A non-classical view of the nature of truth values

Abstract

The paper is devoted to semantics of many-valued logics and deals with the problem of interpretation of values in logical matrices. The starting point of the research is R. Suszko's thesis, according to which every many-valued logic is logically two-valued, as well as the analysis of this thesis in the subsequent literature. As shown by Suszko, any many-valued matrix semantics can be reduced to two values. This raised the question of whether many-valued logics are possible in principle. The positive answer was given by G. Malinowski on the basis of his proposed concept of inferential many-valuedness. The foundation of this concept lies in generalizations of the concepts of logical matrix and consequence, where, along with the class of designated values, other subsets of the matrix universe are involved, considered as logical values. This allows Malinowski to provide examples of many-valued semantics that can not be reduced to two values using Suszko's technique. Based on the results of Malinowski and other authors who developed this topic, we propose our own generalizations of the concepts of logical matrix and consequence. Instead of considering subsets of the matrix universe as logical values, we assign this role to relations, that is, subsets of Cartesian powers of the universe. Such a generalization makes it possible to construct bivalent truth-functional semantics for logics, which, as follows from the results known in the literature, do not have two-valued semantics in the style of Suszko.