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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]

The twist-structure construction is used to represent algebras related to non-classical logics (e.g., Nelson algebras, bilattices) as a special kind of power of better-known algebraic structures (distributive lattices, Heyting algebras). We study a specific type of twist-structure (called implicative twist-structure) obtained as a power of a generalized Boolean algebra, focusing on the implication-negation fragment of the usual algebraic language of twist-structures. We prove that implicative twist-structures form a variety which is semisimple, congruence-distributive, finitely generated, and has equationally definable principal congruences. We characterize the congruences of each algebra in the variety in terms of the congruences of the associated generalized Boolean algebra. We classify and axiomatize the subvarieties of implicative twist-structures. We define a corresponding logic and prove that it is algebraizable with respect to our variety. [ABSTRACT FROM AUTHOR]

Abstract: Trilattices are algebraic structures introduced ten years ago into logic with the aim to provide a uniform framework for the notions of constructive truth and constructive falsity. In more recent years, trilattices have been used to introduce a number of many-valued systems that generalize the Belnap–Dunn logic of first-degree entailment, proposed as logics of how several computers connected together in a network should think in order to deal with incomplete and possibly contradictory information. The aim of the present work is to develop a first purely algebraic study of trilattices, focusing in particular on the problem of representing certain subclasses of trilattices as special products of bilattices. This approach allows to extend the known representation results for interlaced bilattices to the setting of trilattices and to reduce many algebraic problems concerning these new structures to the better-known framework of lattice theory. [Copyright &y& Elsevier]

We introduce a new product bilattice construction that generalizes the well-known one for interlaced bilattices and others that were developed more recently, allowing to obtain a bilattice with two residuated pairs as a certain kind of power of an arbitrary residuated lattice. We prove that the class of bilattices thus obtained is a variety, give a finite axiomatization for it and characterize the congruences of its members in terms of those of their lattice factors. Finally, we show how to employ our product construction to define first-order definable classes of bilattices corresponding to any first-order definable subclass of residuated lattices. [ABSTRACT FROM AUTHOR]

The paper contains some algebraic results on several varieties of algebras having an (interlaced) bilattice reduct. Some of these algebras have already been studied in the literature (for instance bilattices with conflation, introduced by M. Fitting), while others arose from the algebraic study of O. Arieli and A. Avron's bilattice logics developed in the third author's PhD dissertation. We extend the representation theorem for bounded interlaced bilattices (proved, among others, by A. Avron) to unbounded bilattices and prove analogous representation theorems for the other classes of bilattices considered. We use these results to establish categorical equivalences between these structures and well-known varieties of lattices. [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]

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]

Abstract We introduce a proper multi-type display calculus for bilattice logic (with conflation) for which we prove soundness, completeness, conservativity, standard subformula property and cut elimination. Our proposal builds on the product representation of bilattices and applies the guidelines of the multi-type methodology in the design of display calculi. [ABSTRACT FROM AUTHOR]