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This is the second of a series of papers that investigate fragments of quasi-Nelson logic (QNL) from an algebraic logic standpoint. QNL, recently introduced as a common generalization of intuitionistic and Nelson's constructive logic with strong negation, is the axiomatic extension of the substructural logic |$FL_{ew}$| (full Lambek calculus with exchange and weakening) by the Nelson axiom. The algebraic counterpart of QNL (quasi-Nelson algebras) is a class of commutative integral residuated lattices (a.k.a. |$FL_{ew}$| -algebras) that includes both Heyting and Nelson algebras and can be characterized algebraically in several alternative ways. The present paper focuses on the algebraic counterpart (a class we dub quasi-Nelson implication algebras , QNI-algebras) of the implication–negation fragment of QNL, corresponding to the connectives that witness the algebraizability of QNL. We recall the main known results on QNI-algebras and establish a number of new ones. Among these, we show that QNI-algebras form a congruence-distributive variety (Cor. 3.15) that enjoys equationally definable principal congruences and the strong congruence extension property (Prop. 3.16); we also characterize the subdirectly irreducible QNI-algebras in terms of the underlying poset structure (Thm. 4.23). Most of these results are obtained thanks to twist representations for QNI-algebras, which generalize the known ones for Nelson and quasi-Nelson algebras; we further introduce a Hilbert-style calculus that is algebraizable and has the variety of QNI-algebras as its equivalent algebraic semantics. [ABSTRACT FROM AUTHOR]

A recent paper by Jakl, Jung and Pultr (2016, Electron. Notes Theor. Comput. Sci. , 325, 201–219) succeeded for the first time in establishing a very natural link between bilattice logic and the duality theory of d -frames and bitopological spaces. In this paper we further exploit, extend and investigate this link from an algebraic and a logical point of view. In particular, we introduce classes of algebras that extend bilattices, d -frames and N4-lattices (the algebraic counterpart of Nelson's paraconsistent logic) to a setting in which the negation is not necessarily involutive, and we study corresponding logics. We provide product representation theorems for these algebras, as well as completeness, algebraizability (and some non-algebraizability) results for the corresponding logics. [ABSTRACT FROM AUTHOR]

We introduce a family of modal expansions of Belnap-Dunn four-valued logic and related systems, and interpret them in many-valued Kripke structures. Using algebraic logic techniques and topological duality for modal algebras, and generalizing the so-called twist-structure representation, we axiomatize by means of Hilbert-style calculi the least modal logic over the four-element Belnap lattice and some of its axiomatic extensions. We study the algebraic models of these systems, relating them to the algebraic semantics of classical multi-modal logic. This link allows us to prove that both local and global consequence of the least four-valued modal logic enjoy the finite model property and are therefore decidable. [ABSTRACT FROM AUTHOR]

We introduce a new Priestley-style topological duality for N4-lattices, which are the algebraic counterpart of paraconsistent Nelson logic. Our duality differs from the existing one, due to S. Odintsov, in that we only rely on Esakia duality for Heyting algebras and not on the duality for De Morgan algebras of Cornish and Fowler. A major advantage of our approach is that we obtain a simple description for our topological structures, which allows us to extend the duality to other algebraic structures such as N4-lattices with monotonic modal operators, and also to provide a neighbourhood semantics for the non-normal modal logic corresponding to these algebras. [ABSTRACT FROM AUTHOR]

We introduce a class of algebras, called twist-structures, whose members are built as special squares of an arbitrary residuated lattice. We show how our construction relates to and encompasses results obtained by several authors on the algebraic semantics of non-classical logics. We define a logic that corresponds to our twist-structures and show how to expand it with modal operators, obtaining a paraconsistent many-valued modal logic that generalizes existing work on modal expansions of both Belnap–Dunn logic and paraconsistent Nelson logic. [ABSTRACT FROM AUTHOR]

Bilattices, introduced by Ginsberg (1988, Comput. Intell., 265–316) as a uniform framework for inference in artificial intelligence, are algebraic structures that proved useful in many fields. In recent years, Arieli and Avron (1996, J. Logic Lang. Inform., 5, 25–63) developed a logical system based on a class of bilattice-based matrices, called logical bilattices, and provided a Gentzen-style calculus for it. This logic is essentially an expansion of the well-known Belnap–Dunn four-valued logic to the standard language of bilattices. Our aim is to study Arieli and Avron’s logic from the perspective of abstract algebraic logic (AAL). We introduce a Hilbert-style axiomatization in order to investigate the properties of the algebraic models of this logic, proving that every formula can be reduced to an equivalent normal form and that our axiomatization is complete w.r.t. Arieli and Avron’s semantics. In this way, we are able to classify this logic according to the criteria of AAL. We show, for instance, that it is non-protoalgebraic and non-self-extensional. We also characterize its Tarski congruence and the class of algebraic reducts of its reduced generalized models, which in the general theory of AAL is usually taken to be the algebraic counterpart of a sentential logic. This class turns out to be the variety generated by the smallest non-trivial bilattice, which is strictly contained in the class of algebraic reducts of logical bilattices. On the other hand, we prove that the class of algebraic reducts of reduced models of our logic is strictly included in the class of algebraic reducts of its reduced generalized models. Another interesting result obtained is that, as happens with some implicationless fragments of well-known logics, we can associate with our logic a Gentzen calculus which is algebraizable in the sense of Rebagliato and Verdú (1995, Algebraizable Gentzen Systems and the Deduction of Theorem for Gentzen Systems) (even if the logic itself is not algebraizable). We also prove some purely algebraic results concerning bilattices, for instance that the variety of (unbounded) distributive bilattices is generated by the smallest non-trivial bilattice. This result is based on an improvement of a theorem by Avron (1996, Math. Struct. Comput. Sci., 6, 287–299) stating that every bounded interlaced bilattice is isomorphic to a certain product of two bounded lattices. We generalize it to the case of unbounded interlaced bilattices (of which distributive bilattices are a proper subclass). [ABSTRACT FROM PUBLISHER]

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This paper deals with a family of multivalued logics originally proposed by Matthew Ginsberg as a uniform approach to inference in Artificial Intelligence. Ginsberg generalizes Belnap's four-valued logic introducing the notion of bilattices, which are algebraic structures that contain two partial orders simultaneously. Using bilattice-based logics it is possible to unify a variety of existing inference systems, such as McCarthy's Circumscription, De Kleer's ATMS, Reiter's Default Logic, Moore's Autoepistemic Logic, and others. Moreover, it has been shown that Ginsberg's system, when implemented, is at least as computationally efficient as the original ones, in many cases even more. We begin with a brief survey of the applications of multivalued logics to Artificial Intelligence and computer science developed during the last decades (§ 1); afterwards we focus on Ginsberg's system, considering its basic principles and some of the advantages it offers, both from the theoretical and the practical points of view (§ 2). Then we describe in greater detail the structure of bilattices (§ 3) and of the bilattice-based inference systems (§ 4); we present two examples, showing how Ginsberg's formalism deals respectively with classical logic (§ 5) and with Reiter's Default Logic (§ 6). Finally, we consider some further applications which bilattices have found in recent years, both inside and outside the field of Artificial Intelligence, and discuss future perspectives (§ 7). [ABSTRACT FROM AUTHOR]