The Logic of Bunched Implications

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Abstract
We introduce a logicBIin which a multiplicative (or linear) and an additive (or intuitionistic) implication live side-by-side. The propositional version ofBIarises from an analysis of the proof-theoretic relationship between conjunction and implication; it can be viewed as a merging of intuitionistic logic and multiplicative intuitionistic linear logic. The naturality ofBIcan be seen categorically: models of propositionalBI's proofs are given by bicartesian doubly closed categories, i.e., categories which freely combine the semantics of propositional intuitionistic logic and propositional multiplicative intuitionistic linear logic. The predicate version ofBIincludes, in addition to standard additive quantifiers, multiplicative (or intensional) quantifiersandwhich arise from observing restrictions on structural rules on the level of terms as well as propositions. We discuss computational interpretations, based on sharing, at both the propositional and predicate levels.

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