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The logic of linear functors

Published online by Cambridge University Press:  16 September 2002

R. BLUTE
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, 585 King Edward St, Ottawa, ON, K1N6N5, Canada. Email: rblute@mathstat.uottawa.ca
J. R. B. COCKETT
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, 585 King Edward St, Ottawa, ON, K1N6N5, Canada. Email: rblute@mathstat.uottawa.ca
R. A. G. SEELY
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, 585 King Edward St, Ottawa, ON, K1N6N5, Canada. Email: rblute@mathstat.uottawa.ca

Abstract

This paper describes a family of logics whose categorical semantics is based on functors with structure rather than on categories with structure. This allows the consideration of logics that contain possibly distinct logical subsystems whose interactions are mediated by functorial mappings. For example, within one unified framework, we shall be able to handle logics as diverse as modal logic, ordinary linear logic, and the ‘noncommutative logic’ of Abrusci and Ruet, a variant of linear logic that has both commutative and noncommutative connectives.

Although this paper will not consider in depth the categorical basis of this approach to logic, preferring instead to emphasise the syntactic novelties that it generates in the logic, we shall focus on the particular case when the logics are based on a linear functor, in order to give a definite presentation of these ideas. However, it will be clear that this approach to logic has considerable generality.

Type
Research Article
Copyright
2002 Cambridge University Press

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