Salvatore Guccione — Roberto Tortora — Virginia Vaccaro DEDUCTION THEOREMS IN LUKASIEWICZ PROPOSITIONAL CALCULI

As it is well-known, Lukasiewicz developed in [2] the many-valued propositional calculi based on two primitive connectives, negation N and implication C, which were defined semantically. Complete axiomatizations followed later on. Among the works on this argument, we mention Wajsberg [9], Rosser and Turquette [8], Mc Naughton [3], Rose and Rosser [7], Meredith [4], Chang [1], Rose [5], [6]. It is also known that, if two formal systems have the same rules of inference and the same theorems, then the Deduction Theorem holds for the former if and only if it holds for the latter, independently from the axioms of the systems. The aim of the present work is to analyze the rules of inference occurring in the various axiomatizations of the Lukasiewicz propositional calculi, in order to discuss the validity of the Deduction Theorem for them. We emphasize that the Deduction Theorem here discussed is relative to the original Lukasiewicz connective C. The specification is necessary, since in the-