A cut-free sequent calculus for the logic of subset spaces

Following the tradition of labelled sequent calculi for modal logics, we present a one-sided, cut-free sequent calculus for the bimodal logic of subset spaces. In labelled sequent calculi, semantical notions are internalised into the calculus, and we take care to choose them close to the original interpretation of the system. To achieve this, we introduce a variation of the standard method, considering structured labels instead of simple tokens, in our particular case pairs of labels. With this new device, we can formulate a calculus with extremely simple frame rules and good proof-theoretical properties. The logical rules are invertible, structural rules are admissible. We show the admissibility of cut and relate our system to the well-known Hilbert-style axiomatisation of the logic. Finally, we present a direct proof of completeness based on proof search.