A note on Lukasiewicz’s three-valued logic

Abstract

It is well known that Lukasiewicz’s three-valued-logic L3 admits – unlike classical logic – the definition of two non trivial, truth-functional modal operators and ∆. We address the question of finding a convenient syntactic characterization of the “modal content” of L3. To this aim, we consider Wajsberg’s axiomatization of L3 (the calculus W) and prove its equivalence with a modal calculus W_ which, essentially, includes: the BCK+double negation schemas, the characteristic modal schemas of S5 (K;T; 4;B), full contraction for boxed formulas and the “partial collapse” schema α → (α →α). As applications, we obtain a simple and natural completeness proof à la Lindenbaum for W, as well as a considerable simplification of Wajsberg’s original, ingenious completeness proof.