In a previous work we studied, from the perspective ofAlgebraic Logic, the implicationless fragment of a logic introduced by O. Arieli and A. Avron using a class of bilattice-based logical matrices called logical bilattices. Here we complete this study by considering the Arieli-Avron logic in the full language, obtained by adding two implication connectives to the standard bilattice language. We prove that this logic is algebraizable and investigate its algebraic models, which turn out to be distributive bilattices with (...) additional implication operations. We axiomatize and state several results on these new classes of algebras, in particular representation theorems analogue to the well-known one for interlaced bilattices. (shrink)

This paper presents an information-based logic that is applied to the analysis of entailment, implicature and presupposition in natural language. The logic is very fine-grained and is able to make distinctions that are outside the scope of classical logic. It is independently motivated by certain properties of natural human reasoning, namely partiality, paraconsistency, relevance, and defeasibility: once these are accounted for, the data on implicature and presupposition comes quite naturally.The logic is based on the family of semantic spaces known as (...) bilattices, originally proposed by Ginsberg (1988), and used extensively by Fitting (1989, 1992). Specifically, the logic is based on a subset of bilattices that I call evidential bilattices, constructed as the Cartesian product of certain algebras with themselves. The specific details of the epistemic agent approach of the logical system is derived from the work of Belnap (1975, 1977), augmented by the use of evidential links for inferencing. An important property of the system is that it has been implemented using an extension of Fitting's work on bilattice logic programming (1989, 1991) to build a model-based inference engine for the augmented Belnap logic. This theorem prover is very efficient for a reasonably wide range of inferences. (shrink)

This is an exploratory paper whose aim is to investigate the potentialities of bilattice theory for an adequate definition of the deduction apparatus for multi-valued logic. We argue that bilattice theory enables us to obtain a nice extension of the graded approach to fuzzy logic. To give an example, a completeness theorem for a logic based on Boolean algebras is proved.

ABSTRACT A bilattice is said to be regular provided its truth conjunction and disjunction are monotonic with respect to its knowledge ordering. The principal result of this paper is that the following properties of a bilattice B are equivalent: 1. B is regular; 2. the truth conjunction and disjunction of B are definable through the rest of the operations and constants of B; 3. B is isomorphic to a bilattice of the form L 1 · L 2 where L 1 (...) and L 2 are bounded lattices. We also derive from this metaequivalence a number of corollaries concerning distributive bilattices, degenerated bilattices and bilattices with negation. (shrink)

Bilattices, due to M. Ginsberg, are a family of truth value spaces that allow elegantly for missing or conflicting information. The simplest example is Belnap’s four-valued logic, based on classical two-valued logic. Among other examples are those based on finite many-valued logics, and on probabilistic valued logic. A fixed point semantics is developed for logic programming, allowing any bilattice as the space of truth values. The mathematics is little more complex than in the classical two-valued setting, but the result (...) provides a natural semantics for distributed logic programs, including those involving confidence factors. The classical two-valued and the Kripke/Kleene three-valued semantics become special cases, since the logics involved are natural sublogics of Belnap’s logic, the logic given by the simplest bilattice. (shrink)

In this paper we reexamine the question of whether questions areinherently intensional entities. We do so by proposing a novelextensional theory of questions, based on a re-interpretation of thedomain of t as a bilattice rather than the usual booleaninterpretation. We discuss the adequacy of our theory with respect tothe adequacy criteria imposed on the semantics of questionsby (Groenendijk and Stokhof 1997). We show that the theory is able to account in astraightforward manner for some complex issues in the semantics ofquestions (...) including coordinated questions, combined indicative andinterrogative sentences, questions with quantifiers, and theimpossibility of negating questions. (shrink)

Baltag, Moss, and Solecki proposed an expansion of classical modal logic, called logic of epistemic actions and knowledge (EAK), in which one can reason about knowledge and change of knowledge. Kurz and Palmigiano showed how duality theory provides a flexible framework for modeling such epistemic changes, allowing one to develop dynamic epistemic logics on a weaker propositional basis than classical logic (for example an intuitionistic basis). In this paper we show how the techniques of Kurz and Palmigiano can be further (...) extended to define and axiomatize a bilattice logic of epistemic actions and knowledge (BEAK). Our propositional basis is a modal expansion of the well-known four-valued logic of Belnap and Dunn, which is a system designed for handling inconsistent as well as potentially conflicting information. These features, we believe, make our framework particularly promising from a computer science perspective. (shrink)

Bilattices, introduced by M. Ginsberg, constitute an elegant family of multiple-valued logics. Those meeting certain natural conditions have provided the basis for the semantics of a family of logic programming languages. Now we consider further restrictions on bilattices, to narrow things down to logic programming languages that can, at least in principle, be implemented. Appropriate bilattice background information is presented, so the paper is relatively self-contained.

One approach to the paradoxes of self-referential languages is to allow some sentences to lack a truth value (or to have more than one). Then assigning truth values where possible becomes a fixpoint construction and, following Kripke, this is usually carried out over a partially ordered family of three-valued truth-value assignments. Some years ago Matt Ginsberg introduced the notion of bilattice, with applications to artificial intelligence in mind. Bilattices generalize the structure Kripke used in a very natural way, while (...) making the mathematical machinery simpler and more perspicuous. In addition, work such as that of Yablo fits naturally into the bilattice setting. What I do here is present the general background of bilattices, discuss why they are natural, and show how fixpoint approaches to truth in languages that allow self-reference can be applied. This is not new work, but rather is a summary of research I have done over many years. (shrink)

This paper discusses a dualization of Fitting's notion of a "cut-down" operation on a bilattice, rendering a "track-down" operation, later used to represent the idea that a consistent opinion cannot arise from a set including an inconsistent opinion. The logic of track-down operations on bilattices is proved equivalent to the logic d_Sfde, dual to Deutsch's system S_fde. Furthermore, track-down operations are employed to provide an epistemic interpretation for paraconsistent weak Kleene logic. Finally, two logics of sequential combinations of cut-and (...) track-down operations allow settling positively the question of whether bilattice-based semantics are available for subsystems of S_fde. (shrink)

While Kripke's original paper on the theory of truth used a three-valued logic, we believe a four-valued version is more natural. Its use allows for possible inconsistencies in information about the world, yet contains Kripke's development within it. Moreover, using a four-valued logic makes it possible to work with complete lattices rather than complete semi-lattices, and thus the mathematics is somewhat simplified. But more strikingly, the four-valued version has a wide, natural generalization to the family of interlaced bilattices. Thus, (...) with little more work, the theory is extended to a broad class of settings. Indeed, a result like Theorem 6.2 would not even be possible to state without the interlaced bilattice machinery. We hope the notion of interlaced bilattice will make apparent further such connections. (shrink)

The notion of bilattice was introduced by Ginsberg, and further examined by Fitting, as a general framework for many applications. In the present paper we develop proof systems, which correspond to bilattices in an essential way. For this goal we introduce the notion of logical bilattices. We also show how they can be used for efficient inferences from possibly inconsistent data. For this we incorporate certain ideas of Kifer and Lozinskii, which happen to suit well the context of (...) our work. The outcome are paraconsistent logics with a lot of desirable properties.1. (shrink)

We introduce a new product bilattice con- struction 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 lat- tice factors. Finally, we (...) show how to employ our product construction to define first-order definable classes of bi- lattices corresponding to any first-order definable subclass of residuated lattices. (shrink)

We develop a Priestley-style duality theory for different classes of algebras having a bilattice reduct. A similar investigation has already been realized by B. Mobasher, D. Pigozzi, G. Slutzki and G. Voutsadakis, but only from an abstract category-theoretic point of view. In the present work we are instead interested in a concrete study of the topological spaces that correspond to bilattices and some related algebras that are obtained through expansions of the algebraic language.

Bilattices, introduced by Ginsberg as a uniform framework for inference in artificial intelligence, are algebraic structures that proved useful in many fields. In recent years, Arieli and Avron 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 . 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 . We also prove some purely algebraic results concerning bilattices, for instance that the variety of distributive bilattices is generated by the smallest non-trivial bilattice. This result is based on an improvement of a theorem by Avron 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. (shrink)

Lattice logic, bilattice logic, and paraconsistent quantum logic are investigated based on monosequent systems. Paraconsistent quantum logic is an extension of lattice logic, and bilattice logic is an extension of paraconsistent quantum logic. Monosequent system is a sequent calculus based on the restricted sequent that contains exactly one formula in both the antecedent and succedent. It is known that a completeness theorem with respect to a lattice-valued semantics holds for a monosequent system for lattice logic. A completeness theorem with respect (...) to a lattice-valued semantics is proved for paraconsistent quantum logic, and a completeness theorem with respect to a bilattice-valued semantics is proved for bilattice logic. Some syntactical properties, including cut-elimination and duality, are also investigated for the monosequent systems for these logics. (shrink)

A general Gentzen-style framework for handling both bilattice (or strong) negation and usual negation is introduced based on the characterization of negation by a modal-like operator. This framework is regarded as an extension, generalization or re- finement of not only bilattice logics and logics with strong negation, but also traditional logics including classical logic LK, classical modal logic S4 and classical linear logic CL. Cut-elimination theorems are proved for a variety of proposed sequent calculi including CLS (a conservative extension of (...) CL) and CLScw (a conservative extension of some bilattice logics, LK and S4). Completeness theorems are given for these calculi with respect to phase semantics, for SLK (a conservative extension and fragment of LK and CLScw, respectively) with respect to a classical-like semantics, and for SS4 (a conservative extension and fragment of S4 and CLScw, respectively) with respect to a Kripke-type semantics. The proposed framework allows for an embedding of the proposed calculi into LK, S4 and CL. (shrink)

This paper sheds light on the relationship between the logic of generalized truth values and the logic of bilattices. It suggests a definite solution to the problem of axiomatizing the truth and falsity consequence relations, \ and \ , considered in a language without implication and determined via the truth and falsity orderings on the trilattice SIXTEEN 3 . The solution is based on the fact that a certain algebra isomorphic to SIXTEEN 3 generates the variety of commutative and (...) distributive bilattices with conflation. (shrink)

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. Fit- ting), 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 un- bounded 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. (shrink)

This paper focuses on natural dualities for varieties of bilattice-based algebras. Such varieties have been widely studied as semantic models in situations where information is incomplete or inconsistent. The most popular tool for studying bilattices-based algebras is product representation. The authors recently set up a widely applicable algebraic framework which enabled product representations over a base variety to be derived in a uniform and categorical manner. By combining this methodology with that of natural duality theory, we demonstrate how to (...) build a natural duality for any bilattice-based variety which has a suitable product representation over a dualisable base variety. This procedure allows us systematically to present economical natural dualities for many bilattice-based varieties, for most of which no dual representation has previously been given. Among our results we highlight that for bilattices with a generalised conflation operation. Here both the associated product representation and the duality are new. Finally we outline analogous procedures for pre-bilattice-based algebras. (shrink)

The notion of a bilattice was rst introduced by Ginsburg (see Gin]) as a general framework for a diversity of applications (such as truth maintenance systems, default inferences and others). The notion was further investigated and applied for various purposes by Fitting (see Fi1]- Fi6]). The main idea behind bilattices is to use structures in which there are two (partial) order relations, having di erent interpretations. The two relations should, of course, be connected somehow in order for the mathematical (...) structure to be useful. It is not clear, however, what this connection should be. Ginsberg, for example, has made the connection through an extra operation of negation. Fitting, on the other hand, has investigated connections in the form of conditions on the structure (such as being interlaced { see below). These conditions are independent of the existence, or even the possibility to de ne, operations like Ginsberg's negation.* Fitting de nes, accordingly, notions like \an interlaced bilattice", \a distributive bilattice", \a bilattice with negation" and others. He does not provide, however, any de nition of the notion of bilattice itself (without an extra modi er). I was unable to nd anywhere, in fact, a de nition which will cover all the structures which were called \bilattice" in the literature. (shrink)

ABSTRACT In the paper we present a survey of some approaches to the semantics of many-valued propositional systems. These approaches are inspired on one hand by classical problems in the investigations of logical aspects of epistemic activity: knowledge and truth, contradictions, beliefs, reliability of data, etc. On the other hand they reflect contemporary concerns of researchers in Artificial Intelligence (and Cognitive Science in general) with inferences drawn from imperfect information, even from total ignorance. We treat the mathematical apparatus that has (...) emerged recently: algebraic structures related to the new logical systems in the same way Boolean algebras correspond to classical logic. (shrink)

A sequential axiomatization is given for the 16-valued logic that has been proposed by Shramko-Wansing as a candidate for the basic logic of logical bilattices.

Strict/tolerant logic is a formally defined logic that has the same consequence relation as classical logic, though it differs from classical logic at the metaconsequence level. Specifically, it does not satisfy a cut rule. It has been proposed for use in work on theories of truth because it avoids some objectionable features arising from the use of classical logic. Here we are not interested in applications, but in the formal details themselves. We show that a wide range of logics have (...) strict/tolerant counterparts, with the same consequence relations but differing at the metaconsequence level. Among these logics are Kleene’s K3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {K_3}$$\end{document}, Priest’s LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {LP}}$$\end{document}, and first-degree entailment, FDE. The primary tool we use is the bilattice. But it is more than a tool, it seems to be the natural home for this kind of investigation. (shrink)

In Belnap's useful 4-valued logic, the set 2 = {T, F} of classical truth values is generalized to the set 4 = ������(2) = {Ø, {T}, {F}, {T, F}}. In the present paper, we argue in favor of extending this process to the set 16 = ᵍ (4) (and beyond). It turns out that this generalization is well-motivated and leads from the bilattice FOUR₂ with an information and a truth-and-falsity ordering to another algebraic structure, namely the trilattice SIXTEEN₃ with an (...) information ordering together with a truth ordering and a (distinct) falsity ordering. Interestingly, the logics generated separately by the algebraic operations under the truth order and under the falsity order in SIXTEEN₃ coincide with the logic of FOUR₂, namely first degree entailment. This observation may be taken as a further indication of the significance of first degree entailment. In the present setting, however, it becomes rather natural to consider also logical systems in the language obtained by combining the vocabulary of the logic of the truth order and the falsity order. We semantically define the logics of the two orderings in the extended language and in both cases axiomatize a certain fragment comprising three unary operations: a negation, an involution, and their combination. We also suggest two other definitions of logics in the full language, including a bi-consequence system. In other words, in addition to presenting first degree entailment as a useful 16-valued logic, we define further useful 16-valued logics for reasoning about truth and (non-)falsity. We expect these logics to be an interesting and useful instrument in information processing, especially when we deal with a net of hierarchically interconnected computers. We also briefly discuss Arieli's and Avron's notion of a logical bilattice and state a number of open problems for future research. (shrink)

In this paper we study 12 four-valued logics arisen from Belnap's truth and/or knowledge four-valued lattices, with or without constants, by adding one or both or none of two new non-regular operations—classical negation and natural implication. We prove that the secondary connectives of the bilattice four-valued logic with bilattice constants are exactly the regular four-valued operations. Moreover, we prove that its expansion by any non-regular connective (such as, e.g., classical negation or natural implication) is strictly functionally complete. Further, finding axiomatizations (...) of the quasi varieties generated by the 12 logics involved (that prove to be varieties), we find naturell equational axiomatizations of these logics. Finally, applying Pynko's general theory of algebraizable sequential consequence operations, we also find equivalent natural sequentiell axiomatizations of the logics under consideration that expand either of two Pynko's sequential calculi for the constant-free truth-lattice four-valued logic. (shrink)

Łukasiewicz presented two different analyses of modal notions by means of many-valued logics: the linearly ordered systems Ł3,..., Open image in new window,..., \; the 4-valued logic Ł he defined in the last years of his career. Unfortunately, all these systems contain “Łukasiewicz type paradoxes”. On the other hand, Brady’s 4-valued logic BN4 is the basic 4-valued bilattice logic. The aim of this paper is to show that BN4 can be strengthened with modal operators following Łukasiewicz’s strategy for defining truth-functional (...) modal logics. The systems we define lack “Łukasiewicz type paradoxes”. Following Brady, we endow them with Belnap–Dunn type bivalent semantics. (shrink)

The book presents a thoroughly elaborated logical theory of generalized truth-values understood as subsets of some established set of truth values. After elucidating the importance of the very notion of a truth value in logic and philosophy, we examine some possible ways of generalizing this notion. The useful four-valued logic of first-degree entailment by Nuel Belnap and the notion of a bilattice constitute the basis for further generalizations. By doing so we elaborate the idea of a multilattice, and most notably, (...) a trilattice of truth values – a specific algebraic structure with information ordering and two distinct logical orderings, one for truth and another for falsity. Each logical order not only induces its own logical vocabulary, but determines also its own entailment relation. We consider both semantic and syntactic ways of formalizing these relations and construct various logical calculi. (shrink)

In this paper, the notion of degree of inconsistency is introduced as a tool to evaluate the sensitivity of the Full Bayesian Significance Test (FBST) value of evidence with respect to changes in the prior or reference density. For that, both the definition of the FBST, a possibilistic approach to hypothesis testing based on Bayesian probability procedures, and the use of bilattice structures, as introduced by Ginsberg and Fitting, in paraconsistent logics, are reviewed. The computational and theoretical advantages of using (...) the proposed degree of inconsistency based sensitivity evaluation as an alternative to traditional statistical power analysis is also discussed. (shrink)

The relationships between various modal logics based on Belnap and Dunn’s paraconsistent four-valued logic FDE are investigated. It is shown that the paraconsistent modal logic \, which lacks a primitive possibility operator \, is definitionally equivalent with the logic \, which has both \ and \ as primitive modalities. Next, a tableau calculus for the paraconsistent modal logic KN4 introduced by L. Goble is defined and used to show that KN4 is definitionally equivalent with \ without the absurdity constant. Moreover, (...) a tableau calculus is defined for the modal bilattice logic MBL introduced and investigated by A. Jung, U. Rivieccio, and R. Jansana. MBL is a generalization of BK that in its Kripke semantics makes use of a four-valued accessibility relation. It is shown that MBL can be faithfully embedded into the bimodal logic \ over the non-modal vocabulary of MBL. On the way from \ to MBL, the Fischer Servi-style modal logic \ is defined as the set of all modal formulas valid under a modified standard translation into first-order FDE, and \ is shown to be characterized by the class of all models for \. Moreover, \ is axiomatized and this axiom system is proved to be strongly sound and complete with respect to the class of models for \. Moreover, the notion of definitional equivalence is suitably weakened, so as to show that \ and \ are weakly definitionally equivalent. (shrink)

In this paper we consider the theory of predicate logics in which the principle of Bivalence or the principle of Non-Contradiction or both fail. Such logics are partial or paraconsistent or both. We consider sequent calculi for these logics and prove Model Existence. For L4, the most general logic under consideration, we also prove a version of the Craig-Lyndon Interpolation Theorem. The paper shows that many techniques used for classical predicate logic generalise to partial and paraconsistent logics once the right (...) set-up is chosen. Our logic L4 has a semantics that also underlies Belnap’s [4] and is related to the logic of bilattices. L4 is in focus most of the time, but it is also shown how results obtained for L4 can be transferred to several variants. (shrink)

This paper develops a theory of propositional identity which distinguishes necessarily equivalent propositions that differ in subject-matter. Rather than forming a Boolean lattice as in extensional and intensional semantic theories, the space of propositions forms a non-interlaced bilattice. After motivating a departure from tradition by way of a number of plausible principles for subject-matter, I will provide a Finean state semantics for a novel theory of propositions, presenting arguments against the convexity and nonvacuity constraints which Fine (2016, 2017a,b) introduces. I (...) will then move to compare the resulting logic of propositional identity (PI) with Correia’s (2016) logic of generalised identity (GI), as well as the first degree fragment of Angell’s (1989) logic of analytic containment (AC). The paper concludes by extending PI to include axioms and rules for a subject-matter operator, providing a much broader theory of subject-matter than the principles with which I will begin. (shrink)

Various connexive FDE-based modal logics are studied. Some of these logics contain a conditional that is both connexive and strict, thereby highlighting that strictness and connexivity of a conditional do not exclude each other. In particular, the connexive modal logics cBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document}, cKN4, scBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document}, scKN4, cMBL, and scMBL are introduced semantically by means of classes of Kripke models. The logics cBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} (...) \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document} and cKN4 are connexive variants of the FDE-based modal logics BK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document} and KN4 with a weak and a strong implication, respectively. The system cMBL is a connexive variant of the modal bilattice logic MBL. The latter is a modal extension of Arieli and Avron’s logic of logical bilattices and is characterized by a class of Kripke models with a four-valued accessibility relation. In the systems scBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document}, scKN4, and scMBL, the conditional is both connexive and strict. Sound and complete tableau calculi for all these logics are presented and used to show that the entailment relations of the systems under consideration are decidable for finite premise set. Moreover, the logics cBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {cBK}^-$$\end{document} and cMBL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {cMBL}$$\end{document} are shown to be algebraizable. The algebraizability of cMBL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {cMBL}$$\end{document} is derived from proving cMBL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {cMBL}$$\end{document} to be definitionally equivalent to MBL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {MBL}$$\end{document}. All connexive modal logics studied in this paper are decidable, paraconsistent, and inconsistent but non-trivial logics. (shrink)

The trilattice SIXTEEN3 introduced in Shramko & Wansing (2005) is a natural generalization of the famous bilattice FOUR2. Some Hilbert-style proof systems for trilattice logics related to SIXTEEN3 have recently been studied (Odintsov, 2009; Shramko & Wansing, 2005). In this paper, three sequent calculi GB, FB, and QB are presented for Odintsovs coordinate valuations associated with valuations in SIXTEEN3. The equivalence between GB, FB, and QB, the cut-elimination theorems for these calculi, and the decidability of B are proved. In addition, (...) it is shown how the sequent systems for B can be extended to cut-free sequent calculi for Odintsov’s LB, which is an extension of B by adding classical implication and negation connectives. (shrink)

In their useful logic for a computer network Shramko and Wansing generalize initial values of Belnap’s 4-valued logic to the set 16 to be the power-set of Belnap’s 4. This generalization results in a very specific algebraic structure — the trilattice SIXTEEN 3 with three orderings: information, truth and falsity. In this paper, a slightly different way of generalization is presented. As a base for further generalization a set 3 is chosen, where initial values are a — incoming data is (...) asserted, d — incoming data is denied, and u — incoming data is neither asserted nor denied, that corresponds to the answer “don’t know”. In so doing, the power-set of 3, that is the set 8 is considered. It turns out that there are not three but four orderings naturally defined on the set 8 that form the tetralattice EIGHT 4 . Besides three ordering relations mentioned above it is an extra uncertainty ordering. Quite predictably, the logics generated by a –order (truth order) and d –order (falsity order) coincide with first-degree entailment. Finally logic with two kinds of operations ( a –connectives and d –connectives) and consequence relation defined via a –ordering is considered. An adequate axiomatization for this logic is proposed. (shrink)

The aim of this paper is to define the logical system Sm4 characterised by the degree of truth-preserving consequence relation defined on the ordered set of values of Smiley’s four-element matrix MSm4. The matrix MSm4 has been of considerable importance in the development of relevant logics and it is at the origin of bilattice logics. It will be shown that Sm4 is a most interesting paraconsistent logic which encloses a sound theory of logical necessity similar to that of Anderson and (...) Belnap’s logic of entailment E. Intuitively, Sm4 can be described as a four-valued expansion of the positive fragment of Lewis’ S5 or, alternatively, as a four-valued version of S5. (shrink)

A recent paper by Jakl, Jung and Pultr 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 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 results for the corresponding logics. (shrink)

Kleene’s strong three-valued logic extends naturally to a four-valued logic proposed by Belnap. We introduce a guard connective into Belnap’s logic and consider a few of its properties. Then we show that by using it four-valued analogs of Kleene’s weak three-valued logic, and the asymmetric logic of Lisp are also available. We propose an extension of these ideas to the family of distributive bilattices. Finally we show that for bilinear bilattices the extensions do not produce any new equivalences.

In this paper we introduce a Gentzen calculus for (a functionally complete variant of) Belnap's logic in which establishing the provability of a sequent in general requires \emph{two} proof trees, one establishing that whenever all premises are true some conclusion is true and one that guarantees the falsity of at least one premise if all conclusions are false. The calculus can also be put to use in proving that one statement \emph{necessarily approximates} another, where necessary approximation is a natural dual (...) of entailment. The calculus, and its tableau variant, not only capture the classical connectives, but also the `information' connectives of four-valued Belnap logics. This answers a question by Avron. (shrink)

Kleene’s well-known strong three-valued logic is shown to be one of a family of logics with similar mathematical properties. These logics are produced by an intuitively natural construction. The resulting logics have direct relationships with bilattices. In addition they possess mathematical features that lend themselves well to semantical constructions based on fixpoint procedures, as in logic programming.

We define the paraconsistent supra-logic Pσ by a type-shift from the booleans o of propositional logic Po to the supra-booleans σ of the propositional type logic P obtained as the propositional fragment of the transfinite type theory Q defined by Peter Andrews (North-Holland Studies in Logic 1965) as a classical foundation of mathematics. The supra-logic is in a sense a propositional logic only, but since there is an infinite number of supra-booleans and arithmetical operations are available for this and other (...) types, virtually anything can be specified. The supra-logic is a generalization of Lukasiewicz's three-valued logic, with the intermediate value duplicated many times and ordered such that none of the copies of this value imply other ones, but it differs from Lukasiewicz's many-valued logics as well as from logics based on bilattices. There are several automated theorem provers for classical higher order logic (finite type theory) and it should be possible to modify these to our needs. (shrink)

One of the most signi cant drawbacks of classical logic is its being useless in the presence of an inconsistency. Nevertheless, the classical calculus is a very convenient framework to work with. In this work we propose means for drawing conclusions from systems that are based on classical logic, although the informationmightbe inconsistent. The idea is to detect those parts of the knowledge-base that \cause" the inconsistency, and isolate the parts that are \recoverable". We do this by temporarily switching into (...) Ginsberg/Fitting multi-valued framework of bilattices (which is a common framework for logic programmingand nonmonotonic reasoning). Our method is conservative in the sense that it considers the contradictory data as useless and regards all the remaining information una ected. The resulting logic is nonmonotonic, paraconsistent, and a plausibility logic in the sense of Lehmann. (shrink)

All empirical concepts belonging to natural language are vague. The vagueness of empirical discourse is a source of many semantical problems which have been known since ancient times. One of those problems concerns the so called sorites paradoxes. This article is an attempt to show that the paradoxes are either invalid or unsound inferences. Since classical logic is useless for semantical analysis of such inferences, the article makes use of the Belnap four-valued logic based on the bilattice FOUR. Belnap's logic (...) is the base logic for defining a nonmonotonic consequence relation which prefers models with classical valuations among all valuations of the four-valued logic whenever it is possible. The approach outlined in this article is intuitive, and has more advantages than the supervaluational approach. (shrink)

A refutation mechanism is introduced into logic programming, dual to the usual proof mechanism; then negation is treated via refutation. A four-valued logic is appropriate for the semantics: true, false, neither, both. Inconsistent programs are allowed, but inconsistencies remain localized. The four-valued logic is a well-known one, due to Belnap, and is the simplest example of Ginsberg’s bilattice notion. An efficient implementation based on semantic tableaux is sketched; it reduces to SLD resolution when negations are not involved. The resulting system (...) can give reasonable answers to queries that involve both negation and free variables. Also it gives the same results as Prolog when there are no negations. Finally, an implementation in Prolog is given. (shrink)

The work of Arnon Avron and Ofer Arieli has shown a deep relationship between the theory of bilattices and the Belnap-Dunn logic \. This correspondence has been interpreted as evidence that \ is “the” logic of bilattices, a consideration reinforced by the work of Yaroslav Shramko and Heinrich Wansing in which \ is shown to be similarly entrenched with respect to the theories of trilattices and, more generally, multilattices. In this paper, we export Melvin Fitting’s “cut-down” connectives—propositional connectives (...) that “cut down” available evidence—to the case of multilattices and show that two related first degree systems—Harry Deutsch’s four-valued \ and Richard Angell’s \—emerge just as elegantly and are as intimately connected to the theories of bilattices and trilattices as the Belnap-Dunn logic. (shrink)

The trilattice SIXTEEN₃ is a natural generalization of the wellknown bilattice FOUR₂. Cut-free, sound and complete sequent calculi for truth entailment and falsity entailment in SIXTEEN₃, are presented.

Classical fixpoint semantics for logic programs is based on the TP immediate consequence operator. The Kripke/Kleene, three-valued, semantics uses ΦP, which extends TP to Kleene’s strong three-valued logic. Both these approaches generalize to cover logic programming systems based on a wide class of logics, provided only that the underlying structure be that of a bilattice. This was presented in earlier papers. Recently well-founded semantics has become influential for classical logic programs. We show how the well-founded approach also extends naturally to (...) the same family of bilatticebased programming languages that the earlier fixpoint approaches extended to. Doing so provides a natural semantics for logic programming systems that have already been proposed, as well as for a large number that are of only theoretical interest. And finally, doing so simplifies the proofs of basic results about the well-founded semantics, by stripping away inessential details. (shrink)