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We present a uniform syntactical characterisation of the class of quasi-relevant logics which are four-valued extensions of the basic relevant logic B of Meyer and Routley. All these logics are obtained by the addition of suitable quasi-relevant implications to the four-valued logic of First Degree Entailment FDE. So far they were characterised axiomatically and semantically in several ways but did not obtain a special proof-theoretic treatment. To this aim a generalised form of sequent calculus called bisequent calculus (BSC) is applied. In BSC rules operate on the ordered pairs of ordinary sequents. It may be treated as the weakest kind of system in the rich family of generalised sequent calculi operating on items which are some collections of ordinary sequents, like hypersequents or nested sequents. It is shown that all logics under consideration have cut-free characterisation in BSC which satisfies the subformula property and yields decidability. It is also shown that the interpolation theorem holds for these logics if their language is enriched with additional negation. [ABSTRACT FROM AUTHOR]

Ambiguous set has recently been introduced as a new branch of mathematics and computer science dealing with the nature of ambiguity of uncertain events. There is much scope for the development of new theories based on the principles of the ambiguous set. In this direction, an idea of ambiguous logic is proposed, which can be regarded as a family of four-valued logic. In this study, the limitations of existing theories such as fuzzy logic, intuitionistic fuzzy logic, and neutrosophic logic that are used to deal with uncertainty are first discussed. Then an introduction to the ambiguous set and ambiguous logic is given. In ambiguous logic, there are four membership degrees for each proposition, which can be characterized as true, false, true-ambiguous, and false-ambiguous. We then analyze the membership degrees of propositions using various connectives, including negation, conjunction, and disjunction. In this study, we discuss various laws that support the different algebraic operations for comparing propositions in terms of membership degrees. An approach is presented to define a rule using a proposition consisting of membership degrees, called an ambiguous rule. The modus ponens rule of inference is defined by combining various ambiguous rules, and is called ambiguous inference. Finally, to demonstrate the real-time application of ambiguous logic, an ambiguous inference system is designed that can be used in machine control systems. [ABSTRACT FROM AUTHOR]

In this paper, we investigate rule semantics for deductive databases in the context of four-valued logic. In our approach, a database is a pair |$\varDelta =(E,R)$|⁠ , where |$E$| is a set of pairs, each pair associating a ground fact with a truth value (thus, allowing to store true, false or inconsistent facts) and |$R$| is a set of rules generalizing standard Datalog rules in the following sense: (i) the head of a rule can be a positive or a negative atom and (ii) the body can involve any among the connectors of four-valued logic. We define the database semantics as the least fixed point of a monotonic operator and we compare this semantics with that of k-existential programs defined by Fitting and paraconsistent extended logic programs defined by Arieli. Our main contribution is to show that, if we consider rules as implications (i.e. if we view the database as a set of formulas) then the semantics of the database is the unique minimal model of the set of database formulas. Here, minimality is understood with respect to the knowledge ordering of four-valued logic satisfying a monotonicity property whereby the truth value of the head of an instantiated rule is greater than that of the body. Moreover, we characterize databases having finite semantics and then, we address the issue of database updating. We argue that our approach allows for a new kind of updates, in which the update result depends not only on the fact involved in the update but also on its current truth value in the database. [ABSTRACT FROM AUTHOR]

We study the lattice of extensions of four-valued Belnap–Dunn logic, called super-Belnap logics by analogy with superintuitionistic logics. We describe the global structure of this lattice by splitting it into several subintervals, and prove some new completeness theorems for super-Belnap logics. The crucial technical tool for this purpose will be the so-called antiaxiomatic (or explosive) part operator. The antiaxiomatic (or explosive) extensions of Belnap–Dunn logic turn out to be of particular interest owing to their connection to graph theory: the lattice of finitary antiaxiomatic extensions of Belnap–Dunn logic is isomorphic to the lattice of upsets in the homomorphism order on finite graphs (with loops allowed). In particular, there is a continuum of finitary super-Belnap logics. Moreover, a non-finitary super-Belnap logic can be constructed with the help of this isomorphism. As algebraic corollaries we obtain the existence of a continuum of antivarieties of De Morgan algebras and the existence of a prevariety of De Morgan algebras which is not a quasivariety. [ABSTRACT FROM AUTHOR]

Following a proposal by Kooi and Tamminga, we introduce a conservative translation manual for every four-valued truth-functional propositional logic into a modal logic. However, the application of this translation does not preserve the intuitive reading of the truth-values for every four-valued logic. In order to solve this problem, we modify the translation manual and prove its conservativity by exploiting the method of generalized truth-values. [ABSTRACT FROM AUTHOR]

In this paper, we consider a set of quite interesting three- and four-valued logics and prove the normalisation theorem for their natural deduction formulations. Among the logics in question are the Logic of Paradox, First Degree Entailment, Strong Kleene logic, and some of their implicative extensions, including RM3 and RM3⊃. Also, we present a detailed version of Prawitz's proof of Nelson's logic N4 and its extension by intuitionist negation. [ABSTRACT FROM AUTHOR]

Jan Łukasiewicz is known primarily as the founder of the three-valued system of logic. It is also generally renowned that his reason for introducing many-valued systems of logic was an attempt to refute determinism. When he developed the three-valued and n-valued logic, he employed these systems in his arguments against determinism. On the contrary, Łukasiewicz preferred the four-valued system of logic that is not suitable for a refutation of determinism in his latest period. It seems, however, that determinism still interested Łukasiewicz, as he discussed its denial in his last work, the second edition of the book Aristotle's syllogistic from the standpoint of modern formal logic. The aim of my paper is to introduce the issue and describe the relation of these two topics primarily in the latest period of Łukasiewicz's work. There are several possible solutions to the issue. However, Łukasiewicz never addressed the issue clearly in his published works. Therefore, the solutions might be supported or contradicted by his unpublished works in the future. [ABSTRACT FROM AUTHOR]

Three-valued logics are standardly used to formalize gappy languages, i.e., interpreted languages in which sentences can be true, false or neither. A three-valued logic that assigns the same truth value to all gappy sentences is, in our view, insufficient to capture important semantic differences between them. In this paper we will argue that there are two different kinds of pathologies that should be treated separately and we defend the usefulness of a four-valued logic to represent adequately these two types of gappy sentences. Our purpose is to begin the formal exploration of the four-valued logics that could be used to represent the phenomena in question and to show that these phenomena are present in natural language, at least according to some semantic theories of natural language. [ABSTRACT FROM AUTHOR]

Parallel to David Bohm´s development of a realistic interpretation of quantum physics, German philosopher and logician Gotthard Günther worked on a generalization of the classical two-valued logic to satisfy the ontological requirements of quantum physics as well as of cybernetics. Both of these new disciplines introduced information and consciousness into the terminology of science. These terms and concepts need to be reflected in logic, ontology and the theory of science. David Bohm suggested an expansion of his own model by generalization and iteration of the quantum potential to include consciousness and mental states into a new psycho-physical theory. This article proposes Günther´s four-valued logical system of meaning/reflection as a theoretical scientific frame for this expansion of Bohm´s theory and discusses its ontological implications.

In the present paper, current possibilities for further research into the structure of Mendeleev's Periodic System are considered. The four-valued logic of quantum measurements is demonstrated to be applicable for the classification of chemical elements. Quantum scales containing data on the position of a chemical element with a given ordinal number are used in the definition of new metrological research problems leading to the development of new approaches to quantum metrology. [ABSTRACT FROM AUTHOR]

Rough set theory is studied to manage uncertain and inconsistent information. Because Pawlak's decision logic for rough sets is based on the classical two-valued logic, it is inconvenient for handling inconsistent information. We propose a bilattice logic as the deduction basis for the decision logic of rough sets to address inconsistent and ambiguous information. To enhance the decision logic to bilattice semantics, we introduce Variable Precision Rough Set (VPRS). As a deductive basis for bilattice decision logic, we define a consequence relation for Belnap's four-valued semantics and provide a bilattice semantic tableau TB4 for a deduction system. We demonstrate the soundness and completeness of TB4 and enhance it with weak negation. [ABSTRACT FROM AUTHOR]

In this paper, we study logical systems which represent entailment relations of two kinds. We extend the approach of finding 'exactly true' and 'non-falsity' versions of four-valued logics that emerged in series of recent works [Pietz & Rivieccio (2013). Nothing but the truth. Journal of Philosophical Logic, 42(1), 125–135; Shramko (2019). Dual-Belnap logic and anything but falsehood. Journal of Logics and their Applications, 6, 413–433; Shramko et al. (2017). First-degree entailment and its relatives. Studia Logica, 105(6), 1291–1317] to the case of infectious logics, namely to the case of Deutsch's logic S f d e introduced in Deutsch [Relevant analytic entailment. The Relevance Logic Newsletter, 2(1), 26–44; The completeness of S. Studia Logica, 38(2), 137–147]. The particular systems obtained in this way are S e t l and S n f l . We present them in the form of sequent calculi and prove corresponding soundness and completeness theorems. We illuminate the connection between S e t l , S n f l and two well-known systems, strong Kleene three-valued logic K 3 and Priest's Logic of Paradox L P . This connection allows us to investigate the characterisation of the entailment relations associated with S e t l and S n f l as well as to introduce the notion of 'infectious analogue' of a certain logic. We also study implicative extensions of S e t l and S n f l and prove soundness and completeness theorems for them as well. [ABSTRACT FROM AUTHOR]

DOI: http://doi.org/10.26333/sts.xxxiii2.07 Although Łukasiewicz and Quine do not share many common views, they agreed on one important point in the 1950s: they both denied the distinction between empirical and a priori sciences. This agreement might be surprising as this denial was rather controversial at that time. This paper focuses on Quine’s and Łukasiewicz’s denials of the distinction between empirical and a priori sciences, and proposes three possible answers to the question of why both formulated the same conclusion at a similar time. Firstly, it discusses Quine’s possible influence on Łukasiewicz as Łukasiewicz agreed with Quine’s criticism of modality at that time. Secondly, it considers the possibility that Quine was affected by Łukasiewicz via his debates with Łukasiewicz’s student, Tarski. Lastly, it takes into account the possibility that both philosophers were inspired by an external source, namely the rise of quantum mechanics.

Although Łukasiewicz and Quine do not share many common views, they agreed on one important point in the 1950s: they both denied the distinction between empirical and a priori sciences. This agreement might be surprising as this denial was rather controversial at that time. This paper focuses on Quine's and Łukasiewicz's denials of the distinction between empirical and a priori sciences, and proposes three possible answers to the question of why both formulated the same conclusion at a similar time. Firstly, it discusses Quine's possible influence on Łukasiewicz as Łukasiewicz agreed with Quine's criticism of modality at that time. Secondly, it considers the possibility that Quine was affected by Łukasiewicz via his debates with Łukasiewicz's student, Tarski. Lastly, it takes into account the possibility that both philosophers were inspired by an external source, namely the rise of quantum mechanics. [ABSTRACT FROM AUTHOR]

We apply a non-classical four-valued logic in the process of reasoning regarding strategies for cops in a modified game of "Cops and Robber" played on a graph. We extend the game by introducing uncertainty in a form of random failures of detecting devices. This is realized by allowing that a robber can be detected in a node only with the given probability PA. Additionally, with the probability PF, cops can be given a false-positive, i.e., they are informed that the robber is located at some node, whereas it is located somewhere else. Consequently, non-zero PF introduces a measurement noise into the system. All the cops have access to information provided by the detectors and can communicate with each other, so they can coordinate the search. By adjusting the number of detectors, PA, and PF we can achieve a smooth transition between the two well-known variants of the game: "with fully visible robber" and "with invisible robber". We compare a simple probabilistic strategy for cops with the non-parametric strategy based on reasoning with a four-valued paraconsistent logic. It is shown that this novel approach leads to a good performance, as measured by the required mean catch-time. We conclude that this type of reasoning can be applied in real-world applications where there is no knowledge about the underlying source of errors which is particularly useful in robotics. [ABSTRACT FROM AUTHOR]

In nonmonotonic reasoning there are the problems of inconsistency and incoherence in general, and in default reasoning there may be only one trivial extension or no extension in special. We propose the restricted semantics of four-valued logic for default reasoning to resolve the problems of inconsistency and incoherence and in the meantime retain classical extensions in the presence of consistency and coherency. The restricted semantics can maintain both the expressiveness and reasoning ability of default logic. We provide a transformation approach to compute the restricted extensions by reducing them into classical ones. [ABSTRACT FROM AUTHOR]

Bitwise operation is a basic operation in C program. According to the statistics, the bitwise operation accounts for 12% in all numerical operations. There are two problems for the bitwise operation when search technology is used to generate test input: one is the low efficiency for it has no corresponding interval arithmetic rule to reduce the search space; the other is the absence of a unified approach when a variable is simultaneously involved in the bitwise and non-bitwise operations. For problem one, classical two-valued logic is extended to four-valued logic to express the two states of logic variable’s value, ‘uncertain’ and ‘neither true nor false’. According to the operation rules of the four-valued logic, the search space can be reduced. For problem two, the conversion between interval representation and four-valued logic representation is given, and then provided with a unified approach based on the search algorithm. Experiments show that this method can improve the efficiency of test input generation of bitwise operation. [ABSTRACT FROM AUTHOR]

The development of recursion theory motivated Kleene to create regular three-valued logics. Taking his inspiration from the computer science, Fitting later continued to investigate regular three-valued logics and defined them as monotonic ones. Afterwards, Komendantskaya proved that there are four regular three-valued logics and in the three-valued case the set of regular logics coincides with the set of monotonic logics. Next, Tomova showed that in the four-valued case regularity and monotonicity do not coincide. She counted that there are 6400 four-valued regular logics, but only six of them are monotonic. The purpose of this paper is to create natural deduction systems for them. We also describe some functional properties of these logics. [ABSTRACT FROM AUTHOR]

Rough sets theory is used to handle uncertain and inconsistent information. While, Pawlak’s decision logic of rough sets is based on classical bivalence logic, this may cause a limitation for the various reasoning. In this study, we propose four-valued logics, as the deduction basis for the decision logic. To provide four-valued semantics to decision logic of rough set, we introduce a Ziarko’s variable precision rough set. As a deductive system, we adopt tableau calculi and define a consequence relation to construct deductive system based on four-valued semantics. Furthermore, weak-negation is introduced to compensate the deduction property of four-value logics. Finally, we discuss Henkin-type proof of the completeness theorem for the system. [ABSTRACT FROM AUTHOR]

The Belnap-Dunn logic (also known as First Degree Entailment, or FDE) is a well-known and well-studied four-valued logic, but until recently little has been known about its extensions, i.e. stronger logics in the same language, called super-Belnap logics here. We give an overview of several results on these logics which have been proved in recent works by Přenosil and Rivieccio. We present Hilbert-style axiomatizations, describe reduced matrix models, and give a description of the lattice of super-Belnap logics and its connections with graph theory. We adopt the point of view of Abstract Algebraic Logic, exploring applications of the general theory of algebraization of logics to the super-Belnap family. In this respect we establish a number of new results, including a description of the algebraic counterparts, Leibniz filters, and strong versions of super-Belnap logics, as well as the classification of these logics within the Leibniz and Frege hierarchies. [ABSTRACT FROM AUTHOR]

We consider a family of logical systems for representing entailment relations of various kinds. This family has its root in the logic of first-degree entailment formulated as a binary consequence system, i.e. a proof system dealing with the expressions of the form $$\varphi \vdash \psi $$ , where both $$\varphi $$ and $$\psi $$ are single formulas. We generalize this approach by constructing consequence systems that allow manipulating with sets of formulas, either to the right or left (or both) of the turnstile. In this way, it is possible to capture proof-theoretically not only the entailment relation of the standard four-valued Belnap's logic, but also its dual version, as well as some of their interesting extensions. The proof systems we propose are, in a sense, of a hybrid Hilbert-Gentzen nature. We examine some important properties of these systems and establish their completeness with respect to the corresponding entailment relations. [ABSTRACT FROM AUTHOR]

We introduce a novel expansion of the four-valued Belnap-Dunn logic by a unary operator representing reductio ad contradictionem and study its algebraic semantics. This expansion thus contains both the direct, non-inferential negation of the Belnap-Dunn logic and an inferential negation akin to the negation of Johansson's minimal logic. We formulate a sequent calculus for this logic and introduce the variety of reductio algebras as an algebraic semantics for this calculus. We then investigate some basic algebraic properties of this variety, in particular we show that it is locally finite and has EDPC. We identify the subdirectly irreducible algebras in this variety and describe the lattice of varieties of reductio algebras. In particular, we prove that this lattice contains an interval isomorphic to the lattice of classes of finite non-empty graphs with loops closed under surjective graph homomorphisms. [ABSTRACT FROM AUTHOR]

On the sidelines of classical logic, many partial and paraconsistent three-valued logics have been developed. Most of them differ in the notion of logical consequence or in the definition of logical connectives. This article aims, firstly, to provide both a model-theoretic and a proof-theoretic unified framework for these logics and, secondly, to apply these general frameworks to several well-known three-valued logics. The proof-theoretic approach to which we give preference is sequent calculus. In this perspective, several results concerning the properties of functional completeness, cut redundancy, and proof-search procedure are shown. We also provide a general proof for the soundness and the completeness of the three sequent calculi discussed. [ABSTRACT FROM AUTHOR]

Based on the semantic concepts developed by M. Dunn and N. Belnap, a four-valued language containing only two logical symbols is proposed. We show that this language is functionally complete with regard to the given semantics. Specifically, we prove that every truth-function is expressed by a formula of the language. To do this, we define two concepts akin to the disjunctive and conjunctive normal forms. Using these concepts, we establish that every truth-function for a four-valued semantics can be represented by a formula in a disjunctive form or in a conjunctive form. [ABSTRACT FROM AUTHOR]

In their recent paper Bi-facial truth: a case for generalized truth values Zaitsev and Shramko [] distinguish between an ontological and an epistemic interpretation of classical truth values. By taking the Cartesian product of the two disjoint sets of values thus obtained, they arrive at four generalized truth values and consider two 'semi-classical negations' on them. The resulting semantics is used to define three novel logics which are closely related to Belnap's well-known four valued logic. A syntactic characterization of these logics is left for further work. In this paper, based on our previous work on a functionally complete extension of Belnap's logic, we present a sound and complete tableau calculus for these logics. It crucially exploits the Cartesian nature of the four values, which is reflected in the fact that each proof consists of two tableaux. The bi-facial notion of truth of Z&S is thus augmented with a bi-facial notion of proof. We also provide translations between the logics for semi-classical negation and classical logic and show that an argument is valid in a logic for semi-classical negation just in case its translation is valid in classical logic. [ABSTRACT FROM AUTHOR]

This paper discusses an implementation of four speech acts: assert , concede , request and challenge in a paraconsistent framework. A natural four-valued model of interaction yields multiple new cognitive situations. They are analyzed in the context of communicative relations , which partially replace the concept of trust. These assumptions naturally lead to six types of situations, which often require performing conflict resolution and belief revision. The particular choice of a rule-based, Datalog ¬ ¬ - like query language 4QL as a four-valued implementation framework ensures that, in contrast to the standard two-valued approaches, tractability of the model is achieved. [ABSTRACT FROM AUTHOR]

This article proposes to expand first-order Belnap–Dunn logic with a new operator, whose intuitive meaning is ‘- is designated’ or ‘- has designated values’. It amounts to considering all of D'Ottaviano's possibility connective of J3, Baaz's Delta operator of infinite-valued Gödel logic and Kachi's determinate operator, in the different context of Belnap–Dunn logic. As for proof theory, we employ a natural deduction calculus with a slot for weakly definable connectives. Our main theorem is a general completeness result with respect to P({0,1})-valued semantics, which enables us to derive new completeness results for first-order extensions of Kachi's SPL and Omori and Waragai's BS4 and the known completeness results of D'Ottaviano and da Costa's J3 and Carnielli, Marcos and de Amo's LFI1. [ABSTRACT FROM AUTHOR]

Here, I first prove that certain families ofk-valued clones have the Gupta-Belnap fixed-point property. This essentially means that all propositional languages that are interpreted with operators belonging to those clones are such that any net of self-referential sentences in the language can be consistently evaluated. I then focus on two four-valued generalisations of the Kleene propositional operators that generalise the strong and weak Kleene operators: Belnap’s clone and Fitting’s clone, respectively. I apply the theorems from the initial part of the paper to analyse the fixed-point property of Belnap’s and Fitting’s clones when some special operators that reflect the semantics are added. The conclusion of the paper is that Fitting’s clone is better suited than Belnap’s to provide self-referential languages with highly expressive resources. [ABSTRACT FROM AUTHOR]