Pavelka-style completeness in expansions of \L ukasiewicz logic

TL;DR

This paper establishes Pavelka-style completeness for certain extended Lukasiewicz logics using algebraic methods, focusing on the properties of MV-algebras and their expansions with additional connectives and constants.

Contribution

It introduces an algebraic framework demonstrating Pavelka completeness for expansions of Lukasiewicz logic, leveraging the injectivity of the standard MV-algebra.

Findings

Proves Pavelka completeness for logic with product MV-algebras

Establishes completeness for divisible MV-algebras

Uses algebraic properties of MV-algebras to support logical completeness

Abstract

An algebraic setting for the validity of Pavelka style completeness for some natural expansions of \L ukasiewicz logic by new connectives and rational constants is given. This algebraic approach is based on the fact that the standard MV-algebra on the real segment $[0, 1]$ is an injective MV-algebra. In particular the logics associated with MV-algebras with product and with divisible MV-algebras are considered.