How to Obtain Valid Generalized Modal Syllogisms from Valid Generalized Syllogisms

Abstract

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.