Twist Sequent Calculi for S4 and its Neighbors

TL;DR

This paper introduces novel twist sequent calculi for the modal logic S4 that avoid standard negation rules, enabling efficient proofs for negated formulas and extending to other modal logics like S5.

Contribution

The paper presents new twist sequent calculi for S4 and related modal logics, with cut-elimination and subformula properties, and explores their application to negated formulas.

Findings

Cut-elimination theorems established for the calculi.

Calculi enable short proofs for complex negated modal formulas.

Extension of calculi to other modal logics like S5.

Abstract

Two Gentzen-style twist sequent calculi for the normal modal logic S4 are introduced and investigated. The proposed calculi, which do not employ the standard logical inference rules for the negation connective, are characterized by several twist logical inference rules for negated logical connectives. Using these calculi, short proofs can be generated for provable negated modal formulas that contain numerous negation connectives. The cut-elimination theorems for the calculi are proved, and the subformula properties for the calculi are also obtained. Additionally, Gentzen-style twist (hyper)sequent calculi for other normal modal logics including S5 are considered.