Reduction between Aristotelian Modal Syllogisms Based on the Syllogism ◇I□A◇I-3

In order to provide a consistent explanation for Aristotelian modal syllogistic, this paper reveals the reductions between the Aristotelian modal syllogism ◇I□A◇I-3 and the other valid modal syllogisms. Specifically, on the basis of formalizing Aristotelian modal syllogisms, this paper proves the validity of ◇I□A◇I-3 by means of the truth value definition of (modal) categorical propositions. Then in line with classical propositional logic and modal logic, generalized quantifier theory and set theory, this paper deduces the other 47 valid Aristotelian modal syllogisms from the modal syllogism ◇I□A◇I-3. This study shows that the reasons why these syllogisms are reducible are: 1) any of Aristotelian quantifier can be defined by the other three Aristotelian quantifiers; 2) the Aristotelian quantifiers some and no have symmetry; 3) the possible modality ◇ and necessary modality □ can be mutually defined. This formal study of Aristotelian modal syllogistic not only conforms to the needs of formalization transformation of various information in the era of artificial intelligence, but also provides a unified mathematical research paradigm for other kinds of syllogistic.