The approximate strong completeness of the hypersequent calculus $\text{GŁ}\forall$

Abstract

An analytic Gentzen-style proof system of first-order Łukasiewicz logic, hypersequent calculus $\text{GŁ}\forall$, and its approximate completeness have been extensively studied. In this paper, we prove the approximate strong completeness of $\text{GŁ}\forall$ by a labelled tableau method. As applications, we prove a variant of Gentzen's midsequent theorem in $\text{GŁ}\forall$ and an approximate Herbrand's theorem. We also introduce a new cut rule $(\text{s-Cut})$ of $\text{GŁ}\forall$ and show the approximate strong completeness of $\text{GŁ}\forall+(\text{s-Cut})$.