Abstract This paper investigates recovery operators in quasi-Nelson (QN) logic, the algebraizable logical counterpart of QN algebras. These form a variety of three-potent, distributive, but not necessarily involutive residuated lattices that may be regarded as a common generalization of Nelson and Heyting algebras. We consider both consistency and determinedness operators, with a particular focus on logics and algebras that satisfy the prelinearity condition, which is well-known in the area of mathematical fuzzy logics. We show that, essentially, all algebraic and logical results already proved for (prelinear, distributive) involutive residuated lattice-based logics of formal inconsistency/logics of formal undeterminedness can be recovered in the QN setting, where one dispenses with the involutivity assumption. In this setting, consistency and undeterminedness operators are no longer duals of one another, and hence call for a more fine-grained algebraic and logical formalization.

Abstract We consider a family of non-classical three-valued logics proposed to model indicative conditionals in natural language. Among these, systems introduced by B. De Finetti, W.S. Cooper, J. Cantwell and R.J. Farrell, as well as some variants that have not appeared in the literature, but seem nevertheless to be natural objects of interest from a formal point of view. Most of these logics are not easily treatable with the standard techniques of algebraic logic. We therefore resort to non-deterministic structures and multiple-conclusion calculi to provide alternative semantical characterizations and axiomatizations. In the best cases—logics given by a finite monadic matrix—this can be done directly, in a modular way, through a procedure due to Shoesmith and Smiley. In the more involved ones—logics preserving degrees of truth—some ingenuity and more sophisticated techniques are required. We characterize these logics by a partial non-deterministic matrix, and show how to produce analytic (and effective) calculi that are complete with respect to this generalized semantics. In all cases, the calculi thus obtained can be straightforwardly converted, by a uniform procedure, into traditional single-conclusion Hilbert-style axiomatizations.

Abstract Perfect paradefinite algebras are De Morgan algebras expanded with an operation that allows for the full behavior of classical negation to be restored. They form a variety that is term-equivalent to the variety of involutive Stone algebras. Their associated multiple-conclusion (Set-Set) and single-conclusion ( ) order-preserving logics are non-algebraizable self-extensional logics of formal inconsistency and undeterminedness determined by a six-valued matrix. We studied these logics extensively in Gomes et al. ((2022). Electronic Proceedings in Theoretical Computer Science357 56–76.) from both the algebraic and the proof-theoretical perspectives. In the present paper, we continue that study by investigating directions for conservatively expanding these logics with an implication connective (essentially, one that admits the deduction-detachment theorem). We first consider logics given by very simple and manageable non-deterministic semantics whose implication (in isolation) is classical. These, nevertheless, fail to be self-extensional. We then consider the implication realized by the relative pseudo-complement over the six-valued perfect paradefinite algebra. Our strategy is to expand the language of the latter algebra with this connective and study the (self-extensional) Set-Set and order-preserving and $\top$ -assertional logics of the variety induced by the resulting algebra. We provide axiomatizations for such new variety and for such logics, drawing parallels with the class of symmetric Heyting algebras and with Moisil’s “symmetric modal logic.” For the order-preserving Set-Set logic, in particular, we obtain a Set-Set axiomatization that is analytic. We close by studying interpolation properties for these logics and concluding that the new variety has the Maehara amalgamation property.

Abstract The variety of quasi-Nelson algebras (QNAs) has been recently introduced and characterised in several equivalent ways: among others, as (1) the class of bounded commutative integral (but non-necessarily involutive) residuated lattices satisfying the Nelson identity, as well as (2) the class of (0, 1)-congruence orderable commutative integral residuated lattices. Logically, QNAs are the algebraic counterpart of quasi-Nelson logic, which is the (algebraisable) extension of the substructural logic ℱℒew (Full Lambek calculus with Exchange and Weakening) by the Nelson axiom. In the present paper, we collect virtually all the results that are currently known on QNAs, including solutions to certain questions left open in earlier publications. Furthermore, we extend our study to some subreducts of QNAs, that is, classes of algebras corresponding to fragments of the algebraic language obtained by eliding either the implication or the lattice operations.

We take a glimpse at the relation between WNM-algebras (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).

In this paper we present a subvariety of the naBL-algebras based on overlaps functions, as well as a version of the well-known Chinese Remainder Theorem for naBL-algebras. Moreover notions of automorphism and its action on overlaps is used to obtain naBL-conjugated algebras.

Este es el manuscrito aceptado del artículo. La versión registrada fue publicada por primera vez en Journal of Logic and Computation, 27, 2017, pp. 155-199, está disponible en línea en el sitio web del editor: https://doi.org/10.1093/logcom/exv038 This is the accepted manuscript of the article. The registered version was first published in Journal of Logic and Computation, 27, 2017, pp. 155-199, is available online at the publisher's website: https://doi.org/10.1093/logcom/exv038

We employ the well-developed and powerful techniques of algebraic semantics and Priestley duality to set up a Kripke semantics for a modal expansion of Arieli and Avron's bilattice logic, itself based on Belnap's four-valued logic. We obtain soundness and completeness of a Hilbert-style derivation system for this logic with respect to four-valued Kripke frames, the standard notion of model in this setting. The proof is via intermediary relational structures which are analysed through a topological reading of one of the axioms of the logic. Both local and global consequence on the models are covered.