Bulletin of Symbolic Logic 5 (2):215-244 (1999)

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 ∀ new and ∃ new 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
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DOI 10.2307/421090
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References found in this work BETA

Entailment: The Logic of Relevance and Necessity.[author unknown] - 1975 - Studia Logica 54 (2):261-266.
Introduction to Higher Order Categorical Logic.J. Lambek & P. J. Scott - 1989 - Journal of Symbolic Logic 54 (3):1113-1114.
Display Logic.Nuel D. Belnap - 1982 - Journal of Philosophical Logic 11 (4):375-417.
Semantics for Relevant Logics.Alasdair Urquhart - 1972 - Journal of Symbolic Logic 37 (1):159-169.

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Citations of this work BETA

From If to Bi.Samson Abramsky & Jouko Väänänen - 2009 - Synthese 167 (2):207 - 230.
The Ambiguity of Quantifiers.Francesco Paoli - 2005 - Philosophical Studies 124 (3):313-330.
Plans, Affordances, and Combinatory Grammar.Mark Steedman - 2002 - Linguistics and Philosophy 25 (5-6):723-753.

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