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AFFINE LOGIC FOR CONSTRUCTIVE MATHEMATICS

Part of: Proof theory and constructive mathematics

Published online by Cambridge University Press:  21 July 2022

MICHAEL SHULMAN
MICHAEL SHULMAN*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF SAN DIEGO SAN DIEGO, CA 92110, USA E-mail: shulman@sandiego.edu
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Abstract

We show that numerous distinctive concepts of constructive mathematics arise automatically from an “antithesis” translation of affine logic into intuitionistic logic via a Chu/Dialectica construction. This includes apartness relations, complemented subsets, anti-subgroups and anti-ideals, strict and non-strict order pairs, cut-valued metrics, and apartness spaces. We also explain the constructive bifurcation of some classical concepts using the choice between multiplicative and additive affine connectives. Affine logic and the antithesis construction thus systematically “constructivize” classical definitions, handling the resulting bookkeeping automatically.

Keywords

linear logicaffine logicconstructive mathematics

MSC classification

Primary: 03F52: Linear logic and other substructural logics
Secondary: 03F65: Other constructive mathematics

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Type
Articles
Information
Bulletin of Symbolic Logic , Volume 28 , Issue 3 , September 2022 , pp. 327 - 386
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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