Aristotle’s term theory and its underlying logic

George Englebretsen (ed.): New Directions in Term Logic, 2024

Aristotle counts as the founder of formal logic. The logic he develops dominated until Frege and others introduced a new logic. This new logic is taken to be more powerful and better capable of capturing inference patterns. The new logic differs from Aristotelian logic in significant respects. It has been argued by Fred Sommers and Hanoch Ben-Yami that the new logic is not well equipped as a logic of natural language, and that a logic closer to Aristotle's is better suited for this task. Each of them developed their own formalism - Sommers in form of term logic, Ben-Yami in form of his Quantified Argument Calculus (QUARC). I discuss Aristotle's logic - a term logic - and attempt a comparison between Aristotelian logic and (i) the new logic, (ii) Sommers' term logic, and (iii) Ben-Yami's QUARC. I consider the differences between the systems, and show how they are related to and diverge from the new logic.

History and Philosophy of Logic, 2016

Recent formalizations of Aristotle's modal syllogistic have made use of an interpretative assumption with precedent in traditional commentary: That Aristotle implicitly relies on a distinction between two classes of terms. I argue that the way Rini (2011. Aristotle's Modal Proofs: Prior Analytics A8-22 in Predicate Logic, Dordrecht: Springer) employs this distinction undermines her attempt to show that Aristotle gives valid proofs of his modal syllogisms. Rini does not establish that Aristotle gives valid proofs of the arguments which she takes to best represent Aristotle's modal syllogisms, nor that Aristotle's modal syllogisms are instances of any other system of schemata that could be used to define an alternative notion of validity. On the other hand, I argue, Robert Kilwardby's ca. 1240 commentary on the Prior Analytics makes use of a term-kind distinction so as to provide truth conditions for Aristotle's necessity propositions which render Aristotle's conversion rules and first figure modal syllogisms formally valid. I reconstruct a suppositio semantics for syllogistic necessity propositions based on Kilwardby's text, and yield a consequence relation which validates key results in the assertoric, pure necessity and mixed necessity-assertoric syllogistics. of these approaches, see Smith 1989 (p. xxxvi) and Striker 2009 (pp. xvi-xvii). 2 Thom 2007 (p. 11). See Lagerlund 2008 on the assimilation of Aristotle's works into logical tradition of the Latin West. 3 An exception is Thom 1996, who gives a detailed treatment of Aristotle's proof methods alongside the formal systems which he develops.

We investigate the philosophical significance of the existence of different semantic systems with respect to which a given deductive system is sound and complete. Our case study will be Corcoran’s deductive system D for Aristotelian syllogistic and some of the different semantic systems for syllogistic that have been proposed in the literature. We shall prove that they are not equivalent, in spite of D being sound and complete with respect to each of them. Beyond the specific case of syllogistic, the goal is to offer a general discussion of the relations between informal notions—in this case, an informal notion of deductive validity—and logical apparatuses such as deductive systems and (model-theoretic or other) semantic systems that aim at offering technical, formal accounts of informal notions. Specifically, we will be interested in Kreisel’s famous ‘squeezing argument’; we shall ask ourselves what a plurality of semantic systems (understood as classes of mathematical structures) may entail for the cogency of specific applications of the squeezing argument. More generally, the analysis brings to the fore the need for criteria of adequacy for semantic systems based on mathematical structures. Without such criteria, the idea that the gap between informal and technical accounts of validity can be bridged is put under pressure.

2024

Commentaries on Aristotle’s Semantics.

Aristotle 2400 Years, May 23-28 2016, ed. by D. Sfendoni-Mentzou. Aristotle Universuty of Thessaloniki, Interdisciplinary Centre for Aristotle Studies. , 2016

John Corcoran. 1973. A Mathematical Model of Aristotle's Syllogistic, Archiv f"ur Geschichte der Philosophie 55, 191–219. This article presents a mathematical model designed to reflect certain structural aspects of Aristotle's logic. Accompanying the presentation is an interpretation of certain scattered parts of the Prior and Posterior Analytics. Although our interpretation does not agree in all respects with those previously put forth, the present work would have been impossible without the enormous ground work of previous scholars—especially Łukasiewicz and Ross—to whom we are deeply grateful. Our interpretation restores Aristotle's reputation as a logician of consummate imagination and skill. Several attributions of shortcomings and logical errors to Aristotle are seen to be without merit. Aristotle's logic is found to be self-sufficient in several senses. In the first place, his theory of deduction is logically sound in every detail. (His indirect deductions have been criticized, but incorrectly on our account.) In the second place, Aristotle's logic presupposes no other logical concepts, not even those of propositional logic. (His deductions were falsely alleged to have gaps only correctable using propositional logic.) In the third place, the Aristotelian system is seen to be complete in the sense that every valid argument statable in his system admits of a deduction within his deductive system, i. e. every conclusion that follows from given premises is deducible from them using Aristotle’s explicitly described methods: in short, every valid argument is deducible. This result, stated but not proved by Aristotle, connects logical ontology to logical epistemology: every argument that in fact is valid can be known to be valid. It is not clear whether Aristotle appreciated the epistemic significance of his own completeness claim.

Anuario Filosófico

Advances in Modal Logic, 2010

The first systematic study of reasoning and inference in the West was done by Aristotle. However, while his assertoric theory of syllogistic reasoning is provably sound and complete for the class of models validating the inferences in the traditional square of opposition [5, p. 100], his modal syllogistic, developed in chapters 3 and 8–22 of the Prior Analytics [1], has the rather dubious honor of being one of the most difficult to understand logical systems in history. Starting with some of his own students, many have considered Aristotle's modal ...

The Review of Symbolic Logic, 2010

Since Frege’s predicate logical transcription of Aristotelian categorical logic, the standard semantics of Aristotelian logic considers terms as standing for sets of individuals. From a philosophical standpoint, this extensional model poses problems: There exist serious doubts that Aristotle’s terms were meant to refer always to sets, that is, entities composed of individuals. Classical philosophy up to Leibniz and Kant had a different view on this question—they looked at terms as standing for concepts (“Begriffe”). In 1972, Corcoran presented a formal system for Aristotelian logic containing a calculus of natural deduction, while, with respect to semantics, he still made use of an extensional interpretation. In this paper we deal with a simple intensional semantics for Corcoran’s syntax—intensional in the sense that no individuals are needed for the construction of a complete Tarski model of Aristotelian syntax. Instead, we view concepts as containing or excluding other, “higher” c...

Aristotle's Syllogistic Underlying Logic, 2022

Table of Contents (with dedication) for "Aristotle's Syllogistic Underlying Logic: His Model with his Proofs of Soundness and Completeness"