Negation and Implication in Quasi-Nelson Logic

Quasi-Nelson logic is a recently-introduced generalization of Nelson’s constructive logic with strong negation to a non-involutive setting. In the present paper we axiomatize the negation-implication fragment of quasi-Nelson logic (QNI-logic), which constitutes in a sense the algebraizable core of quasi-Nelson logic. We introduce a finite Hilbert-style calculus for QNI-logic, showing completeness and algebraizability with respect to the variety of QNI-algebras. Members of the latter class, also introduced and investigated in a recent paper, are precisely the negation-implication subreducts of quasi-Nelson algebras. Relying on our completeness result, we also show how the negation-implication fragments of intuitionistic logic and Nelson’s constructive logic may both be obtained as schematic extensions of QNI-logic.