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arXiv · 2606.03133

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

Kai Duo
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Abstract

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.

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Kai Duo. 2026-06-02. The approximate strong completeness of the hypersequent calculus $\text{G\L}\forall$. https://arxiv.org/abs/2606.03133

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