On the Rosser-Turquette method of constructing axiom systems for finitely many-valued propositional logics of Łukasiewicz

摘要

A method of constructing Hilbert-type axiom systems for standard many-valued propositional logics was offered by Rosser and Turquette. Although this method is considered to be a solution of the problem of axiomatisability of a wide class of many-valued logics, the article demonstrates that it fails to produce adequate axiom systems. The article concerns finitely many-valued propositional logics of Łukasiewicz. It proves that if standard propositional connectives of the Rosser-Turquette axiom systems are definable in terms of the propositional connectives of Lukasiewicz's logics, and thus, they are normal ones, then every Rosser-Turquette axiom system for a finite-valued Lukasiewicz's logic is semantically incomplete.