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