Algebraic Analysis of Many Valued Logics^)

This paper is an attempt at developing a theory of algebraic systems that would correspond in a natural fashion to the N0-valued prepositional calculus^). For want of a better name, we shall call these algebraic systems MV-algebras where MV is supposed to suggest many-valued logics. It is known that the classical two-valued logic gives rise to the study of Boolean algebras and, as can be expected, every Boolean algebra will be an MValgebra whereas the converse does not hold. However, many results for Boolean algebras can be appropriately carried over to MV-algebras, although in some cases the proofs become more subtle and delicate. The motivation behind the present study is to find a proof of the completeness of the Novalued logic by using some algebraic results concerning MV-algebras; more specifically, it is known that the completeness of the two-valued logic is a consequence of the Boolean prime ideal theorem and we wish to exploit just some such corresponding result for MV-algebras(3). It will be seen that our effort in duplicating this result is only partially successful. In the first four sections of this paper we present various theorems concerning both the arithmetic in MV-algebras and the structure of these algebras. In the last section we give some applications of our results to the study of completeness of No-valued logic and some related topics. We point out here that the treatment of MV-algebras as given here is not meant to be complete and exhaustive. 1. Axioms of MV-algebras and some elementary consequences. An MValgebra is a system (A, +, •, ~, 0, 1) where A is a nonempty set of elements, 0 and 1 are distinct constant elements of A, + and • are binary operations on elements of A, and is a unary operation on elements of A obeying the following axioms. (We assume here, of course, that A is closed under the operations +, -, and ~.)