How to Obtain Other Valid Generalized Modal Syllogisms from the Syllogism ▯EF◊O-1

Abstract

For the sake of obtaining valid generalized modal syllogisms, the article first proves the validity of the generalized modal syllogism ▯ EF◊O-1 by means of set theory and modal logic, and then deduces the other 22 valid generalized modal syllogisms from the syllogism ▯ EF◊O-1 in accordance with modern modal logic, generalized quantifier theory, and so on. The reason why there are reducibilities between different generalized modal syllogisms is that: (1) any of the Aristotelian quantifiers is definable by the other three Aristotelian quantifiers; (2) any of the four generalized quantifiers mentioned in this article is definable by the other three generalized quantifiers; (3) the transformation relationship between necessity and possibility; (4) the symmetry of some and no. The article presents a formal research method for generalized modal syllogistic, which not only provides a unified mathematical research paradigm for other generalized modal syllogisms and even other kinds of syllogisms, but also meet with the demands for formalization transformation of modern logic in the era of artificial intelligence. Therefore, this study has considerable theoretical and practical values.