The Reducibility of Generalized Modal Syllogisms Based on ☐AM◊I-1
Jing Li Xu, Xiaojun Zhang · SCIREA Journal of Philosophy · 2023
There is the reducibility between the generalized modal syllogism ☐AM◊I-1 and the other 20 valid generalized modal syllogisms. This paper first proves the validity of the generalized modal syllogism based on the truth value definitions of sentences with quantification, set theory and modal logic, then derives the other 20 valid generalized modal syllogisms from the syllogism ☐AM◊I-1 in line with some facts and inference rules. The reason why these syllogisms are reducible is that: (1) any of the Aristotelian quantifiers can be defined by the other three Aristotelian quantifiers; (2) any of the four generalized quantifiers in this paper (that is, most, at most half of the, fewer than half of the and at least half of the) can be defined by the other three generalized quantifiers; (3) the Aristotelian quantifiers some and no have symmetry; (4) a necessary modality ☐ and a possible modality ◊ can be mutually defined. And the process of these reductions are ultimately presented in a structured formalization way. Thus, this paper provides a fragmentary research approach for other generalized modal syllogisms including four generalized quantifiers with transformation relations. There are many generalized modal syllogisms in natural language. Therefore, this study has practical significance and theoretical value for knowledge representation and reasoning in artificial intelligence.