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 Retoré 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.

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In this paper we investigate Hughes' combinatorial proofs as a notion of proof identity for classical logic. We show for various syntactic formalisms including sequent calculus, analytic tableaux, and resolution, how they can be translated into combinatorial proofs, and which notion of identity they enforce. This allows the comparison of proofs that are given in different formalisms.

Hughes' combinatorial proofs give canonical representations for classical logic proofs. In this paper we characterize classical combinatorial proofs which also represent valid proofs for relevant logic with and without the mingle axiom. Moreover, we extend our syntax in order to represent combinatorial proofs for the more restrictive framework of entailment logic.