Some Problems Concerning Axiom Systems for Finitely Many-Valued Propositional Logics

Abstract

In this chapter, we will examine some problems concerning axiomatization of finitely many-valued propositional logics. It has been recently demonstrated that both a particular axiom system for the functionally complete three-valued logic and a certain general method of constructing axiom systems for finitely many-valued logics do not satisfy some salient metalogical requirements. Firstly, we will examine an axiom system for the functionally complete three-valued logic based on the well-known Mordchaj Wajsberg axiom system for the three-valued logic of Jan Łukasiewicz. The examined axiom system was introduced in 1936, by Jerzy Slupecki. As we will see, this axiom system is not semantically complete. Then, we will investigate a method of constructing axiom systems for standard many-valued propositional logics introduced by John B. Rosser and Atwell R. Turquette. The Rosser-Turquette method is considered to be a solution of the problem of axiomatizability of a particular class of many-valued logics, i.e., standard many-valued propositional logics, including finitely many-valued propositional logics of Łukasiewicz and Emil Post. However, we will demonstrate that the Rosser-Turquette method fails to produce adequate axiom systems for a particular class of many-valued propositional logics, i.e., for finitely many-valued propositional logics of Łukasiewicz. The presented investigations concern the mutual definability of the Rosser-Turquette standard connectives and the connectives of finitely many-valued propositional logics of Łukasiewicz. The conclusion is that every Rosser-Turquette axiom system for finitely many-valued propositional logics of Łukasiewicz that satisfies a necessary condition of being a sound system, is semantically incomplete.