A Completeness Theorem for a Functionally Complete Łukasiewicz Logic

Abstract

We present a new axiomatization for a truth-functionally complete version of Ł3, a three-valued propositional logic, using two propositional constants and the Łukasiewicz implication as primitive symbols. We develop a corresponding proof system that incorporates Łukasiewicz’s axioms, a variant of Słupecki’s postulates for a 0-ary connective, and some properties of the Baaz delta operation. Using a weakened Deduction Theorem and adapting Henkin’s method to this three-valued setting, we establish a Completeness Theorem for the system.