23 October 2012, Logic Tea, Kohei Kishida

Abstract

It is a textbook fact that Aristotle's (non-modal) logic of syllogisms is sound with respect to the modern semantics of classical first-order logic (with the existential import assumed of certain universal predications). Moreover, since Lukasiewicz's (1957) seminal work, there have been several attempts made to reconstruct Aristotle's logic in modern fashions: for instance, Corcoran's (1972, 1974) and Smiley's (1973) natural-deduction systems, Martin's (1997) sequent calculus, and more recently, Pratt-Hartmann's and Moss's (2009) natural deduction in the framework of natural logic. Not only does classical logic, but each of these systems also proves everything Aristotle deemed good syllogisms. It should be noted, nevertheless, that they all manage to prove things Aristotle did not endorse as good syllogisms: for instance, the inference from sentence phi to conclude phi itself, or from no premise to conclude that all x are x. The goal of this paper is to provide a formal system that captures not just all but moreover only the things Aristotle endorsed in his syllogistic.

For this goal, I propose a sequent calculus that has a simple syntax and a simple set of axioms and rules. I investigate the proof theory of our calculus; one of the notable facts I show is that Aristotle's own proof-theoretic observations are naturally reflected in my calculus. Another is that the calculus proves all and essentially only the syllogisms Aristotle endorsed. I also investigate semantics with respect to which the calculus is sound and complete. (My semantics is obtained by modifying and extending Smiley's results.) These results help us to characterize, from a modern perspective, in what essential way Aristotle's syllogistic differs from classical first-order logic or from the above-mentioned, existent reconstructions. I conclude that Aristotle's syllogistic is characterized by a conception of good inference that is similar to the one that motivated relevance logic, and that can be extended to a much broader range of logic.

For more information on this talk and future events please refer to the website at http://www.illc.uva.nl/logic_tea/ or contact Virginie Fiutek (fiutek.virginie at gmail.com), Johannes Marti (johannes.marti at gmail.com), Sebastian Speitel (sebastian.speitel at gmail.com).