How to Deduce the Other 91 Valid Aristotelian Modal Syllogisms from the Syllogism IAI-3

Abstract

This paper firstly formalizes Aristotelian modal syllogisms by taking advantage of the trisection structure of (modal) categorical propositions. And then making full use of the truth value definition of (modal) categorical propositions, the transformable relations between an Aristotelian quantifier and its three negative quantifiers, the reasoning rules of classical propositional logic, and the symmetry of the two Aristotelian quantifiers (i.e. some and no), this paper shows that at least 91 valid Aristotelian modal syllogisms can be deduced from IAI-3 on the basis of possible world semantics and set theory. The reason why these valid Aristotelian modal syllogisms can be reduced is that any Aristotelian quantifier can be defined by the other three Aristotelian quantifiers, and the necessary modality ( ) and possible modality ( ) can also be defined mutually. This research method is universal. This innovative study not only provides a unified mathematical research paradigm for the study of generalized modal syllogistic and other kinds of syllogistic, but also makes contributions to knowledge representation and knowledge reasoning in computer science.