Generalized Syllogism Reasoning with the Quantifiers in Modern Square{no} and Square{most}

Abstract

A modern Square{Q}={Q, Q_, Q_, _Q_} is composed of a generalized quantifier Q and its three types of negative quantifiers: inner, outer and dual negative one. This paper mainly discusses the non-trivial generalized syllogisms reasoning with the quantifiers in Square{no} and Square{most}. To this end, this paper firstly gives formalizes generalized syllogisms, then proves the validity of the syllogism AMM-1 with the generalized quantifier most, and further deduces the other 24 valid syllogisms. The reason why these valid generalized syllogisms studied in this paper can be mutually reduced is because: (1) any of the four Aristotelian quantifiers in Square{no} can define the other three ones; (2) so can any of the four generalized quantifiers in Square{most}. This study is undoubtedly beneficial not only for the development of modern logic, but also for the development of inference machines in artificial intelligence.