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

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

How to Obtain Valid Generalized Modal Syllogisms from Valid Generalized Syllogisms

Making full use of the truth value definitions of sentences with quantification, possible world semantics and/or fuzzy logic, one can prove the validity of generalized modal syllogisms. This paper shows that the proof of the validity of a generalized modal syllogism can be transformed into that of its corresponding generalized syllogism, and that the generalized syllogism obtained by removing all modalities in any valid generalized modal syllogism is still valid. Therefore, the simplest way to screen out valid generalized modal syllogisms is to add modalities to valid generalized syllogisms, and then to delete all invalid syllogisms by means of the basic rules with which valid generalized modal syllogisms should meet. And then the remainders are valid. This paper illustrates how to obtain 12 valid generalized modal syllogisms by adding necessary modalities and/or possible modalities to any valid generalized syllogism. The two kinds of syllogisms discussed in this paper are composed of sentences with quantification which is the largest number of sentences in natural language. Hence, this innovative research can provide theoretical support for linguistics, logic, artificial intelligence, and among other fields

Reduction between Categorical Syllogisms Based on the Syllogism EIO-2

Syllogism reasoning is a common and important form of reasoning in human thinking from Aristotle onwards. To overcome the shortcomings of previous studies, this article makes full use of set theory and classical propositional logic, and deduces the remaining 23 valid syllogisms only on the basis of the syllogism EIO-2 from the perspective of mathematical structuralism, and then successfully establishes a concise formal axiom system for categorical syllogistic logic. More specifically, the article takes advantage of the trisection structure of categorical propositions such as Q(a, b), the transformation relations between an Aristotelian quantifier and its inner and outer negation, the symmetry of the two Aristotelian quantifier (that is, no and some), and some inference rules in classical propositional logic, and derives the remaining 23 valid syllogisms from the syllogism EIO-2, so as to realize the reduction between different valid categorical syllogisms

How to Deduce the Other 91 Valid Aristotelian Modal Syllogisms from the Syllogism IAI-3

This paper firstly formalizes Aristotelian modal syllogisms by taking advantage of the trisection structure of (modal) categorical propositions. And then making full use of the truth value definition of (modal) categorical propositions, the transformable relations between an Aristotelian quantifier and its three negative quantifiers, the reasoning rules of classical propositional logic, and the symmetry of the two Aristotelian quantifiers (i.e. some and no), this paper shows that at least 91 valid Aristotelian modal syllogisms can be deduced from IAI-3 on the basis of possible world semantics and set theory. The reason why these valid Aristotelian modal syllogisms can be reduced is that any Aristotelian quantifier can be defined by the other three Aristotelian quantifiers, and the necessary modality ( ) and possible modality ( ) can also be defined mutually. This research method is universal. This innovative study not only provides a unified mathematical research paradigm for the study of generalized modal syllogistic and other kinds of syllogistic, but also makes contributions to knowledge representation and knowledge reasoning in computer science

Deductive Systems in Traditional and Modern Logic

The book provides a contemporary view on different aspects of the deductive systems in various types of logics including term logics, propositional logics, logics of refutation, non-Fregean logics, higher order logics and arithmetic

Syllogistic logic with “most”

This paper presents a sound and complete proof system for the logical system whose sentences are of the form All X are Y, Some X are Y and Most X are Y, where we interpret these sentences on finite models, with the meaning of “most” being “strictly more than half.” Our proof system is syllogistic; there are no individual variables

Syllogistic logic with Most

We add Most X are Y to the syllogistic logic of All X are Y and Some X are Y. We prove soundness, completeness, and decidability in polynomial time. Our logic has infinitely many rules, and we prove that this is unavoidable