Studia Logica

, Volume 88, Issue 3, pp 325-348

First online:

Constructive Logic with Strong Negation is a Substructural Logic. I

  • Matthew SpinksAffiliated withDepartment of Education, University of Cagliari Email author 
  • , Robert VeroffAffiliated withDepartment of Computer Science, University of New Mexico

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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 .


Constructive logic strong negation substructural logic Nelson algebra FL ew -algebra residuated lattice