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