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.
We take a glimpse at the relation between WNMalgebras (algebraic models of the well-known Weak Nilpotent Minimum logic) and quasi-Nelson algebras, a non-involutive generalisation of Nelson algebras (models of Nelson's constructive logic with strong negation) that was introduced in a recent paper. We show that the two varieties can be related via the twist-structure construction, obtaining a new representation for a subvariety of WNM-algebras that includes the involutive ones (i.e. NM-algebras). Our results imply, in particular, that every pre-linear quasi-Nelson algebra is a WNM-algebra; we thus generalize the known result that the class of pre-linear Nelson algebras coincides with that of NM-algebras (models of Nilpotent Minimum logic).
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