Constructive logic with strong negation as a substructural logic

Abstract

Spinks and Veroff have shown that constructive logic with strong negation (CLSN for short), can be considered as a substructural logic. We use algebraic tools developed to study substructural logics to investigate some axiomatic extensions of CLSN. For instance, we prove that Nilpotent minimum logic is the extension of CLSN by the prelinearity axiom. This generalizes the well-known result by Monteiro and Vakarelov that three-valued ukasiewicz logic is an extension of CLSN. A Glivenko-like theorem relating CLSN and three-valued ukasiewicz logic is proved.