Strong standard completeness theorems for S5-modal Lukasiewicz logics

TL;DR

This paper establishes strong completeness theorems for S5-modal Lukasiewicz logics by developing a finitary calculus and extending it with an infinitary rule, supported by algebraic properties.

Contribution

It introduces a finitely strongly complete propositional calculus for S5-modal Lukasiewicz logic and extends it to achieve full strong completeness using algebraic methods.

Findings

Finitary propositional calculus is finitely strongly complete.

Extension with an infinitary rule achieves strong completeness.

Results are based on properties of monadic MValgebras.

Abstract

We study the S5-modal expansion of the logic based on the Lukasiewicz t-norm. We exhibit a finitary propositional calculus and show that it is finitely strongly complete with respect to this logic. This propositional calculus is then expanded with an infinitary rule to achieve strong completeness. These results are derived from properties of monadic MValgebras: functional representations of simple and finitely subdirectly irreducible algebras, as well as the finite embeddability property. We also show similar completeness theorems for the extension of the logic based on models with bounded universe.