Simple sequent systems for the modal logics K,D,T, and S4

Proof theorists have long been struggling to provide simple accounts of various modal logics. Drawing on the recent literature on metainferences, I develop in the present paper a novel approach to this challenge: regular sequent systems augmented with natural deduction-like rules for assuming and discharging sequents. Based on this approach, I introduce an elegant calculus for the modal logic. Various discharging conditions on assumptions are shown to yield the weaker logics, and. The rules in these calculi have attractive proof-theoretic properties: they are explicit, symmetrical, and non-circular, and they enjoy the subformula property.