.
Deinde cum dicit: secundum quidem igitur etc., ostendit quantum ad praedicta differentiam inter syllogismum demonstrativum et syllogismum dialecticum. Quia enim syllogismus dialecticus ad hoc tendit, ut opinionem faciat, hoc solum est de intentione dialectici, ut procedat ex his, quae sunt maxime opinabilia, et haec sunt ea, quae videntur vel pluribus, vel maxime sapientibus. Et ideo si dialectico in syllogizando occurrat aliqua propositio, quae secundum rei veritatem habeat medium, per quod possit probari, sed tamen non videatur habere medium, sed propter sui probabilitatem videatur esse per se nota; hoc sufficit dialectico, nec inquirit aliud medium, licet propositio sit mediata, et, ex ea syllogizans, sufficienter perficit dialecticum syllogismum. Then (81b18), apropos of what has been said, he shows the difference between a demonstrative and a dialectical syllogism. For since the latter aims at producing opinion, the sole intent of a dialectician is to proceed from things that are most probable, and these are things that appear to the majority or to the very wise. Hence if a dialectician in syllogizing happens upon a proposition which really has a middle through which it could be proved, but it seems not to have a middle because it appears to be per se known on account of its probability, this is enough for the dialectician: he does not search for a middle, even though the proposition is mediate. Rather he syllogizes from it and completes the dialectical syllogism satisfactorily.
Sed syllogismus demonstrativus ordinatur ad scientiam veritatis; et ideo ad demonstratorem pertinet, ut procedat ex his, quae sunt secundum rei veritatem immediata. Et si occurrat ei mediata propositio, necesse est quod probet eam per medium proprium, quousque deveniat ad immediata, nec est contentus probabilitate propositionis. The demonstrative syllogism, on the other hand, is ordained to scientific knowledge of the truth; accordingly, it pertains to the demonstrator to proceed from truths which are really immediate. Hence if he happens upon a mediate proposition, he must prove it through its proper middle until he reaches something immediate, because he is not content with the probability of a proposition.
Deinde cum dicit: habet autem se sic etc., manifestat quod dixerat, dicens quod hoc, quod dictum est, quod demonstrator ad veritatem ex his quae sunt procedit, sic se habet ut dicetur. Invenitur enim aliquid, quod de alio praedicatur, non secundum accidens, et hoc exponit per affirmativam, ostendens quid praedicetur secundum accidens. Then (81b24) he elucidates what he has said, asserting that his claim that the demonstrator proceeds “to the truth from things that are” is supported by the fact that it is possible to find something “which is not predicated of a thing accidentally.” What this means he explains by showing how in the case of affirmative statements something is predicated accidentally.
Dupliciter enim aliquid praedicatur secundum accidens: uno modo, quando subiectum praedicatur de accidente, puta cum dicimus, album est homo; alio modo dissimiliter, quando accidens praedicatur de subiecto, sicut cum dicitur, homo est albus. Et differt hic modus a primo, quoniam hic, quando accidens praedicatur de subiecto, dicitur, homo est albus, non quia aliquid alterum sit album, sed quia ipse homo est albus: et tamen est propositio per accidens, quia album non convenit homini secundum propriam rationem. Non enim ponitur in definitione eius, neque e converso. Sed quando dicitur, album est homo, hoc non dicitur, quia esse hominem insit albo, sed quia esse hominem inest subiecto albi, cui scilicet accidit esse album. Unde hic modus est magis remotus a praedicatione per se, quam primus. For something is predicated accidentally in two ways: in one way, when the subject is predicated of an accident, as when we say, “The white thing is a man”; in another way, when the accident is predicated of the subject, as when we say, “The man is white.” Now this way differs from the first, because when the accident is predicated of the subject, it is stated that the man is white not because something else is white but because the man himself is white. Yet it is a per accidens proposition, because “white” does not belong to man according to the specific nature of man. For neither is placed in the definition of the other. But when it is stated that the white thing is a man, it is not so stated because being a man is in the whiteness, but because being a man is in the subject of whiteness, which subject happens to be white. Hence this way is further removed from per se predication than the first.
Sunt autem quaedam, quae neutro istorum modorum per accidens praedicantur; et ista dicuntur per se. Et talia sunt, ex quibus demonstrator procedit. Sed hoc dialecticus non requirit, et ideo quaestio, quae infra proponitur de huiusmodi quae per se praedicantur, non habet locum in syllogismis dialecticis, sed solum in syllogismo demonstrativo. But there are certain things which are not predicated per accidens in either of these ways: these are said to be per se. Such are the things from which the demonstrator proceeds. But the dialectician is not so demanding; consequently, the question concerning such things as are predicated per se is not relevant to the dialectical syllogism but only to the demonstrative syllogism.
Deinde cum dicit: sit igitur c huiusmodi etc., movet quaestiones intentas. Et circa hoc duo facit: primo, movet quaestiones in quibus locum habent; secundo, ostendit in quibus locum non habent; ibi: sed in convertentibus et cetera. Circa primum duo facit: primo, movet quaestiones in demonstrationibus affirmativis; secundo, ostendit quod hae quaestiones similiter locum habent in demonstrationibus negativis; ibi: similiter autem dico et in privativis et cetera. Circa primum duo facit: primo, movet quaestiones; secundo, ostendit ad quid huiusmodi quaestiones pertineant; ibi: est autem hoc intendere et cetera. Then (81b30) he raises the questions he intended. Concerning this he does two things. First, he raises the questions in regard to things to which they are relevant. Secondly, he shows the cases in which they are not relevant (82a15). Concerning the first he does two things. First, he raises questions in affirmative demonstrations. Secondly, he shows that these questions also have relevance in negative demonstrations (82a9). In regard to the first he does two things. First, he raises the questions. Secondly, he shows where such questions are relevant (82a7).
Circa primum movet tres quaestiones secundum tres terminos syllogismi. Et primo, movet quaestionem ex parte maioris extremitatis, utrum sit abire in infinitum ascendendo. Et in hac quaestione supponitur ultimum subiectum, quod non praedicatur de alio, et alia praedicantur de ipso. Sit ergo hoc c, et in c primo et immediate sit b, et in b sit e quasi de eo universaliter praedicatum, et iterum f sit in e similiter de eo universaliter praedicatum. Est ergo quaestio: utrum iste ascensus alicubi stet, ita scilicet quod sit devenire ad aliquid quod praedicetur de aliis universaliter, et nihil aliud praedicetur de ipso; aut hoc non sit necesse, sed contingat ascendere in infinitum? In regard to the first (81b30) he raises three questions corresponding to the three terms of a syllogism. First, he raises a question concerning the major extreme, namely, whether one can go to infinity in ascending order? In this question an ultimate subject is supposed which is not predicated of any other, but other things are predicated of it. Let this subject be C, and let B be in C first and immediately, and let E be in B, as universally predicated of B; furthermore, let F be in E as universally predicated of it. The question is this: Should this ascending process come to a halt somewhere, so that something is reached which is predicated universally of other things but nothing else is predicated of it, or is that not necessary but a process to infinity occurs?
Secundo, ibi: et iterum si de a quidem etc., movet quaestionem ex parte minoris termini, utrum scilicet sit ire in infinitum descendendo. Et in hac quaestione supponitur esse aliquod primum praedicatum universale, quod de aliis praedicetur, et nihil sit universalius eo, quod praedicetur de ipso. Sit ergo a tale, quod nihil de eo praedicetur sicut totum universale de parte, a vero praedicetur de c primo et immediate, et c de I, et I de b. Est ergo quaestio: utrum necesse sit hic descendendo stare, aut contingat in infinitum ire? Secondly (81b34) he raises the question on the part of the minor term, namely, whether one can go to infinity in descending. In this question some first universal predicate is supposed which is predicated of other things and nothing is more universal than it so as to be predicated of it. Thus let A be such that nothing is predicated of it as a universal whole of a part, but A is predicated of H both first and immediately, and H of G, and G of B. The question then is this: Is it necessary to come to a halt in this descending process, or may it proceed to infinity?
Et ostendit consequenter differentiam harum duarum quaestionum, quia in prima quaestione quaerebatur: si aliquis incipiat a particularissimo subiecto, quod nulli inest per modum quo totum universale inest parti, sed alia insunt ei, utrum contingat procedere in infinitum ascendendo? Secunda vero quaestio est: si aliquis incipiat ab universalissimo praedicato, quod praedicatur de aliis sicut totum universale de parte, et nihil hoc modo praedicatur de illo, utrum contingat descendendo procedere in infinitum? Then he shows the difference between these two questions. For in the first one we asked: If someone begins from a most particular subject which is in nothing else the way a universal whole is in a part but other things are in it, does an infinite ascending process occur? But in the second we are asking: If someone begins with a most universal predicate, which is predicated of other things as a universal whole of its parts but nothing is predicated of it in this way, does an infinite descending process occur?
Tertio, ibi: amplius media etc., movet tertiam quaestionem ex parte medii termini. Et in hac quaestione supponuntur duo extrema determinata, scilicet universalissimum praedicatum, et particularissimum subiectum; et quaeritur cum hoc, utrum possint esse infinita media: puta, si a sit universalissimum praedicatum, et c sit particularissimum subiectum, et inter a et c sit medium b, et inter a et b iterum sit aliud, et similiter inter b et c, et horum etiam mediorum sint alia media, inter ipsa scilicet et extrema, tam ascendendo quam descendendo. Est ergo quaestio: utrum hoc possit procedere in infinitum, aut hoc sit impossibile? Thirdly (82a2) he raises the third question on the part of the middle term. In this question two determinate extremes are supposed, namely, a most universal predicate and a most particular subject. The question is whether under these conditions there can be an infinity of middles. Thus, if A is the most universal predicate and C the most particular subject, and if between A and C there is the middle, B, and again between A and B another middle, and likewise between B and C; furthermore, if there are other middles of these middles between them and the extremes both in ascending and in descending order. The question then is this: May these processes go on to infinity or is that impossible?
Deinde cum dicit: est autem hoc intendere etc., ostendit ad quid tendant huiusmodi quaestiones; in quo declaratur quod huiusmodi quaestiones pertinent ad materiam, de qua nunc agitur, scilicet ad demonstrationes. Dicit ergo quod intendere inquisitioni veritatis in istis quaestionibus idem est ac si quaeratur, utrum demonstrationes procedant in infinitum, vel ascendendo vel descendendo. Ascendendo quidem, ita quod quaelibet propositio, ex qua demonstratio procedit, sit demonstrabilis per aliam priorem demonstrationem; et hoc est quod subiungit, et si est demonstratio omnis, idest cuiuslibet propositionis. Quod quidam existimantes, circa principia erraverunt, ut dicitur in IV metaphysicae. Descendendo autem, si ex qualibet propositione demonstrata contingat iterum ad aliam demonstrationem posteriorem procedere. Et hoc est unum membrum dubitationis, si demonstrationes in infinitum procedunt, vel descendendo vel ascendendo. Aliud autem membrum dubitationis est, si demonstrationes ad invicem terminantur, ita scilicet quod una demonstratio confirmetur per aliam ascendendo, et ex una demonstratione procedat alia descendendo, et hoc usque ad aliquem terminum. Then (82a7) he shows what is the tenor of these questions. And he says that these questions pertain to the matter now under discussion, namely, to demonstrations. He says, therefore, that the attempt to reach true answers to these questions is the same as trying to settle the question whether demonstrations proceed to infinity by ascending or descending. By ascending, i.e., so that each proposition from which a demonstration proceeds would be demonstrable by another prior proposition. This is what he means when he asks: “Is there a demonstration of everything,” i.e., of every proposition? By so thinking, some have erred in regard to principles, as is stated in Metaphysics IV. And by descending, i.e., whether it is possible from any demonstrated proposition to proceed again to another demonstration subsequent to it. Thus, one element of the doubt is whether demonstrations proceed to infinity either by ascending or by descending. The other element is whether demonstrations are mutually limiting, so that one demonstration may be confirmed by another in an ascending process, and from one demonstration another may proceed by a descending process: and this until a limit is reached.
Deinde cum dicit: similiter autem dico etc., ostendit quod praedictae dubitationes habent locum etiam in demonstrationibus negativis, quia demonstratio negativa oportet quod utatur propositione affirmativa, in qua subiectum conclusionis contineatur sub medio, a quo praedicatum conclusionis removeatur. Secundum ergo quod est ascensus et descensus in affirmativis, oportet quod sit ascensus et descensus in negativis syllogismis, et propositionibus; ut puta si conclusio demonstrativi syllogismi sit, nullum c est a, et accipiatur sicut medium b, a quo a removeatur. Est ergo primo considerandum utrum a removeatur a b primo, sive immediate, aut sit aliquod medium accipere, a quo primo removeatur a quam a b, puta si prius removeatur ab I, quod oportet universaliter praedicari de b; et iterum erit considerandum utrum a removeatur ab aliquo per prius quam ab I, scilicet a t, quod praedicatur universaliter de I. Ita ergo et in his potest procedi in infinitum in removendo, ut semper sit aliquid accipere, a quo per prius removeatur, vel oportet alicubi stare. Then (82a9) he shows that these questions are also relevant to negative demonstrations, because a negative demonstration must employ an affirmative proposition in which the subject of the conclusion is contained under the middle and from which the predicate of the conclusion is removed. Therefore, to the extent that there is ascent and descent in affirmative, there must be ascent and descent in negative syllogisms and propositions. For example, if the conclusion of a demonstrative syllogism is, “No C is A,” and the middle taken is B, from which A is removed: the first thing to be considered, therefore, is whether A is removed from B first and immediately, or whether there is another middle G to be taken, from which A would be removed before it would be removed from B. In that case it would be necessary to consider whether A would be removed from something else before G, namely, from H which is predicated universally of G. Therefore in these also the question arises whether one can proceed to infinity in removing (so that something would always remain from which it would have to be removed), or must one stop somewhere.
Deinde cum dicit: sed in convertibilibus etc., ostendit in quibus praedictae quaestiones locum non habeant. Quia in his, quae aequaliter de se invicem praedicantur et convertuntur ad invicem, non est accipere aliquod prius et posterius secundum illum modum, quo prius est, a quo non convertitur consequentia essendi, prout universalia sunt priora; quia sive praedicata sint infinita, ita scilicet quod procedatur in infinitum in praedicando, sive sint infinita ex utraque parte, idest tam ex parte praedicati quam ex parte subiecti, omnia huiusmodi infinita similiter se habebunt ad omnia; quia quodlibet eorum poterit praedicari de quolibet, et subiici cuilibet convertibilium. Nisi solum quod potest esse talis differentia, quod unum eorum praedicatur ut accidens, et aliud praedicatur sicut praedicamentum, idest sicut substantiale praedicatum. Et haec est differentia proprii et definitionis, quorum utrumque est convertibile; et tamen definitio est praedicatum essentiale, et propter hoc est prius naturaliter proprio, quod est praedicatum accidentale. Et inde est quod in demonstrationibus utimur definitione quasi medio ad demonstrandum propriam passionem de subiecto. Then (82a15) he shows the cases in which these questions have no relevance: for in cases in which there is mutual predication and mutual conversion there is no prior and subsequent to be taken in the sense in which the prior [notion] is that with which a subsequent [notion] is not convertible, as universals are prior; because no matter whether the predicates be infinite, so that one might proceed to infinity in predicating, or whether there be infinity on both sides, i.e., on the side of the predicate as well as of the subject, all such infinites bear a like relationship to all, because any of them could be predicated of any other and be the subject of any of the convertibles. However, there can be this difference: one of them might be predicated as an accident and another as a predicament, i.e., as a substantial predicate. And this is the difference between a property, and a definition: although the two are convertible with the subject ,nevertheless the definition is an essential predicate and therefore naturally prior to the property, which is an accidental predicate. That is why in demonstrations we use the definition as the middle to demonstrate a proper attribute of the subject.

Lectio 32
Caput 20
Ὅτι μὲν οὖν τὰ μεταξὺ οὐκ ἐνδέχεται ἄπειρα εἶναι, εἰ ἐπὶ τὸ κάτω καὶ τὸ ἄνω ἵστανται αἱ κατηγορίαι, δῆλον. λέγω δ' ἄνω μὲν τὴν ἐπὶ τὸ καθόλου μᾶλλον, κάτω δὲ τὴν ἐπὶ τὸ κατὰ μέρος. a2l. Now, it is clear that if the predications terminate in both the upward and the downward direction (by 'upward' I mean the ascent to the more universal, by 'downward' the descent to the more particular), the middle terms cannot be infinite in number.
εἰ γὰρ τοῦ Α κατηγορουμένου κατὰ τοῦ Ζ ἄπειρα τὰ μεταξύ, ἐφ' ὧν Β, δῆλον ὅτι ἐνδέχοιτ' ἂν ὥστε καὶ ἀπὸ τοῦ Α ἐπὶ τὸ κάτω ἕτερον ἑτέρου κατηγορεῖσθαι εἰς ἄπειρον (πρὶν γὰρ ἐπὶ τὸ Ζ ἐλθεῖν, ἄπειρα τὰ μεταξύ) καὶ ἀπὸ τοῦ Ζ ἐπὶ τὸ ἄνω ἄπειρα, πρὶν ἐπὶ τὸ Α ἐλθεῖν. ὥστ' εἰ ταῦτα ἀδύνατα, καὶ τοῦ Α καὶ Ζ ἀδύνατονἄπειρα εἶναι μεταξύ. a24. For suppose that A is predicated of F, and that the intermediates — call them B B' B"... — are infinite, then clearly you might descend from and find one term predicated of another ad infinitum, since you have an infinity of terms between you and F; and equally, if you ascend from F, there are infinite terms between you and A. It follows that if these processes are impossible there cannot be an infinity of intermediates between A and F.
οὐδὲ γὰρ εἴ τις λέγοι ὅτι τὰ μέν ἐστι τῶν Α Β Ζ ἐχόμενα ἀλλήλων ὥστε μὴ εἶναι μεταξύ, τὰ δ' οὐκ ἔστι λαβεῖν, οὐδὲν διαφέρει. ὃ γὰρ ἂν λάβω τῶν Β, ἔσται πρὸς τὸ Α ἢ πρὸς τὸ Ζ ἢ ἄπειρα τὰ μεταξὺ ἢ οὔ. ἀφ' οὗ δὴ πρῶτον ἄπειρα, εἴτ' εὐθὺς εἴτε μὴ εὐθύς, οὐδὲν διαφέρει· τὰ γὰρ μετὰ ταῦτα ἄπειρά ἐστιν. a30. Nor is it of any effect to urge that some terms of the series AB...F are contiguous so as to exclude intermediates, while others cannot be taken into the argument at all: whichever terms of the series B...I take, the number of intermediates in the direction either of A or of F must be finite or infinite: where the infinite series starts, whether from the first term or from a later one, is of no moment, for the succeeding terms in any case are infinite in number.
Chapter 21
Φανερὸν δὲ καὶ ἐπὶ τῆς στερητικῆς ἀποδείξεως ὅτι στήσεται, εἴπερ ἐπὶ τῆς κατηγορικῆς ἵσταται ἐπ' ἀμφότερα. ἔστω γὰρ μὴ ἐνδεχόμενον μήτε ἐπὶ τὸ ἄνω ἀπὸ τοῦ ὑστάτου εἰς ἄπειρον ἰέναι (λέγω δ' ὕστατον ὃ αὐτὸ μὲν ἄλλῳ (82b.) μηδενὶ ὑπάρχει, ἐκείνῳ δὲ ἄλλο, οἷον τὸ Ζ) μήτε ἀπὸ τοῦ πρώτου ἐπὶ τὸ ὕστατον (λέγω δὲ πρῶτον ὃ αὐτὸ μὲν κατ' ἄλλου, κατ' ἐκείνου δὲ μηδὲν ἄλλο). εἰ δὴ ταῦτ' ἔστι, καὶ ἐπὶ τῆς ἀποφάσεως στήσεται. a37. Further, if in affirmative demonstration the series terminates in both directions, clearly it will terminate too in negative demonstration. Let us assume that we cannot proceed to infinity either by ascending from the ultimate term (by 'ultimate term' I mean a term such as was, not itself attributable to a subject but itself the subject of attributes), or by descending towards an ultimate from the primary term (by 'primary term' I mean a term predicable of a subject but not itself a subject). If this assumption is justified, the series will also terminate in the case of negation.
τριχῶς γὰρ δείκνυται μὴ ὑπάρχον. ἢ γὰρ ᾧ μὲν τὸ Γ, τὸ Β ὑπάρχει παντί, ᾧ δὲ τὸ Β, οὐδενὶ τὸ Α. τοῦ μὲν τοίνυν Β Γ, καὶ ἀεὶ τοῦ ἑτέρου διαστήματος, ἀνάγκη βαδίζειν εἰς ἄμεσα· κατηγορικὸν γὰρ τοῦτο τὸ διάστημα. τὸ δ' ἕτερον δῆλον ὅτι εἰ ἄλλῳ οὐχ ὑπάρχει προτέρῳ, οἷον τῷ Δ, τοῦτο δεήσει τῷ Β παντὶ ὑπάρχειν. καὶ εἰ πάλιν ἄλλῳ τοῦ Δ προτέρῳ οὐχ ὑπάρχει, ἐκεῖνο δεήσει τῷ Δ παντὶ ὑπάρχειν. ὥστ' ἐπεὶ ἡ ἐπὶ τὸ ἄνω ἵσταται ὁδός, καὶ ἡ ἐπὶ τὸ Α στήσεται, καὶ ἔσται τι πρῶτον ᾧ οὐχ ὑπάρχει. b4. For a negative conclusion can be proved in all three figures. In the first figure it is proved thus: no B is A, all C is B. In packing the interval B-C we must reach immediate propositions — as is always the case with the minor premiss — since B-C is affirmative. As regards the other premiss it is plain that if the major term is denied of a term D prior to B, D will have to be predicable of all B, and if the major is denied of yet another term prior to D, this term must be predicable of all D. Consequently, since the ascending series is finite, the descent will also terminate and there will be a subject of which A is primarily non-predicable.
Πάλιν εἰ τὸ μὲν Β παντὶ τῷ Α, τῷ δὲ Γ μηδενί, τὸ Α τῶν Γ οὐδενὶ ὑπάρχει. πάλιν τοῦτο εἰ δεῖ δεῖξαι, δῆλον ὅτι ἢ διὰ τοῦ ἄνω τρόπου δειχθήσεται ἢ διὰ τούτου ἢ τοῦ τρίτου. ὁ μὲν οὖν πρῶτος εἴρηται, ὁ δὲ δεύτερος δειχθήσεται. οὕτω δ' ἂν δεικνύοι, οἷον τὸ Δ τῷ μὲν Β παντὶ ὑπάρχει, τῷ δὲ Γ οὐδενί, εἰ ἀνάγκη ὑπάρχειν τι τῷ Β. καὶ πάλιν εἰ τοῦτο τῷ Γ μὴ ὑπάρξει, ἄλλο τῷ Δ ὑπάρχει, ὃ τῷ Γ οὐχ ὑπάρχει. οὐκοῦν ἐπεὶ τὸ ὑπάρχειν ἀεὶ τῷ ἀνωτέρω ἵσταται, στήσεται καὶ τὸ μὴ ὑπάρχειν. b13. In the second figure the syllogism is, all A is B, no C is B,..no C is A. If proof of this is required, plainly it may be shown either in the first figure as above, in the second as here, or in the third. The first figure has been discussed, and we will proceed to display the second, proof by which will be as follows: all B is D, no C is D..., since it is required that B should be a subject of which a predicate is affirmed. Next, since D is to be proved not to belong to C, then D has a further predicate which is denied of C. Therefore, since the succession of predicates affirmed of an ever higher universal terminates, the succession of predicates denied terminates too.
Ὁ δὲ τρίτος τρόπος ἦν· εἰ τὸ μὲν Α τῷ Β παντὶ ὑπάρχει, τὸ δὲ Γ μὴ ὑπάρχει, οὐ παντὶ ὑπάρχει τὸ Γ ᾧ τὸ Α. πάλιν δὲ τοῦτο ἢ διὰ τῶν ἄνω εἰρημένων ἢ ὁμοίως δειχθήσεται. ἐκείνως μὲν δὴ ἵσταται, εἰ δ' οὕτω, πάλιν λήψεται τὸ Β τῷ Ε ὑπάρχειν, ᾧ τὸ Γ μὴ παντὶ ὑπάρχει. καὶ τοῦτο πάλιν ὁμοίως. ἐπεὶ δ' ὑπόκειται ἵστασθαι καὶ ἐπὶ τὸ κάτω, δῆλον ὅτι στήσεται καὶ τὸ Γ οὐχ ὑπάρχον. b23. The third figure shows it as follows: all B is A, some B is not C. Therefore some A is not C. This premiss, i.e. C-B, will be proved either in the same figure or in one of the two figures discussed above. In the first and second figures the series terminates. If we use the third figure, we shall take as premisses, all E is B, some E is not C, and this premiss again will be proved by a similar prosyllogism. But since it is assumed that the series of descending subjects also terminates, plainly the series of more universal non-predicables will terminate also.
Φανερὸν δ' ὅτι καὶ ἐὰν μὴ μιᾷ ὁδῷ δεικνύηται ἀλλὰ πάσαις, ὁτὲ μὲν ἐκ τοῦ πρώτου σχήματος, ὁτὲ δὲ ἐκ τοῦ δευτέρου ἢ τρίτου, ὅτι καὶ οὕτω στήσεται· πεπερασμέναι γάρ εἰσιν αἱ ὁδοί, τὰ δὲ πεπερασμένα πεπερασμενάκις ἀνάγκη πεπεράνθαι πάντα. Ὅτι μὲν οὖν ἐπὶ τῆς στερήσεως, εἴπερ καὶ ἐπὶ τοῦ ὑπάρχειν, ἵσταται, δῆλον. b28. Even supposing that the proof is not confined to one method, but employs them all and is now in the first figure, now in the second or third — even so the regress will terminate, for the methods are finite in number, and if finite things are combined in a finite number of ways, the result must be finite. Thus it is plain that the regress of middles terminates in the case of negative demonstration, if it does so also in the case of affirmative demonstration.
Postquam philosophus movit quaestiones, hic incipit eas determinare; et dividitur in duas partes. In prima parte, ostendit quod quarumdam dubitationum solutio reducitur ad solutionem aliarum. In secunda, solvit dubitationem quantum ad illa, in quibus per se et principaliter difficultas consistit; ibi: quod autem in illis, si logice et cetera. Circa primum duo facit: primo enim ostendit quod dubitatio, quae potest esse circa media, reducitur ad dubitationem, quae movetur de extremis, et, ea soluta, solvitur; secundo, ostendit quod dubitatio, quae est circa negativas demonstrationes, reducitur ad dubitationem, quae est de affirmativis; ibi: manifestum est autem in privativis et cetera. Circa primum tria facit: primo, proponit quod intendit; secundo, probat propositum; ibi: si enim a praedicante etc.; tertio, excludit quamdam obviationem; ibi: nec si aliquis dicat et cetera. After raising the questions, the Philosopher here begins to settle them. And his treatment is divided into two parts. In the first he shows that the solution of some of the doubts is reduced to the solution of others. In the second he settles the doubt as to those items in which the difficulty lies per se as in its source (82b34) [L. 33]. Concerning the first he does two things. First, he shows that the doubt bearing on the middles is reduced to the one which is concerned with the extremes and is solved by the solution of the latter. Secondly, he shows that the doubt bearing on negative demonstrations is reduced to the one which is concerned with affirmative demonstrations (82a37). In regard to the first he does three things. First, he states his intended proposition. Secondly, he proves this proposition (82a24). Thirdly, he excludes a subterfuge (82a30).
Dicit ergo primo quod manifestum est, si quis rationem sequentem consideret, quod non contingit esse media infinita, si praedicationes tam in sursum quam in deorsum stent in aliquibus terminis, scilicet in summo praedicato et in infimo subiecto. Et exponit quid sit procedere praedicationes sursum, et deorsum; et dicit quod sursum ascenditur, quando proceditur ad magis universale, de cuius ratione est quod praedicetur: deorsum autem proceditur, quando itur ad magis particulare, de cuius ratione est quod subiiciatur. He says therefore first (82a21) that it will be plain to anyone who considers the following reason that “an infinity of middles does not occur,” if the predications both upwards and downwards stop at certain terms, namely, at the highest predicate and the lowest subject. And he explains what upward and downward predication consists in, saying that one proceeds upwards when there is movement to the more universal, one of whose marks it is that it be predicated; but one proceeds downwards when there is movement to the most particular, one of whose marks it is that it functions as a subject.
Deinde cum dicit: si enim a praedicante etc., ostendit propositum per hunc modum. Sit ita quod a sit summum praedicatum, et c sit infimum subiectum, et sint infinita media, quorum quodlibet vocetur b. Quia igitur a erat primum praedicatum, praedicabitur de aliquo medio sibi propinquiori, et iterum illud medium de alio medio inferiori; et cum media sint infinita, sequitur quod in infinitum procedet praedicatio in descendendo, quod est contra positum. Ponebatur enim quod non descendat praedicatio in infinitum. Similiter etiam si incipiamus a c, quod est infimum subiectum, procedetur ascendendo in infinitum antequam perveniatur ad a, quod etiam est contrarium posito. Si ergo haec sint impossibilia, scilicet quod procedatur praedicando in infinitum sursum vel deorsum, sequetur quod impossibile sit media esse infinita. Et ita patet quod quaestio de infinitate mediorum reducitur ad quaestionem de infinitate extremorum. Then (82a24) he shows what he has proposed in the following way: Let the case be that A is the highest predicate and F the lowest subject, and that there is an infinitude of middles, each of which we shall call B. Now since A was the first predicate, it will be predicated of some middle near it, and that middle of another middle below it. Since the middles are infinite, it follows that the predication will proceed downwards to infinity—which is contrary to what we are assuming. For it was assumed that the predications do not proceed downwards to infinity. The result is the same if we start at F, which is the lowest subject, and proceed upwards to infinity before A is reached—this too would be against our assumption. Therefore, if these are impossible, namely, that one may proceed to infinity by ascending or by descending, it will be impossible for the middles to be infinite. Thus it is clear that the question of an infinity of middles is reduced to a question of the infinity of the extremes.
Deinde cum dicit: neque enim si aliquis etc., excludit quamdam obviationem. Posset enim aliquis obviare, dicens quod praedicta probatio procedebat, ac si a b c, idest medium et extrema, ita se haberent, quod essent habita ad invicem, ita scilicet, quod inter ea non esset aliquod medium: sic enim definitur habitum in V physicorum, quod scilicet consequenter se habet, cum tangat; et hoc videbatur in praedicta probatione supponi, scilicet quod a praedicaretur de aliquo medio quasi habito, idest immediate sequenti. Sed ille qui ponit media infinita, dicet quod hoc non contingit accipere. Dicit enim quod inter quoscunque terminos acceptos est aliquod medium. Then (82a30) he excludes an objection. For someone might object saying that the aforesaid proof would hold if ABF, i.e., the middle and the extremes, were so related as to be “had” to one another, i.e., so that there would be no middle between them: for this is the way the Philosopher defines “had” in Physics V, namely, that it is next to something without anything between. And this seemed to be supposed in the above proof, namely, that A is predicated of some middle as though “had” to it, i.e., following it immediately. But one who posits infinite middles will say that this cannot be supposed. For he will say that between any two terms that are taken there is a middle.
Sed philosophus dicit quod nihil differt, sive sic accipiantur infinita media quod sint habita ad invicem, sicut contingit in discretis; puta, in civitate domus domui est habita, et in numeris unitas unitati: sive non possit inveniri in mediis aliquid habitum, sed semper inter duo media sit aliquod medium accipere; sicut accidit in continuis, in quibus inter quaelibet duo signa, sive inter duo puncta, semper est aliquod medium accipere. But the philosopher says that it makes no difference whether the infinitude be of middles that are “had” to one another, the way discrete things are (for example, in a city. house is consecutive to house, and in numbers, unity to unity), or whether something “had” cannot be found in the middles although between any two middles it is always possible to find another, as happens in continua, in which, between any two signs, i.e., between two points, another can be found between them.
Et quod hoc nihil differat ad propositum, sive uno modo, sive alio, sic manifestat subdens: quia supposito quod sint infinita media inter a et c, quorum quodlibet vocatur b, quodcunque horum accipio, necesse est quod inter illud et a et c sint infinita media, vel non sint infinita respectu alterius eorum. Verbi gratia: ponamus quod media sint habita ad invicem, sicut accidit in discretis, et accipiamus aliquod medium quod sit habitum ad ipsum a; necesse erit quod inter illud medium et c sint adhuc infinita media. Et similiter si ponantur quaedam finita media inter a et illud medium acceptum. Et eadem ratio est si ponatur medium acceptum immediate coniungi ipsi c, vel per finita media ab eo distare. Ex quo igitur semper a medio accepto oportet accipere infinita media ad alterum extremorum, non differt utrum statim coniungatur alii extremorum, idest sine medio, vel non statim, idest per aliqua media: quia etiam si coniungatur uni extremo sine medio, necesse est quod postea inveniantur infinita media respectu alterius; et ita semper oportebit, si est infinitum in mediis, quod inveniatur infinitum in praedicationibus vel ascendendo vel descendendo, sicut praedicta probatio procedebat. That this makes no difference one way or the other to the matter at hand he manifests in the following way: Granted that between A and F there is an infinitude of middles, each of which is called B, yet no matter which of these I employ, there is either an infinitude of middles between it and A and F, or there is not an infinitude of them between it and one or the other of the extremes. For example, let us suppose that the middles are mutually “had,” as happens in discrete things, and let us take a middle which is “had” to A; then it will be necessary that between that middle and F there is still an infinitude of middles; and similarly, if we assume a certain finitude of middles between that middle and A. And the same reasoning holds if the middle which is taken be joined immediately to for is distant from it by a finitude of middles. From the fact, therefore, that from any given middle one must take an infinitude of middles to one or other of the extremes, it makes no difference whether it is joined to either extreme immediately, i.e., without a middle, or not immediately, i.e., through other middles: because even if it be joined to one extreme without a middle, it will still be necessary later to find an infinitude of middles in relation to the other. Consequently, it will always be required, if there is an infinitude of middles, to proceed to infinity in predications either by ascending or by descending, as the above proof showed.
Deinde cum dicit: manifestum est autem etc., ostendit quod si in affirmativis demonstrationibus non proceditur in infinitum, neque in privativis in infinitum proceditur; et sic quaestio de demonstrationibus negativis reducitur ad quaestionem de affirmativis. Et circa hoc tria facit: primo, proponit quod intendit; secundo, probat propositum; ibi: tripliciter enim demonstratur etc.; tertio, excludit quamdam obviationem; ibi: manifestum est autem et cetera. Then (82a37) he shows that if there is no process to infinity in affirmadve demonstrations, then neither in negative demonstrations: and thus the question of negative demonstrations is reduced to the question of affirmative ones. He does three things in regard to this point. First, he proposes what he intends. Secondly, he proves what he proposed (82b4). Thirdly, he excludes an objection (82b28).
Dicit ergo primo, quod manifestum erit ex sequentibus, quod si in praedicativa, idest affirmativa demonstratione statur utrinque, idest in sursum et deorsum, necesse erit quod stetur in negativa demonstratione. He says therefore first (82a37) that it will be clear from what follows that if in the predicative, i.e., in the affirmative demonstration, a stop is made at both, i.e., upwards and downwards, it will be necessary that a stop be made in the negative demonstration.
Et ad exponendum hoc quod propositum est, dicit: sit ita quod non contingat ab ultimo, idest ab infimo subiecto, ire in sursum in infinitum versus praedicata universalia. Et exponit quid est ultimum, scilicet illud quod non inest alicui alii tanquam minus particulari, sed aliud sit in illo, et sit illud z. Et sit etiam quod incipiendo a primo versus ultimum non procedatur in infinitum. Et exponit quid sit primum illud, scilicet quod praedicatur de aliis, et nihil aliud praedicatur de eo, quasi eo universalius; ut sic primum intelligatur universalissimum, ultimum autem particularissimum. Si igitur ex utraque parte stetur in demonstrationibus affirmativis, dicit consequens esse quod etiam stetur in demonstrationibus negativis. To elucidate what he is proposing he says: Let the case be such that from the ultimate, i.e., from the lowest subject one cannot go in ascending order to infinity toward universal predicates. And he explains that “ultimate” means that which is not in any other as in a less particular, but something else is in it, and let it be F. And let the case also be that one does not go to infinity when proceeding from the first to the ultimate. And he explains that the “first” means that which is predicated of others but nothing else is predicated of it as more universal than it. Thus the “first” is understood to be the most universal, and the “ultimate” the most particular. If, therefore, on both sides there be a stop in affirmative demonstrations, he says that as a consequence, there is also a stop in negative demonstrations.
Deinde cum dicit: tripliciter enim etc., probat propositum. Et primo in prima figura; secundo in secunda; ibi: iterum sit b quidem etc.; tertio in tertia; ibi: tertius autem est et cetera. In tribus enim figuris contingit negativam concludi. Then (82b4) he proves his proposition. First, in the first figure. Secondly, in the second (8513). Thirdly, in the third (82b23). For a negative can be concluded in three figures.
Dicit ergo primo quod tripliciter potest demonstrari propositio negativa, per quam significatur aliquid non esse. Uno quidem modo in prima figura, secundum hunc modum, quod b insit c universaliter, minori existente universali affirmativa; a vero insit nulli b, maiori existente universali negativa. Quia igitur supponitur quod in affirmativis stetur et in sursum et in deorsum, necesse est quod ista propositio, quae est b-c, affirmativa, si non sit immediata, et quodcunque aliud spatium accipitur, existente aliquo medio inter b et c, necesse erit reducere in immediata; quia ista distantia, quae attenditur secundum habitudinem medii ad minorem extremitatem, est affirmativa, in qua supponitur esse status. Si autem accipiamus alterum spatium, quod est inter b et a, manifestum est quod, si haec propositio, nullum b est a, non est immediata, necesse est quod a removeatur ab aliquo alio per prius quam a b, et illud sit d; quod si accipiatur ut medium inter a et b, necesse est quod praedicetur universaliter de b, quia oportet minorem esse affirmativam. Et iterum si haec non sit immediata, nullum d est a, oportet quod a negetur ab aliquo alio per prius quam a d, puta sit illud e; quod eadem ratione oportebit universaliter praedicari de d. Quia ergo ascendendo statur in affirmativis, ut supponitur, sequitur per consequens quod sit devenire ad aliquid, de quo primo et immediate negetur ipsum a. Alioquin adhuc procederetur amplius in affirmativis, sicut ex praedictis patet. He says therefore first (82b4) that there are three ways of demonstrating a negative proposition through which something is signified not to be. In one way in the first figure according to the mode that B is universally in C in the universal affirmative minor, but A is in no B in the universal negative major. Now since we are supposing that there is a stop in affirmatives, both upwards and downwards, it is necessary that the proposition, BC, which is affirmative, if it is not immediate but a space exists with middles between B and C, be reduced to immediates, because that space which exists between the middle term and the extreme is affirmative, in which a stop is supposed. But if we take the other space, which is between B and A, it is clear that if this proposition, “No B is A,” is not immediate, it is necessary that A be removed from something else before being removed from B. Let this be D. But if this D be taken as a middle between A and B, it is necessary that it be universally predicated of B, because the minor must be affirmative. And if this too is not immediate, i.e., “No D is A ‘ “ then A has to be denied of something prior to D, say E, which again will be predicated universally of D for the same reason. Therefore, since there is a stop in affirmatives when we ascend, as supposed, it follows that something is reached of which A should be denied first and immediately; otherwise one would go still further in affirmatives, as is clear from the foregoing.
Deinde cum dicit: iterum si b quidem etc., probat idem in negativa, quae concluditur in secunda figura. Sit enim ita quod b, quod est medium, praedicetur universaliter de a et negetur universaliter de c, et ex his concludatur quod, nullum c sit a. Si autem negativam iterum demonstrari oporteat, propter hoc quod est mediata, necesse est quod vel demonstretur in prima figura, de quo modo demonstrationis iam ostensum est quod habet statum, si in affirmativis sit status; aut oportet quod demonstretur per hunc modum, idest per secundam figuram; aut per tertium, idest per tertiam figuram. Dictum est autem in prima figura, quod habet statum in negativis, si sit status in affirmativis. Sed hoc quidem demonstrabitur nunc quantum ad secundam figuram. Then (82b13) he proves the same thing for the negative which is concluded in the second figure. For let the case be that B, which is the middle, is predicated universally of A and denied universally of C, so that the conclusion is “No C is A.” Now if the negative premise needs to be demonstrated because it is mediate, it must be demonstrated either in the first figure in the mode of demonstrating concerning which we have shown that there is a stop, if there is a stop in affirmatives; or it must be demonstrated through this mode, i.e., in the second figure, or through a third mode, i.e., in the third figure. Now it has been established that there is a stop in negatives of the first figure, if there is a stop in affirmatives. Consequently, the same will now be demonstrated as to the second figure.
Demonstretur ergo haec propositio, nullum c est b, sic quod d universaliter praedicetur de b, maiori existente universali affirmativa, et negetur universaliter de c, minori existente universali negativa. Si iterum haec propositio, nullum c est d, est mediata, necesse erit accipere aliquod aliud medium, quod etiam praedicetur de d universaliter, et universaliter removeatur a c. Et ita, sicut proceditur in negativis demonstrationibus, oportebit procedere in affirmativis, scilicet quod b praedicabitur de a, et d de b, et aliquid aliud de d; et sic procedetur in infinitum in affirmativis. Quia ergo supponitur quod in affirmativis stetur in sursum, necesse est etiam quod stetur in negativis, secundum hunc modum, quo negativa demonstratur in secunda figura. Therefore, let this proposition, “No C is B,” be demonstrated in such a way that D is universally predicated of B in the universal affirmative major and denied universally of C in the universal negative minor. Now if the proposition, “No C is D,” is mediate, it will be required to take some other middle which will be predicated universally of D and universally removed from C. Continuing thus, it will be necessary to proceed in negative demonstrations just as we do in affirmatives, namely, B will be predicated of A and D of B, and something else of D, and so on to infinity in affirmatives. But because we are supposing an upward stop in the affirmatives, it is also necessary to come to a stop in the negatives according to this mode in which a negative is demonstrated in the second figure.
Deinde cum dicit: tertius autem est etc., ostendit idem in tertia figura. Sit ergo medium, ut b, de quo a universaliter praedicetur, c vero ab eo removeatur: sequitur particularis negativa, scilicet quod c negetur a quodam a. Et quod quidem in praemissa affirmativa, quae est, omne b est a, stetur, habetur ex suppositione; quod autem necesse sit stare etiam in hac negativa, nullum b est c, quae est maior, patet, quia si hoc debeat demonstrari, necesse est quod vel demonstretur per superius dicta, idest per primam et secundam figuram, vel demonstrabitur similiter sicut concludebatur conclusio, scilicet per tertiam figuram: ita tamen quod haec maior non assumatur ut universalis, sed ut particularis. Illo autem modo statur, scilicet si procedatur in prima et in secunda figura. Si autem procedatur in tertia figura ad concludendum, quoddam b non esse c, accipiatur medium e, de quo quidem b universaliter affirmetur, c vero ab eo particulariter negetur. Et hoc iterum similiter continget, quod secundum hoc procedetur in demonstratione negativa semper secundum augmentum praedicationis affirmativae in inferius: quia b, quod erat primum medium, praedicabitur de e, et e de quodam alio, et sic in infinitum. Quia igitur supponitur statum esse in affirmativis in deorsum, manifestum est quod stabitur in negativis ex parte ipsius c. Then (82b23) he shows the same thing in the third figure. Therefore, let B be a middle of which A is universally predicated, but C is universally denied of it: the conclusion will be a particular negative, namely, C is denied of some A. Now that there is a stop in the affirmative premise, “Every B is A,” is granted by our supposition. Furthermore, that there must be a stop in the negative, “No B is C,” which is the major, is evident, because if it had to be demonstrated, it would be done either “through what was said above,” i.e., through the first and second figure or in the way that the conclusion was concluded, namely, through the third figure, in which case this minor is not affirmed as universal but as particular. “But there is a stop in that way,” i.e., if one proceeds in the first and second figure. But if one proceeds in the third figure to conclude that “Some B is not C,” let a middle, E, be taken such that B is universally affirmed of it but C is particularly denied of it. “Then this happens once more in like manner,” i.e., according to this, one will always proceed in the negative demonstration by accumulating affirmative predications in descending order, because B which was the first middle will be predicated of E, and E of something else, and so on to infinity. But since we are supposing that there is a stop in the descending order in affirmatives, it is clear that there will be a stop in the negatives on the part of C.
Deinde cum dicit: manifestum autem est etc., excludit quamdam obviationem. Posset enim aliquis dicere quod necesse est stare in demonstrationibus negativis, statu existente in affirmativis, si semper syllogizetur secundum eamdem figuram; sed potest in infinitum procedi, si nunc demonstretur per unam figuram, nunc per aliam. Et dicit, manifestum est quod si non procedatur in demonstrationibus una via, sed omnibus, aliquando quidem ex prima figura, aliquando autem ex secunda vel tertia, sic etiam oportebit statum esse in negativis, statu existente in affirmativis. Huiusmodi enim viae diversae demonstrandi sunt finitae, et quaelibet earum multiplicatur non in infinitum, sed finite ascendendo vel descendendo, ut ostensum est. Si autem finita finities accipiantur, necesse est totum esse finitum. Unde relinquitur quod omnibus modis necesse sit in demonstrationibus negativis esse statum, si sit status in affirmativis. Then (82b28) he excludes an objection. For someone could say that it is necessary to stop in negative propositions when there is a stop in the affirmatives, provided that one always syllogizes according to the same figure; but if one demonstrates now in one figure and now in another, one can go to infinity. And he say’~ that “it is obvious” that if one does not limit himself to one figure\in demonstrating but uses all, proceeding now in the first figure and now in the second and third, there must still be a stop in the negatives if there is one in the affirmatives. For these various ways of demonstrating are finite, and each of them will be enlarged not to infinity but finitely by ascending or descending, as was shown. Now if, finite things be taken a finite number of times, the result is finite. Hence it remains that in all the modes there must be a stop in negative demonstrations, if there is a stop in the affirmatives.

Lectio 33
Caput 21 cont.
ὅτι δ' ἐπ' ἐκείνων, λογικῶς μὲν θεωροῦσιν ὧδε φανερόν. Ἐπὶ μὲν οὖν τῶν ἐν τῷ τί ἐστι κατηγορουμένων δῆλον· εἰ γὰρ ἔστιν ὁρίσασθαι ἢ εἰ γνωστὸν τὸ τί ἦν εἶναι, τὰ δ' ἄπειρα μὴ ἔστι διελθεῖν, ἀνάγκη πεπεράνθαι τὰ ἐν τῷ τί (83a.) ἐστι κατηγορούμενα. b34. That in fact the regress terminates in both these cases may be made clear by the following dialectical considerations. In the case of predicates constituting the essential nature of a thing, it clearly terminates, seeing that if definition is possible, or in other words, if essential form is knowable, and an infinite series cannot be traversed, predicates constituting a thing's essential nature must be finite in number.
καθόλου δὲ ὧδε λέγομεν. ἔστι γὰρ εἰπεῖν ἀληθῶς τὸ λευκὸν βαδίζειν καὶ τὸ μέγα ἐκεῖνο ξύλον εἶναι, καὶ πάλιν τὸ ξύλον μέγα εἶναι καὶ τὸν ἄνθρωπον βαδίζειν. ἕτερον δή ἐστι τὸ οὕτως εἰπεῖν καὶ τὸ ἐκείνως. ὅταν μὲν γὰρ τὸ λευκὸν εἶναι φῶ ξύλον, τότε λέγω ὅτι ᾧ συμβέβηκε λευκῷ εἶναι ξύλον ἐστίν, ἀλλ' οὐχ ὡς τὸ ὑποκείμενον τῷ ξύλῳ τὸ λευκόν ἐστι· καὶ γὰρ οὔτε λευκὸν ὂν οὔθ' ὅπερ λευκόν τι ἐγένετο ξύλον, ὥστ' οὐκ ἔστιν ἀλλ' ἢ κατὰ συμβεβηκός. ὅταν δὲ τὸ ξύλον λευκὸν εἶναι φῶ, οὐχ ὅτι ἕτερόν τί ἐστι λευκόν, ἐκείνῳ δὲ συμβέβηκε ξύλῳ εἶναι, οἷον ὅταν τὸ μουσικὸν λευκὸν εἶναι φῶ (τότε γὰρ ὅτι ὁ ἄνθρωπος λευκός ἐστιν, ᾧ συμβέβηκεν εἶναι μουσικῷ, λέγω), ἀλλὰ τὸ ξύλον ἐστὶ τὸ ὑποκείμενον, ὅπερ καὶ ἐγένετο, οὐχ ἕτερόν τι ὂν ἢ ὅπερ ξύλον ἢ ξύλον τί. εἰ δὴ δεῖ νομοθετῆσαι, ἔστω τὸ οὕτω λέγειν κατηγορεῖν, τὸ δ' ἐκείνως ἤτοι μηδαμῶς κατηγορεῖν, ἢ κατηγορεῖν μὲν μὴ ἁπλῶς, κατὰ συμβεβηκὸς δὲ κατηγορεῖν. ἔστι δ' ὡς μὲν τὸ λευκὸν τὸ κατηγορούμενον, ὡς δὲ τὸ ξύλον τὸ οὗ κατηγορεῖται. ὑποκείσθω δὴ τὸ κατηγορούμενον κατηγορεῖσθαι ἀεί, οὗ κατηγορεῖται, ἁπλῶς, ἀλλὰ μὴ κατὰ συμβεβηκός· οὕτω γὰρ αἱ ἀποδείξεις ἀποδεικνύουσιν. a1. But as regards predicates generally we have the following prefatory remarks to make. (1) We can affirm without falsehood 'the white (thing) is walking', and that big (thing) is a log'; or again, 'the log is big', and 'the man walks'. But the affirmation differs in the two cases. When I affirm 'the white is a log', I mean that something which happens to be white is a log — not that white is the substratum in which log inheres, for it was not qua white or qua a species of white that the white (thing) came to be a log, and the white (thing) is consequently not a log except incidentally. On the other hand, when I affirm 'the log is white', I do not mean that something else, which happens also to be a log, is white (as I should if I said 'the musician is white,' which would mean 'the man who happens also to be a musician is white'); on the contrary, log is here the substratum — the substratum which actually came to be white, and did so qua wood or qua a species of wood and qua nothing else. If we must lay down a rule, let us entitle the latter kind of statement predication, and the former not predication at all, or not strict but accidental predication. 'White' and 'log' will thus serve as types respectively of predicate and subject. We shall assume, then, that the predicate is invariably predicated strictly and not accidentally of the subject, for on such predication demonstrations depend for their force.
ὥστε ἢ ἐν τῷ τί ἐστιν ἢ ὅτι ποιὸν ἢ ποσὸν ἢ πρός τι ἢ ποιοῦν τι ἢ πάσχον ἢ ποὺ ἢ ποτέ, ὅταν ἓν καθ' ἑνὸς κατηγορηθῇ. a21. It follows from this that when a single attribute is predicated of a single subject, the predicate must affirm of the subject either some element constituting its essential nature, or that it is in some way qualified, quantified, essentially related, active, passive, placed, or dated.
Chapter 22
Ἔτι τὰ μὲν οὐσίαν σημαίνοντα ὅπερ ἐκεῖνο ἢ ὅπερ ἐκεῖνό τι σημαίνει καθ' οὗ κατηγορεῖται· ὅσα δὲ μὴ οὐσίαν σημαίνει, ἀλλὰ κατ' ἄλλου ὑποκειμένου λέγεται ὃ μὴ ἔστι μήτε ὅπερ ἐκεῖνο μήτε ὅπερ ἐκεῖνό τι, συμβεβηκότα, a24. (2) Predicates which signify substance signify that the subject is identical with the predicate or with a species of the predicate. Predicates not signifying substance which are predicated of a subject not identical with themselves or with a species of themselves are accidental or coincidental;
οἷον κατὰ τοῦ ἀνθρώπου τὸ λευκόν. οὐ γάρ ἐστιν ὁ ἄνθρωπος οὔτε ὅπερ λευκὸν οὔτε ὅπερ λευκόν τι, ἀλλὰ ζῷον ἴσως· ὅπερ γὰρ ζῷόν ἐστιν ὁ ἄνθρωπος. ὅσα δὲ μὴ οὐσίαν σημαίνει, δεῖ κατά τινος ὑποκειμένου κατηγορεῖσθαι, καὶ μὴ εἶναί τι λευκὸν ὃ οὐχ ἕτερόν τι ὂν λευκόν ἐστιν. a27. e.g. white is a coincident of man, seeing that man is not identical with white or a species of white, but rather with animal, since man is identical with a species of animal. These predicates which do not signify substance must be predicates of some other subject, and nothing can be white which is not also other than white.
τὰ γὰρ εἴδη χαιρέτω· τερετίσματά τε γάρ ἐστι, καὶ εἰ ἔστιν, οὐδὲν πρὸς τὸν λόγον ἐστίν· αἱ γὰρ ἀποδείξεις περὶ τῶν τοιούτων εἰσίν. a33. The Forms we can dispense with, for they are mere sound without sense; and even if there are such things, they are not relevant to our discussion, since demonstrations are concerned with predicates such as we have defined.
Postquam philosophus ostendit quod si sit status in extremis, necesse est esse statum in mediis, et si sit status in affirmativis, necesse est esse statum in negativis; hic intendit ostendere quod sit status in affirmativis in sursum et deorsum. Et dividitur in duas partes: in prima parte, ostendit propositum logice, idest per rationes communes omni syllogismo, quae accipiuntur secundum praedicata communiter sumpta; in secunda, ostendit idem analytice, idest per rationes proprias demonstrationi, quae accipiuntur secundum praedicata per se, quae sunt demonstrationi propria; ibi: analytice autem manifestum et cetera. Prima autem pars dividitur in duas partes: in prima, ostendit quod non sit procedere in infinitum in praedicatis, quae praedicantur in eo quod quid; in secunda, ostendit quod non sit procedere in infinitum universaliter in praedicatis affirmativis; ibi: universaliter autem sic dicimus et cetera. After showing that if there is a stop in the extremes there must be a stop in the middles, and if there is a stop in affirmatives there must be a stop in negatives, the Philosopher now shows that there is a stop in affirmatives both upwards and downwards. And his treatment is divided into two parts. In the first part he shows his thesis “logically,” i.e., through characteristics common to every syllogism, which are based on predicates considered commonly. In the second he shows the same thing analytically, i.e., through characteristics proper to demonstration, which are based on per se predicates which are proper to demonstration (84a8) [L. 35]. The first is divided into two parts. In the first he shows that one does not proceed to infinity in predicates which are predicated in eo quod quid (i.e., pertaining to the essence of the subject]. In the second he shows universally that one does not proceed to infinity in affirmative predicates (83a1).
Dicit ergo primo, quod cum ostensum sit quod in privativis non est ire in infinitum, si stetur in affirmativis; hic iam manifestum erit quomodo aliqui speculantur in illis, idest in affirmativis, esse statum per logicas rationes. Et dicuntur hic logicae rationes, quae procedunt ex quibusdam communibus, quae pertinent ad considerationem logicae. He says therefore first (82b34) that since we have established that an infinite process does not occur in negatives if there is a stop in affirmatives, our present task will be to show how one speculates through logical reasons that there is a stop “in those,” i.e., in affirmatives. (These reasons are called “logical,” because they proceed from certain common notions that pertain to the considerations of logic).
Haec autem veritas manifesta est in his, quae praedicantur in eo quod quid est, idest in praedicatis, ex quibus quod quid est, idest definitio constituitur. Si enim huiusmodi praedicata dentur esse infinita, sequitur et quod non contingat definire aliquid, et quod si definitur aliquid, eius definitio non possit esse nota. Et hoc ideo, quia infinita non est pertransire. Non autem contingit definiri, neque definitionem cognosci, nisi descendendo perveniatur usque ad ultimum, et ascendendo perveniatur usque ad primum. Si ergo contingit aliquid definire, vel si contingit definitionem alicuius esse notam, ex utroque antecedenti sequitur hoc consequens, quod in praedictis praedicatis non sit procedere in infinitum, sed in eis contingat stare. Now this truth is clear in regard to things predicated as constituting the essence of a thing, namely, the predicates from which the quod quid est, i.e., the definition, is formed. For if such predicates were agreed to be infinite, the result would be that nothing can be defined, and that if something is defined, its definition cannot be known: and all because the infinite cannot be traversed. For a thing cannot be defined, or its definition known, except by reaching the ultimate through descent and the first through ascent. Therefore, if something can be defined, or if a thing’s definition can be known, then in either case this consequence will follow, namely, there is no infinite process in the aforementioned predicates, but there is a stop in them.
Deinde cum dicit: universaliter autem sic etc., ostendit universaliter quod in praedicatis affirmativis non sit procedere in infinitum. Et circa hoc duo facit: primo, quaedam praemittit, quae sunt necessaria ad propositum ostendendum; secundo, ostendit propositum; ibi: amplius si non est et cetera. Circa primum duo facit: primo, distinguit praedicata per accidens a praedicatis per se; secundo, distinguit praedicata per se ad invicem; ibi: quare autem in eo quod quid est et cetera. Then (83a1) he shows universally that there is not a process to infinity in affirmative predicates. In regard to this he does two things. First, he prefaces certain things needed for establishing the thesis. Secondly, he establishes it (83a36) [L. 34]. In regard to the first he does two things. First, he distinguishes per accidens from per se predicates. Secondly, he distinguishes among the per se predicates (83a21).
Dicit ergo primo, quod cum ostensum sit in quibusdam praedicatis, quod in eis non est procedere in infinitum, scilicet in his, quae praedicantur in quod quid est, ostendendum est hoc universaliter in omnibus praedicatis affirmativis. Et incipit suam considerationem a praedicatis per accidens, in quibus est triplex modus verae praedicationis. Unus quidem modus est, quando accidens praedicatur de accidente; puta, cum dicimus, album ambulat. Secundus modus est, quando subiectum praedicatur de accidente; puta, cum dicimus, hoc magnum est lignum. Tertius modus est, quando accidens praedicatur de subiecto; puta, cum dicimus, lignum est album: vel cum dicimus, homo ambulat. He says therefore first (83a1) that since it has been established in regard to certain predicates that there is no infinite process in them, namely, in those which are predicated as pertaining to the essence, our task is to show that this is universally so in all affirmative predicates. And he begins his consideration with per accidens predicates in which there are three modes of true predication. One mode is when an accident is predicated of an accident, as when we say, “Something white is walking.” The second mode is when the subject is predicated of an accident, as when we say, “Something white is wood.” The third mode is when an accident is predicated of a subject, as when we say, “Wood is white,” or “Man walks.”
Isti autem modi praedicandi sunt alteri et diversi ad invicem: quia cum subiectum praedicatur de accidente, puta, cum dicitur, album est lignum, hoc significatur, quod illud universale praedicatum, quod est lignum, praedicatur de subiecto, cui accidit esse album, scilicet de hoc particulari ligno, in quo est albedo. Idem enim est sensus cum dico, album est lignum, ac si dicerem, hoc lignum, cui accidit esse album, est lignum; non autem est sensus quod album sit subiectum ligni. Now these modes of predicating are mutually other and diverse, because when its subject is predicated of an accident, say “Something white is wood,” what is signified is that the universal predicate, “wood,” is being predicated of a subject which happens to be wood, i.e., this particular wood in which there is whiteness. For when I say, “Something white is wood,” the meaning is the same as “This wood which happens to be white is wood”; in other words, the sense is not that “white” is the subject of wood.
Et hoc probat, quia subiectum fit hoc quod praedicatur de ipso sicut de subiecto, vel secundum totum, vel secundum partem, sicut homo fit albus: sed neque album, neque aliqua pars albi, quae vere sit album, idest quae sit de substantia ipsius albedinis, fit lignum; non enim accidens est subiectum transmutationis, qua de non ligno fit lignum. Omne autem quod incipit esse hoc, fit hoc; si igitur non fit hoc, non est hoc, nisi detur quod semper hoc fuerit; non autem semper fuit verum dicere, album est lignum, quia aliquando non simul fuerunt albedo et lignum. Cum ergo non sit verum dicere quod album fiat lignum, manifestum est quod album non est lignum proprie et per se loquendo: sed si hoc concedatur, album est lignum, intelligitur per accidens, quia scilicet illud particulare subiectum, cui accidit album, est lignum. Iste ergo est sensus huiusmodi praedicationis, in qua subiectum praedicatur de accidente. And he proves this on the ground that it is either according to its totality or according to a part that a subject comes to be that which is predicated of it as of a subject, as a man comes to be white. But neither the white nor its white part, which is really white, i.e., which pertains to the very essence of whiteness, becomes wood: for an accident is not the subject of change whereby wood comes to be from non-wood. But whatever begins to be such and such, comes to be it; therefore, if it does not come to be this, it is not it-unless it is granted that it always was this. But it was not always true to say that the white [object] is wood, because at some time the whiteness and the wood were not together. Therefore, since it is not true to say that the white [object] becomes wood, it is obvious that the white [object], properly and per se speaking, is not wood. Yet if it be granted that something white is wood, it is understood per accidens, namely, because that particular subject, which happens to be white, is wood. This, therefore, is the sense of any predication in which a subject is predicated of an accident.
Sed cum dico, lignum est album, praedicando accidens de subiecto, non significo sicut in praedicto modo praedicationis, quod alterum aliquid sit substantialiter album, cui accidit esse lignum. Quod quidem significatur tam in praedicto modo, quo subiectum praedicatur de accidente, quam etiam in alio modo, quo accidens praedicatur de accidente, ut cum dico, musicum est album: hic enim nihil aliud significo, nisi quod ille homo particularis, puta Socrates, cui accidit esse musicum, est albus. Sed quando dico, lignum est album, significo quod ipsum lignum vere factum est subiectum albi, non quod aliquid aliud a ligno, vel a parte ligni, quae est lignum aliquod, sit factum album. But when I say, “Wood is white,” predicating an accident of the subject, I do not signify, as I did in the previous mode of predication, that there is something else substantially white, such that to be wood happens to it: which, of course, is signified both in the previous mode, where a subject is predicated of an accident, and in the other mode, where an accident is predicated of an accident, as when I say, “The musician is white”: for this signifies nothing but the fact that this particular man, say Socrates, who happens to be a musician, is white. But when I say, “The wood is white,” I signify that the wood itself has become the subject of white, and not that something other than the wood or other than a section of the wood has become white.
Est ergo differentia in tribus modis praedictis: quia cum praedicatur accidens de subiecto, non praedicatur per aliquod aliud subiectum; cum autem praedicatur subiectum de accidente, vel accidens de accidente, fit praedicatio ratione eius quod subiicitur termino posito in subiecto; de quo quidem praedicatur aliud accidens accidentaliter, ipsa vero species subiecti essentialiter. Therefore, there is a difference among these three modes of predicating, because when an accident is predicated of a subject, it is not predicated in virtue of some other subject; but when the subject is predicated of an accident or an accident of an accident, the predication is made in virtue of that which is subjected to the terms acting as the subject, of which another accident is predicated accidentally in the second case, and the species of the subject is predicated essentially in the first case.
Et quia in quolibet praedictorum modorum utimur nomine praedicationis, et sicut possumus nomina ponere, ita possumus ea restringere; imponamus sic nomina in probatione sequenti, ut praedicari solum dicamus illud, quod dicitur hoc modo, scilicet non ratione alterius subiecti. Illud vero quod dicitur illo modo, scilicet ratione alterius subiecti, velut cum subiectum praedicatur de accidente, vel accidens de accidente, non dicatur praedicari, vel si dicatur praedicari, non dicatur praedicari simpliciter, sed secundum accidens. Et accipiamus semper illud, quod se habet per modum albi, ex parte praedicati, id autem, quod se habet per modum ligni, accipiatur ex parte subiecti. Hoc ergo supponamus praedicari semper, in probatione sequenti, quod praedicatur de eo, de quo praedicatur, simpliciter, et non secundum accidens. Et ratio quare debemus sic uti vocabulo praedicationis, haec est: quia loquimur in materia demonstrativa, demonstrationes autem non utuntur nisi talibus praedicationibus. Now since in each of the above modes we use the name, “predication,” and since it lies within our power to employ names as well as to restrict them, let us agree so to use our names in the following proof that only those things are said to be predicated which are said in this way, namely, not in virtue of some other subject. Consequently, whatever is said in another way, namely, by reason of another subject, as when a subject is predicated of an accident or an accident of an accident, shall not be said to be predicated—or if it is said to be predicated, it shall be said to be predicated not absolutely but per accidens. Furthermore, that which is after the manner of “white,” let us always take on the part of the predicate, and that which is after the manner of “wood,” let it be taken on the part of the subject. Therefore, in the following proof let us suppose this to be predicated which is predicated not per accidens, but absolutely, of that of which it is predicated. And the reason why we should use the word “predication” in this way is that we are speaking of demonstrative matters, and demonstrations use only such predications.
Deinde cum dicit: quare autem in quod etc. ostendit differentiam praedicatorum per se ad invicem. Et circa hoc duo facit: primo, distinguit praedicata ad invicem secundum diversa genera; secundo, ostendit differentiam praedicatorum; ibi: amplius substantiam quidem et cetera. Then (83a21) he shows the differences among per se predicates. In regard to this he does two things. First, he distinguishes these predicates into diverse genera. Secondly, he shows the differences among them (83a24).
Dicit ergo primo, quod quia nos praedicari dicimus solum illud, quod praedicatur non secundum aliud subiectum, hoc autem diversificatur secundum decem praedicamenta; sequitur quod omne quod sic praedicatur, praedicetur aut in quod quid est, idest per modum substantialis praedicati, aut per modum qualis, vel quanti, vel alicuius alterius praedicamentorum, de quibus actum est in praedicamentis. Et addit cum unum de uno praedicetur: quia si praedicatum non sit unum sed multa, non poterit praedicatum simpliciter dici quid vel quale; sed forte dicetur simul quale quid, puta si dicam, homo est animal album. Fuit autem necessaria haec additio; quia si multa praedicentur de uno, ita quod multa accipiantur in ratione unius praedicati, poterunt in infinitum praedicationes multiplicari, secundum infinitos modos combinandi praedicata ad invicem. Unde cum quaeritur status in his quae praedicantur, necesse est accipere unum de uno praedicari. He says therefore first (83a21) that because we are saying that those things alone are being predicated which are predicated not in virtue of some other subject, and this is diversified according to the ten predicaments, it follows that whatever is thus predicated is predicated “either in quod quid est,” i.e., after the manner of an essential predicate, or after the manner of quality or quantity or one of the other predicaments discussed in the Categories. And he adds, “when one is predicated of one,” because if a predicate is not one but several, it cannot be said to be predicated precisely as quid or quale, but perhaps jointly as quale quid, as when I say “Man is a white animal.” Now it was necessary to add this because if several things are predicated of one in such a way that they function as one predicate, predications will be multipliable to infinity according to the infinite modes of combining predicates one to another. Hence when there is question of a stop in things that are predicated, it is necessary to take one thing predicated of one thing.
Deinde cum dicit: amplius substantiam quidem etc., ostendit differentiam praedictorum praedicatorum. Et circa hoc tria facit: primo, proponit differentiam; secundo, manifestat per exempla; ibi: ut de homine et albo etc.; tertio, excludit quamdam obviationem; ibi: species enim gaudeant et cetera. Then (83a24) he indicates the difference among the aforesaid predicates. In regard to this he does three things. First, he proposes the difference. Secondly, he clarifies it with examples (83a27). Thirdly, he excludes an objection (83a33).
Dicit ergo primo, quod illa quae substantiam significant, oportet quod significent respectu eius de quo praedicantur, quod vere illud est, aut quod vere illud aliquid. Quod potest dupliciter intelligi. Uno modo, ut ostendatur distinctio ex parte praedicati, quod vel significat totam essentiam subiecti, sicut definitio: et hoc significat cum dicit: quod vere illud est; vel significat partem essentiae, sicut genus, vel differentia: et hoc significat cum dicit; aut quod vere illud aliquid. Alio modo, et melius, ut ostendatur distinctio ex parte subiecti, quod quandoque est convertibile cum praedicato essentiali, sicut definitum cum definitione: et hoc significat cum dicit: quod vere illud est; quandoque vero est pars subiectiva praedicati, sicut homo animalis: et hoc significat cum dicit: aut quod vere illud aliquid. Homo enim aliquod animal est. Sed illa quae non significant substantiam, sed dicuntur de aliquo subiecto, quod quidem subiectum nec vere, idest essentialiter, est illud praedicatum, neque aliquid eius; omnia huiusmodi praedicata sunt accidentalia. He says therefore first (83a24) that those which signify substance must signify, in respect to that of which they are predicated, “what it truly is or something that it truly is.” This can be understood in two ways: in one way, as showing a distinction on the part of the predicate which might signify either the entire essence of the subject, as a definition does (and he signifies this when he says, “what it truly is”), or part of the essence, as a genus or a difference does (and he signifies this when he says, “or something that it truly is.”) In another way, and better, as showing a distinction on the part of the subject, which is sometimes convertible with an essential predicate, as the definitum with the definition (and he signifies this, when he says, “what it truly is”), and sometimes is a subjective part of the predicate, as man is of animal (and he signifies this when he says, “or something that it truly is,” for man is a certain anima1). But those which do not signify substance but are said of some subject (which subject is not truly, i.e., essentially, that predicate nor something of it), all such predicates are accidental.
Deinde cum dicit: ut de homine est album etc., manifestat praemissam differentiam per exempla: et dicit quod cum dicimus, homo est albus, praedicatum illud est accidentale, quia homo non est quod vere album est, idest esse album non est essentia hominis; neque quod vere album aliquid, ut supra expositum est. Sed cum dicitur, homo est animal, forsan homo est quod vere est animal: animal enim significat essentiam hominis, quia illud ipsum quod est homo, est essentialiter animal. Et quamvis illa, quae non significant substantiam, sint accidentia, non tamen per accidens praedicantur. Praedicantur enim de quodam subiecto non propter aliquod aliud subiectum: puta cum dico, homo est albus, praedicatur album de homine, non ea ratione, quod aliquod aliud subiectum sit album, ratione cuius homo dicatur albus; sicut supra dictum est in his, quae praedicantur per accidens. Then (83a27) he clarifies the aforesaid difference with examples, saying that when we say, “Man is white,” the predicate is accidental, because man is not that which white truly is, i.e., to be white is not the essence of man “or anything of what white truly is,” as was explained above. But , when it is stated, “Man is an animal,” perhaps man is that which animal truly is. For animal signifies the essence of man, because that which is man is essentially animal. And although predicates that do not signify substance are accidents, they are not predicated per accidens. For they are not predicated of some subject in virtue of some other subject; for when I say, “Man is white,” “white” is not predicated of man by reason of some other subject’s being white in virtue of which man is called white as was explained above in regard to things predicated per accidens.
Deinde cum dicit: species enim gaudeant etc., excludit quamdam obviationem. Posset enim aliquis dicere quod praedicata, quae significant substantiam, non sunt vere et essentialiter id, de quo praedicantur, vel aliquid eius: neque accidentia, quae sunt in individuis sicut in subiectis, conveniunt huiusmodi communibus praedicatis essentialibus; quia huiusmodi praedicata universalia significant quasdam essentias semper separatas per se subsistentes, sicut Platonici dicebant. Then (83a33) he excludes an objection. For someone could say that predicates which signify the substance are not truly and essentially that which they are predicated of, nor are they something of it; and that accidents which exist in individuals as in subjects, do not correspond to any common essential predicates, because such universal predicates signify certain separated essences always subsisting by themselves, as the Platonists say.
Sed ipse respondet quia, si supponantur species, idest ideae, esse, debent gaudere, quia secundum Platonicos habent aliquod nobilius esse, quam res nobis notae naturales. Huiusmodi enim res sunt particulares et materiales, illae autem sunt universales et immateriales. Sunt enim quaedam praemonstrationes respectu naturalium, idest quaedam exemplaria horum: ut accipiantur hic monstra vel praemonstrationes sicut praemonstratur aliquid ad aliquid probandum. Quia ergo sunt praemonstrationes vel exemplaria rerum naturalium, necesse est quod in istis rebus naturalibus inveniantur aliquae participationes illarum specierum, quae pertinent ad essentiam harum rerum naturalium. Et ideo si sint huiusmodi species separatae, sicut Platonici posuerunt, nihil pertinent ad rationem praesentem. Nos enim intendimus de huiusmodi rebus, de quibus in nobis scientia per demonstrationem acquiritur. Et huiusmodi sunt res in natura existentes nobis notae, de quibus demonstrationes fiunt. Et ideo si detur quod animal sit quoddam separatum, quasi praemonstratio existens animalium naturalium, tunc cum dico, homo est animal, secundum quod hac propositione utimur in demonstrando, ly animal significat essentiam rei naturalis, de qua fit demonstratio. But he answers that if Forms, i.e., Ideas, are assumed to exist, they should be happy, because according to the Platonists they have a nobler existence than the material things known to us. For the latter are particular and material, but the former universal and immaterial. For they are “premonstrations,” i.e., certain exemplars, of material things (taking “premonstrations” here as above, when we spoke of something being shown beforehand in order to prove something). Therefore, since they are the premonstrations or exemplars of natural things, it is necessary that in these natural things there be found certain participations of those Forms which pertain to the essences of these natural things. Hence if such separated Forms exist, as the Platonists contend, they have nothing to do with the present matter. For we are concerned with things, the science of which is produced in us through demonstration. And these are things existing in matter and known to us and concerning which demonstrations deal. Consequently, if it be granted that “animal” is something separated, an existing premonstration, as it were, of natural animals, then when I say, “Man is an animal,” in the sense that we use this preposition in demonstrating, “animal” signifies the essence of the natural thing concerning which the demonstration is made.

Lectio 34
Caput 22 cont.
Ἔτι εἰ μὴ ἔστι τόδε τοῦδε ποιότης κἀκεῖνο τούτου, μηδὲ ποιότητος ποιότης, a36. (3) If A is a quality of B, B cannot be a quality of A — a quality of a quality.
ἀδύνατον ἀντικατηγορεῖσθαι ἀλλήλων οὕτως, ἀλλ' ἀληθὲς μὲν ἐνδέχεται εἰπεῖν, ἀντικατηγορῆσαι δ' ἀληθῶς οὐκ ἐνδέχεται. a37. Therefore A and B cannot be predicated reciprocally of one another in strict predication: they can be affirmed without falsehood of one another, but not genuinely predicated of each other.
ἢ γάρ τοι ὡς οὐσία κατηγορηθήσεται, (83b.) οἷον ἢ γένος ὂν ἢ διαφορὰ τοῦ κατηγορουμένου. a39. For one alternative is that they should be substantially predicated of one another, i.e. B would become the genus or differentia of A — the predicate now become subject.
ταῦτα δὲ δέδεικται ὅτι οὐκ ἔσται ἄπειρα, οὔτ' ἐπὶ τὸ κάτω οὔτ' ἐπὶ τὸ ἄνω (οἷον ἄνθρωπος δίπουν, τοῦτο ζῷον, τοῦτο δ' ἕτερον· οὐδὲ τὸ ζῷον κατ' ἀνθρώπου, τοῦτο δὲ κατὰ Καλλίου, τοῦτο δὲ κατ' ἄλλου ἐν τῷ τί ἐστιν), τὴν μὲν γὰρ οὐσίαν ἅπασαν ἔστιν ὁρίσασθαι τὴν τοιαύτην, τὰ δ' ἄπειρα οὐκ ἔστι διεξελθεῖν νοοῦντα. ὥστ' οὔτ' ἐπὶ τὸ ἄνω οὔτ' ἐπὶ τὸ κάτω ἄπειρα· ἐκείνην γὰρ οὐκ ἔστιν ὁρίσασθαι ἧς τὰ ἄπειρα κατηγορεῖται. b1. But it has been shown that in these substantial predications neither the ascending predicates nor the descending subjects form an infinite series; e.g. neither the series, man is biped, biped is animal, etc., nor the series predicating animal of man, man of Callias, Callias of a further. subject as an element of its essential nature, is infinite. For all such substance is definable, and an infinite series cannot be traversed in thought: consequently neither the ascent nor the descent is infinite, since a substance whose predicates were infinite would not be definable.
ὡς μὲν δὴ γένη ἀλλήλων οὐκ ἀντικατηγορηθήσεται· ἔσται γὰρ αὐτὸ ὅπερ αὐτό τι. b8. Hence they will not be predicated each as the genus of the other; for this would equate a genus with one of its own species.
οὐδὲ μὴν τοῦ ποιοῦ ἢ τῶν ἄλλων οὐδέν, ἂν μὴ κατὰ συμβεβηκὸς κατηγορηθῇ· πάντα γὰρ ταῦτα συμβέβηκε καὶ κατὰ τῶν οὐσιῶν κατηγορεῖται. b10. Nor (the other alternative) can a quale be reciprocally predicated of a quale, nor any term belonging to an adjectival category of another such term, except by accidental predication; for all such predicates are coincidents and are predicated of substances.
ἀλλὰ δὴ ὅτι οὐδ' εἰς τὸ ἄνω ἄπειρα ἔσται· ἑκάστου γὰρ κατηγορεῖται ὃ ἂν σημαίνῃ ἢ ποιόν τι ἢ ποσόν τι ἤ τι τῶν τοιούτων ἢ τὰ ἐν τῇ οὐσίᾳ· ταῦτα δὲ πεπέρανται, καὶ τὰ γένη τῶν κατηγοριῶν πεπέρανται· ἢ γὰρ ποιὸν ἢ ποσὸν ἢ πρός τι ἢ ποιοῦν ἢ πάσχον ἢ ποὺ ἢ ποτέ. Ὑπόκειται δὴ ἓν καθ' ἑνὸς κατηγορεῖσθαι, αὐτὰ δὲ αὑτῶν, ὅσα μὴ τί ἐστι, μὴ κατηγορεῖσθαι. συμβεβηκότα γάρ ἐστι πάντα, ἀλλὰ τὰ μὲν καθ' αὑτά, τὰ δὲ καθ' ἕτερον τρόπον· ταῦτα δὲ πάντα καθ' ὑποκειμένου τινὸς κατηγορεῖσθαί φαμεν, τὸ δὲ συμβεβηκὸς οὐκ εἶναι ὑποκείμενόν τι· οὐδὲν γὰρ τῶν τοιούτων τίθεμεν εἶναι ὃ οὐχ ἕτερόν τι ὂν λέγεται ὃ λέγεται, ἀλλ' αὐτὸ ἄλλου καὶ τοῦτο καθ' ἑτέρου. οὔτ' εἰς τὸ ἄνω ἄρα ἓν καθ' ἑνὸς οὔτ' εἰς τὸ κάτω ὑπάρχειν λεχθήσεται. b13. On the other hand — in proof of the impossibility of an infinite ascending series — every predication displays the subject as somehow qualified or quantified or as characterized under one of the other adjectival categories, or else is an element in its substantial nature: these latter are limited in number, and the number of the widest kinds under which predications fall is also limited, for every predication must exhibit its subject as somehow qualified, quantified, essentially related, acting or suffering, or in some place or at some time. I assume first that predication implies a single subject and a single attribute, and secondly that predicates which are not substantial are not predicated of one another. We assume this because such predicates are all coincidents, and though some are essential coincidents, others of a different type, yet we maintain that all of them alike are predicated of some substratum and that a coincident is never a substratum — since we do not class as a coincident anything which does not owe its designation to its being something other than itself, but always hold that any coincident is predicated of some substratum other than itself, and that another group of coincidents may have a different substratum.
καθ' ὧν μὲν γὰρ λέγεται τὰ συμβεβηκότα, ὅσα ἐν τῇ οὐσίᾳ ἑκάστου, ταῦτα δὲ οὐκ ἄπειρα· ἄνω δὲ ταῦτά τε καὶ τὰ συμβεβηκότα, ἀμφότερα οὐκ ἄπειρα. ἀνάγκη ἄρα εἶναί τι οὗ πρῶτόν τι κατηγορεῖται καὶ τούτου ἄλλο, καὶ τοῦτο ἵστασθαι καὶ εἶναί τι ὃ οὐκέτι οὔτε κατ' ἄλλου προτέρου οὔτε κατ' ἐκείνου ἄλλο πρότερον κατηγορεῖται. b24. Subject to these assumptions then, neither the ascending nor the descending series of predication in which a single attribute is predicated of a single subject is infinite. For the subjects of which coincidents are predicated are as many as the constitutive elements of each individual substance, and these we have seen are not infinite in number, while in the ascending series are contained those constitutive elements with their coincidents — both of which are finite. We conclude that there is a given subject (D) of which some attribute (C) is primarily predicable; that there must be an attribute (B) primarily predicable of the first attribute, and that the series must end with a term (A) not predicable of any term prior to the last subject of which it was predicated (B), and of which no term prior to it is predicable.
Εἷς μὲν οὖν τρόπος λέγεται ἀποδείξεως οὗτος, ἔτι δ' ἄλλος, εἰ ὧν πρότερα ἄττα κατηγορεῖται, ἔστι τούτων ἀπόδειξις, ὧν δ' ἔστιν ἀπόδειξις, οὔτε βέλτιον ἔχειν ἐγχωρεῖ πρὸς αὐτὰ τοῦ εἰδέναι, οὔτ' εἰδέναι ἄνευ ἀποδείξεως, εἰ δὲ τόδε διὰ τῶνδε γνώριμον, τάδε δὲ μὴ ἴσμεν μηδὲ βέλτιον ἔχομεν πρὸς αὐτὰ τοῦ εἰδέναι, οὐδὲ τὸ διὰ τούτων γνώριμον ἐπιστησόμεθα. εἰ οὖν ἔστι τι εἰδέναι δι' ἀποδείξεως ἁπλῶς καὶ μὴ ἐκ τινῶν μηδ' ἐξ ὑποθέσεως, ἀνάγκη ἵστασθαι τὰς (84a.) κατηγορίας τὰς μεταξύ. εἰ γὰρ μὴ ἵστανται, ἀλλ' ἔστιν ἀεὶ τοῦ ληφθέντος ἐπάνω, ἁπάντων ἔσται ἀπόδειξις· ὥστ' εἰ τὰ ἄπειρα μὴ ἐγχωρεῖ διελθεῖν, ὧν ἔστιν ἀπόδειξις, ταῦτ' οὐκ εἰσόμεθα δι' ἀποδείξεως. εἰ οὖν μηδὲ βέλτιον ἔχομεν πρὸς αὐτὰ τοῦ εἰδέναι, οὐκ ἔσται οὐδὲν ἐπίστασθαι δι' ἀποδείξεως ἁπλῶς, ἀλλ' ἐξ ὑποθέσεως. Λογικῶς μὲν οὖν ἐκ τούτων ἄν τις πιστεύσειε περὶ τοῦ λεχθέντος, b33. The argument we have given is one of the so-called proofs; an alternative proof follows. Predicates so related to their subjects that there are other predicates prior to them predicable of those subjects are demonstrable; but of demonstrable propositions one cannot have something better than knowledge, nor can one know them without demonstration. Secondly, if a consequent is only known through an antecedent (viz. premisses prior to it) and we neither know this antecedent nor have something better than knowledge of it, then we shall not have scientific knowledge of the consequent. Therefore, if it is possible through demonstration to know anything without qualification and not merely as dependent on the acceptance of certain premisses — i.e. hypothetically — the series of intermediate predications must terminate. If it does not terminate, and beyond any predicate taken as higher than another there remains another still higher, then every predicate is demonstrable. Consequently, since these demonstrable predicates are infinite in number and therefore cannot be traversed, we shall not know them by demonstration. If, therefore, we have not something better than knowledge of them, we cannot through demonstration have unqualified but only hypothetical science of anything. As dialectical proofs of our contention these may carry conviction,
Praemissis his, quae necessaria sunt ad propositum demonstrandum, de distinctione praedicatorum ad invicem, hic accedit ad propositum ostendendum, scilicet quod non procedatur in praedicatis in infinitum. Et dividitur haec pars in partes duas, secundum duos modos quibus ostendit propositum. Secunda pars incipit ibi: adhuc autem alius et cetera. Circa primum duo facit: primo, ostendit quod non est procedere in infinitum in praedicatis per modum circulationis; secundo, quod non procedatur in infinitum in eis secundum rectitudinem in sursum neque in deorsum; ibi: sed utique quod neque in sursum et cetera. Circa primum tria facit: primo, praemissis suppositis, addit quaedam adhuc necessaria ad propositum ostendendum; secundo, ex his et aliis praemissis concludit propositum; ibi: impossibile est aeque praedicari etc.; tertio, probat; ibi: aut enim sicut substantia et cetera. Having set forth the distinction of predicates from one another as a necessary preliminary to demonstrating his proposition, the Philosopher now undertakes to show his proposition, namely, that there is no infinite process in predicates. And his treatment falls into two parts, according to the two ways in which he shows his proposition, the second part beginning at (83b33). Concerning the first he does two things. First, he shows that one does not proceed to infinity in predicates after the manner of circularity (see Lect. 8). Secondly, that one does not proceed to infinity in a direct line upwards or downwards (83b13). Concerning the first he does three things. First, granting what has gone before, he adds certain things required for showing his proposition. Secondly, from these and other established facts he concludes the proposition (83a37). Thirdly, he proves it (8309).
Primo ergo proponit duo: quorum unum est, quia cum praedicatum, quod significat accidens, significet aliquod genus accidentis, puta qualitatem, non potest esse quod duo se habeant hoc modo ad invicem, quod primum sit qualitas secundi, et secundum sit qualitas primi; alia est enim ratio qualitatis, et eius cui qualitas inest. Secundum est, quod universaliter non est possibile quod qualitas habeat quamcunque aliam qualitatem sibi inhaerentem; quia nullum accidens est subiectum alterius accidentis per se loquendo. Soli enim substantiae convenit proprie ratio subiecti. First, therefore (83a6) he proposes two things: one of these is that since a predicate which signifies an accident signifies some genus of accident, such as quality, it cannot occur that two things be so related to one another that the first is a quality of the second and the second a quality of the first: for the nature of a quality and that of which it is the quality are diverse. The second is that universally it is not possible that a quality have some other quality inhering in it, because no accident is the subject of another accident, absolutely speaking. For to substance alone does the notion of subject properly belong.
Deinde cum dicit: impossibile est aeque etc., proponit, quasi ex praemissis concludens, quod intendit probare, et dicit: si ista sunt vera, quae praemissa sunt, impossibile est quod fiat mutua praedicatio ad invicem sic, idest secundum aliquem praedictorum modorum. Non autem ita hoc dicitur, quin contingat vere praedicari unum de alio, et e converso. Dicimus enim vere quod homo est albus, et album est homo. Sed hoc non fit aeque, idest secundum aequalem rationem praedicandi. Et similiter est in praedicatis essentialibus. Then (83a37) he proposes, as though concluding from these antecedents, what he intends to prove. He says, “If our preliminary rules are true, it is impossible that there be equal reciprocal predication,” i.e., according to any of the above-mentioned ways. But this does not exclude the possibility of one thing’s being truly predicated of another and conversely. For we say truly that man is white and that something white is a man. However, this is not done equally, i.e., according to an equal manner of predicating. And it is the same in essential predicates.
Deinde cum dicit: aut enim sicut substantia etc., ostendit propositum: et primo, in praedicatis essentialibus; secundo, in accidentalibus; ibi: verum etiam ipsius qualis et cetera. Circa primum tria facit: primo, ponit quamdam divisionem essentialium praedicatorum; secundo, resumit quoddam, quod supra probatum est; ibi: haec autem ostensa etc.; tertio, probat propositum; ibi: si quidem igitur genera et cetera. Then (83a39) he shows the proposition. First, in essential predicates Secondly, in accidental ones (83b10). In regard to the first he does three things. First, he lays down a division of essential predicates. Secondly, he recalls something established above (83b1). Thirdly, he proves the proposition (83b8).
Dicit ergo primo, quod ad ostendendum quod non sit ad invicem aeque praedicari, primo oportet hoc considerare in essentialibus praedicatis. Aut enim quod aeque praedicatur praedicabitur sicut substantia, aut alio modo. Et si sicut substantia, aut sicut genus aut sicut differentia. Haec enim duo sunt partes definitionis, quae significant essentiam. He says therefore first (83a39) that in order to show that they are not reciprocally predicated equally, it will be necessary to consider this in essential predicates. For that which is predicated equally will be predicated either as a substance or some other way: and if as a substance, then either as a genus or as a difference. For these are the two parts of a definition which signify the essence.
Deinde cum dicit: haec autem ostensa sunt etc., resumit quod supra probaverat, scilicet huiusmodi praedicata non esse infinita: quia si in infinitum procederent, non haberet in eis locum reciprocatio, seu circulatio. Dicit ergo quod sicut supra de his ostensum est, in huiusmodi non contingit procedere in infinitum, neque in sursum neque in deorsum: sicut si bipes praedicatur de homine, et animal de bipede, et de animali aliquid alterum, non est hoc procedere in infinitum in sursum; neque in deorsum, ut si animal dicatur de homine in eo quod quid est, et homo de Callia, et hoc de quodam alio (supposito quod homo esset genus continens sub se multas species, quarum una esset Callias), non posset sic procedi in infinitum. Et resumit rationem ad ostendendum quod supra posuit, quia omnem huiusmodi substantiam, quae scilicet habet aliquid universalius, quod de ipsa praedicetur, et quae potest de alio inferiori praedicari, contingit definire: genera vero generalissima, de quibus alia universaliora non praedicantur, et singularia, quae non praedicantur de aliquibus inferioribus, non contingit definire. Solum ergo substantiam mediam definire contingit. Illam vero substantiam non contingit definire, de qua infinita praedicantur: quia oportet definientem intelligendo pertransire omnia illa, quae substantialiter praedicantur de definito; cum omnia cadant in definitione, vel sicut genus, vel sicut differentia. Infinita autem non contingit pertransire. Ergo oportet omnem universalem substantiam, quae non est supremum genus, neque infimum subiectum, non habere infinita, quae de ipsa substantialiter praedicentur. Sic ergo non est procedere in infinitum, neque in sursum neque in deorsum. Then (83b1) he recalls what he had proved above at the beginning of the previous lecture, namely, that predicates of this kind are not infinite; because if they were to proceed to infinity, reciprocity or circularity would find no place therein. He says, therefore, that as has been shown above, an infinite process does not take place in such predicates either by ascending or by descending: for example, if “two-footed” be predicated of man, and “animal” of two-footed, something else of animal, there is no process to infinity either upwards or downwards. Thus, if “animal” were said essentially of man, and “man” of Callias, and this of somone else (supposing that there were a genus containing under it species, one of which would be Callias), it would not be possible to go on in this way to infinity. And he recalls the reason he used earlier to prove this: for every such substance which has something more universal that can be predicated of it and which can be predicated of some inferior is capable of being defined; but the most general genera of which other more universal things are not predicated, and singulars which are not predicated of any inferiors, are not capable of definition. Only the substance between these can be defined. But a substance of which an infinite number of things is predicated turns out to be indefinable, because one who would define it must go through and understand all the items which are substantially predicated of the defined, since all of them occur in the definition either as a genus or as a difference. But the infinite cannot be gone through. Therefore, it is required that neither a universal substance, which is not a supreme genus, nor a lowest subject, can have an infinitude of predicates which are predicated of it substantially. Consequently, there is no infinite process either upwards or downwards.
Deinde cum dicit: si quidem igitur etc., ostendit quod in substantialibus praedicatis non possit esse processus in infinitum per modum circulationis. Et dicit quod si aliqua praedicata substantialia praedicantur de aliquo ut genera, non praedicantur ad invicem aequaliter, idest convertibiliter, ita quod unum sit genus alterius et e converso. Et ad hoc probandum subdit: erit enim ipsum quod vere ipsum aliquid; quasi diceret: si aliquid praedicatur de aliquo ut genus, illud de quo praedicatur, est aliquid, quod vere est ipsum, idest est aliquid particulariter, quod substantialiter recipit praedicationem ipsius. Si ergo hoc praedicetur de illo ut genus, sequetur quod ipsum, quod particulariter conveniebat alicui, e converso particulariter recipiat praedicationem illius; quod est idem respectu eiusdem esse partem et totum, quod est impossibile. Et eadem ratio est de differentiis. Unde et in I topicorum dicitur quod problema de differentia reducitur ad problema de genere. Then (83b8) he shows that there is no process to infinity after the manner of circularity in substantial predicates. And he says that if certain substantial predicates are predicated as genera of something, they are not predicated of one another equally, i.e., convertibly, i.e., so that one would be the genus of another and vice versa. To prove this he continues, “for the one will be what something truly is.” As if to say: if something is predicated of something as a genus, that of which it is predicated is something which truly is that, i.e., something particular which receives that predication substantially. Therefore, if this be predicated of that as a genus, it will follow that something which belongs to something particularly would conversely receive the predication of it, which is tantamount to saying that a same thing is both a whole and a part in relation to the same thing—which is impossible. And the same reasoning applies to differences. Hence in Topics I it is stated that problems concerning a difference are reduced to problems concerning the genus.
Deinde cum dicit: neque tamen qualis etc., ostendit quod non potest esse processus in infinitum per modum circulationis in praedicationibus, in quibus praedicatur accidens de subiecto. Et dicit quod neque etiam ipsius qualis potest esse conversio cum suo subiecto, aut aliorum nullum potest habere huiusmodi praedicationem, quae accidentaliter praedicantur, nisi fiat praedicatio per accidens; secundum quod dictum est quod accidentia non praedicantur de subiectis nisi per accidens. Qualitas enim et omnia alia huiusmodi accidunt substantiae: unde praedicantur de substantiis sicut accidens de subiecto. Then (83b10) he shows that there cannot be an infinite circular process in predications in which an accident is predicated of a subject. And he says that there cannot be conversion of a quality with its subject; furthermore, none of the others which are predicated accidentally can have that sort of predication, unless the predication be made per accidens in the sense described above, namely, that accidents are not predicated of their subjects except per accidens. For quality and all these others are accidental to the subject; hence they are predicated of their substances as an accident of a subject.
Deinde cum dicit: sed utique neque in sursum etc., ostendit universaliter quod in nullo genere praedicationis sit procedere in infinitum in sursum aut deorsum. Et dicit quod non solum non est procedere in praedicationibus in infinitum secundum circulationem, sed neque etiam procedendo in sursum infinita erunt praedicata, et similiter nec in deorsum. Et ad hoc probandum, primo resumit quaedam supra posita; secundo ex his probat intentum; ibi: neque in sursum ergo et cetera. Then (83b13) he shows universally that in no genus of predication is i liere an infinite process upwards or downwards. And he says that not only is there no infinite process in predications according to circulation, but also that in proceeding upwards or downwards the predicates will not be infinite. To prove this: First, he recalls certain things which were established above. Secondly, from these he concludes his proposition 83b24).
Circa primum, primo resumit quod de unoquoque possunt aliqua praedicari, quidquid significent: sive sit quale, sive quantum, vel quodcunque aliud genus accidentis, vel etiam quae intrant substantiam rei, quae sunt essentialia praedicata. In regard to the first he reaffirms (83b13) that certain things can be predicated of a subject, whether they signify quality or quantity or any other genus of accident, or even items which constitute the substance of a thing, i.e., essential predicates.
Secundo, resumit quod haec, scilicet substantialia praedicata, sunt finita. Secondly, he reiterates, that the latter, i.e., the substantial predicates, are finite.
Tertio, resumit quod genera praedicamentorum sunt finita; scilicet quale et quantum et cetera. Si enim aliquis dicat quod quantitas praedicetur de substantia, et qualitas de quantitate, et sic in infinitum; hoc excludit per hoc, quod genera praedicamentorum sunt finita. Thirdly, he reaffirms that the genera of predicaments are finite, namely, quality, quantity and so on. For if someone were to say that quantity is predicated of substance, and quality of quantity, and so on to infinity, he excludes this on the ground that the genera of predicaments are finite.
Quarto, resumit quod, sicut supra expositum est, unum de uno praedicatur in simplici praedicatione. Et hoc ideo inducit, quia posset aliquis dicere quod primo praedicabitur unum de uno, puta de homine animal; et ista praedicatio multiplicabitur quousque poterit inveniri aliquod unum, quod de homine praedicetur. Quibus finitis, praedicabuntur duo de uno: puta, dicetur quod homo est animal album; et sic multo plura praedicata invenirentur secundum diversas combinationes praedicatorum. Rursus, praedicabuntur tria de uno: puta, dicetur quod homo est animal album magnum; et sic semper addendo ad numerum, magis multiplicabuntur praedicata, et erit procedere in infinitum in praedicatis, sicut etiam in additione numerorum. Sed hoc excludit per praedicationem unius de uno. Fourthly, he reaffirms that, as stated above, one thing is predicated of one thing in simple predication. And he mentions this because someone might say that one thing may well be first predicated of one thing, as “animal” of man, and this predication will be multiplied until something else can be found predicable of man, and when this is found, two things, will be predicated of one, so that it will be said that man is a white animal. Thus, many more predicates might be found according to various combinations of predicates. And so by continually adding to this number, the predicates will be increased more and more, so that there will be a process to infinity in predicates, just as there is in the succession of numbers. But he excludes this by predicating one of one.
Quinto, resumit ut non dicamus aliqua simpliciter praedicari de ipsis, quae non aliquid sunt, idest de accidentibus, quorum nullum est aliquid subsistens. De accidente enim neque subiectum neque accidens proprie praedicatur, ut supra dictum est. Omnia enim huiusmodi, quae non sunt aliquid substantiale, sunt accidentia, et de his nihil praedicatur simpliciter loquendo: sed haec quidem praedicantur per se, scilicet de subiectis, vel substantialia praedicata vel accidentalia. Illa vero secundum alium modum, idest per accidens, scilicet cum praedicantur de accidentibus, aut subiecta, aut accidentia. Haec enim omnia, scilicet accidentia, habent de sui ratione quod dicantur de subiecto: illud autem quod est accidens, non est subiectum aliquod; unde nihil proprie loquendo potest de eo praedicari, quia nihil talium, scilicet accidentium, ponimus esse tale, quod dicatur id, quod dicitur, idest quod suscipiat praedicationem eius, quod de eo praedicatur, non quasi aliquid alterum existens, sicut accidit in substantiis. Homo enim dicitur animal vel album, non quia aliquid aliud sit animal vel album, sed quia ipsummet quod est homo, est animal vel album: sed album ideo dicitur homo vel musicum, quia aliquid alterum, scilicet subiectum albi, est homo vel musicum. Sed ipsum accidens inest aliis; et alia, quae praedicantur de accidente, praedicantur de altero, idest de subiecto accidentis; et propter hoc praedicantur de accidente, ut dictum est. Hoc autem introduxit, quia si accidens praedicatur de subiecto, et e converso, et omnia quae accidunt subiecto, praedicentur de se invicem, sequetur quod praedicatio procedat in infinitum, quia uni infinita accidunt. Fifthly, he repeats that we should not say that certain items are predicated absolutely “of things which are not something,” i.e., of accidents, none of which is a subsistent being. For, as shown above, neither the, subject nor an accident is properly predicated of an accident. For all things of this sort that are not substantial are accidents, and nothing is predicated, simply speaking, of such things. Yet they whether they be substantial predicates or accidental, they are predicated per se of their subjects; but if they are subjects or accidents being predicated of an accident, they are predicated in another way, i.e., per accidens. For it belongs to the very notion of all accidents that they be said of a subject; and since an accident is not a subject, nothing can, properly speaking, be predicated of it, because “none of such,” i.e., no accidents, “are stated to be such that they are said to be that which is said,” i.e., stated to be such that they themselves, rather than something else distinct from them, receives the predication of that which is predicated of them, as happens in the case of substances. For man is not called “animal” or “white” because something else is “animal” or “white,” but because the very thing which is a man is animal or white; but a white thing is called “man” or “musician” because something else, namely, the subject of white is the man or musician. But the accident itself is in something distinct from it; and items which are predicated of an a accident are predicated of something other than that accident, namely, of the subject of the accident: it is on this ground that they are predicated of the accident, as has been stated. (Now he mentioned this because if an accident is predicated of a subject and vice versa, and all accidents of a subject can be mutually predicated of one another, it will follow that predication could proceed to infinity, because an infinitude of things can happen to one thing).
Deinde cum dicit: neque in sursum ergo etc., ostendit propositum ex praemissis, scilicet quod in praedicatione, qua praedicatur unum de uno, non proceditur in infinitum, neque in sursum neque in deorsum; quia omnia accidentia praedicantur de his, quae pertinent ad substantiam rei, quod erat quinta suppositio. Substantialia autem praedicata non sunt infinita (quod erat secunda suppositio); et ita ex parte subiectorum non proceditur in infinitum in huiusmodi praedicationibus, quasi in deorsum. In sursum autem neutra sunt infinita, scilicet neque substantialia praedicata, neque accidentalia: quia et genera accidentium sunt finita, et in unoquoque generum non est procedere in infinitum, neque in sursum neque in deorsum, sicut neque in substantialibus praedicatis; quia in quolibet praedicamento genus praedicatur de specie in eo quod quid est. Unde concludi potest universaliter quod necesse est esse aliquod primum subiectum, de quo aliquid praedicetur, existente statu praedicationis in deorsum: et de hoc aliquid aliud praedicabitur, et hoc habebit statum in sursum: et erit invenire aliquid quod non amplius praedicabitur de alio, neque sicut posterius praedicatur de priori per accidens, neque sicut prius praedicatur de posteriori per se. Hic igitur est unus modus logice demonstrandi propositum, qui sumitur secundum diversos modos praedicationis. Then (83b24) from these premises he shows his proposition, namely, that in a predication in which one thing is predicated of one thing there is no upward or downward process to infinity, because, as the fifth supposition states, all accidents are predicated of items which pertain to the substance of the thing. Furthermore, according to the second supposition, substantial predicates are not infinite; consequently, on the part of the subjects there is no infinite process downwards in these predications. Again, neither of these is infinite in the upward movement, i.e., neither the substantial nor the accidental predicates, both because the genera of accidents are finite, and because there is no infinite process upwards or downwards in any of these genera any more than there is in substantial predicates, because in each predicament the genus is predicated of a species in regard to something essential. Hence we can conclude universally that there must be some first subject of which something is predicated, thus establishing a stop in downward predication; then something else will be predicated of this, but it will come to a stop in the ascending process, so that something will be found which is not found predicated of another, either as the subsequent is predicated of its prior per accidens, or as the prior is predicated per se of its subsequent. This, therefore, is one way of demonstrating the proposition logically, and it is based on the diverse modes of predicating.
Deinde cum dicit: adhuc autem alius etc., ponit secundum modum probationis, et dicit quod quando est aliqua talis propositio, in qua praedicatur aliquid de subiecto, si aliqua possunt per prius praedicari de illo subiecto, talis propositio demonstrabilis erit: puta haec propositio, homo est substantia, demonstratur per hanc, animal est substantia, quia de animali per prius praedicatur substantia quam de homine. Si autem aliqua propositio est demonstrabilis, non possumus eam melius cognoscere quam sciendo: sicut principia indemonstrabilia melius cognoscimus quam sciendo, quia cognoscimus ea ut per se nota. Et iterum huiusmodi demonstrabilia non possumus scire nisi per demonstrationem; quia demonstratio est syllogismus faciens scire, ut supra dictum est. Item considerandum est quod si aliqua propositio est nota per aliam, si illam per quam nota est nescimus, neque cognoscimus eam meliori modo quam sciendo, consequens est quod nec sciamus illam propositionem, quae per eam cognoscitur. Then (83b33) he sets forth the second way of proving. And he says that when a proposition in which something is predicated of a subject is such that certain things can be predicated per prius of that subject, that proposition will be demonstrable: for example, this proposition, “Man is a substance,” is demonstrated by the proposition, “Animal is a substance,” because “substance” is predicated of animal before it is predicated of man. But if a proposition is demonstrable, there is no better way of knowing it than to know it by demonstration, just as we know indemonstrable principles better than by way of demonstration because we know them as self-evident. Besides, we cannot scientifically know such demonstrable propositions except through demonstration, because a demonstration is a syllogism productive of scientific knowledge, as we showed above. Furthermore, if a proposition is known through another one, and if we do not scientifically know the one through which it is known, or know it in the way which is better than knowing scientifically, then we do not scientifically know that proposition which is made known through it.
His igitur tribus suppositis, procedit sic. Si contingit aliquid simpliciter scire per demonstrationem, et non ex aliquibus nec ex suppositione, necesse est quod sit status in praedicatis, quae accipiuntur ut media. Dicit autem simpliciter, et non ex aliquibus, ad excludendum demonstrationes ducentes ad impossibile; in quibus proceditur contra positiones aliquas ex aliquibus propositionibus datis. Dicit autem neque ex suppositione, ad excludendum demonstrationes quia, quales fiunt in scientiis subalternis; quae supponunt conclusiones superiorum scientiarum, ut supra habitum est. With these three suppositions in mind he proceeds thus: If one does know something purely through demonstration “and not from something or from supposition,” it is necessary that there be a stop in the predicates which are-taken as middles. (He says, “purely and not from something,” to exclude) demonstrations leading to the impossible, in which one proceeds, against certain positions by arguing from propositions that have been agreed upon. Furthermore, he says, “or from supposition,” to exclude such demonstrations as are formed in subalternate sciences, which suppose the conclusion of higher sciences, as explained above).
Est ergo simpliciter per demonstrationem scire, quando quaelibet propositionum praemissarum, si sit demonstrabilis, scitur per demonstrationem; et si non est demonstrabilis, intelligitur per seipsam. Et hoc supposito, necesse est quod sit status in praedicationibus, quia si non fuerit status, sed semper potest accipi aliquid superius, sequitur quod omnium sit demonstratio, ut primo dicebatur. Si ergo aliqua conclusio demonstratur, oportet quod quaelibet praemissarum sit demonstrabilis. Sic ergo ad eius cognitionem nullo modo possumus melius nos habere, quam sciendo eam per demonstrationem: ergo oportebit eam demonstrare per aliquas alias propositiones, et illas iterum per alias, et sic in infinitum. Quia igitur infinita non est transire, non poterimus ea cognoscere per demonstrationem, neque melius ea cognoscere possumus, cum omnia sint demonstrabilia. Ergo sequetur quod nihil contingat scire per demonstrationem simpliciter, sed solum ex suppositione. Therefore, one knows purely through demonstration when each one of the premised propositions, if it is demonstrable, is known through demonstration; and if it is not demonstrable, is known in virtue of itself. Under these suppositions it is necessary that there be a stop in predications, because if there is no stop but something prior can always be taken, it follows that there is demonstration of everything, as was said above. Therefore, if a. conclusion is demonstrated, each of the premises must be demonstrable. But if we can have knowledge of it in no better way than by knowing it through demonstration, and if it will be necessary to demonstrate it through other propositions, and those through others again, and so on to infinity, then, since it is not possible to go through those infinites, we shall not be able to make it known through demonstration or through the method better than demonstration, since all things are demonstrable. The consequence will be that one knows nothing purely through demonstration, but only from supposition.
Ultimo autem epilogando concludit principale propositum. Finally, by way of summary he concludes the main proposition.

Lectio 35
Caput 22 cont.
ἀναλυτικῶς δὲ διὰ τῶνδε φανερὸν συντομώτερον, ὅτι οὔτ' ἐπὶ τὸ ἄνω οὔτ' ἐπὶ τὸ κάτω ἄπειρα τὰ κατηγορούμενα ἐνδέχεται εἶναι ἐν ταῖς ἀποδεικτικαῖς ἐπιστήμαις, περὶ ὧν ἡ σκέψις ἐστίν. a8. but an analytic process will show more briefly that neither the ascent nor the descent of predication can be infinite in the demonstrative sciences which are the object of our investigation.
ἡ μὲν γὰρ ἀπόδειξίς ἐστι τῶν ὅσα ὑπάρχει καθ' αὑτὰ τοῖς πράγμασιν. a10. Demonstration proves the inherence of essential attributes in things.
καθ' αὑτὰ δὲ διττῶς· ὅσα τε γὰρ [ἐν] ἐκείνοις ἐνυπάρχει ἐν τῷ τί ἐστι, καὶ οἷς αὐτὰ ἐν τῷ τί ἐστιν ὑπάρχουσιν αὐτοῖς· οἷον τῷ ἀριθμῷ τὸ περιττόν, ὃ ὑπάρχει μὲν ἀριθμῷ, ἐνυπάρχει δ' αὐτὸς ὁ ἀριθμὸς ἐν τῷ λόγῳ αὐτοῦ, καὶ πάλιν πλῆθος ἢ τὸ διαιρετὸν ἐν τῷ λόγῳ τῷ τοῦ ἀριθμοῦ ἐνυπάρχει. a11. Now attributes may be essential for two reasons: either because they are elements in the essential nature of their subjects, or because their subjects are elements in their essential nature. An example of the latter is odd as an attribute of number — though it is number's attribute, yet number itself is an element in the definition of odd; of the former, multiplicity or the indivisible, which are elements in the definition of number.
τούτων δ' οὐδέτερα ἐνδέχεται ἄπειρα εἶναι, οὔθ' ὡς τὸ περιττὸν τοῦ ἀριθμοῦ (πάλιν γὰρ ἂν τῷ περιττῷ ἄλλο εἴη ᾧ ἐνυπῆρχεν ὑπάρχοντι· τοῦτο δ' εἰ ἔστι, πρῶτον ὁ ἀριθμὸς ἐνυπάρξει ὑπάρχουσιν αὐτῷ· εἰ οὖν μὴ ἐνδέχεται ἄπειρα τοιαῦτα ὑπάρχειν ἐν τῷ ἑνί, οὐδ' ἐπὶ τὸ ἄνω ἔσται ἄπειρα· a18. In neither kind of attribution can the terms be infinite. They are not infinite where each is related to the term below it as odd is to number, for this would mean the inherence in odd of another attribute of odd in whose nature odd was an essential element: but then number will be an ultimate subject of the whole infinite chain of attributes, and be an element in the definition of each of them. Hence, since an infinity of attributes such as contain their subject in their definition cannot inhere in a single thing, the ascending series is equally finite.
ἀλλὰ μὴν ἀνάγκη γε πάντα ὑπάρχειν τῷ πρώτῳ, οἷον τῷ ἀριθμῷ, κἀκείνοις τὸν ἀριθμόν, ὥστ' ἀντιστρέφοντα ἔσται, ἀλλ' οὐχ ὑπερτείνοντα)· a23. Note, moreover, that all such attributes must so inhere in the ultimate subject — e.g. its attributes in number and number in them — as to be commensurate with the subject and not of wider extent.
οὐδὲ μὴν ὅσα ἐν τῷ τί ἐστιν ἐνυπάρχει, οὐδὲ ταῦτα ἄπειρα· οὐδὲ γὰρ ἂν εἴη ὁρίσασθαι. ὥστ' εἰ τὰ μὲν κατηγορούμενα καθ' αὑτὰ πάντα λέγεται, ταῦτα δὲ μὴ ἄπειρα, ἵσταιτο ἂν τὰ ἐπὶ τὸ ἄνω, ὥστε καὶ ἐπὶ τὸ κάτω. a25. Attributes which are essential elements in the nature of their subjects are equally finite: otherwise definition would be impossible. Hence, if all the attributes predicated are essential and these cannot be infinite, the ascending series will terminate, and consequently the descending series too.
Εἰ δ' οὕτω, καὶ τὰ ἐν τῷ μεταξὺ δύο ὅρων ἀεὶ πεπερασμένα. a28. If this is so, it follows that the intermediates between any two terms are also always limited in number.
εἰ δὲ τοῦτο, δῆλον ἤδη καὶ τῶν ἀποδείξεων ὅτι ἀνάγκη ἀρχάς τε εἶναι, καὶ μὴ πάντων εἶναι ἀπόδειξιν, ὅπερ ἔφαμέν τινας λέγειν κατ' ἀρχάς. εἰ γὰρ εἰσὶν ἀρχαί, οὔτε πάντ' ἀποδεικτὰ οὔτ' εἰς ἄπειρον οἷόν τε βαδίζειν· τὸ γὰρ εἶναι τούτων ὁποτερονοῦν οὐδὲν ἄλλο ἐστὶν ἢ τὸ εἶναι μηδὲν διάστημα ἄμεσον καὶ ἀδιαίρετον, ἀλλὰ πάντα διαιρετά. τῷ γὰρ ἐντὸς ἐμβάλλεσθαι ὅρον, ἀλλ' οὐ τῷ προσλαμβάνεσθαι ἀποδείκνυται τὸ ἀποδεικνύμενον, ὥστ' εἰ τοῦτ' εἰς ἄπειρον ἐνδέχεται ἰέναι, ἐνδέχοιτ' ἂν δύο ὅρων ἄπειρα μεταξὺ εἶναι μέσα. ἀλλὰ τοῦτ' ἀδύνατον, εἰ ἵστανται αἱ κατηγορίαι (84b.) ἐπὶ τὸ ἄνω καὶ τὸ κάτω. ὅτι δὲ ἵστανται, δέδεικται λογικῶς μὲν πρότερον, ἀναλυτικῶς δὲ νῦν. a29. An immediately obvious consequence of this is that demonstrations necessarily involve basic truths, and that the contention of some — referred to at the outset — that all truths are demonstrable is mistaken. For if there are basic truths, (a) not all truths are demonstrable, and (b) an infinite regress is impossible; since if either (a) or (b) were not a fact, it would mean that no interval was immediate and indivisible, but that all intervals were divisible. This is true because a conclusion is demonstrated by the interposition, not the apposition, of a fresh term. If such interposition could continue to infinity there might be an infinite number of terms between any two terms; but this is impossible if both the ascending and descending series of predication terminate; and of this fact, which before was shown dialectically, analytic proof has now been given.
Postquam philosophus ostendit logice quod non sit procedere in infinitum in praedicatis in sursum aut deorsum, hic ostendit idem analytice. Et dividitur in duas partes: in prima ostendit principale propositum; in secunda infert quaedam corollaria ex dictis; ibi: monstratis autem his manifestum et cetera. Circa primum duo facit: primo, proponit quod intendit: secundo, probat propositum; ibi: demonstratio quidem enim et cetera. After showing logically that there is no infinite process upwards or downwards in predicates, the Philosopher now shows the same thing analytically. And his treatment falls into two parts. In the first he shows the principal proposition. In the second he infers certain corrolaries from the aforesaid (84b3) [L. 36]. Concerning the first he does two things. First, he proposes what he intends. Secondly, he proves his proposition (84a10).
Dicit ergo primo, quod hoc quod non contingit in demonstrativis scientiis, de quibus intendimus, praedicationes in infinitum procedere, neque in sursum neque in deorsum, brevius et citius poterit manifestari analytice quam manifestatum sit logice. Ubi considerandum est quod analytica, idest demonstrativa scientia, quae resolvendo ad principia per se nota iudicativa dicitur, est pars logicae, quae etiam dialecticam sub se continet. Ad logicam autem communiter pertinet considerare praedicationem universaliter, secundum quod continet sub se praedicationem quae est per se, et quae non est per se. Sed demonstrativae scientiae propria est praedicatio per se. Et ideo supra logice probavit propositum, quia ostendit universaliter in omni genere praedicationis non esse processum in infinitum; hic autem intendit ostendere analytice, quia hoc probat solum in his, quae praedicantur per se. Et haec est via expeditior: et ideo sufficit ad propositum, quia in demonstrationibus non utimur nisi tali modo praedicationis. He says therefore first (84a8) that the fact an infinite process upwards or downwards does not occur in the demonstrative sciences with which we are concerned can be more briefly and quickly manifested analytically than it was logically. Here we might note that analytic, i.e., demonstrative, science which is called judicative, because it resolves to self-evident principles, is a part of logic which even contains dialectics under it. However, it pertains to logic in general to consider predication universally, i.e., as containing under it predication which is per se and predication which is not per se. But predication per se is proper to demonstrative science. Therefore, above he proves his proposition logically, because he showed universally in every genus of predication that there is no infinite process. But here he intends to show it analytically, because he proves it only in things which are predicated per se. And this is a more efficient way; furthermore, it suffices for our purpose, because that is the only mode of predication we use in demonstration.
Deinde cum dicit: demonstratio quidem etc., ostendit propositum. Et circa hoc tria facit: primo, proponit qua praedicatione analytica, idest demonstrativa scientia, utatur, quia praedicatione per se; secundo, resumit quot sunt modi talis praedicationis; ibi: per seipsa vero etc.; tertio, ostendit quod in nullo modo praedicationis per se possit procedi in infinitum; ibi: horum autem neutra contingunt et cetera. Then (84a10) he shows his proposition concerning which he does three things. First, he proposes which predication analytic, i.e., demonstrative, science employs, for it uses per se predication. Secondly, he recalls how many modes there are of such predication (84al 1). Thirdly, he shows that there cannot be an infinite process in any mode of per se predication (8408).
Dicit ergo primo, quod demonstratio est solum circa illa, quae per se insunt rebus. Tales enim sunt eius conclusiones, et ex talibus demonstrat, ut supra habitum est. He says therefore first (84a10) that demonstration is concerned exclusively with items that are per se in things. For such are its conclusions and from such does it demonstrate, as was established above.
Deinde cum dicit: secundum seipsa autem etc., ponit duos modos praedicandi per se. Nam primo quidem praedicantur per se quaecunque insunt subiectis in eo quod quid est, scilicet cum praedicata ponuntur in definitione subiecti. Secundo, quando ipsa subiecta insunt praedicatis in eo quod quid est, idest quando subiecta ponuntur in definitione praedicatorum. Et exemplificat de utroque modo. Nam impar praedicatur de numero per se secundo modo, quia numerus ponitur in definitione ipsius imparis. Est enim impar numerus medio carens. Multitudo autem vel divisibile praedicatur de numero, et ponitur in definitione eius. Unde huiusmodi praedicantur per se de numero primo modo. Alii autem modi, quos supra posuit, reducuntur ad istos. Then (84a 11) he lays down two modes of predicating per se. For in the first place those things are predicated per se which are present in their subjects as constituting their essence, namely, when predicates are placed in the definition of a subject. Secondly, when the subjects themselves are in the essence of the predicate, i.e., when the subjects are placed in the definition of the predicate. And he gives examples of each of these ways: for “odd” is predicated per se of number in the second way, because “number” is placed in the definition of odd. For the odd is a number not divisible by two. Multitude or divisible, however, are predicated of number and are present in its definition; hence these are predicated per se of number in the first way. The other ways, which he mentioned previously, are reduced to these.
Deinde cum dicit: horum autem neutra etc., ostendit quod in utroque modo praedicationis per se necesse est esse statum. Et circa hoc tria facit: primo, ostendit quod necessarium est esse statum in utroque modo praedicationis per se, tam in sursum quam in deorsum; secundo, concludit quod non possit esse infinitum in mediis; ibi: si autem sic est etc.; tertio, concludit quod non potest procedi in infinitum in demonstrationibus; ibi: si vero hoc et cetera. Circa primum duo facit: primo, ostendit propositum in secundo modo dicendi per se, quando scilicet subiectum ponitur in definitione praedicati; secundo, in primo modo, quando praedicatum ponitur in definitione subiecti; ibi: neque etiam quaecunque et cetera. Then (84a18) he shows that there must be a stop in each of these modes of per se predication. In regard to this he does three things. First, he shows that there must be a stop both upwards and downwards in both of these modes of per se predication. Secondly, he concludes that there cannot be an infinitude of middles (84a28). Thirdly, he concludes that one cannot proceed to infinity in demonstrations (84a29). Concerning the first he does two things. First, he shows his proposition in regard to the second way of saying per se,” namely, when the subject is placed in the definition of the predicate. Secondly, in the first way, when the predicate is placed in the definition of the subject (84a25). In regard to the first he gives two reasons, in the first of which he proceeds in the following way.
Circa primum ponit duas rationes. Circa quarum primam sic procedit: primo quidem praemittit propositum, scilicet quod in neutro modo dicendi per se contingit in infinitum procedere; deinde probat hoc in secundo modo, puta cum impar praedicatur de numero. Si enim procedatur ulterius, quod aliquid aliud praedicetur per se de impari secundum istum modum dicendi per se, sequitur quod impar insit in definitione eius. Numerus autem ponitur in definitione imparis: unde sequeretur quod etiam numerus ponatur in definitione illius tertii, quod per se inest impari. Sed hic non contingit abire in infinitum, ut scilicet infinita insint in definitione alicuius, sicut supra probatum est. Relinquitur ergo quod in talibus per se praedicationibus non contingit procedere in infinitum in sursum, idest ex parte praedicati. First (84a18) he states ‘his proposition, namely, that in neither of these modes of saying per se does an infinite process occur. Then he proves this of the second mode, as when “odd” is predicated of number. For if one goes further and states that something else should be predicated per se of “odd” according to that mode of saying per se, it follows that “odd” is present in its definition. But “number” is placed in the definition of odd; hence it will follow that “number” is also present in the definition of that third thing which is present per se in “odd.” However, this cannot go on to infinity, so that an infinitude of things would be in the definition of something, as was established above. It remains, therefore, that in such per se predications an upward process to infinity does not occur, i.e., on the side of the predicate.
Secundam rationem ponit ibi: at vero necesse est omnia etc., et dicit quod quantumcunque procedatur in huiusmodi per se praedicationibus secundi modi, oportebit quod omnia praedicata per ordinem accepta insint primo subiecto, puta numero, quasi praedicata de eo: quia si impar per se praedicatur de numero, oportet quod quidquid per se praedicatur de impari, etiam per se praedicetur de numero. Et iterum oportet quod numerus omnibus illis insit; quia si numerus ponitur in definitione imparis, oportet quod ponatur in definitione omnium eorum, quae definiuntur per impar. Et ita sequitur quod mutuo sibi invicem insint. Ergo erunt convertibilia et non se invicem excedentia; sic enim propriae passiones se habent ad sua subiecta. Unde si etiam sint infinita per se praedicata secundum hunc modum, non erit ad propositum, quo aliquis intendit ponere infinita in praedicatis esse, vel in sursum vel in deorsum. He presents the second reason (84a23) and says that no matter how fair one advances in these per se predications of the second mode, it will be required that all the predicates taken in order be in their first subject, say in number, as predicated of it, because if “odd” is predicated per se of number, it will be required that whatever is predicated of odd be also predicated of number. And it is further required that “number” be in all of them, because if “number” is placed in the definition of odd, it has to be placed in the definition of all those things which are defined by “odd.” And thus it follows that they are mutually in one another. Therefore, they will be convertible and none of wider extent than another: for this is the way proper attributes are related to their subjects. Hence, even though there be an infinitude of per se predicates in this way, it offers nothing to the purpose of one who intends to establish that there is an infinitude of predicates upwards or downwards.
Deinde cum dicit: neque etiam quaecunque sunt etc., probat propositum in primo modo dicendi per se: et dicit quod illa, quae praedicantur in eo quod quid est, idest quasi posita in definitione subiecti, non possunt esse infinita, quia non contingeret definire, ut supra probatum est. Ex hoc ergo concludit quod si omnia, quae praedicantur in demonstrationibus, per se praedicantur, et in praedicatis per se non est procedere in infinitum in sursum, necesse est quod praedicata in demonstrationibus stent in sursum. Et ex hoc etiam sequitur quod stent in deorsum, quia ex quacunque parte ponatur infinitum, tollitur scientia et definitio, ut ex supra dictis patet. Then (84a25) he proves his own point in regard to the first mode of saying per se, and he states that those things which are predicated in essence, i.e., as pertaining to the definition of the subject, cannot be infinite; otherwise, definition would be impossible, as we showed above. From this, therefore, he concludes that if all the items predicated in demonstrations are predicated per se, and if there is no infinite upward process in per se predicates, it is necessary that the predicates in demonstrations stop in the upward movement. And from this it also follows that they must stop in the downward movement, because no matter on which side infinity is posited, science and definition are destroyed, as is evident from what has been said above.
Deinde cum dicit: si autem sic est etc., concludit ex praemissis quod si est status in sursum et deorsum, quod media non contingit esse infinita. Supra enim ostendit quod extremis existentibus determinatis, media non possunt esse infinita. Then (84a28) he concludes from the foregoing that if there is a stop upwards and downwards, the middles cannot be infinite. For it has been established above that if the extremes are determinate, there cannot be an infinitude of middles.
Deinde cum dicit: si vero hoc est etc., concludit ulterius quod in demonstrationibus non proceditur in infinitum: et dicit quod si praedicta sunt vera, necesse est esse aliqua prima principia demonstrationum, quae non demonstrantur; et sic non omnium erit demonstratio, secundum quod quidam dicunt, ut in principio huius libri dictum est. Then (84a29) he further concludes that there is no infinite process in demonstrations. And he says that if the above statements are true, it is necessary that there be certain first principles of demonstration that are not demonstrated; consequently, there will not be demonstration of everything, as some claim, as was stated in the beginning of this book.
Et quod hoc sequatur ostendit. Posito enim quod sint aliqua principia demonstrationum, necesse est quod illa sint indemonstrabilia; quia cum omnis demonstratio sit ex prioribus, ut supra habitum est, si principia demonstrarentur, sequeretur quod aliquid esset prius principiis; quod est contra rationem principii. Et ita, si non sunt omnia demonstrabilia, sequetur quod non procedant demonstrationes in infinitum. Then he goes on to show that his consequence follows. For if it is granted that there are certain principles of demonstrations, it is necessary that they be indemonstrable: for since every demonstration proceeds from things that are prior, as has been established above, then if the principles are demonstrated, it will follow that something would be prior to the principles, and this is contrary to the notion of a principle. And so, if not all things are demonstrable, it will follow that demonstrations do not proceed to infinity.
Omnia autem praedicta consequuntur ex hoc quod ostensum est, quod non proceditur in infinitum in mediis: quia nihil est aliud ponere verum esse quodcunque praedictorum, scilicet vel quod demonstrationes procedant in infinitum, vel quod omnia sint demonstrabilia, vel quod nulla sint demonstrationum principia, quam ponere nullum spatium esse immediatum et indivisibile; idest ponere duos terminos sibi invicem non cohaerere in aliqua propositione affirmativa vel negativa, nisi per medium. Si enim aliqua propositio sit immediata, sequitur quod sit indemonstrabilis; quia cum aliquid demonstratur, oportet sumere terminum immittendo, idest, quod sit infra praedicatum et subiectum; de quo scilicet per prius praedicetur praedicatum quam de subiecto, vel a quo prius removeatur. Non autem in demonstrationibus accipitur medium assumendo extrinsecus: hoc enim esset assumere extraneum medium, et non proprium, quod contingit in litigiosis et dialecticis syllogismis. Si ergo demonstrationes contingit in infinitum procedere, sequitur quod sint media infinita inter duos terminos. Sed hoc est impossibile, si praedicationes steterint in sursum et deorsum, ut supra probatum est. Et quod stent praedicationes in sursum et deorsum, prius ostendimus logice, et postea analytice, ut expositum est. Per hanc igitur conclusionem ultimo inductam manifestat intentionem totius capituli, et quare quaelibet propositio sit inducta. But all these follow from what has been established, namely, from the fact that there is no infinite process in middles, because the position that any of the foregoing statements is true, i.e., that demonstrations proceed to infinity, or that all things are demonstrable, or that there are no principles of demonstrations is tantamount to the position that no distance is immediate and indivisible’ i.e., to the position that the two terms of any affirmative or negative proposition belong together only in virtue of a middle. For if any proposition is immediate, it follows that it is indemonstrable; because when something is demonstrated, it is necessary to take a term by interposing, i.e., by setting it between the subject and predicate, so that the predicate will be predicated of that term before being predicated of the subject-or removed from it. But the middle in demonstrations is not taken by assuming extraneously; for this would be to assume an extraneous middle and not a proper middle-which occurs in contentious and dialectical syllogisms. Therefore, if demonstrations were to proceed to infinity, it would follow that there is an infinitude of middles between two extremes. But this is impossible if, as has been established above, the predications stop in the upward and downward process. But as we have shown, first logically and then analytically, these predications do stop both upwards and downwards, as explained. Therefore, in virtue of this conclusion finally induced, he manifests the intent of the entire chapter and why each proposition was introduced.

Lectio 36
Caput 23
Δεδειγμένων δὲ τούτων φανερὸν ὅτι, ἐάν τι τὸ αὐτὸ δυσὶν ὑπάρχῃ, οἷον τὸ Α τῷ τε Γ καὶ τῷ Δ, μὴ κατηγορουμένου θατέρου κατὰ θατέρου, ἢ μηδαμῶς ἢ μὴ κατὰ παντός, ὅτι οὐκ ἀεὶ κατὰ κοινόν τι ὑπάρξει. b3. It is an evident corollary of these conclusions that if the same attribute A inheres in two terms C and D predicable either not at all, or not of all instances, of one another, it does not always belong to them in virtue of a common middle term.
οἷον τῷ ἰσοσκελεῖ καὶ τῷ σκαληνεῖ τὸ δυσὶν ὀρθαῖς ἴσας ἔχειν κατὰ κοινόν τι ὑπάρχει (ᾗ γὰρ σχῆμά τι, ὑπάρχει, καὶ οὐχ ᾗ ἕτερον), τοῦτο δ' οὐκ ἀεὶ οὕτως ἔχει. b6. Isosceles and scalene possess the attribute of having their angles equal to two right angles in virtue of a common middle; for they possess it in so far as they are both a certain kind of figure, and not in so far as they differ from one another. But this is not always the case:
ἔστω γὰρ τὸ Β καθ' ὃ τὸ Α τῷ Γ Δ ὑπάρχει. δῆλον τοίνυν ὅτι καὶ τὸ Β τῷ Γ καὶ Δ κατ' ἄλλο κοινόν, κἀκεῖνο καθ' ἕτερον, ὥστε δύο ὅρων μεταξὺ ἄπειροι ἂν ἐμπίπτοιεν ὅροι. ἀλλ' ἀδύνατον. κατὰ μὲν τοίνυν κοινόν τι ὑπάρχειν οὐκ ἀνάγκη ἀεὶ τὸ αὐτὸ πλείοσιν, εἴπερ ἔσται ἄμεσα διαστήματα. b9. for, were it so, if we take B as the common middle in virtue of which A inheres in C and D, clearly B would inhere in C and D through a second common middle, and this in turn would inhere in C and D through a third, so that between two terms an infinity of intermediates would fall — an impossibility. Thus it need not always be in virtue of a common middle term that a single attribute inheres in several subjects, since there must be immediate intervals.
ἐν μέντοι τῷ αὐτῷ γένει καὶ ἐκ τῶν αὐτῶν ἀτόμων ἀνάγκη τοὺς ὅρους εἶναι, εἴπερ τῶν καθ' αὑτὸ ὑπαρχόντων ἔσται τὸ κοινόν· οὐ γὰρ ἦν ἐξ ἄλλου γένους εἰς ἄλλο διαβῆναι τὰ δεικνύμενα. b15. Yet if the attribute to be proved common to two subjects is to be one of their essential attributes, the middle terms involved must be within one subject genus and be derived from the same group of immediate premisses; for we have seen that processes of proof cannot pass from one genus to another.
Φανερὸν δὲ καὶ ὅτι, ὅταν τὸ Α τῷ Β ὑπάρχῃ, εἰ μὲν ἔστι τι μέσον, ἔστι δεῖξαι ὅτι τὸ Α τῷ Β ὑπάρχει, καὶ στοιχεῖα τούτου ἔστι ταὐτὰ καὶ τοσαῦθ' ὅσα μέσα ἐστίν· αἱ γὰρ ἄμεσοι προτάσεις στοιχεῖα, ἢ πᾶσαι ἢ αἱ καθόλου. εἰ δὲ μὴ ἔστιν, οὐκέτι ἔστιν ἀπόδειξις, ἀλλ' ἡ ἐπὶ τὰς ἀρχὰς ὁδὸς αὕτη ἐστίν. b19. It is also clear that when A inheres in B, this can be demonstrated if there is a middle term. Further, the 'elements' of such a conclusion are the premisses containing the middle in question, and they are identical in number with the middle terms, seeing that the immediate propositions — or at least such immediate propositions as are universal — are the 'elements'. If, on the other hand, there is no middle term, demonstration ceases to be possible: we are on the way to the basic truths.
ὁμοίως δὲ καὶ εἰ τὸ Α τῷ Β μὴ ὑπάρχει, εἰ μὲν ἔστιν ἢ μέσον ἢ πρότερον ᾧ οὐχ ὑπάρχει, ἔστιν ἀπόδειξις, εἰ δὲ μή, οὐκ ἔστιν, ἀλλ' ἀρχή, καὶ στοιχεῖα τοσαῦτ' ἔστιν ὅσοι ὅροι· αἱ γὰρ τούτων προτάσεις ἀρχαὶ τῆς ἀποδείξεώς εἰσιν. καὶ ὥσπερ ἔνιαι ἀρχαί εἰσιν ἀναπόδεικτοι, ὅτι ἐστὶ τόδε τοδὶ καὶ ὑπάρχει τόδε τῳδί, οὕτω καὶ ὅτι οὐκ ἔστι τόδε τοδὶ οὐδ' ὑπάρχει τόδε τῳδί, ὥσθ' αἱ μὲν εἶναί τι, αἱ δὲ μὴ εἶναί τι ἔσονται ἀρχαί. b24. Similarly if A does not inhere in B, this can be demonstrated if there is a middle term or a term prior to B in which A does not inhere: otherwise there is no demonstration and a basic truth is reached. There are, moreover, as many 'elements' of the demonstrated conclusion as there are middle terms, since it is propositions containing these middle terms that are the basic premisses on which the demonstration rests; and as there are some indemonstrable basic truths asserting that 'this is that' or that 'this inheres in that', so there are others denying that 'this is that' or that 'this inheres in that' — in fact some basic truths will affirm and some will deny being.
Ὅταν δὲ δέῃ δεῖξαι, ληπτέον ὃ τοῦ Β πρῶτον κατηγορεῖται. ἔστω τὸ Γ, καὶ τούτου ὁμοίως τὸ Δ. καὶ οὕτως ἀεὶ βαδίζοντι οὐδέποτ' ἐξωτέρω πρότασις οὐδ' ὑπάρχον λαμβάνεται τοῦ Α ἐν τῷ δεικνύναι, ἀλλ' ἀεὶ τὸ μέσον πυκνοῦται, ἕως ἀδιαίρετα γένηται καὶ ἕν. ἔστι δ' ἓν ὅταν ἄμεσον γένηται, καὶ μία πρότασις ἁπλῶς ἡ ἄμεσος. b32. When we are to prove a conclusion, we must take a primary essential predicate — suppose it C — of the subject B, and then suppose A similarly predicable of C. If we proceed in this manner, no proposition or attribute which falls beyond A is admitted in the proof: the interval is constantly condensed until subject and predicate become indivisible, i.e. one. We have our unit when the premiss becomes immediate, since the immediate premiss alone is a single premiss in the unqualified sense of 'single'.
καὶ ὥσπερ ἐν τοῖς ἄλλοις ἡ ἀρχὴ ἁπλοῦν, τοῦτο δ' οὐ ταὐτὸ πανταχοῦ, ἀλλ' ἐν βάρει μὲν μνᾶ, ἐν δὲ μέλει δίεσις, ἄλλο δ' ἐν ἄλλῳ, οὕτως ἐν συλλογισμῷ τὸ ἓν (85a.) πρότασις ἄμεσος, ἐν δ' ἀποδείξει καὶ ἐπιστήμῃ ὁ νοῦς. b38. And as in other spheres the basic element is simple but not identical in all — in a system of weight it is the mina, in music the quarter-tone, and so on — so in syllogism the unit is an immediate premiss, and in the knowledge that demonstration gives it is an intuition.
ἐν μὲν οὖν τοῖς δεικτικοῖς συλλογισμοῖς τοῦ ὑπάρχοντος οὐδὲν ἔξω πίπτει, a2. In syllogisms, then, which prove the inherence of an attribute, nothing falls outside the major term.
ἐν δὲ τοῖς στερητικοῖς, ἔνθα μὲν ὃ δεῖ ὑπάρχειν, οὐδὲν τούτου ἔξω πίπτει, οἷον εἰ τὸ Α τῷ Β διὰ τοῦ Γ μή (εἰ γὰρ τῷ μὲν Β παντὶ τὸ Γ, τῷ δὲ Γ μηδενὶ τὸ Α)· πάλιν ἂν δέῃ ὅτι τῷ Γ τὸ Α οὐδενὶ ὑπάρχει, μέσον ληπτέον τοῦ Α καὶ Γ, καὶ οὕτως ἀεὶ πορεύσεται. a3. In the case of negative syllogisms on the other hand, (1) in the first figure nothing falls outside the major term whose inherence is in question; e.g. to prove through a middle C that A does not inhere in B the premisses required are, all B is C, no C is A. Then if it has to be proved that no C is A, a middle must be found between and C; and this procedure will never vary.
ἐὰν δὲ δέῃ δεῖξαι ὅτι τὸ Δ τῷ Ε οὐχ ὑπάρχει τῷ τὸ Γ τῷ μὲν Δ παντὶ ὑπάρχειν, τῷ δὲ Ε μηδενί [ἢ μὴ παντί], τοῦ Ε οὐδέποτ' ἔξω πεσεῖται· τοῦτο δ' ἐστὶν ᾧ δεῖ ὑπάρχειν. a7. (2) If we have to show that E is not D by means of the premisses, all D is C; no E, or not all E, is C; then the middle will never fall beyond E, and E is the subject of which D is to be denied in the conclusion.
ἐπὶ δὲ τοῦ τρίτου τρόπου, οὔτε ἀφ' οὗ δεῖ οὔτε ὃ δεῖ στερῆσαι οὐδέποτ' ἔξω βαδιεῖται. E a10. (3) In the third figure the middle will never fall beyond the limits of the subject and the attribute denied of it.
Postquam philosophus ostendit quod non contingit procedere in infinitum in demonstrationibus, hic inducit quaedam corollaria ex dictis. Et circa hoc duo facit: primo, ostendit quod necesse est accipere aliquas primas propositiones; secundo, quomodo illis primis sit utendum in demonstrationibus; ibi: cum autem oportet demonstrare et cetera. Circa primum duo facit: primo, ostendit quod necesse est devenire ad aliquod primum, quando unum de pluribus praedicatur; secundo, quando unum praedicatur de uno; ibi: manifestum autem et cetera. Circa primum quatuor facit: primo, proponit intentum; secundo, manifestat propositum; ibi: ut scaleno etc.; tertio, probat; ibi: sit autem b etc.; quarto, excludit quamdam obviationem; ibi: in eodem quidem genere et cetera. After showing that a process to infinity does not occur in demonstrations, the Philosopher here adduces certain corollaries from what has been established. In regard to this he does two things. First, he shows that it is necessary to accept certain first propositions. Secondly, how those first things are to be used in demonstrations (8032). Concerning the first he does two things. First, he shows that it is necessary to arrive at a first, when one thing is predicated of several. Secondly, when one thing is predicated of one (84b19). In regard to the first he does four things. First, he states the intended proposition. Secondly, he manifests the proposition (84b6). Thirdly, he proves it (84b9). Fourthly, he excludes an objection (84b15).
Dicit ergo primo, quod demonstratis praemissis, scilicet quod non sit procedere in infinitum in praedicationibus et demonstrationibus, manifestum est quod si aliquid praedicatur de duobus, puta a de c et d, ita scilicet quod unum eorum non praedicetur de altero, aut nullo modo, sicut animal praedicatur de homine et bove, quorum unum nullo modo de alio praedicatur, aut non de omni, puta animal praedicatur de homine et masculo, quorum neutrum de altero universaliter praedicatur; sic, inquam, se habentibus terminis, manifestum est quod non oportet quod illud praedicatum, quod de utroque praedicatur, insit utrique secundum aliquod commune, et hoc semper, idest in infinitum procedendo. He says therefore first (84b3) that having demonstrated the aforesaid, namely, that there is no process to infinity in predications and demonstrations, it is clear that if something is predicated of two things, say A of C and of D, such that one of them is not predicated of the other, i.e., either not at all, as animal is predicated of man and ox, but neither of them is predicated of the other in any way; or not in all cases, as animal is predicated of man and male, neither of which is universally predicated of the other: when the terms, I repeat, are thus related, it is clear that there is no need for that predicate, which is predicated of both, to be in them in virtue of something common to both and this again in virtue of something else, and so on to infinity.
Deinde cum dicit: ut scaleno et isosceli etc., manifestat propositum per exemplum. Sunt enim duae species trianguli, quarum una vocatur scalenon, vel triangulus gradatus, cuius sunt tria latera inaequalia; alia est isosceles, cuius sunt duo latera aequalia: unum autem horum non praedicatur de altero; utrique autem inest haec passio, habere tres angulos aequales duobus rectis. Inest autem hoc eis secundum aliquid commune, scilicet secundum quod uterque horum est figura quaedam, scilicet triangulus. Hoc autem non semper sic se habet, scilicet quod in infinitum conveniat secundum aliquid aliud; puta quod habere tres conveniat triangulo iterum secundum aliquid aliud, et sic in infinitum. Then (806) he cites an example to clarify what he is proposing. For there are two species of triangle, one of which is scalene (none of whose three sides is equal to any other) and the other is isosceles (having two sides that are equal). But neither of these species is predicated of the other, and yet this proper attribute of having three angles equal to two right angles is present in both. Furthermore, this attribute is present in them in virtue of something common, namely, that each is a certain figure, namely, a triangle. However, this process does not continue forever so that something would always belong to it in virtue of something else and so on to infinity, so that “having three” would belong to triangle in virtue of something else again, and so on to infinity.
Deinde cum dicit: sit enim b secundum etc., probat propositum et dicit: sit ita quod b praedicetur de c et de d secundum hoc commune, quod est a. Manifestum est ergo quod b erit in c et in d secundum illud commune, quod est a; et si iterum insit a secundum aliquod commune, et iterum illi communi secundum aliquid aliud, procedetur in infinitum in mediis. Sequitur igitur quod inter duo extrema, quae sunt c et b, cadant infiniti termini medii. Hoc autem est impossibile: ergo non necesse est, si idem insit pluribus, quod semper in infinitum insit eis secundum aliquid commune; quia necesse est quod deveniatur ad aliqua spatia immediata, idest ad aliquas immediatas praedicationes, quas appellat spatia, ut supra dictum est. Then (84b9) he proves his proposition and says: Let the case be that B is predicated of C and of D in virtue of this common feature A. It will then be clear that B will be in C and in D in virtue of that common feature which is A. Now if A in turn is in them in virtue of something common, which again is in them in virtue of some other common item, there will be an infinite process in the middles. It follows, therefore, that between the two extremes C and B there falls an infinitude of middle terms. But this is impossible. Therefore, if a same thing is in several things, it is not necessary that it be in them always in virtue of something else ad infinitum, because it is necessary to reach certain “immediate distances,” i.e., certain immediate predications, which he calls “distances,” as explained above.
Quantum igitur videtur ex hac probatione Aristotelis, non est suus intellectus, quod hoc non semper sit verum, quod quando aliquid praedicatur de pluribus, quae de se invicem non praedicantur, quod illud non insit illis pluribus secundum aliquid commune. Hoc enim verum est in omni quod praedicatur sicut passio: oportet enim si inest pluribus, quod insit eis secundum aliquid commune, licet forte illud sit innominatum, sicut supra dictum est cum de universali ageretur. Sed in illo communi non proceditur in infinitum, ut haec ratio inducta a philosopho evidenter probat. Si autem accipiatur aliquid, quod insit pluribus sicut genus speciebus, non semper oportebit aliquid prius accipere, secundum quod insit, puta si vivum insit homini et asino secundum aliquod prius, scilicet secundum animal; animali autem et plantae non inest secundum aliquod prius, quia haec sunt primae species corporis vivi, sive animati. As can be seen from this proof of Aristotle’s, it is not his understanding that it is not always true that when some item is predicated of several, which are not in turn predicated of one another, that item is not in the several in virtue of something common. For this is true in everything predicated as a proper attribute: for if it is in several, it is required that it be in them in virtue of something common which might even be nameless, as we explained above when we treated concerning the universal. But there is no infinite process in that which is common, as this reason introduced by the Philosopher clearly proves. However, if something be taken which is in several as a genus in its species, it will not always be necessary to find something prior in virtue of which it is in them. For example, “living” is in man and in ass in virtue of something prior, namely, in virtue of “animal,” but it is not in animal and plant in virtue of something prior, because these are the first species of “living,” i.e., of animate, body.
Deinde cum dicit: in eodem quidem genere etc., excludit quamdam obviationem. Posset enim aliquis dicere quod semper accipitur secundum aliquid commune, quia potest accipi commune alterius generis: puta si dicamus quod esse seipsum movens inest homini et asino secundum hoc commune, quod est animal, et secundum aliud commune, quod est habens quantitatem, vel habens colorem, aut aliquid aliud huiusmodi; quae possunt accipi in infinitum. Then (84b15) he excludes an objection. For someone could say that it is always taken in virtue of something common, in the sense that something from another genus might be found common to them: for example, if we say that “to be self-movent” is in man and in ass in virtue of this common feature which is “animal” and also in virtue of some other common feature such as “having quantity” or having color or other things of this sort that can be taken ad infinitum.
Et ad hoc excludendum dicit, quod necesse est terminos medios, qui accipiuntur, accipi ex eodem genere et ex eisdem atomis, idest indivisibilibus. Et appellat atomos, ipsos terminos extremos: inter quos oportet accipi medium, si illud commune, quod accipitur ut medius terminus, sit de numero eorum, quae praedicantur per se. Quare autem oporteat ex eodem genere assumere terminos medios, ostendit per hoc, quod sicut supra habitum est, non contingit demonstrationem transire de uno genere in aliud. In order to exclude this he says that it is required of the middle terms that they be taken from the same genus and “from the same atoms,” i.e., indivisibles. (By “atoms” he means those extreme terms between which the middle must be taken, if that common item which is taken as a middle term is to be numbered among things which are predicated per se). Why the middle terms must be taken from the same genus he shows from the fact that, as stated above, a demonstration does not cross from one genus into another.
Deinde cum dicit: manifestum autem est etc., ostendit quod necesse est devenire ad aliquod unum in praedicabilibus, in quibus praedicatur unum de uno. Et primo, in affirmativis; secundo, in negativis; ibi: similiter autem et si a et cetera. Then (84b19) he shows that it is necessary to arrive at a first among the predicables in which one thing is predicated of one. First, in affirmatives. Secondly, in negatives (84b24).
Dicit ergo primo, manifestum esse quod cum a praedicatur de b, si horum sit aliquod medium, quod illo medio uti possumus ad demonstrandum quod a sit in b: et haec sunt principia huiusmodi conclusionis. Et quaecunque accipiuntur ut media, sunt principia conclusionum mediatarum, quae per ea concluduntur. Nihil enim aliud sunt elementa, sive principia demonstrationum, quam propositiones immediatae. Et hoc dico vel omnes, vel universales: He says therefore first (84b19) that it is clear that when A is predicated of B, if there is a middle between them, we can use that middle to demonstrate that A is in B: and these are the principles of this kind of conclusion. And what things soever be taken as middles are the principles of the mediate conclusions which are concluded through them. For the “elements” or principles of demonstrations are none but immediate propositions: and I mean “either all such propositions or universal ones.”
quod quidem potest dupliciter intelligi. Uno modo, ut propositio universalis accipiatur secundum quod dividitur contra singularem. Nam species specialissima non praedicatur de singulari per aliquod medium. Unde haec propositio est immediata: Socrates est homo, non tamen est principium demonstrationis, quia demonstrationes non sunt de singularibus, cum eorum non sit scientia: et ita non omnis propositio immediata est demonstrationis principium, sed solum universalis. Alio modo potest intelligi secundum quod propositiones universales dicuntur propositiones communes in omnibus propositionibus alicuius scientiae, sicut, omne totum est maius sua parte: unde huiusmodi sunt simpliciter demonstrationum principia, et omnibus per se nota. Haec autem propositio, homo est animal, vel, isosceles est triangulus, non est principium demonstrationis in tota scientia, sed solum aliquarum particularium demonstrationum; neque etiam huiusmodi propositiones sunt omnibus per se notae. Now this can be understood in two ways: in one way so that universal proposition is taken as set off against singular. For it is not in virtue of a middle that the most special species is predicated of a singular. Hence this proposition, “Socrates is a man,” is immediate, although it is not a principle of demonstration, because demonstrations are not concerned with singulars, since there is no science of such. Consequently, not every immediate proposition, but only one that is universal, is a principle of demonstration. In another way it can be understood in the sense that of all the propositions of any science, the universal propositions are the common propositions, such as “Every whole is greater than its part.” Hence these are absolutely the principles of demonstrations and self-evident to all. But the proposition, “Man is an animal,” or “The isosceles is a triangle,” is not a principle of demonstration throughout the science but only for some particular demonstrations; hence such propositions are not self-evident to all.
Sic igitur si sit aliquod medium propositionis datae, erit demonstrare per aliquod medium, quousque deveniatur ad aliquod immediatum. Si vero non sit aliquod medium propositionis datae, non poterit demonstrari. Sed haec est via ad inveniendum prima principia demonstrationum, scilicet procedere a mediatis ad immediata resolvendo. Consequently, if there is a middle for a given proposition, one will demonstrate through a middle until something immediate is readied. But if there is no middle for a given proposition, it cannot be demonstrated. But ihis is the way to find the chief principles of demonstrations, namely, to proceed by analysis from mediates to immediates.
Deinde cum dicit: similiter autem erit etc., ostendit quod sit accipere primum in negativis; et dicit quod si a negetur de b, si sit accipere aliquod medium, a quo scilicet per prius removeatur a quam a b, tunc haec propositio, b non est a, erit demonstrabilis. Si autem non sit aliquod tale medium accipere, non erit haec propositio demonstrabilis, sed principium demonstrationis. Et tot erunt elementa, idest principia demonstrationis, quot erunt termini; ad quos scilicet statur, ut ultra non sit invenire medium. Propositiones enim quae fiunt ex huiusmodi terminis, sunt principia demonstrationis. Puta si c immediate praedicetur de b, et a immediate removeatur a b, aut praedicetur de eo immediate, b erit terminus ad quem ultimo pervenitur in mediis sumendis: unde utraque propositio erit immediata, et demonstrationis principium. Then (84b24) he shows that it is necessary to arrive at a first in negatives, saying that if A is denied of B, if there is a middle to be taken from which A is removed prior to being removed from B, then this proposition, “B is not A,” will be demonstrable. But if there is no such middle to be taken, that proposition will not be demonstrable but will be a principle of demonstration. And there are as many “elements,” i.e., principles of demonstrations as there are terms at which a stop is reached in such a way that there is no further middle to find. For propositions formed of such terms are principles of demonstrations. Thus, if C be predicated immediately of B, and A be immediately removed from B or immediately predicated of it, B is the final term reached among the middles to be taken; hence each proposition will be immediate and a principle of demonstration.
Et patet ex praemissis quod sicut sunt quaedam principia indemonstrabilia affirmativa, in quibus unum de alio praedicatur, significando quod hoc essentialiter est illud, sicut cum genus praedicatur de proxima specie, vel hoc sit in illo, sicut cum passio praedicatur de proprio et immediato subiecto; ita etiam sunt principia indemonstrabilia in negativis, negando vel essentiale praedicatum, vel etiam propriam passionem. Ex quo patet quod quaedam sunt principia demonstrationis ad demonstrandum conclusionem affirmativam, quam oportet concludere ex omnibus affirmativis; et quaedam sunt principia demonstrationis ad probandum conclusionem negativam, ad cuius illationem oportet assumere aliquam negativam. Furthermore, it is clear from the foregoing that just as there are certain indemonstrable affirmative principles in which one thing is predicated of one thing by signifying that this is essentially that (as when a genus is predicated of a proximate species) or that this is in that (as when a proper attribute is predicated of its proper and immediate subject), so there are indemonstrable principles in negatives, namely, by denying that something is an essential predicate or a proper attribute. From this it is evident that there are certain principles of demonstration for demonstrating an affirmative conclusion (which must be concluded from all affirmative propositions), and there are certain principles of demonstration for proving a negative conclusion (to infer which requires that something negative be taken).
Deinde cum dicit: cum autem oporteat etc., ostendit quomodo utendum sit primis propositionibus in demonstrando. Et primo, in demonstrationibus affirmativis; secundo, in negativis; ibi: in privativis autem et cetera. Circa primum tria facit: primo, ostendit qualiter oporteat sumere propositiones primas et immediatas in demonstrationibus; secundo, ostendit quomodo huiusmodi propositiones se habeant ad demonstrationes; ibi: et quemadmodum in aliis etc.; tertio, epilogat; ibi: in ostensivis quidem igitur et cetera. Then (84b32) he shows how first propositions are to be used in demonstrating. First, in affirmative demonstrations. Secondly, in negatives (85a3). In regard to the first he does three things. First, he shows how first and immediate propositions should be taken in demonstrations. Secondly, how such propositions are related to demonstrations (8038). Thirdly, he summarizes (85a2).
Dicit ergo primo, quod quando oportet demonstrare aliquam conclusionem affirmativam, puta, omne b est a, necesse est accipere aliquid quod primo praedicetur de b quam a, et de quo a etiam praedicetur, et sit illud c; et si iterum aliquid sit, de quo a per prius praedicetur quam de c, sic semper procedendo; sic nec propositio nec terminus significans aliquod ens accipietur in demonstrando extra ipsum a, quia oportebit quod a praedicetur de eo per se, et ita quod contineatur sub eo et non sit ab eo extrinsecum; sed oportebit semper condensare media. Et loquitur ad similitudinem hominum, qui videntur esse condensati sedentes in aliqua sede, quando inter sedentes nullus potest intercidere medius. Ita et media in demonstratione dicuntur densata, quando inter terminos acceptos nihil cadit medium. Et hoc est quod dicit, quod medium densatur quousque perveniatur ad hoc quod spatia fiant indivisibilia; idest, distantiae inter duos terminos sint tales, quod non possint dividi in plures huiusmodi distantias, sed sit unum spatium tantum. Et hoc contingit, quando propositio est immediata. Tunc enim vere est una propositio non solum actu, sed etiam potentia, quando est immediata. Si enim sit mediata, quamvis sit una in actu, quia unum praedicatur de uno, tamen est multa in potentia, quia accepto medio formantur duae propositiones. Sicut etiam linea, quae est una in actu in quantum est continua, est tamen multa in potentia, in quantum est divisibilis per punctum medium. Et ideo dicit quod propositio immediata est una sicut simplex indivisibilis. He says therefore first (84b32) that when one is required to demonstrate an affirmative conclusion, for example, “Every B is A,” it is first necessary to take something which is first predicated of B before A is, and of which A is also predicated, say C. Furthermore, if there is something of which A is predicated before A is predicated of C, and something ahead of that, and so on, then no proposition and no term signifying a being outside of A will be taken in the demonstration, because it will be necessary that A be predicated of it per se in such a way that it is contained under it and is not extraneous to it; but it will always be necessary to condense (crowd together] the middles. (Here he is speaking according to an example of men who appear crowded together when they are sitting on a bench and there is no room for anyone else to sit between any two who are seated; in like fashion, the middles in demonstrations are condensed when no middle falls between the terms taken). And this is his meaning when he says, “the middle is condensed until the spaces become indivisible,” i.e., the distances between two terms are such that they cannot be divided into several such distances, but there is only one space. And this occurs when a proposition is immediate. For it is then that a proposition is one, not only actually but potentially, when it is immediate. For if it is mediate, then although it is actually one (because one thing is predicated of one thing), it is in fact several in potency, because when the middle is taken, two propositions are formed. In the same way, a line which is actually one, inasmuch as it is continuous, is several in potency, inasmuch as it is divisible at an intermediate point. Hence he says that an immediate proposition is one as a simple indivisible.
Deinde cum dicit: et quemadmodum in aliis etc., ostendit quomodo se habeat propositio immediata ad demonstrationem. Ubi considerandum est quod, sicut habetur in X Metaphys., in quolibet genere oportet esse unum primum, quod est simplicissimum in genere illo, et mensura omnium quae sunt illius generis. Et quia mensura est homogenea mensurato, secundum diversitatem generum oportet esse huiusmodi prima indivisibilia diversa. Unde hoc non est idem in omnibus: sed in gravitate ponderum accipitur ut unum indivisibile uncia, sive mna, idest quoddam minimum pondus; quod tamen non est simplex omnino, quia quodlibet pondus est divisibile in minora pondera, sed accipitur ut simplex per suppositionem. In melodiis autem accipitur ut unum principium tonus, qui consistit in sesquioctava proportione, vel diesis, quae est differentia toni et semitonii. Et in diversis generibus sunt diversa principia indivisibilia. Then (8038) he shows the relation between an immediate proposition and a demonstration. Here we should note that, as it is stated in Metaphysics X, in every genus there must be one first thing which is the most simple in that genus and is the measure of all the things in that genus. And because a measure is homogeneous to the thing measured, such first indivisibles will vary according to the diversity of genera. Hence these will not be the same in all genera: but in regard to weights one indivisible, the ounce or mina, is taken as the minimum weight (even though it is not the absolute minimum, because every weight is further divisible into smaller weights, but it is taken as a minimum by supposition). Again, in melodies there is taken as the one principle “a tone,” which consists in a proportion of an octave and a half or a “diesis,” which is the difference between a tone and a semi-tone. Similarly, in diverse genera there are diverse indivisible principles.
Syllogismi autem principia sunt propositiones; unde oportet quod propositio simplicissima, quae est immediata, sit unum, quod est mensura syllogismorum. Demonstratio autem addit supra syllogismum quod facit scientiam. Comparatur autem intellectus ad scientiam sicut unum et indivisibile ad multa. Nam scientia est per decursum a principiis ad conclusiones; intellectus autem est absoluta et simplex acceptio principii per se noti. Unde intellectus respondet immediatae propositioni; scientia autem conclusioni, quae est propositio mediata. Sic igitur demonstrationis, in quantum est syllogismus, unum indivisibile est propositio immediata. Ex parte autem scientiae, quam causat, unum eius est intellectus. Now the principles of a syllogism are propositions; hence it is required that the most simple proposition, which is immediate, be the unit which is the measure of syllogisms. But demonstration has, over and above a syllogism, the added feature that it causes science. Now “understanding” and science are related as the indivisible unit is related to the many. For science is effected by going from principles to conclusions, whereas “understanding” is the absolute and simple acceptance of a self-evident principle. Hence “understanding” corresponds to the immediate proposition, and science to a conclusion, which is a mediate proposition. Consequently, the indivisible unit of a demonstration regarded as a syllogism is the immediate proposition. But on the part of the science which it causes, the unit is “understanding.”
Deinde cum dicit: in demonstrativis quidem etc., epilogando concludit quod supra ostensum est, scilicet quod in affirmativis syllogismis medium non cadit extra extrema. Then (85a2) he sums up and concludes what was established above, namely, that the middle in affirmative syllogisms does not fall outside the extremes.
Deinde cum dicit: in privativis autem ubi quidem etc., ostendit quomodo utendum sit propositionibus immediatis in syllogismis negativis. Et primo, in prima figura; secundo, in secunda; ibi: si vero oporteat monstrare etc.; tertio, in tertia; ibi: in tertio autem modo et cetera. Then (850) he shows how to use immediate propositions in negative syllogisms. First, in the first figure. Secondly, in the second (85a7). Thirdly, in the third (8500).
Dicit ergo primo, quod in negativis syllogismis, nihil mediorum acceptorum procedendo ad immediata cadit extra genus terminorum affirmativae propositionis in prima figura: puta si demonstrandum sit quod nullum b est a, et accipiatur medium c, tali existente syllogismo, nullum c est a; omne b est c; ergo nullum b est a. Si ergo oporteat iterum probare quod in nullo c sit a, oportet accipere medium ipsius c et a, quod scilicet praedicetur de c, et per consequens de b, et sic pertinebit ad genus terminorum affirmativae propositionis: et ita semper procedetur quod media accepta non cadent extra affirmativam propositionem; cadent tamen extra genus praedicati negativi, puta extra genus a. He says therefore first (85a3) that in negative syllogisms in the first figure, none of the middles taken while proceeding to immediates will fall outside the genus of the terms of an affirmative proposition; for example, if it is to be demonstrated that “No B is A,” and the middle taken is C, we have the following syllogism: “No C is A; but every B is C: therefore, No B is A.” Now if it should be necessary to prove that “No C is A,” one will be required to take a middle of C and A, i.e., one that will be predicated of C and consequently of B; thus it will belong to the genus of the terms of the affirmative proposition. And so it will always turn out that the middles taken will not fall outside the affirmative proposition, although they will fall outside the genus of the negative predicate, i.e., outside the genus of A.
Deinde cum dicit: si vero oporteat demonstrare etc., ostendit qualiter hoc se habeat in secunda figura; et dicit quod si oporteat demonstrare quod nullum e sit d, in secunda figura, accipiendo medium c, ut fiat talis syllogismus, omne d est c; nullum e est c, aut, quoddam e non est c; ergo nullum, vel, non omne e est d; nunquam medius terminus acceptus cadet extra e. Quia si oportebit iterum demonstrare quod nullum e est c, oportebit iterum accipere aliquod medium inter e et c; quia oportebit in secunda figura semper probare negativam; affirmativa enim in hac figura probari non potest. Unde sicut in prima figura media accepta semper accipiuntur ex parte propositionis affirmativae, ita oportet in secunda figura semper media accipi ex parte propositionis negativae. Then (85a7) he shows how it is in the second figure, saying that if one is required to demonstrate in the second figure that “No E is D,” by taking C as middle, so as to form the syllogism, “Every D is C; but No E is C or Some E is not C: therefore, No E is D or Not every E is D,” the middle term taken will never fall outside of E. For if it should be necessary to demonstrate that “no E is C,” one will have to find a middle between E and C, because in the second figure it will always be necessary to prove a negative, since an affirmative cannot be concluded in this figure. Hence, just as in the first figure the middles are always taken on the side of the affirmative proposition, so in the second figure the middles will always be taken on the side of the negative proposition.
Deinde cum dicit: in tertio autem modo etc., ostendit qualiter hoc se habeat in tertia figura; et dicit quod in tertia figura media accepta non erunt neque extra praedicatum, quod negatur, neque extra subiectum, a quo negatur. Et hoc ideo quia medium subiicitur affirmativae vel negativae utrique; unde si oportet accipere adhuc aliquod medium, oportet iterum illud medium subiici utrique affirmando vel negando; et sic media accepta nunquam accipientur neque extra praedicatum negatum nec extra subiectum, de quo negatur. Then (85a10) he shows how it is in the third figure, saying that in the third figure the middles which are taken will not be outside the predicate which is denied, nor outside the subject of which it is denied. The reason for this is that the middle is the subject in both propositions, whether affirmative or negative: hence if it is necessary to take yet another middle, it will again have to act as the subject of both, whether affirming or denying. And so the middles which are taken will never be taken outside the predicate which is denied, or outside the subject of which it is denied.

Lectio 37
Caput 24
Οὔσης δ' ἀποδείξεως τῆς μὲν καθόλου τῆς δὲ κατὰ μέρος, καὶ τῆς μὲν κατηγορικῆς τῆς δὲ στερητικῆς, ἀμφισβητεῖται ποτέρα βελτίων· ὡς δ' αὔτως καὶ περὶ τῆς ἀποδεικνύναι λεγομένης καὶ τῆς εἰς τὸ ἀδύνατον ἀγούσης ἀποδείξεως. a12. Since demonstrations may be either commensurately universal or particular, and either affirmative or negative; the question arises, which form is the better? And the same question may be put in regard to so-called 'direct' demonstration and reductio ad impossibile.
πρῶτον μὲν οὖν ἐπισκεψώμεθα περὶ τῆς καθόλου καὶ τῆς κατὰ μέρος· δηλώσαντες δὲ τοῦτο, καὶ περὶ τῆς δεικνύναι λεγομένης καὶ τῆς εἰς τὸ ἀδύνατον εἴπωμεν. a17. Let us first examine the commensurately universal and the particular forms, and when we have cleared up this problem proceed to discuss 'direct' demonstration and reductio ad impossibile.
Δόξειε μὲν οὖν τάχ' ἄν τισιν ὡδὶ σκοποῦσιν ἡ κατὰ μέρος εἶναι βελτίων. εἰ γὰρ καθ' ἣν μᾶλλον ἐπιστάμεθα ἀπόδειξιν βελτίων ἀπόδειξις (αὕτη γὰρ ἀρετὴ ἀποδείξεως), μᾶλλον δ' ἐπιστάμεθα ἕκαστον ὅταν αὐτὸ εἰδῶμεν καθ' αὑτὸ ἢ ὅταν κατ' ἄλλο (οἷον τὸν μουσικὸν Κορίσκον ὅταν ὅτι ὁ Κορίσκος μουσικὸς ἢ ὅταν ὅτι ἅνθρωπος μουσικός· ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων), ἡ δὲ καθόλου ὅτι ἄλλο, οὐχ ὅτι αὐτὸ τετύχηκεν ἐπιδείκνυσιν (οἷον ὅτι τὸ ἰσοσκελὲς οὐχ ὅτι ἰσοσκελὲς ἀλλ' ὅτι τρίγωνον), ἡ δὲ κατὰ μέρος ὅτι αὐτό· —εἰ δὴ βελτίων μὲν ἡ καθ' αὑτό, τοιαύτη δ' ἡ κατὰ μέρος τῆς καθόλου μᾶλλον, καὶ βελτίων ἂν ἡ κατὰ μέρος ἀπόδειξις εἴη. a20. The following considerations might lead some minds to prefer particular demonstration. (1) The superior demonstration is the demonstration which gives us greater knowledge (for this is the ideal of demonstration), and we have greater knowledge of a particular individual when we know it in itself than when we know it through something else; e.g. we know Coriscus the musician better when we know that Coriscus is musical than when we know only that man is musical, and a like argument holds in all other cases. But commensurately universal demonstration, instead of proving that the subject itself actually is x, proves only that something else is x — e.g. in attempting to prove that isosceles is x, it proves not that isosceles but only that triangle is x — whereas particular demonstration proves that the subject itself is x. The demonstration, then, that a subject, as such, possesses an attribute is superior. If this is so, and if the particular rather than the commensurately universal forms demonstrates, particular demonstration is superior.
ἔτι εἰ τὸ μὲν καθόλου μὴ ἔστι τι παρὰ τὰ καθ' ἕκαστα, ἡ δ' ἀπόδειξις δόξαν ἐμποιεῖ εἶναί τι τοῦτο καθ' ὃ ἀποδείκνυσι, καί τινα φύσιν ὑπάρχειν ἐν τοῖς οὖσι ταύτην, οἷον τριγώνου παρὰ τὰ τινὰ καὶ σχήματος παρὰ τὰ τινὰ καὶ ἀριθμοῦ παρὰ τοὺς τινὰς ἀριθμούς, βελτίων δ' ἡ περὶ ὄντος ἢ μὴ ὄντος καὶ δι' ἣν μὴ ἀπατηθήσεται ἢ δι' ἥν, a31. (2) The universal has not a separate being over against groups of singulars. Demonstration nevertheless creates the opinion that its function is conditioned by something like this — some separate entity belonging to the real world; that, for instance, of triangle or of figure or number, over against particular triangles, figures, and numbers. But demonstration which touches the real and will not mislead is superior to that which moves among unrealities and is delusory.
ἔστι δ' ἡ μὲν καθόλου τοιαύτη (προϊόντες γὰρ δεικνύουσιν ὥσπερ περὶ τοῦ ἀνὰ λόγον, οἷον ὅτι ὃ ἂν ᾖ τι τοιοῦτον ἔσται ἀνὰ λόγον ὃ οὔτε γραμμὴ οὔτ' ἀριθμὸς οὔτε στερεὸν οὔτ' ἐπίπεδον, (85b.) ἀλλὰ παρὰ ταῦτά τι)· —εἰ οὖν καθόλου μὲν μᾶλλον αὕτη, περὶ ὄντος δ' ἧττον τῆς κατὰ μέρος καὶ ἐμποιεῖ δόξαν ψευδῆ, χείρων ἂν εἴη ἡ καθόλου τῆς κατὰ μέρος. Now commensurately universal demonstration is of the latter kind: if we engage in it we find ourselves reasoning after a fashion well illustrated by the argument that the proportionate is what answers to the definition of some entity which is neither line, number, solid, nor plane, but a proportionate apart from all these. Since, then, such a proof is characteristically commensurate and universal, and less touches reality than does particular demonstration, and creates a false opinion, it will follow that commensurate and universal is inferior to particular demonstration.
Ἢ πρῶτον μὲν οὐδὲν μᾶλλον ἐπὶ τοῦ καθόλου ἢ τοῦ κατὰ μέρος ἅτερος λόγος ἐστίν; εἰ γὰρ τὸ δυσὶν ὀρθαῖς ὑπάρχει μὴ ᾗ ἰσοσκελὲς ἀλλ' ᾗ τρίγωνον, ὁ εἰδὼς ὅτι ἰσοσκελὲς ἧττον οἶδεν ᾗ αὐτὸ ἢ ὁ εἰδὼς ὅτι τρίγωνον. ὅλως τε, εἰ μὲν μὴ ὄντος ᾗ τρίγωνον εἶτα δείκνυσιν, οὐκ ἂν εἴη ἀπόδειξις, εἰ δὲ ὄντος, ὁ εἰδὼς ἕκαστον ᾗ ἕκαστον ὑπάρχει μᾶλλον οἶδεν. b4. We may retort thus. (1) The first argument applies no more to commensurate and universal than to particular demonstration. If equality to two right angles is attributable to its subject not qua isosceles but qua triangle, he who knows that isosceles possesses that attribute knows the subject as qua itself possessing the attribute, to a less degree than he who knows that triangle has that attribute. To sum up the whole matter: if a subject is proved to possess qua triangle an attribute which it does not in fact possess qua triangle, that is not demonstration: but if it does possess it qua triangle the rule applies that the greater knowledge is his who knows the subject as possessing its attribute qua that in virtue of which it actually does possess it.
εἰ δὴ τὸ τρίγωνον ἐπὶ πλέον ἐστί, καὶ ὁ αὐτὸς λόγος, καὶ μὴ καθ' ὁμωνυμίαν τὸ τρίγωνον, καὶ ὑπάρχει παντὶ τριγώνῳ τὸ δύο, οὐκ ἂν τὸ τρίγωνον ᾗ ἰσοσκελές, ἀλλὰ τὸ ἰσοσκελὲς ᾗ τρίγωνον, ἔχοι τοιαύτας τὰς γωνίας. ὥστε ὁ καθόλου εἰδὼς μᾶλλον οἶδεν ᾗ ὑπάρχει ἢ ὁ κατὰ μέρος. βελτίων ἄρα ἡ καθόλου τῆς κατὰ μέρος. Since, then, triangle is the wider term, and there is one identical definition of triangle — i.e. the term is not equivocal — and since equality to two right angles belongs to all triangles, it is isosceles qua triangle and not triangle qua isosceles which has its angles so related. It follows that he who knows a connexion universally has greater knowledge of it as it in fact is than he who knows the particular; and the inference is that commensurate and universal is superior to particular demonstration.
ἔτι εἰ μὲν εἴη τις λόγος εἷς καὶ μὴ ὁμωνυμία τὸ καθόλου, εἴη τ' ἂν οὐδὲν ἧττον ἐνίων τῶν κατὰ μέρος, ἀλλὰ καὶ μᾶλλον, ὅσῳ τὰ ἄφθαρτα ἐν ἐκείνοις ἐστί, b15. (2) If there is a single identical definition i.e. if the commensurate universal is unequivocal — then the universal will possess being not less but more than some of the particulars, inasmuch as it is universals which comprise the imperishable, particulars that tend to perish.
τὰ δὲ κατὰ μέρος φθαρτὰ μᾶλλον, ἔτι τε οὐδεμία ἀνάγκη ὑπολαμβάνειν τι εἶναι τοῦτο παρὰ ταῦτα, ὅτι ἓν δηλοῖ, οὐδὲν μᾶλλον ἢ ἐπὶ τῶν ἄλλων ὅσα μὴ τὶ σημαίνει ἀλλ' ἢ ποιὸν ἢ πρός τι ἢ ποιεῖν. εἰ δὲ ἄρα, οὐχ ἡ ἀπόδειξις αἰτία ἀλλ' ὁ ἀκούων. b18. (3) Because the universal has a single meaning, we are not therefore compelled to suppose that in these examples it has being as a substance apart from its particulars — any more than we need make a similar supposition in the other cases of unequivocal universal predication, viz. where the predicate signifies not substance but quality, essential relatedness, or action. If such a supposition is entertained, the blame rests not with the demonstration but with the hearer.
Postquam philosophus determinavit de syllogismo demonstrativo, hic agit de comparatione demonstrationum ad invicem. Et quia scientia ex demonstratione causatur, ideo dividitur pars ista in duas partes: in prima, agit de comparatione demonstrationis; in secunda, de comparatione scientiae; ibi: certior autem scientia est et cetera. Circa primum tria facit: primo, movet dubitationem de comparatione demonstrationum; secundo, dicit quo ordine sit procedendum; ibi: primo quidem igitur etc.; tertio, prosequitur dubitationes motas; ibi: videbitur quidem igitur et cetera. After determining about the demonstrative syllogism, the Philosopher now treats of the comparison of demonstrations one to another. And because science is caused by demonstration, his treatment is divided into two parts. In the first he treats of the comparison of demonstrations. In the second of the comparison of sciences (87a31) [L. 41]. In regard to the first he does three things. First, he raises a doubt concerning the comparison of demonstrations. Secondly, he lays down the order of procedure (85a17). Thirdly, he deals with the doubts raised (85a20).
Dicit ergo primo quod demonstratio tripliciter dividitur: uno enim modo dividitur in universalem et particularem; alio autem modo dividitur in categoricam et privativam, idest affirmativam et negativam; tertio modo dividitur in eam quae demonstrat ostensive, et in eam quae ducit ad impossibile. Est ergo quaestio in singulis divisionibus qualis potior sit. He says therefore first (85a12) that demonstration is divided in three ways: for in one way it is divided into universal and particular; in another way into categorical and privative, i.e., affirmative and negative; in a third way into that which demonstrates ostensively and that which leads to the impossible. In each division, therefore, the question arises as to which is the stronger.
Deinde cum dicit: primum quidem igitur etc., ostendit quo ordine sit agendum; et dicit quod primo agendum est de comparatione universalis et particularis demonstrationis. Et cum hoc fuerit ostensum, tunc dicemus et de demonstratione, quae demonstrat aliquid affirmative, et de ea quae demonstrat ad impossibile; utrum scilicet affirmativa sit potior, et utrum ea quae est ad impossibile sit potior. Then (85a17) he shows what order should be followed, saying that the comparison of universal to particular demonstration should be treated first. And when this has been done, we shall speak of demonstrations which demonstrate something affirmatively and of those which demonstrate to the impossible, namely, whether the affirmative is stronger and whether the demonstration to the impossible is stronger.
Deinde cum dicit: videbitur quidem igitur etc., prosequitur dubitationes propositas. Et primo, de comparatione demonstrationis particularis et universalis; secundo, de comparatione affirmativae et negativae; ibi: quod autem affirmativa etc.; tertio, de comparatione ostensivae et ducentis ad impossibile; ibi: quoniam autem categorica et cetera. Circa primum tria facit: primo, proponit rationes ad ostendendum quod particularis demonstratio sit potior quam universalis; secundo, solvit eas; ibi: aut primum quidem etc.; tertio, ponit rationes in contrarium; ibi: amplius si demonstratio et cetera. Then (85a20) he deals with the problems he has proposed. First, of the comparison of the particular with the universal. Secondly, of the comparison of the affirmative with the negative (86a32) [L. 39]. Thirdly, of the comparison of the ostensive with that which leads to the impossible (87a1) [L. 40]. Concerning the first he does three things. First, he proposes reasons to show that the particular demonstration is more powerful than the universal. Secondly, he solves them (85b4). Thirdly, he gives reasons to the contrary (85b22) [L. 38].
Circa primum ponit tres rationes, dicens quod quibusdam forte videbitur per has rationes immediate ponendas, quod particularis demonstratio sit dignior quam universalis. In regard to the first he sets forth three reasons, remarking that in virtue of these reasons to be set down forthwith, it will perhaps seem to some that a particular demonstration is of more value than a universal one.
Et prima ratio talis est. Illa demonstratio est potior, per quam maxime scimus. Et hoc sic probat, quia virtus demonstrationis est scire. Dicitur enim virtus uniuscuiusque id quod ultimum potest, sicut hominis qui potest ferre centum libras, virtus non est quod ferat decem, sed quod ferat centum, quod est ultimum suae potentiae, ut dicitur in I de coelo et mundo. Hoc autem est maximum quod potest facere demonstratio, scilicet quod faciat scire. Unde haec est virtus demonstrationis. Unumquodque autem tanto perfectius est, quanto magis attingit ad propriam virtutem, ut patet in VII Physic. Unde manifeste patet haec propositio, quod tanto est demonstratio potior, quanto magis facit scire. Assumit autem quod magis scimus unumquodque cum cognoscimus ipsum secundum se, quam quando cognoscimus ipsum secundum aliud: ut puta, cum cognoscimus de Corisco quod ipse Coriscus est musicus, magis hoc scimus quam si sciamus solum quod homo est musicus. Et ista etiam propositio simpliciter vera est, quia semper id quod est per se, prius est eo quod est per aliud et causa eius, ut habetur in VIII Physic. The first reason (85a20) is this: That demonstration is stronger through which we know best in a scientific way. And he proves it on the ground that the strength of a demonstration consists in knowing in a scientific way. For the most that a thing can do is called its strength: for the strength of a man able to carry 100 pounds is not that he can carry ten but that he can carry 100, which is the limit of his power, as it is stated in On the Heavens I. Now the most that a demonstration can do is to cause scientific knowledge; consequently, that is the strength of a demonstration. But a thing is more perfect to the extent that it attains the strength appropriate to it, as is clear from Physics VII. Hence this proposition is quite evident, namely, that the better a demonstration causes scientific knowledge, the stronger it is. (He assumes that we know a thing better when we know it according to itself than when we know it according to something else: thus we know more in regard to Coriscus when we know that Coriscus himself is a musician than when we merely know that some man is a musician). And this proposition is true, absolutely speaking, because that which is per se is always prior to and the cause of that which is through something else, as it is stated in Physics VIII.
Ex his autem subintelligitur conclusio, quod potior est demonstratio, quae facit scire aliquid secundum se, quam quae facit scire aliquid secundum aliud. Demonstratio autem universalis demonstrat aliquid et facit scire non secundum ipsum, sed secundum aliud, scilicet secundum universale; sicut quod triangulus duorum aequalium laterum, qui est isosceles, habet tres, non quia est isosceles, sed quia est triangulus. Particularis autem demonstratio demonstrat de aliqua re particulari secundum seipsam. Unde sequitur, secundum praemissa, quod particularis demonstratio sit potior quam universalis. From these facts the conclusion is gathered that a demonstration which makes one know something according to itself is stronger than one which makes one know something according to something else. But a universal demonstration demonstrates and makes one know something not according to itself, but according to something else, namely, according to the universal, as that a triangle with two equal sides, i.e., an isosceles, “has three,” not because it is isosceles but because it is a triangle. A particular demonstration, on the other hand, demonstrates about a particular thing according to itself. Hence according to this it follows that a particular demonstration is stronger than a universal.
Secundam et tertiam rationem ponit ibi: amplius si universale quidem etc., quae talis est. Universale non est aliquid praeter singularis, ut probatur in VII Metaphys. Demonstratio autem universalis facit opinionem, ex ipso modo suae demonstrationis, quod sit aliquid et quaedam natura in entibus; puta cum demonstrat aliquid de triangulo praeter particulares triangulos, et de figura praeter particulares figuras, et de numero praeter particulares numeros. Then (85a31) he gives a second reason. It is this: The universal is not something apart from singulars, as was proved in Metaphysics VII. But a universal demonstration leads one to think, from the very manner of its demonstration, that the universal is “something,” i.e., a certain nature in the realm of beings: for example, when it demonstrates something of triangle apart from particular triangles, and of number apart from particular numbers.
Praemissis autem duabus propositionibus addit alias duas. Nam primae propositioni, quae dicebat quod universale non est aliquid praeter singularia, addit hanc propositionem, quod potior est demonstratio, quae est de ente, quam illa quae est de non ente. Secundae autem propositioni, quae dicebat quod demonstratio universalis facit opinionem quod universale sit aliquid in rerum natura, addit aliam propositionem, scilicet quod demonstratio, quae non facit errare, est potior quam ea per quam erratur. To these two propositions he adds two others: to the first one, which stated that the universal is not something apart from the singulars, he adds this proposition, namely, that a demonstration concerned with being is stronger than one concerned with non-being. But to the second proposition, which stated that a universal demonstration leads one to think that a universal is something existing as a real nature, he adds another proposition, namely, that a demonstration which does not cause error is stronger than one which leads to error.
Et ostendit quod propter demonstrationem universalem erratur, quia procedentes secundum demonstrationem universalem demonstrant de aliquo universali sicut de quodam analogo; idest sicut de quodam communi, quod proportionaliter se habet ad multa, quasi sit aliquid commune, quod neque est linea, neque numerus, neque solidum, idest corpus, neque planum, idest superficies, sed aliquid praeter haec, idest ipsa quantitas universalis; vel, aliquid propter haec, idest quod necesse est ponere ad hoc quod ista habeant rationem quantitatis. Then he shows that one is led into error on account of a universal demonstration, because one who proceeds according to a universal demonstration demonstrates concerning some universal as of some analogue, i.e., as of something common which is referred to many things proportionally, as though there existed something common which is neither a line nor a number nor a solid, i.e., a body, nor a plane, i.e., a surface, but “something apart from these,” i.e., a universal quantity; or “something owing to them,” i.e., something which must be posited, if they are to have the formality of quantity.
Sic igitur secundum duo media, quasi duplici ratione concludit unam conclusionem, dicens quod si universalis demonstratio ita se habet, quod minus est de ente quam particularis, et magis facit opinionem falsam quam particularis; sequitur ex his duobus mediis quod universalis sit indignior quam particularis. Thus, therefore, in virtue of two middles, equivalent as it were to two arguments, he concludes to one conclusion, saying that if the universal demonstration is as described, i.e., is less an entity than the particular, and more likely to create a false opinion than the particular, it follows from these two middles that the universal ranks lower than the particular.
Deinde cum dicit: aut primum quidem nihil etc., solvit praedictas rationes per ordinem. Et primo primam, dicens quod primum quidem, idest secundum quod procedebat prima ratio, non habet aliam rationem in universali quam in particulari; quia utrobique invenitur secundum se et secundum aliud. Et manifestat quod in universali inveniatur secundum se. Habere enim tres angulos aequales duobus rectis non convenit isosceli secundum se, idest secundum quod isosceles est, sed secundum quod est triangulus; et ideo qui cognoscit quemdam triangulum habere tres, scilicet isoscelem, minus habet cognitionem de eo quod est per se, quam si cognoscat quod triangulus habet tres. Et hoc est universaliter dicendum, quod si aliquid non insit triangulo secundum quod est triangulus, et demonstretur de eo, quidquid sit illud, non erit vera demonstratio. Si autem insit ei secundum quod est triangulus, cognoscens in universali de triangulo secundum quod huiusmodi, perfectiorem cognitionem habet. Then (85b4) he solves these reasons in order. First, he solves the first one, saying that the first, i.e., the ground on which the first reason rests, is no different in the universal than in the particular, because in both cases we find something which is according to itself and something according to something else. And he shows that something which is according to itself is found in the universal. For “to have three angles equal to two right angles” does not belong to isosceles according to itself, i.e., precisely as isosceles, but according as it is a triangle. Consequently, one who knows that a certain triangle, namely, the isosceles, “has three,” has less knowledge of that which is per se than if he knew that a triangle “has three.” And it must be admitted universally that if there be any characteristic which does not belong to triangle as triangle, but that characteristic is nevertheless demonstrated of it, the demonstration will not be true. But if it is in it precisely as it is a triangle, then by knowing it in a universal way of triangle precisely as of triangle, he has a more perfect knowledge.
Ex his igitur concludit quamdam conditionalem, in cuius antecedenti tria ponuntur. Quorum unum est quod triangulus sit in plus quam isosceles; secundum est quod triangulus praedicetur de isoscele et aliis secundum eamdem rationem et non aequivoce; tertium est quod habere tres angulos aequales duobus rectis insit omni triangulo. Et his tribus suppositis, consequens est quod habere tres non conveniat triangulo in quantum est isosceles, sed e converso. From these facts, therefore, he concludes a conditional statement in whose antecedent three things are placed: one is that “triangle” is in more things than isosceles is; the second is that “triangle” is predicated of isosceles and of those others according to the same formality and not equivocally; the third is that “having three angles equal to two right angles” is present in every triangle. With these three suppositions, the consequent is that “the having of three” does not belong to triangle precisely as it is isosceles, but vice versa.
Apposuit autem prima duo in antecedente, quia si triangulus non esset in plus, vel si aequivoce praedicaretur de pluribus, non compararetur ad isoscelem sicut universale ad particulare. Tertium autem addit, quia si habere tres non conveniret omni triangulo, non conveniret ei in quantum triangulus, sed in quantum aliquis triangulus. Sicut hoc ipsum quod est habere tres, quia non convenit omni figurae, non convenit figurae in quantum est figura, sed in quantum est figura quaedam, quae est triangulus. Now the first two were put in the antecedent on the ground that if “triangle” were not the wider term or if it were predicated equivocally of its several inferiors, it would not be compared to “isosceles” as universal to particular. But he added the third, because if “having three” did not belong to every triangle, it would not belong to isosceles precisely as triangle, but in virtue of being a certain triangle; just as the characteristic of “having three” does not belong to every figure precisely as figure, but because it is a certain figure which is a triangle.
Ex his igitur concludit oppositum eius quod obiectio supponebat, scilicet quod ille qui scit in universali, magis cognoscit rem per se et in quantum huiusmodi, quam ille qui cognoscit in particulari. Et ex hoc ulterius concludit principale propositum, scilicet quod potior sit demonstratio universalis quam particularis. From these statements, therefore, he concludes to the opposite of that which the objection presupposed, namely, he concludes that one who knows in the universal knows the thing per se and as such in a better way than one who knows in the particular. And from this he further concludes his chief proposition, namely, that universal demonstration is stronger than particular.
Secundam rationem solvit ibi: amplius si quidem sit quaedam etc., et dicit quod si universale praedicatur de pluribus secundum unam rationem et non aequivoce, universale quantum ad id quod rationis est, idest quantum ad scientiam et demonstrationem, non erit minus ens quam particulare sed magis: quia incorruptibile est magis ens quam corruptibile; ratio autem universalis est incorruptibilis; particularia autem sunt corruptibilia, quibus accidit corruptio secundum principia individualia, non secundum rationem speciei, quae communis est omnibus et conservatur per generationem. Sic igitur quantum ad id quod rationis est, universalia magis sunt entia quam particularia. Quantum vero ad naturalem subsistentiam, particularia magis sunt entia, quae dicuntur primae et principales substantiae. Then (85b15) he answers the second reason, saying that if the universal is predicated of several according to one formality and not equivocally, the universal, so far as its formality is concerned, i.e., so far as science and demonstration are concerned, will not be less of an entity than the particular, but more. For the incorruptible is more a being than the corruptible; but the formality of a universal is incorruptible, whereas particulars are corruptible, in that they are subject to corruption as to their individual principles, although not as to the formality of the species, which is common to all and conserved through generation. And so in regard to that which pertains to their formality, the universals are beings to a higher degree than particulars. Nevertheless, in regard to natural subsistence the particulars, which are called the first and chief substances, have more being.
Tertiam rationem solvit ibi: amplius neque una necessitas etc., et dicit quod quamvis in propositionibus vel demonstrationibus universalibus significetur aliquid unum secundum se, puta triangulus, nulla tamen necessitas est quod propter hoc aliquis opinetur quod triangulus sit quoddam unum praeter multa; sicut in his quae non significant substantiam, sed aliquod genus accidentis, cum ea absolute significamus, puta dicendo albedinem, vel paternitatem, non propter hoc cognoscimus aliquem opinari quod huiusmodi sint praeter substantiam. Intellectus enim potest intelligere aliquid eorum, quae sunt coniuncta secundum rem, sine hoc quod actu intelligat aliud, nec tamen intellectus est falsus. Sicut si album sit musicum, possum intelligere album et aliquid attribuere ei et demonstrare de ipso, puta quod sit disgregativum visus, nulla consideratione habita de musico. Si tamen aliquis intelligeret album non esse musicum, esset intellectus falsus. Sic igitur cum dicimus aut intelligimus quod albedo est color, nulla mentione facta de subiecto, verum dicimus. Esset autem falsum si diceremus, albedo, quae est color, non est in subiecto. Et similiter cum dicimus homo est animal, vere loquimur, non facta mentione de aliquo particulari homine. Esset tamen falsum si diceremus, homo est animal, existens separatus a particularibus hominibus. Si autem hoc est, ergo sequitur quod demonstratio non sit causa falsae opinionis, qua quis opinatur universale esse extra singularia, sed audiens, qui male intelligit. Unde ex hoc nihil derogatur universali demonstrationi. Then (85b18) he answers the third reason, saying that although in propositions or demonstrations that are universal, something which is one according to itself, say “triangle,” is signified, nevertheless there is no need for anyone to suppose on this account that “triangle” is some one thing apart from the many, any more than there is need in the case of things which do not signify substance but some genus of accident (when we signify them absolutely, as when we say “whiteness” or “fatherhood”), to suppose that such things exist apart from the substance. For the intellect is able to understand one of the things which are joined in reality without actually thinking of some other one; yet the intellect is not false. Thus, if something white is musical, I am able to think of the white and attribute something to it and demonstrate something of it, say, that it disperses the vision, without adverting at all to musical. However, if one were to understand that the white one is not musical, then the intellect would be false. And so when we say or understand that whiteness is a color, no mention being made of the subject, we are saying something true. But it would be false, were we to say that the whiteness which is a color is not in a subject. In like fashion, when we say that every man is an animal, we are speaking truly, even though no particular man is mentioned. But it would be false were we to say that man is “an animal existing apart from particular men.” And if this is so, it follows that demonstration is not the cause of the false opinion according to which someone supposes that the universal is some thing outside the singulars, but it is rather the hearer who understands incorrectly. Hence this does not detract at all from universal demonstration.

Lectio 38
Caput 24 cont.
Ἔτι εἰ ἡ ἀπόδειξις μέν ἐστι συλλογισμὸς δεικτικὸς αἰτίας καὶ τοῦ διὰ τί, τὸ καθόλου δ' αἰτιώτερον (ᾧ γὰρ καθ' αὑτὸ ὑπάρχει τι, τοῦτο αὐτὸ αὑτῷ αἴτιον· τὸ δὲ καθόλου πρῶτον· αἴτιον ἄρα τὸ καθόλου)· ὥστε καὶ ἡ ἀπόδειξις βελτίων· μᾶλλον γὰρ τοῦ αἰτίου καὶ τοῦ διὰ τί ἐστιν. b22. (4) Demonstration is syllogism that proves the cause, i.e. the reasoned fact, and it is rather the commensurate universal than the particular which is causative (as may be shown thus: that which possesses an attribute through its own essential nature is itself the cause of the inherence, and the commensurate universal is primary; hence the commensurate universal is the cause). Consequently commensurately universal demonstration is superior as more especially proving the cause, that is the reasoned fact.
Ἔτι μέχρι τούτου ζητοῦμεν τὸ διὰ τί, καὶ τότε οἰόμεθα εἰδέναι, ὅταν μὴ ᾖ ὅτι τι ἄλλο τοῦτο ἢ γινόμενον ἢ ὄν· τέλος γὰρ καὶ πέρας τὸ ἔσχατον ἤδη οὕτως ἐστίν. οἷον τίνος ἕνεκα ἦλθεν; ὅπως λάβῃ τἀργύριον, τοῦτο δ' ὅπως ἀποδῷ ὃ ὤφειλε, τοῦτο δ' ὅπως μὴ ἀδικήσῃ· καὶ οὕτως ἰόντες, ὅταν μηκέτι δι' ἄλλο μηδ' ἄλλου ἕνεκα, διὰ τοῦτο ὡς τέλος φαμὲν ἐλθεῖν καὶ εἶναι καὶ γίνεσθαι, καὶ τότε εἰδέναι μάλιστα διὰ τί ἦλθεν. b28. (5) Our search for the reason ceases, and we think that we know, when the coming to be or existence of the fact before us is not due to the coming to be or existence of some other fact, for the last step of a search thus conducted is eo ipso the end and limit of the problem. Thus: 'Why did he come?' 'To get the money — wherewith to pay a debt — that he might thereby do what was right.' When in this regress we can no longer find an efficient or final cause, we regard the last step of it as the end of the coming — or being or coming to be — and we regard ourselves as then only having full knowledge of the reason why he came.
εἰ δὴ ὁμοίως ἔχει ἐπὶ πασῶν τῶν αἰτιῶν καὶ τῶν διὰ τί, ἐπὶ δὲ τῶν ὅσα αἴτια οὕτως ὡς οὗ ἕνεκα οὕτως ἴσμεν μάλιστα, καὶ ἐπὶ τῶν ἄλλων ἄρα τότε μάλιστα ἴσμεν, ὅταν μηκέτι ὑπάρχῃ τοῦτο ὅτι ἄλλο. ὅταν μὲν οὖν γινώσκωμεν ὅτι τέτταρσιν αἱ ἔξω ἴσαι ὅτι ἰσοσκελές, ἔτι λείπεται διὰ (86a.) τί τὸ ἰσοσκελές—ὅτι τρίγωνον, καὶ τοῦτο, ὅτι σχῆμα εὐθύγραμμον. εἰ δὲ τοῦτο μηκέτι διότι ἄλλο, τότε μάλιστα ἴσμεν. καὶ καθόλου δὲ τότε· ἡ καθόλου ἄρα βελτίων. If, then, all causes and reasons are alike in this respect, and if this is the means to full knowledge in the case of final causes such as we have exemplified, it follows that in the case of the other causes also full knowledge is attained when an attribute no longer inheres because of something else. Thus, when we learn that exterior angles are equal to four right angles because they are the exterior angles of an isosceles, there still remains the question 'Why has isosceles this attribute?' and its answer 'Because it is a triangle, and a triangle has it because a triangle is a rectilinear figure.' If rectilinear figure possesses the property for no further reason, at this point we have full knowledge — but at this point our knowledge has become commensurately universal, and so we conclude that commensurately universal demonstration is superior.
Ἔτι ὅσῳ ἂν μᾶλλον κατὰ μέρος ᾖ, εἰς τὰ ἄπειρα ἐμπίπτει, ἡ δὲ καθόλου εἰς τὸ ἁπλοῦν καὶ τὸ πέρας. ἔστι δ', ᾗ μὲν ἄπειρα, οὐκ ἐπιστητά, ᾗ δὲ πεπέρανται, ἐπιστητά. ᾗ ἄρα καθόλου, μᾶλλον ἐπιστητὰ ἢ ᾗ κατὰ μέρος. ἀποδεικτὰ ἄρα μᾶλλον τὰ καθόλου. τῶν δ' ἀποδεικτῶν μᾶλλον μᾶλλον ἀπόδειξις· ἅμα γὰρ μᾶλλον τὰ πρός τι. βελτίων ἄρα ἡ καθόλου, ἐπείπερ καὶ μᾶλλον ἀπόδειξις. a3. (6) The more demonstration becomes particular the more it sinks into an indeterminate manifold, while universal demonstration tends to the simple and determinate. But objects so far as they are an indeterminate manifold are unintelligible, so far as they are determinate, intelligible: they are therefore intelligible rather in so far as they are universal than in so far as they are particular. From this it follows that universals are more demonstrable: but since relative and correlative increase concomitantly, of the more demonstrable there will be fuller demonstration. Hence the commensurate and universal form, being more truly demonstration, is the superior.
Ἔτι εἰ αἱρετωτέρα καθ' ἣν τοῦτο καὶ ἄλλο ἢ καθ' ἣν τοῦτο μόνον οἶδεν· ὁ δὲ τὴν καθόλου ἔχων οἶδε καὶ τὸ κατὰ μέρος, οὗτος δὲ τὴν καθόλου οὐκ οἶδεν· ὥστε κἂν οὕτως αἱρετωτέρα εἴη. Ἔτι δὲ ὧδε. a11. (7) Demonstration which teaches two things is preferable to demonstration which teaches only one. He who possesses commensurately universal demonstration knows the particular as well, but he who possesses particular demonstration does not know the universal. So that this is an additional reason for preferring commensurately universal demonstration. And there is yet this further argument:
τὸ γὰρ καθόλου μᾶλλον δεικνύναι ἐστὶ τὸ διὰ μέσου δεικνύναι ἐγγυτέρω ὄντος τῆς ἀρχῆς. ἐγγυτάτω δὲ τὸ ἄμεσον· τοῦτο δ' ἀρχή. εἰ οὖν ἡ ἐξ ἀρχῆς τῆς μὴ ἐξ ἀρχῆς, ἡ μᾶλλον ἐξ ἀρχῆς τῆς ἧττον ἀκριβεστέρα ἀπόδειξις. ἔστι δὲ τοιαύτη ἡ καθόλου μᾶλλον· κρείττων <ἄρ' > ἂν εἴη ἡ καθόλου. οἷον εἰ ἔδει ἀποδεῖξαι τὸ Α κατὰ τοῦ Δ· μέσα τὰ ἐφ' ὧν Β Γ· ἀνωτέρω δὴ τὸ Β, ὥστε ἡ διὰ τούτου καθόλου μᾶλλον. Ἀλλὰ τῶν μὲν εἰρημένων ἔνια λογικά ἐστι· a14. (8) Proof becomes more and more proof of the commensurate universal as its middle term approaches nearer to the basic truth, and nothing is so near as the immediate premiss which is itself the basic truth. If, then, proof from the basic truth is more accurate than proof not so derived, demonstration which depends more closely on it is more accurate than demonstration which is less closely dependent. But commensurately universal demonstration is characterized by this closer dependence, and is therefore superior. Thus, if A had to be proved to inhere in D, and the middles were B and C, B being the higher term would render the demonstration which it mediated the more universal. Some of these arguments, however, are dialectical.
μάλιστα δὲ δῆλον ὅτι ἡ καθόλου κυριωτέρα, ὅτι τῶν προτάσεων τὴν μὲν προτέραν ἔχοντες ἴσμεν πως καὶ τὴν ὑστέραν καὶ ἔχομεν δυνάμει, οἷον εἴ τις οἶδεν ὅτι πᾶν τρίγωνον δυσὶν ὀρθαῖς, οἶδέ πως καὶ τὸ ἰσοσκελὲς ὅτι δύο ὀρθαῖς, δυνάμει, καὶ εἰ μὴ οἶδε τὸ ἰσοσκελὲς ὅτι τρίγωνον· ὁ δὲ ταύτην ἔχων τὴν πρότασιν τὸ καθόλου οὐδαμῶς οἶδεν, οὔτε δυνάμει οὔτ' ἐνεργείᾳ. a2l. Some of these arguments, however, are dialectical. The clearest indication of the precedence of commensurately universal demonstration is as follows: if of two propositions, a prior and a posterior, we have a grasp of the prior, we have a kind of knowledge — a potential grasp — of the posterior as well. For example, if one knows that the angles of all triangles are equal to two right angles, one knows in a sense — potentially — that the isosceles' angles also are equal to two right angles, even if one does not know that the isosceles is a triangle; but to grasp this posterior proposition is by no means to know the commensurate universal either potentially or actually.
καὶ ἡ μὲν καθόλου νοητή, ἡ δὲ κατὰ μέρος εἰς αἴσθησιν τελευτᾷ. a28. Moreover, commensurately universal demonstration is through and through intelligible; particular demonstration issues in sense-perception.
Caput 25
Ὅτι μὲν οὖν ἡ καθόλου βελτίων τῆς κατὰ μέρος, τοσαῦθ' ἡμῖν εἰρήσθω· The preceding arguments constitute our defence of the superiority of commensurately universal to particular demonstration.
Postquam philosophus solvit rationes, quae sunt ad partem falsam, hic inducit rationes ad partem veram, scilicet ad ostendendum quod demonstratio universalis sit potior. Et circa hoc ponit septem rationes, annectens eas praemissis solutionibus, ex quibus etiam propositum concludi potest, ut supra patuit. After answering the arguments which favored the false side, the Philosopher now introduces arguments for the true side, namely, to show that universal demonstration is the more powerful. In regard to this he sets down seven reasons, adding them to the previous solutions from which the proposition can also be concluded, as was clear from the above.
Prima ergo ratio talis est. Demonstratio est syllogismus ostendens causam et propter quid: sic enim contingit scire, sicut supra habitum est. Sed universale est magis tale quam particulare. Iam enim ostensum est in prima solutione quod universali magis inest per se aliquid quam particulari. Illud autem cui inest aliquid per se, est causa eius: subiectum enim est causa propriae passionis, quae ei per se inest. Universale autem est primum cui propria passio inest, ut ex supra dictis patet: unde patet quod proprie causa est id quod est universale. Ex quo concludit propositum, scilicet quod demonstratio universalis sit dignior, utpote magis declarans causam et propter quid. The first reason (85b22), is this: Demonstration is a syllogism showing the cause and propter quid: for this is the way scientific knowing takes place, as stated above. But the universal does this better than the particular. For as was shown in the first solution, something is per se in the universal more than in the particular. But that in which something is per se is the cause of this something: for the subject is the cause of the proper attribute which is in it per se. However, the first thing in which the proper attribute is present is the universal, as is clear from what has been stated above. Hence it is plain that, properly speaking, the cause is that which is universal. From this he concludes the proposition, namely, that universal demonstration is more valuable as better declaring the cause and propter quid.
Secundam rationem ponit ibi: amplius usque ad hoc etc., et sumitur haec ratio a causis finalibus. Ubi considerandum est quod aliquid est finis alterius et quantum ad fieri, et quantum ad esse: quantum ad fieri quidem, sicut generatio est propter formam; quantum ad esse autem, sicut domus est propter habitationem. Then (85b28) he gives the second reason. This reason is taken from the final causes. Here it should be pointed out that something is the end of another thing both in regard to becoming and in regard to being: in regard to becoming, as generation is for the sake of form; in regard to being, as a house is for the sake of habitation.
Dicit ergo quod usque ad illum terminum quaerimus propter quid fiat aliquid, aut propter quid sit aliquid, quousque non sit aliquid aliud assignare quam hoc ad quod perventum est, propter quod fiat vel sit illud, de quo quaeritur propter quid. Et quando hoc invenimus, tunc opinamur nos scire propter quid; et hoc ideo quia illud quod iam sic est ultimum ut non sit aliquid aliud ulterius quaerendum, est id quod est vere finis et terminus, qui quaeritur cum quaerimus propter quid. Et ponit exemplum, puta si quaeramus cuius causa aliquis venit, et respondeatur, ut accipiat argentum: hoc autem propter quid? Ut scilicet reddat debitum: et hoc propter hoc aliud, ut scilicet non iniuste agat. Et sic semper procedentes, quando iam non erit amplius propter aliquid aliud sicut propter finem, puta cum pervenerimus ad ultimum finem, qui est beatitudo, dicemus quod propter hoc venit sicut propter finem. Et similiter est in omnibus aliis, quae sunt vel fiunt propter finem, et quando ad hoc pervenerimus, sciemus propter quid venit. He says, therefore, that we search for the “reason why” something is done or why something is, until we reach the point where there is nothing further to assign, beyond the point reached, as the reason why that comes to be or is, whose reason why is sought. And when we find this we think that we know the propter quid: the reason being that that which is ultimate in this way, i.e., leaving nothing further to be sought, is truly the end and terminus which is being sought when we seek the propter quid. And he gives this example: if we should ask why someone went out, and the answer is given, “to get money,” and this in order to pay a debt, and “this for this other reason,” namely, lest he be guilty of injustice; and we continue in this way until there is nothing further for the sake of which as for an end—as when we arrive at the ultimate end, which is happiness—we will say that it was for this, as for an end, that he went. And it is the same in all other things that are or are done for the sake of an end: when we arrive at it, we will know the reason why he went.
Si igitur ita se habet in aliis causis sicut in causis finalibus, quod tunc maxime scimus quando ad ultimum fuerit perventum; ergo in aliis tunc maxime sciemus, quando perveniemus ad hoc, quod hoc inest huic non amplius propter aliquid aliud: et hoc contingit cum pervenerimus ad universale. Now if things are so related in the other causes as they are in the final cause, namely, that we know fully when the last one has been reached, then we will know best in the others, when we shall have arrived at the fact that “this is in this” no longer because of something further. And this happens when we shall have reached the universal.
Et hoc manifestat in tali exemplo. Si enim quaeramus de isto triangulo particulari, quare anguli eius extrinseci sunt aequales quatuor rectis; respondebitur quod hoc contingit huic triangulo quia est isosceles; isosceles autem est talis quia est triangulus; triangulus autem est talis quia est figura rectilinea talis. Si ergo amplius non possit procedi, tunc maxime scimus: hoc autem est, quando pervenitur ad universale. Ergo universalis demonstratio potior est particulari. To elucidate this he offers the following example: If we should ask concerning this particular triangle, why its exterior angles are equal to four right angles, the answer will be that this happens to this triangle because it is isosceles; and it is so for the isosceles, because it is a triangle; and it is so for a triangle, because it is such and such a rectilinear figure. If no further step can be taken, then we know in the best way. But this happens when the universal has been reached. Therefore the universal demonstration is stronger than the particular.
Tertiam rationem ponit ibi: amplius quantocumque etc., et dicit quod quanto magis proceditur versus particularia, tanto magis itur versus infinitum; quia, ut dicitur in III Physic., infinitum congruit materiae, quae est individuationis principium. Sed quanto magis proceditur versus universale, tanto magis itur in aliquid simplex et in ipsum finem; quia ratio universalis sumitur ex parte formae, quae est simplex, et habet rationem finis, in quantum terminat infinitatem materiae. Manifestum est autem quod infinita in quantum huiusmodi non sunt scibilia, sed in quantum aliqua sunt finita in tantum sunt scibilia; quia materia non est principium cognoscendi rem, sed magis forma. Manifestum est ergo quod universalia sunt magis scibilia quam particularia. Ergo etiam sunt magis demonstrabilia, quia demonstratio est syllogismus faciens scire. Sed magis demonstrabilium est potior demonstratio: simul enim intenduntur ea, quae dicuntur ad invicem; demonstratio autem ad demonstrabile dicitur. Et sic cum universalia sint magis demonstrabilia, demonstratio universalis erit potior. Then (86a3) he gives the third reason, saying that the more one proceeds toward particulars, the nearer one gets to what is infinite; because, as it is stated in Physics III, the infinite is appropriate to matter which is the principle of individuation. On the other hand, the more one proceeds toward the universal, the nearer he gets to what is simple and to the end itself; because the universal reason is taken on the part of the form, ic is simple and which has the character of an end insofar as it terminates the infinitude of matter. Now it is obvious that infinite things as such are not scientifically knowable; rather to the extent that they are finite, to that extent are they knowable; because the principle of knowing a thing is not the matter but the form. Therefore, it is obvious that universals are scientifically more knowable than particulars. Consequently, they are also more demonstrable, because a demonstration is a syllogism that makes one know scientifically. But a demonstration of the more demonstrable is more powerful; for things which are described one in terms of the other grow apace. But demonstration is described in terms of the demonstrable. Consequently, since universals are more demonstrable, universal demonstration will be more powerful.
Quartam rationem ponit ibi: amplius si magis praeponenda etc., quae talis est. Cum demonstrationis finis sit scientia, quanto demonstratio plura facit scire, tanto potior est. Et hoc est quod dicit, quod magis praeferenda est demonstratio, secundum quam homo cognoscit hoc et aliud, quam illa secundum quam homo cognoscit unum solum. Sed ille qui habet cognitionem de universali, cognoscit etiam particulare, dummodo sciat quod sub universali contineatur particulare; sicut qui cognoscit omnem mulam esse sterilem, scit hoc animal, quod cognoscit esse mulam, esse sterile: sed ille qui cognoscit particulare, non propter hoc cognoscit universale. Non enim si cognosco hanc mulam esse sterilem, propter hoc cognosco omnem mulam esse sterilem. Relinquitur ergo quod demonstratio universalis, per quam cognoscitur universale et particulare, sit potior quam particularis, per quam cognoscitur solum particulare. Then (86a11) he gives the fourth reason and it is this: Since the end of demonstration is scientific knowledge, the more things a demonstration enables one to know, the more powerful it is. And this is what he states, namely, that a demonstration, according to which a man knows one thing and something additional, is preferable to one according to which a man knows only the one. But a man who has knowledge of the universal knows also the particular, so long as he knows that the particular is contained under the universal. Thus, one who knows that every mule is sterile, knows that this animal, which he recognizes to be a mule, is sterile. But one who knows the particular does not on that account know the universal. For if I know that this mule is sterile, I do not on that account know that every mule is sterile. It remains, therefore, that universal demonstration, through which the universal is known, is stronger than particular, through which the particular is known.
Quintam rationem ponit ibi: amplius autem et sic etc., quae talis est. Quanto medium demonstrationis est propinquius primo principio, tanto demonstratio est potior. Et hoc probat, quia si illa demonstratio, quae procedit ex principio immediato, est certior ea quae non procedit ex principio immediato, sed ex mediato, necesse est quod quanto aliqua demonstratio procedit ex medio propinquiori principio immediato, tanto sit potior. Sed universalis demonstratio procedit ex medio propinquiori principio, quod est propositio immediata. Et hoc manifestat in terminis. Si enim oporteat demonstrare a, quod est universalissimum, de d, quod est particularissimum, puta substantiam de homine, et accipiantur media b et c, puta animal et vivum, ita quod b sit superius quam c, sicut vivum quam animal; manifestum est quod b, quod est universalius, erit immediatum ipsi a, et per hoc magis cognoscetur quam per c, quod est minus universale. Unde relinquitur quod demonstratio universalis potior sit quam particularis. Then (86a14) he gives the fifth reason, and it is this: The nearer to the first principle the middle of a demonstration is, the more powerful is the demonstration. He proves this on the ground that if a demonstration which proceeds from an immediate principle is more certain than one which does not proceed from an immediate, but from a mediate, principle, then it is necessary that to the extent that a demonstration proceeds from a middle nearer an immediate principle, the more powerful it is. But a universal demonstration proceeds from a middle nearer the principle which is an immediate proposition. And he exemplifies this with terms. For if one is required to demonstrate A, which is the most universal, of B, which is the most particular, say “substance” of man, and B and C, say “living” and “animal” be taken, such that B is more general than C, as “living” than “animal,” it is obvious that B, which is the more universal, will be immediate to A; consequently, more will be known through it than through C, which is less universal. Hence it remains that universal demonstration is more powerful than particular.
Addit autem quasdam praedictarum rationum logicas esse: quia scilicet procedunt ex communibus principiis, quae non sunt demonstrationi propria; sicut praecipue tertia et quarta, quae accipiunt pro medio id quod est commune omni cognitioni. Aliae vero tres praedictarum rationum, scilicet prima, secunda et quinta, magis videntur esse analyticae, utpote procedentes ex propriis principiis demonstrationis. But he remarks that some of the above reasons are “logical,” because, namely, they proceed from common principles which are not proper to demonstration; especially the third and fourth, which take as their middle something which is common to all knowledge. However, the other three, namely, the first, second and fifth, seem to be more analytic, proceeding as they do from principles proper to demonstration.
Sextam rationem ponit ibi: maxime autem manifestum est etc., et dicit quod maxime evidens est universalem demonstrationem principaliorem esse ex ipsis propositionibus, ex quibus utraque demonstratio procedit. Nam universalis demonstratio procedit ex universalibus propositionibus. Particularis autem demonstratio procedit ex aliqua particulari propositione. Propositionum autem universalis et particularis talis est comparatio, quod ille qui habet cognitionem de priori, scilicet de universali, cognoscit quodammodo posteriorem, scilicet in potentia. Nam in universali sunt in potentia particularia, sicut in toto sunt in potentia partes. Puta si aliquis cognoscit quod omnis triangulus habet tres angulos aequales duobus rectis, iam in potentia cognoscit hoc de isoscele. Sed ille qui cognoscit aliquid in particulari, non propter hoc cognoscit in universali, neque in actu neque in potentia. Non enim universalis propositio continetur in particulari, neque in actu neque in potentia. Si igitur demonstratio est potior, quae ex potioribus propositionibus procedit, sequitur quod demonstratio universalis sit potior. Then (86a21) he gives the sixth reason and says that the primacy of universal demonstration over particular is evident from the very propositions from which the two demonstrations proceed. For the universal demonstration proceeds from universal propositions, whereas the particular demonstration proceeds from a particular proposition. Now the comparison between universal and particular propositions is such that one who has knowledge of the former, i.e., of the universal, somehow knows the latter, namely, in potency. For the particulars are potentially in the universal, as the parts are potentially in a whole. Thus, if one knows that every triangle has three angles equal to two right angles, he already knows it potentially of isosceles. But one who knows something in a particular way does not on that account know it universally either potentially or actually. For the universal proposition is neither potentially nor actually contained in the particular. Therefore, if that demonstration is more powerful which proceeds from stronger propositions, it follows that universal demonstration is more powerful.
Est autem attendendum quod haec ratio non differt a quarta supraposita, nisi quod ibi fiebat comparatio conclusionum, quae cognoscuntur per demonstrationem, hic autem fit comparatio propositionum, ex quibus demonstratio procedit. It should be noted that this reason does not differ from the fourth one given above, except that in the fourth one the comparison was made between the conclusions which are known through demonstration, whereas here the comparison is between the propositions from which the demonstration proceeds.
Septimam rationem ponit ibi: et universalis quidem etc., quae talis est. Universalis demonstratio intelligibilis est, idest in ipso intellectu terminatur, quia finitur ad universale, quod solo intellectu cognoscitur. Sed demonstratio particularis in intellectu incipiens terminatur ad sensum, quia concludit particulare, quod directe per sensum cognoscitur; et per quamdam applicationem, seu reflexionem, ratio demonstrans usque ad particulare producitur. Cum igitur intellectus sit potior sensu, sequitur quod demonstratio universalis potior sit quam particularis. Then (86a28) he gives the seventh reason and it is this: The universal demonstration is intelligible, i.e., is terminated in the intellect, because it finishes in”a universal which is known only by the intellect. On the other. hand, a particular demonstration, although it begins in the intellect is terminated in sense, because it concludes a particular which is directly known by sense, and the reason demonstrating reaches out to the particular through a certain application or reflexion. Now since intellect is more powerful than sense, it follows that universal demonstration is stronger than particular.
Ultimo epilogando concludit hoc esse manifestum per omnia supra dicta. Finally, he concludes that this is clear in virtue of all that has been said above.

Lectio 39
Caput 25 cont.
ὅτι δ' ἡ δεικτικὴ τῆς στερητικῆς, ἐντεῦθεν δῆλον. a32. That affirmative demonstration excels negative may be shown as follows.
ἔστω γὰρ αὕτη ἡ ἀπόδειξις βελτίων τῶν ἄλλων τῶν αὐτῶν ὑπαρχόντων, ἡ ἐξ ἐλαττόνων αἰτημάτων ἢ ὑποθέσεων ἢ προτάσεων. εἰ γὰρ γνώριμοι ὁμοίως, τὸ θᾶττον γνῶναι διὰ τούτων ὑπάρξει· τοῦτο δ' αἱρετώτερον. λόγος δὲ τῆς προτάσεως, ὅτι βελτίων ἡ ἐξ ἐλαττόνων, καθόλου ὅδε· εἰ γὰρ ὁμοίως εἴη τὸ γνώριμα εἶναι τὰ μέσα, τὰ δὲ πρότερα γνωριμώτερα, ἔστω ἡ μὲν διὰ μέσων ἀπόδειξις τῶν (86b.) Β Γ Δ ὅτι τὸ Α τῷ Ε ὑπάρχει, ἡ δὲ διὰ τῶν Ζ Η ὅτι τὸ Α τῷ Ε. ὁμοίως δὴ ἔχει τὸ ὅτι τὸ Α τῷ Δ ὑπάρχει καὶ τὸ Α τῷ Ε. τὸ δ' ὅτι τὸ Α τῷ Δ πρότερον καὶ γνωριμώτερον ἢ ὅτι τὸ Α τῷ Ε· διὰ γὰρ τούτου ἐκεῖνο ἀποδείκνυται, πιστότερον δὲ τὸ δι' οὗ. καὶ ἡ διὰ τῶν ἐλαττόνων ἄρα ἀπόδειξις βελτίων τῶν ἄλλων τῶν αὐτῶν ὑπαρχόντων. ἀμφότεραι μὲν οὖν διά τε ὅρων τριῶν καὶ προτάσεων δύο δείκνυνται, ἀλλ' ἡ μὲν εἶναί τι λαμβάνει, ἡ δὲ καὶ εἶναι καὶ μὴ εἶναί τι· διὰ πλειόνων ἄρα, ὥστε χείρων. (1) We may assume the superiority ceteris paribus of the demonstr