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Constructive Logic with Strong Negation is a Substructural Logic. I

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Abstract

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL ew of the substructural logic FL ew . In this paper, it is shown that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFL ew (namely, a certain variety of FL ew -algebras) are term equivalent. This answers a longstanding question of Nelson [30]. Extensive use is made of the automated theorem-prover Prover9 in order to establish the result.

The main result of this paper is exploited in Part II of this series [40] to show that the deductive systems N and NFL ew are definitionally equivalent, and hence that constructive logic with strong negation is a substructural logic over FL ew .

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Authors and Affiliations

  1. Department of Education, University of Cagliari, Cagliari, 09123, Italy

    Matthew Spinks

  2. Department of Computer Science, University of New Mexico, Albuquerque, NM, 87131, USA

    Robert Veroff

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  1. Matthew Spinks
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  2. Robert Veroff
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Spinks, M., Veroff, R. Constructive Logic with Strong Negation is a Substructural Logic. I. Stud Logica 88, 325–348 (2008). https://doi.org/10.1007/s11225-008-9113-x

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