On the Methods of Constructing Hilbert-type Axiom Systems for Finite-valued Propositional Logics of Łukasiewicz

Abstract

The article explores the following question: which among the most often examined in the literature method of constructing Hilbert-type axiom systems for finite-valued propositional logics of Łukasiewicz – the Rosser-Turquette method, the Tuziak method or the Grigolia system of axiom schemata – is in fact applicable in the researches on many-valued logics? Although the method offered by Rosser and Turquette is considered to be a solution of the problem of axiomatizability of finite-valued logics of Łukasiewicz, it has a serious limitation in producing adequate axiom systems. If the Rosser-Turquette axiom schemata are expressed just in terms of the standard propositional connectives which are definable in terms of the propositional connectives of Łukasiewicz’s logics, then every Rosser-Turquette axiom system for a Łukasiewicz’s logic is semantically incomplete. The article also examines the Tuziak axiom systems that actually axiomatize Łukasiewicz’s finite-valued logics and can be applied in the logical researches. The article compares the Tuziak axiom systems with the Grigolia set of axiom schemata, and it demonstrates that since a crucial in the Grigolia method definition of connectives is in fact invalid in Łukasiewicz’s logics, the Grigolia method does not provide sound axiom systems for Łukasiewicz’s logics.