A Logic for Aristotle's Modal Syllogistic

TL;DR

This paper introduces a new modal logic inspired by Aristotle's modal syllogistic, providing a formal system with semantics that clarifies and validates Aristotle's reasoning while connecting it to modern modal logic.

Contribution

It presents a novel deductive system for Aristotle's modal syllogistic, including semantics and validation of syllogisms, bridging ancient logic with modern modal logic frameworks.

Findings

Validated all modal syllogisms considered by Aristotle

Connected Aristotle's logic with modern propositional modal axioms

Provided a sound and consistent formal system for Aristotle's modal reasoning

Abstract

We propose a new modal logic endowed with a simple deductive system to interpret Aristotle's theory of the modal syllogism. While being inspired by standard propositional modal logic it is also a logic of terms that admits a (sound) extensional semantics involving possible states-of-affairs in a given world. Applied to the analysis of Aristotle's modal syllogistic as found in the \emph{Prior Analytics} A8-22 it sheds light on various fine-grained distinctions which when made allow us to clarify some ambiguities and obtain a completely consistent system and prove all of the modal syllogisms considered valid by Aristotle.This logic allows us also to make a connection with the axioms of modern propositional modal logic and to perceive to what extent these are implicit in Aristotle's reasoning. Further work wil involve addressing the question of the completeness of this logic (or variants thereof) together with the extension of the logic to include a calculus of relations (for instance the relational syllogistic treated in Galen's \emph{Introduction to Logic}) which Slomkowsky has argued is already found in the \emph{Topics}.