Super - Lukasiewicz Propositional Logics

In [8] (1920), Lukasiewicz introduced a 3-valued propositional calculus with one designated truth-value and later in [9], Lukasiewicz and Tarski generalized it to an m-valued propositional calculus (where m is a natural number or ^ 0 ) with one designated truth-value. For the original 3-valued propositional calculus, an axiomatization was given by Wajsberg [16] (1931). In a case of m Φ ^0> Rosser and Turquette gave an axiomatization of the m-valued propositional calculus with an arbitrary number of designated truth-values in [13] (1945). In [9], Lukasiewicz conjectured that the ^o-valued propositional calculus is axiomatizable by a system with modus ponens and substitution as inference rules and the following five axioms: p =) q Z> p, (p ID q) D (q ID r) 3 p ID r, p V q 3 q V p, (p ID q) V (q Z> p), (~p Z) ~ q) ID q ID p. Here we use P V Q as the abbreviation of (P 3 Q) Z) Q. We associate to the right and use the convention that 3 binds less strongly than V. In [15] p. 51, it is stated as follows: "This conjecture has proved to be correct; see Wajsberg [17] (1935) p. 240. As far as we know, however, Wajsberg's proof has not appeared in print." Rose and Rosser gave the first proof of it in print in [12] (1958). Their proof was essentially due to McNaughton's theorem [10], so it was metamathematical in nature. An algebraic proof was given by Chang [1] [2] (1959). On the other hand, Rose [11] (1953) showed that the cardinality of the set of all super-Lukasiewicz propositional logics is ^ 0 . Surprisingly it was before Rose and Rosser's completeness theorem [12]. The proof in Rose [11] was also due to McNaughton's theorem. Some of our theorems in this paper have already been obtained by Rose [11], But our proofs are completely algebraic. In our former paper [5], we gave a complete description of super-