History of Formal Logic

Description

A collection of passages from logical writers from Plato to Gödel, with interstitial comments by the author, arranged to illustrate the thesis that logic has not progressed along a single line, but has developed independently in the West and in India, and in the West has had three periods of growth (the Greek, the medieval and the modern-mathematical) with intervening periods of decay and oblivion. Within each of the sections thus arising, the arrangement is partly chronological, partly according to topics and problems, so that it is clear at a glance that, for example, in each of the three Western periods logicians were worried about self-referential paradoxes, about the nature and varieties of implication and about the relation between propositional logic and term-logic.
The Greek period of growth subdivides into an earlier part dominated by Aristotle and a later by Chrysippus the Stoic, much as modern mathematical logic was dominated first by Boole and then by Frege (of the latest period, dominated by Gödel, the author gives only an introductory glimpse: extracts from the incompleteness proof). In both cases, we find a class- or term-logic developed first and a propositional logic later, but in the modern period we have a rich logic of relations and of multiple quantification, of which both the Greek and the intervening medieval epochs have only rudiments. There was much more of it in the medieval epoch, all the same, than the author suggests, and higher-order quantification, and quantification into intensional contexts, were handled in the Middle Ages with some skill and freedom, and with an awareness of the traps. While the author misses much of this, he has enough from the medieval writers to make it worth advising mathematicians to look at them—especially at their theories of reference (supposition') and their treatment of antinomies—and not just at the more obviously relevant modern section. There are hints that the Indians also handled higher-order quantifications with skill, and even had the essence of Frege's definition of number, but there is not enough given (or known) of Indian logic for us yet to disentangle its real subtleties from idealistic confusions. The extent to which later Western periods built on earlier is left problematic, but the selections and comment make it obvious that medieval logic is vastly less merely imitative than was thought, and modern logicians (Peirce apart) have only begun to be aware of their precursors. The author has an introductory section on logical historiography; the subject is plainly growing very rapidly, and this collection of sources amounts to aninterim report'. Much has been done in the field even since its original publication in 1956, and the translator adds to it a section on Abelard. Reviewed by A. N. Prior

Innocent-Marie Bocheński's History of Formal Logic cited in Deely's edition of John of St. Thomas (João Poinsot, O.P.)'s Tractatus de Signis

Discusses supposition on §27 pp. 162-173 (DjVu pp. 189ff.). The following § discusses various types of supposition, like ampliation and analogy. St. Vincent Ferrer's work on supposition is quoted. On DjVu p. 205, Fr. Bocheński mentions "structure" (à la structural realism?) in the context of analogy and St. Thomas's example of 6:3 = 4:2, which Fr. B. seems to think is a bad example because it's a comparison of relations (ratios), but why would that make the example bad? cf. also isomorphy (47.41, DjVu p. 420). Moody also assisted Fr. B. here, too.

I first heard about Bocheński during my slower re-study of John of St. Thomas's Tractatus de Signis (from the Ars Logica volume of his Cursus Philosophica) with Deely's beautifully-typeset bilingual edition (now in its 2nd ed.). Deely cites Bocheński on p. 18 (Deely's "Second Semiotic Marker") & p. 36 (his "Fourth Semiotic Marker"). Both of those two-page "semiotic markers" are worth reading because they show the immense relevance of John of St. Thomas.

I am absolutely astounded by João Poinsot's Tractatus de Signis! He treats the most exigent questions still lingering unanswered today, primarily:

• What is the relationship between ens rationis and ens reale?

  • The first three articles of the Tractatus are probably the most important:

• Quid Sit Ens Rationis in Communi et Quotuplex?

• Quid Sit Secunda Intentio et Relatio Rationis Logica et Quotuplex?
• Per Quam Potentiam et Per Quos Actus Fiant Entia Rationis?

The subset of this question is the relationship between mathematics or a "systéme abstrait " and the real beings they help make intelligible. ("qui a pour but de résumer et de classer logiquement un ensemble de lois expérimentales " as Duhem defined physical theory)

To think that Poinsot answered these questions in 1634, right after Galileo's 2nd condemnation, and very few people took heed! From what I've read of Poinsot's material logic a few years ago, he has a very well-developed exposition of the logic of scientific demonstration.

I used to quasi-relegate semiotics into the "literary criticism" category, far from philosophy of science, but now semiotics (in the Peircean sense) seems vital to post-post-modern philosophy.

Regarding Bocheński:

An excellent quote from his History of Formal Logic (quoted on the first page of Deely's "The relation of logic to semiotics"):

Now it is hard for a logician trained in the contemporary variety of logic to think himself into another. In other words, it is hard for him to find a criterion of comparison. He is constantly tempted to find what is valuable only what fits into the categories of his own logic. Impressed by our technique, which is not by itself properly logic, having only superficial knowledge of past forms, judging from a particular standpoint, we too often risk misunderstanding and under-rating other forms. … The modern mathematical logician certainly has a strong support in his calculus, but all too frequently that same calculus leads him to dispense with thought just where it may be most required.

Amen!

pp. 41-44 (DjVu pp. 67-70): Very neat arguments regarding the Organon 's chronological arrangement based on intrinsic criteria, as "We have no extrinsic criteria to help us establish the chronological sequence of the different parts of the Organon." (p. 41, DjVu p. 67), which helps shed light on how Aristotle discovered his logic. Cf. pp. 22-4 (DjVu pp. 27-9) of his Ancient Formal Logic, which is smiliar to ibid. , but with the addition of "4 E. Survey of the evolution" (pp. 23-4, DjVu pp. 28-9).

How did Aristotle discover his logic?
This question was prompted by the passage about Aristotle in Stephen Wolfram, What Is ChatGPT Doing … and Why Does It Work? , § "What Really Lets ChatGPT Work?"

Bocheński, O.P., History of Formal Logic pp. 41-44 notes that although "We have no extrinsic criteria to help us establish the chronological sequence of the different parts of the Organon ", there are intrinsic criteria (ibid. p. 42):

The presence or absence of:

1. the syllogism
2. variables
3. a "technical level of the thought"
4. modal vs. assertoric logic
   Modal logic is more consonant with his hylemorphism than the assertoric logic that appeared in Plato's dialogues; thus, assertoric logic in Aristotle is an indicator the Aristotelian work is youthful, while modal logic would indicate a more mature work.

Bocheński concludes (ibid. p. 44):

It is only certain that the Topics and Sophistic Refutations contain a different and earlier logic than the Analytics , and that the Hermeneia exhibits an intermediate stage. For the rest we have well-founded hypotheses which can lay claim at least to great probability.

Bocheński, O.P., Ancient Formal Logic pp. 23-24:

4 E. Survey of the evolution

In the fight of the above chronology the evolution of Aristotle’s formal logic may be stated in the following way:

He first elaborated the Platonic λόγους (Top. , Met. Γ , De Int.), considerably developing and explicitly stating the rules or laws on which they are based. By doing so he stated a wealth of interesting logical principles of which, however, none is an analytic syllogism. This period may coincide with that of travels (348/7—335/4).

Later on he made his two great discoveries: that of the analytical syllogism and that of the variable. He then declared that the other (non-analytic) laws and rules are of lesser importance and concentrated on syllogism, first assertoric, then modal.

By analyzing the axiomatic system of the former (he did not have time, as it seems, to do so with the latter) he discovered several metalogical rules and even some laws of the logic of propositions. These last discoveries were, however, not systematized by him.