A proof for completeness of \L ukasiewicz logic

TL;DR

This paper presents a new Hilbert-style proof demonstrating the completeness of infinite valued propositional ukasiewicz logic, emphasizing the role of maximal consistent extensions that do not necessarily include all formulas or their negations.

Contribution

It introduces a novel proof technique for ukasiewicz logic's completeness, expanding understanding of maximal consistent extensions in many-valued logic.

Findings

New proof of completeness for ukasiewicz logic

Examples illustrating non-inclusion of all formulas in maximal extensions

Clarification of maximal consistent extensions in many-valued logic

Abstract

In this paper we give a new proof for the completeness of infinite valued propositional \L ukasiewicz logic introduced by \L ukasiewicz and Tarski in 1930. Our approach employs a Hilbert-style proof that relies on the concept of maximal consistent extensions, and unlike classical logic, in this context, the maximal extensions are not required to include all formulas or their negations. To illustrate this point, we provide examples of such formulas.