An Algebraic Analysis of Categorical Syllogisms by Using Caroll’s Diagrams

Abstract

In this paper, we analyze the algebraic properties of categorical syllogisms by constructing a logical calculus system and by using Caroll Diagrammatic Method(SLCD). We prove that any categorical syllogism is valid if and only if it is provable in this system. For this purpose, we explain firstly the
quantitative relation between two terms by means of bilateral diagrams and we clarify premises via bilateral diagrams. Afterwards, we input the data taken from bilateral diagrams, on the trilateral diagram. With the help of the elimination method, we obtain a conclusion that is transformed from trilateral diagram to bilateral diagram. Subsequently, we study a syllogistic conclusion mapping which gives us a conclusion obtained from premises. Finally, we allege valid forms of syllogisms using algebraic methods, and we examine their algebraic properties, and also by using syllogisms, we construct algebraic structures, such as lattices, Boolean algebras, Boolean rings, and MV-algebras.