A proof of axiomatizability of łukasiewicz’s three-valued implicational propositional calculus

Abstract

LetL 3 c be the smallest set of propositional formulas, which contains and is closed with respect to substitution and detachment. Let $$\mathfrak{M}_3^c $$ be Łukasiewicz’s three-valued implicational matrix defined as follows:cxy=min (1,1−x+y), where $$x,y \in \{ 0,\tfrac{1}{2},1\}$$ . In this paper the following theorem is proved: $$L_3^c = E( \mathfrak{M}_3^c )$$ The idea used in the proof is derived from Asser’s proof of completeness of the two-valued propositional calculus. The proof given here is based on the Pogorzelski’s deduction theorem fork-valued propositional calculi and on Lindenbaum’s theorem.