Displaying Modal Logic

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This is an introduction to display logic in the context of generalized Gentzen calculi. After reviewing several standard and non-standard sequent-style proof systems for modal logics, the author carefully motivates and develops display logic, an important refinement of Gentzen’s sequent calculus devised by N.Belnap. The text proves a general strong cut-elimination theorem that covers a large class of display sequent calculi. Moreover, a proof-theoretic semantics of the modal operators is developed. Proof-theoretic characterizations are also obtained for the logical operations of systems associated with Tarskian structured consequence relations. These systems include constructive logics with strong negation. Using the embedding of intuitionistic logic in S4, display calculi are presented for certain subintuitionistic logics that may be used as monotonic base systems for semantics-based non-monotonic reasoning. Eventually, a first-order display calculus is defined. Its modal extension is general enough to avoid the provability of both the Barcan formula and its converse.