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

The hypersequent calculus $\text{GŁ}\mathrm{\forall }$, an analytic Gentzen-style proof system of first-order Łukasiewicz logic, and its approximate completeness have been extensively studied. In this paper, we prove the approximate strong completeness of $\text{GŁ}\mathrm{\forall }$ by a labelled tableau method. Then we introduce a sequent-level cut rule (s-Cut) and show the approximate strong completeness of $\text{GŁ}\mathrm{\forall }+(\text{s-Cut})$. As applications, we establish the compactness of approximate $[0,1]$-consequence, a variant of Gentzen’s mid-sequent theorem of $\text{GŁ}\mathrm{\forall }$ and an approximate Herbrand’s theorem of first-order Łukasiewicz logic.