Twist Sequent Calculi for S4 and its Neighbors

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.