Refutation calculi for lattice-based logics: from display to tableaux

2026-05-23

AI Summary

This study addresses the absence of refutation calculi tailored to lattice-based logics (LE-logics), particularly the lack of display-style refutation systems. The paper presents the first refutation display calculus framework specifically designed for LE-logics, integrating techniques from structural proof theory—namely display calculi, refutation system design, and proof analysis—to establish a sound and complete refutation calculus. Building on this foundation, the authors further develop a systematic translation mechanism from display calculi to semantic tableau calculi, thereby deriving a terminating tableau system. This contribution effectively fills a significant theoretical gap in the proof-theoretic treatment of LE-logics.

Abstract

Refutation calculi are formal systems developed to derive the invalid formulas of a given logic. While the notion of refutation calculi has played a key role in the development of tableaux calculi, a refutation approach to display calculi has not yet been attempted. In this paper, we introduce refutation display calculi for basic LE-logics, i.e., those logics canonically associated with basic normal lattice expansions of any signature. In particular, we prove soundness and completeness via proof-analysis results on derivable sequents. Finally, we obtain terminating tableaux calculi from these refutation display calculi.

Problem

• refutation calculi
• display calculi
• lattice-based logics
• tableaux calculi
• LE-logics

Innovation

• refutation calculi
• display calculi
• LE-logics
• tableaux calculi
• proof analysis