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

Summary

LetL ^c₃ be the smallest set of propositional formulas, which contains

• 1. CpCqp

• 2. CCCpqCrqCCqpCrp

• 3. CCCpqCCqrqCCCpqpp

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.