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The Logic of Bunched Implications

Published online by Cambridge University Press:  15 January 2014

Peter W. O'Hearn  and
David J. Pym
Peter W. O'Hearn
Affiliation:
Department of Computer Science, Queen Mary & Westfield College, University of LondonLondon E1 4NS, UKE-mail:ohearn@dcs.qmw.ac.uk
David J. Pym
Affiliation:
Department of Computer Science, Queen Mary & Westfield College, University of London LondonE1 4NS, UKE-mail:pym@dcs.qmw.ac.uk
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Abstract

We introduce a logic BI in which a multiplicative (or linear) and an additive (or intuitionistic) implication live side-by-side. The propositional version of BI arises 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 of BI can be seen categorically: models of propositional BI'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 of BI includes, in addition to standard additive quantifiers, multiplicative (or intensional) quantifiers and which 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.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 5 , Issue 2 , June 1999 , pp. 215 - 244
Copyright
Copyright © Association for Symbolic Logic 1999

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