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Aristotle's Syllogistic from the Standpoint of Modern Formal Logic (2nd ed.)

Aristotle's Syllogistic from the Standpoint of Modern Formal Logic (2nd ed.)

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AuthorsŁukasiewicz, Jan
PublisherClarendon Press
Published01 gen 1957
Date23 ago 2016
Languageseng
Identifiersoclc: 289729
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Description

This important book presents a detailed analysis of Aristotle's theory of the syllogism in the light of modern logic. The author first distinguishes the features of the doctrine which are due to Aristotle rather than to later writers. He is careful to note what now appear as inadequacies in Aristotle's treatment. His analysis is both textual and structural, a careful exposition of Aristotle's text being given before the theory is put into a rigorous, deductive formalism. Łukasiewicz's formalization of the theory is especially neat and simple. It contains axioms and rules of rejection as well as the more familiar kind of axioms and rules of assertion. This is in keeping with a rule of rejection stated by Aristotle himself [Prior analytics, i. 5, 27b 12–23 (Bekker edition, Reimer, Berlin, 1831)]. Finally, by the addition of a new rule of rejection, it is shown how, following Słupecki [Trav. Soc. Sci. Lett. Wrocław. Ser. B. no. 6 (1948)], every significant expression of Aristotle's syllogistic is either asserted or rejected. This constitutes the solution of a kind of decision problem for this area of logical theory. Reviewed by R. M. Martin

Łukasiewicz is a famous Polish, Catholic logician.
cf. my question Can all mathematical reasoning be translated into traditional logic?:

Can all mathematical reasoning be translated into traditional (Aristotelian, syllogistic) logic?

It would seem not ∵ one cannot syllogistically establish the validity of the reasoning in the following argument:

This doesn't work because "greater than 2" ≠ "2", or 2 ≯ 2.

The form of the following syllogism is valid, but it shows how a false mathematical premise can lead to a true conclusion:

Thus, it doesn't seem traditional logic can handle mathematical reasoning. Didn't Aristotle, the medieval logicians, et al. realize this?

Poincaré thought that mathematical induction consisted in an ∞ number of syllogisms. Is that true?
(cf. Pierre Duhem's article contra Poincaré: "The Nature of Mathematical Reasoning" from "La nature du raisonnement mathématique," Revue de philosophie 21 (1912): 531-543.)

my answer:

mentions

Smiley's abstract:

Anyone who reads Aristotle, knowing something about

You’re correct about Duhem doing a better job on Poincaré. He surely was in a better position to do so and did it well in La nature du raisonnement mathématique. Il a raison (p. 542) :

« Mais les Mathématiques ne sont pas virtuellement contenues dans les seuls axiomes [something many reductionist logicians and computer scientists to this day don’t want to realize] ; elles sont le résultat produit par l’application des axiomes aux définitions. »

The Search for Mathematical Roots is written by a Russellian Bristish mathematical logician, who was also a fairly narrow mathematical historian (however prolific his writings). Symbolic logicians typically rank the piece as a reference, for their purposes.

“Aristotelian logic,” in part at least (I remember that we conversed about this matter before). Even within a deductive approach, modern logic has usefully extended reasonings to non-syllogistic forms (used, for example, to prove things about propositional proof systems themselves, by way of both deductive and inductive/recurrence reasonings, à la Gödel). Also, the complexification and replacement of the ‘subject’ and ‘predicate’ terms of categorical syllogisms by way of ‘functions’ and ‘arguments,’ which Frege brought about, allowed him to analyse the logical connection of statements involving manifold generalities (the analysis of which uses rules of inference and cannot simply assume a merely validating form).

Suffice it to recall that already (before modern extensions) the Stoics had begun pondering and constructing non -syllogistic deductive reasonings, i.e. non-syllogistic conclusive reasonings. modern logic and nothing about its history, must ask himself why the syllogistic cannot be translated as it stands into the logic of quantification. It is now more than twenty years since the invention of the requisite framework, the logic of many-sorted quantification.

He concludes:

If the Aristotelian logic, after a long pre-eminence and a shorter period of disrepute, is now more temperately regarded, the change is surely due to Lukasiewicz' formalisation of the traditional syllogistic in the 1930's, and his bringing modern techniques and ideas to bear on the resulting system. But the price paid for a rehabilitation of the traditional logic through an algebra of the Łukasiewicz type is a certain divorce from the main current of modern logic: Łukasiewicz was even led to conclude (op. cit. [Łukasiewicz, Jan. 1957. Aristotle's syllogistic from the standpoint of modern formal logic. Oxford: Clarendon Press. ], p. 130) that the syllogistic of Aristotle "exists apart from other deductive systems, having its own axiomatic and its own problems." The result is a certain ambivalence in the current attitude towards the old logic - when we compile our World Team of logicians we tend to include Aristotle as (non-playing) captain. This attitude, at once admiring and dismissive, is well illustrated in Łukasiewicz' conclusion that "The syllogistic of Aristotle is a system the exactness of which surpasses even the exactness of a mathematical theory, and this is its everlasting merit. But it is a narrow system and cannot be applied to all kinds of reasoning, for instance to mathematical arguments. … The logic of the Stoics, the inventors of the ancient form of the propositional calculus, was much more important than all the syllogisms of Aristotle. We realize today that the theory of deduction and the theory of quantifiers are the most fundamental branches of logic." (p. 131.)

It would of course be absurd and anachronistic for me to try to vindicate Aristotle's choice of subject-matter by suggesting that he was consciously guided by anything like the modern idea of quantification. But without committing this mistake there are two observations which I think may properly be made. One is that if it is anachronistic to suggest that Aristotle's logic is 'really' a theory of quantification then it is equally anachronistic to suggest that it is 'really' a theory of primitive functors A, I, etc. As Łukasiewicz himself remarks in his book, "the logic of Aristotle is formal without being formalistic"; and what I have for the sake of convenience called the 'traditional' theory in § 2 is, both in its conscious conception as an algebra of non-empty classes and in its formalistic vocabulary and axiomatisation, as distinctively 'modern' as the logic of quantification. The other remark to be made is that the logic of many-sorted quantification is in no sense something existing "apart from other deductive systems". Not only is it formally no more than a systematic reduplication of the standard single-sorted logic, but it is also the obvious framework for the formalisation of a whole range of mathematical theories: any branch of geometry will furnish one example and Russell's or von Neumann's set theories another. I should like therefore to think that the translations introduced above would help to counter the suggestion of even a residual incompatibility between the modern and the Aristotelian formal logic.

Duhem, Pierre Maurice Marie. “La nature du raisonnement mathématique.” Revue de Philosophie 21 (1912): 531–43.

Duhem concludes in favor of syllogistic reasoning (Aristotelian logic?):

Nous pensons avoir suffisamment établi, dans ce qui précède, que la démonstration mathématique se poursuit par voie syllogistique exactement de la même manière que n'importe quelle autre science déductive. Ce qui la distingue des autres sciences déductives, ce n'est pas la forme du raisonnement qu'elle emploie; c'est la nature des notions et propositions auxquelles elle applique ce raisonnement.

You’re correct about Duhem doing a better job on Poincaré. He surely was in a better position to do so and did it well in La nature du raisonnement mathématique. Il a raison (p. 542) :

« Mais les Mathématiques ne sont pas virtuellement contenues dans les seuls axiomes [something many reductionist logicians and computer scientists to this day don’t want to realize] ; elles sont le résultat produit par l’application des axiomes aux définitions. »

The Search for Mathematical Roots is written by a Russellian Bristish mathematical logician, who was also a fairly narrow mathematical historian (however prolific his writings). Symbolic logicians typically rank the piece as a reference, for their purposes.

“Aristotelian logic,” in part at least (I remember that we conversed about this matter before). Even within a deductive approach, modern logic has usefully extended reasonings to non-syllogistic forms (used, for example, to prove things about propositional proof systems themselves, by way of both deductive and inductive/recurrence reasonings, à la Gödel). Also, the complexification and replacement of the ‘subject’ and ‘predicate’ terms of categorical syllogisms by way of ‘functions’ and ‘arguments,’ which Frege brought about, allowed him to analyse the logical connection of statements involving manifold generalities (the analysis of which uses rules of inference and cannot simply assume a merely validating form).

Suffice it to recall that already (before modern extensions) the Stoics had begun pondering and constructing non -syllogistic deductive reasonings, i.e. non-syllogistic conclusive reasonings.

cf. Veatch Two Logics

and: "Can all mathematical reasoning be translated into traditional logic?"