Approximate Completeness of Hypersequent Calculus for First-Order {\L}ukasiewicz Logic

TL;DR

This paper extends the hypersequent calculus G{ extbackslash}Lorall for first-order { extbackslash}Lukasiewicz logic, proving its approximate completeness for all formulas, not just prenex, by generalizing previous proofs.

Contribution

It provides a generalized proof of approximate completeness of G{ extbackslash}Lorall for arbitrary first-order formulas, expanding prior results limited to prenex formulas.

Findings

Proves approximate completeness for all first-order formulas

Generalizes previous completeness proofs to hypersequents

Enhances understanding of hypersequent calculus in fuzzy logic

Abstract

Hypersequent calculus G{\L}$\forall$ for first-order {\L}ukasiewicz logic was first introduced by Baaz and Metcalfe, along with a proof of its approximate completeness with respect to standard $[0,1]$-semantics. The completeness result was later pointed out by Gerasimov that it only applies to prenex formulas. In this paper, we will present our proof of approximate completeness of G{\L}$\forall$ for arbitrary first-order formulas by generalizing the original completeness proof to hypersequents.