Reduction between Categorical Syllogisms Based on the Syllogism EIO-2

Abstract

Syllogism reasoning is a common and important form of reasoning in human thinking from Aristotle onwards. To overcome the shortcomings of previous studies, this article makes full use of set theory and classical propositional logic, and deduces the remaining 23 valid syllogisms only on the basis of the syllogism EIO-2 from the perspective of mathematical structuralism, and then successfully establishes a concise formal axiom system for categorical syllogistic logic. More specifically, the article takes advantage of the trisection structure of categorical propositions such as Q(a, b), the transformation relations between an Aristotelian quantifier and its inner and outer negation, the symmetry of the two Aristotelian quantifier (that is, no and some), and some inference rules in classical propositional logic, and derives the remaining 23 valid syllogisms from the syllogism EIO-2, so as to realize the reduction between different valid categorical syllogisms.