Purity through Unravelling
نویسندگان
چکیده
We divide attempts to give the structural proof theory of modal logics into two kinds, those pure formulationswhose inference rules characterise modality completely by means of manipulations of boxes and diamonds, and those labelled formulations that leverage the use of labels in giving inference rules. The widespread adoption of labelled formulations is driven by their ability to model features of the model theory of modal logic in its proof theory. We describe here an approach to the structural proof theory of modal logic that aims to bring under one roof the benefits of both the pure and the labelled formulations. We introduce two proof calculi, one labelled sequent formulation and one pure formulation in the calculus of structures that are shown to be in a systematic correlation, where the latter calculus uses deep inference with shaped modal rules to capture in a pure manner the manipulations that the former calculations mediates through the use of labels. We situate this work within a larger investigation into the proof theory of modal logic that solves problems that existed with the earlier investigation based on prefix modal rules. We hold this development provides yet stronger evidence justifying the claim that good, pure proof theory for modal logic needs deep inference.
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