With the help of the definitions of three negative quantifiers of Aristotelian quantifiers (i.e. all, no, some and not all), the symmetry of no and some, and some basic inference rules in propositional logic, one can deduce the remaining 23 valid syllogisms only from the syllogism OAO-3. In other words, there is reducible relations between/among different forms and different figures of valid traditional syllogisms. And these reducible relations actually reflect the transformation relations between the monotonicity of the four Aristotelian quantifiers. This paper provides a computational level of reasoning for syllogistic logic and an important theoretical basis for knowledge representation and knowledge reasoning in computers.

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

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.

Syllogistic reasoning plays a crucial part in natural language information processing. For the purpose of providing a consistent interpretation for Aristotelian modal syllogistic, this paper firstly proves the validity of the syllogism EI◇O-2, and then takes it as the basic axiom to derive the other 38 valid modal syllogisms by taking advantage of some reasoning rules in classical propositional logic, the symmetry of two Aristotelian quantifiers (i.e. some and no), the transformation between any one of Aristotelian quantifiers and its three negative quantifiers, as well as some facts in first order logic. In other words, there are reducible relations between the modal syllogism EI◇O-2 and the other 38 valid modal syllogisms.There are infinitely many instances in natural language corresponding to any valid modal syllogism. Therefore, this study has theoretical value and practical significance for natural language information processing in computer science.