On Combinatorial Proofs for Modal Logic

In this paper we extend Hughes' combinatorial proofs to modal logics. The crucial ingredient for modeling the modalities is the use of a self-dual non-commutative operator that has first been observed by Retore through pomset logic. Consequently, we had to generalize the notion of skew fibration from cographs to Guglielmi's relation webs. Our main result is a sound and complete system of combinatorial proofs for all normal and non-normal modal logics in the S4-tesseract. The proof of soundness and completeness is based on the sequent calculus with some added features from deep inference.