How to Derive the Other 37 Valid Modal Syllogisms from the Syllogism ◇A£I◇I-1

Abstract

Syllogistic reasoning plays an important role in natural language information processing. In order to provide a consistent interpretation for Aristotelian modal syllogistic, this paper firstly proves the validity of the syllogism ◇A£I◇I-1, and then takes it as the basic axiom to derive the other 37 valid modal syllogisms on the basis of some reasoning rules in classical propositional logic, the transformation between any one of Aristotelian quantifiers and its three negative quantifiers, the symmetry of the Aristotelian quantifier some and no, and some relevant definitions and facts. In other words, there are reducibility between the modal syllogism ◇A£I◇I-1 and the other 37 valid modal syllogisms. There are infinitely many modal syllogism instances in natural language corresponding to every valid modal syllogism, thus this study has important practical significance and theoretical value for natural language information processing in computer science.