Fitch-Style Modal Necessity as a Substructural Sequent-Style System

The question of defining a Jaskowski-Fitch natural deduction system for modal logic has been settled since the very introduction of such formalisms in the middle of the last century. In contrast, a sequent-style formulation of this approach has only been discussed since the turn of this century but exclusively from the point of view of type theories. In this paper we propose a substructural sequent-style deductive system, based on previous ideas by Borghuis and Clouston, which captures Fitch-style for modal logic in a faithful way, meaning that the features of the original diagrammatic proofs are enforced by the sequent rules. This answers the question of what is a sequent-style version of Fitch-style natural deduction in the case of the necessity fragment of minimal modal logic S4.