Articles de revues : « Three-valued logic »

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Kooi, Barteld, and Allard Tamminga. "Three-valued Logics in Modal Logic." Studia Logica 101, no. 5 (2012): 1061–72. http://dx.doi.org/10.1007/s11225-012-9420-0.

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Oliveira, Kleidson Êglicio Carvalho da Silva. "Paraconsistent Logic Programming in Three and Four-Valued Logics." Bulletin of Symbolic Logic 28, no. 2 (2022): 260. http://dx.doi.org/10.1017/bsl.2021.34.

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AbstractFrom the interaction among areas such as Computer Science, Formal Logic, and Automated Deduction arises an important new subject called Logic Programming. This has been used continuously in the theoretical study and practical applications in various fields of Artificial Intelligence. After the emergence of a wide variety of non-classical logics and the understanding of the limitations presented by first-order classical logic, it became necessary to consider logic programming based on other types of reasoning in addition to classical reasoning. A type of reasoning that has been well studied is the paraconsistent, that is, the reasoning that tolerates contradictions. However, although there are many paraconsistent logics with different types of semantics, their application to logic programming is more delicate than it first appears, requiring an in-depth study of what can or cannot be transferred directly from classical first-order logic to other types of logic.Based on studies of Tarcisio Rodrigues on the foundations of Paraconsistent Logic Programming (2010) for some Logics of Formal Inconsistency (LFIs), this thesis intends to resume the research of Rodrigues and place it in the specific context of LFIs with three- and four-valued semantics. This kind of logics are interesting from the computational point of view, as presented by Luiz Silvestrini in his Ph.D. thesis entitled “A new approach to the concept of quase-truth” (2011), and by Marcelo Coniglio and Martín Figallo in the article “Hilbert-style presentations of two logics associated to tetravalent modal algebras” [Studia Logica (2012)]. Based on original techniques, this study aims to define well-founded systems of paraconsistent logic programming based on well-known logics, in contrast to the ad hoc approaches to this question found in the literature.Abstract prepared by Kleidson Êglicio Carvalho da Silva Oliveira.E-mail: kecso10@yahoo.com.brURL: http://repositorio.unicamp.br/jspui/handle/REPOSIP/322632

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Томова, Н. Е. "Natural three-valued logics and classical logic." Logical Investigations 19 (April 9, 2013): 344–52. http://dx.doi.org/10.21146/2074-1472-2013-19-0-344-352.

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In this paper implicative fragments of natural three- valued logic are investigated. It is proved that some fragments are equivalent by set of tautologies to implicative fragment of classical logic. It is also shown that some natural three-valued logics verify all tautologies of classical propositional logic.

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Vauzeilles, J., and A. Strauss. "Intuitionistic three-valued logic and logic programming." RAIRO - Theoretical Informatics and Applications 25, no. 6 (1991): 557–87. http://dx.doi.org/10.1051/ita/1991250605571.

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Giuntini, Roberto. "Three-valued Brouwer-zadeh logic." International Journal of Theoretical Physics 32, no. 10 (1993): 1875–87. http://dx.doi.org/10.1007/bf00979508.

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Delahaye, J. P., and V. Thibau. "Programming in three-valued logic." Theoretical Computer Science 78, no. 1 (1991): 189–216. http://dx.doi.org/10.1016/0304-3975(51)90008-4.

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Тамминга, А. "Correspondence analysis for strong three-valued logic." Logical Investigations 20 (May 8, 2014): 253–66. http://dx.doi.org/10.21146/2074-1472-2014-20-0-253-266.

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I apply Kooi and Tamminga’s (2012) idea of correspondence analysis for many-valued logics to strong three-valued logic (K3). First, I characterize each possible single entry in the truth-table of a unary or a binary truth-functional operator that could be added to K3 by a basic inference scheme. Second, I define a class of natural deduction systems on the basis of these charac- terizing basic inference schemes and a natural deduction system for K3. Third, I show that each of the resulting natural deduction systems is sound and complete with respect to its particular semantics. Among other things, I thus obtain a new proof system for _ukasiewicz’s three-valued logic.

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Morikawa, Osamu. "Some modal logics based on a three-valued logic." Notre Dame Journal of Formal Logic 30, no. 1 (1988): 130–37. http://dx.doi.org/10.1305/ndjfl/1093635000.

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SEGERBERG, KRISTER. "Some Modal Logics based on a Three-valued Logic." Theoria 33, no. 1 (2008): 53–71. http://dx.doi.org/10.1111/j.1755-2567.1967.tb00610.x.

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Takagi, Tsubasa. "Translation from Three-Valued Quantum Logic to Modal Logic." International Journal of Theoretical Physics 60, no. 1 (2021): 366–77. http://dx.doi.org/10.1007/s10773-020-04701-z.

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AbstractWe translate the three-valued quantum logic into modal logic, and prove 3-equivalence between the valuation of the three-valued logic and a kind of Kripke model in regard to this translation. To prove 3-equivalence, we introduce an observable-dependent logic, which is a fragment of the many-valued quantum logic. Compared to the Birkhoff and von Neumann’s quantum logic, some notions about observables, the completeness relation for example, in quantum mechanics can be utilized if the observable-dependent logic is employed.

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Ma, Minghui, and Yuanlei Lin. "A Three-Valued Fregean Quantification Logic." Journal of Philosophical Logic 48, no. 2 (2018): 409–23. http://dx.doi.org/10.1007/s10992-018-9469-y.

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Pkhakadze, Sopo, and Hans Tompits. "Sequent-Type Calculi for Three-Valued and Disjunctive Default Logic." Axioms 9, no. 3 (2020): 84. http://dx.doi.org/10.3390/axioms9030084.

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Default logic is one of the basic formalisms for nonmonotonic reasoning, a well-established area from logic-based artificial intelligence dealing with the representation of rational conclusions, which are characterised by the feature that the inference process may require to retract prior conclusions given additional premisses. This nonmonotonic aspect is in contrast to valid inference relations, which are monotonic. Although nonmonotonic reasoning has been extensively studied in the literature, only few works exist dealing with a proper proof theory for specific logics. In this paper, we introduce sequent-type calculi for two variants of default logic, viz., on the one hand, for three-valued default logic due to Radzikowska, and on the other hand, for disjunctive default logic, due to Gelfond, Lifschitz, Przymusinska, and Truszczyński. The first variant of default logic employs Łukasiewicz’s three-valued logic as the underlying base logic and the second variant generalises defaults by allowing a selection of consequents in defaults. Both versions have been introduced to address certain representational shortcomings of standard default logic. The calculi we introduce axiomatise brave reasoning for these versions of default logic, which is the task of determining whether a given formula is contained in some extension of a given default theory. Our approach follows the sequent method first introduced in the context of nonmonotonic reasoning by Bonatti, which employs a rejection calculus for axiomatising invalid formulas, taking care of expressing the consistency condition of defaults.

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Konikowska, Beata, Andrzej Tarlecki, and Andrzej Blikle. "A Three-Valued Logic for Software Specification and Validation. Tertium tamen datur." Fundamenta Informaticae 14, no. 4 (1991): 411–53. http://dx.doi.org/10.3233/fi-1991-14403.

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Different calculi of partial or three-valued predicates have been used and studied by several authors in the context of software specification, development and validation. This paper offers a critical survey on the development of three-valued logics based on such calculi. In the first part of the paper we review two three-valued predicate calculi, based on, respectively, McCarthy’s and Kleene’s propositional connectives and quantifiers, and point out that in a three-valued logic one should distinguish between two notions of validity: strong validity (always true) and weak validity (never false). We define in model-theoretic terms a number of consequence relations for three-valued logics. Each of them is determined by the choice of the underlying predicate calculus and of the weak or strong validity of axioms and of theorems. We discuss mutual relationships between consequence relations defined in such a way and study some of their basic properties. The second part of the paper is devoted to the development of a formal deductive system of inference rules for a three-valued logic. We use the method of semantic tableaux (slightly modified to deal with three-valued formulas) to develop a Gentzen-style system of inference rules for deriving valid sequents, from which we then derive a sound and complete system of natural deduction rules. We have chosen to study the consequence relation determined by the predicate calculus with McCarthy’s propositional connectives and Kleene’s quantifiers and by the strong interpretation of both axioms and theorems. Although we find this choice appropriate for applications in the area of software specification, verification and development, we regard this logic merely as an example and use it to present some general techniques of developing a sequent calculus and a natural deduction system for a three-valued logic. We also discuss the extension of this logic by a non-monotone is-true predicate.

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Font, Josep M., and Massoud Moussavi. "Note on a six-valued extension of three-valued logic." Journal of Applied Non-Classical Logics 3, no. 2 (1993): 173–87. http://dx.doi.org/10.1080/11663081.1993.10510806.

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García Olmedo, Francisco M., and Antonio J. Rodríguez Salas. "Algebraization of the Three-valued BCK-logic." MLQ 48, no. 2 (2002): 163–78. http://dx.doi.org/10.1002/1521-3870(200202)48:2<163::aid-malq163>3.0.co;2-b.

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Feldman, Norman. "The cylindric algebras of three-valued logic." Journal of Symbolic Logic 63, no. 4 (1998): 1201–17. http://dx.doi.org/10.2307/2586647.

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In this paper we consider the three-valued logic used by Kleene [6] in the theory of partial recursive functions. This logic has three truth values: true (T), false (F), and undefined (U). One interpretation of U is as follows: Suppose we have two partially recursive predicates P(x) and Q(x) and we want to know the truth value of P(x) ∧ Q(x) for a particular x0. If x0 is in the domain of definition of both P and Q, then P(x0) ∧ Q(x0) is true if both P(x0) and Q(x0) are true, and false otherwise. But what if x0 is not in the domain of definition of P, but is in the domain of definition of Q? There are several choices, but the one chosen by Kleene is that if Q(X0) is false, then P(x0) ∧ Q(x0) is also false and if Q(X0) is true, then P(x0) ∧ Q(X0) is undefined.What arises is the question about knowledge of whether or not x0 is in the domain of definition of P. Is there an effective procedure to determine this? If not, then we can interpret U as being unknown. If there is an effective procedure, then our decision for the truth value for P(x) ∧ Q(x) is based on the knowledge that is not in the domain of definition of P. In this case, U can be interpreted as undefined. In either case, we base our truth value of P(x) ∧ Q(x) on the truth value of Q(X0).

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Bigaj, Tomasz. "Three-valued Logic, Indeterminacy and Quantum Mechanics." Journal of Philosophical Logic 30, no. 2 (2001): 97–119. http://dx.doi.org/10.1023/a:1017571731461.

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TÖRNEBOHM, HÅKAN. "On truth, implication, and three-valued logic." Theoria 22, no. 3 (2008): 185–98. http://dx.doi.org/10.1111/j.1755-2567.1956.tb01181.x.

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FISHER, MARK. "A three-valued calculus for deontic logic." Theoria 27, no. 3 (2008): 107–18. http://dx.doi.org/10.1111/j.1755-2567.1961.tb00019.x.

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Marchenkov, S. S. "Positively closed classes of three-valued logic." Journal of Applied and Industrial Mathematics 8, no. 2 (2014): 256–66. http://dx.doi.org/10.1134/s1990478914020124.

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Libkin, Leonid. "SQL’s Three-Valued Logic and Certain Answers." ACM Transactions on Database Systems 41, no. 1 (2016): 1–28. http://dx.doi.org/10.1145/2877206.

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NAISH, LEE. "A three-valued semantics for logic programmers." Theory and Practice of Logic Programming 6, no. 5 (2006): 509–38. http://dx.doi.org/10.1017/s1471068406002742.

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This paper describes a simpler way for programmers to reason about the correctness of their code. The study of semantics of logic programs has shown strong links between the model theoretic semantics (truth and falsity of atoms in the programmer's interpretation of a program), procedural semantics (for example, SLD resolution) and fixpoint semantics (which is useful for program analysis and alternative execution mechanisms). Most of this work assumes that intended interpretations are two-valued: a ground atom is true (and should succeed according to the procedural semantics) or false (and should not succeed). In reality, intended interpretations are less precise. Programmers consider that some atoms “should not occur” or are “ill-typed” or “inadmissible”. Programmers don't know and don't care whether such atoms succeed. In this paper we propose a three-valued semantics for (essentially) pure Prolog programs with (ground) negation as failure which reflects this. The semantics of Fitting is similar but only associates the third truth value with non-termination. We provide tools to reason about correctness of programs without the need for unnatural precision or undue restrictions on programming style. As well as theoretical results, we provide a programmer-oriented synopsis. This work has come out of work on declarative debugging, where it has been recognised that inadmissible calls are important.

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Bergstra, Jan A., and Alban Ponse. "Kleene's three-valued logic and process algebra." Information Processing Letters 67, no. 2 (1998): 95–103. http://dx.doi.org/10.1016/s0020-0190(98)00083-0.

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Leszczyńska-Jasion, Dorota, and Paweł Łupkowski. "Erotetic Search Scenarios and Three-Valued Logic." Journal of Logic, Language and Information 25, no. 1 (2015): 51–76. http://dx.doi.org/10.1007/s10849-015-9233-4.

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Radzikowska, Anna. "A three-valued approach to default logic." Journal of Applied Non-Classical Logics 6, no. 2 (1996): 149–90. http://dx.doi.org/10.1080/11663081.1996.10510876.

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Nota, G., S. Orefice, G. Pacini, F. Ruggiero, and G. Tortora. "Legality concepts for three-valued logic programs." Theoretical Computer Science 120, no. 1 (1993): 45–68. http://dx.doi.org/10.1016/0304-3975(93)90244-n.

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Teusink, Frank. "Three-valued completion for abductive logic programs." Theoretical Computer Science 165, no. 1 (1996): 171–200. http://dx.doi.org/10.1016/0304-3975(96)00044-8.

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Bezerra, Edson Vinícius. "Society semantics for four-valued Łukasiewicz logic." Logic Journal of the IGPL 28, no. 5 (2018): 892–911. http://dx.doi.org/10.1093/jigpal/jzy066.

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AbstractWe argue that many-valued logics (MVLs) can be useful in analysing informational conflicts by using society semantics (SSs). This work concentrates on four-valued Łukasiewicz logic. SSs were proposed by Carnielli and Lima-Marques (1999, Advances in Contemporary Logic and Computer Science, 235, 33–52) to deal with conflicts of information involving rational agents that make judgements about propositions according to a given logic within a society, where a society is understood as a collection $\mathcal{A}$ of agents. The interesting point of such semantics is that a new logic can be obtained by combining the logic of the agents under some appropriate rules. Carnielli and Lima-Marques (1999, Advances in Contemporary Logic and Computer Science, 235, 33–52) defined SSs for the three-valued logics $I^{1}$ and $P^{1}$. In this kind of semantics, all the agents reason according to classical logic (CL) and the molecular formulas behave in the same way as in CL (the non-classical character of these logics only appears at the propositional level). Marcos (unpublished data) provided SSs with classical agents for the three-valued Łukasiewicz logic Ł$_{3}$, but in this case, the molecular formulas do not behave classically. We prove here that one can characterize Ł$_{4}^{\prime}$, a conservative extension of Ł$_{4}$ obtained by adding a connective $\blacktriangledown$, by means of a closed society where the agents reason according to Ł$_{3}$. We shall emphasize the importance of recovery operators in the construction of this class of societies. Moreover, we shall relate this semantics to Suszko’s view on the ‘two-valuedness’ of logic.

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Blikle, Andrzej. "Three-Valued Predicates for Software Specification and Validation." Fundamenta Informaticae 14, no. 4 (1991): 387–410. http://dx.doi.org/10.3233/fi-1991-14402.

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Partial functions, hence also partial predicates, cannot be avoided in algorithms. However, in spite of the fact that partial functions have been formally introduced into the theory of software very early, partial predicates are still not quite commonly recognized. In many programming- and software-specification languages partial Boolean expressions are treated in a rather simplistic way: the evaluation of a Boolean sub-expression to an error leads to the evaluation of the hosting Boolean expression to an error and, in the consequence, to the abortion of the whole program. This technique is known as an eager evaluation of expressions. A more practical approach to the evaluation of expressions – gaining more interest today among both theoreticians and programming-language designers – is lazy evaluation. Lazily evaluated Boolean expressions correspond to (non-strict) three-valued predicates where the third value represents both an error and an undefinedness. On the semantic ground this leads to a three-valued propositional calculus, three-valued quantifiers and an appropriate logic. This paper is a survey-essay devoted to the discussion and the comparison of a few three-valued propositional and predicate calculi and to the discussion of the author’s claim that a two-valued logic, rather than a three-valued logic, is suitable for the treatment of programs with three-valued Boolean expressions. The paper is written in a formal but not in a formalized style. All discussion is carried on a semantic ground. We talk about predicates (functions) and a semantic consequence relation rather than about expressions and inference rules. However, the paper is followed by more formalized works which carry our discussion further on a formalized ground, and where corresponding formal logics are constructed and discussed.

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Martínez-Fernández, José, and Genoveva Martí. "The representation of gappy sentences in four-valued semantics." Semiotica 2021, no. 240 (2021): 145–63. http://dx.doi.org/10.1515/sem-2021-0011.

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Abstract 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.

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Przymusinski, Teodor. "Well-Founded Semantics Coincides with Three-Valued Stable Semantics1." Fundamenta Informaticae 13, no. 4 (1990): 445–63. http://dx.doi.org/10.3233/fi-1990-13404.

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We introduce 3-valued stable models which are a natural generalization of standard (2-valued) stable models. We show that every logic program P has at least one 3-valued stable model and that the well-founded model of any program P [Van Gelder et al., 1990] coincides with the smallest 3-valued stable model of P. We conclude that the well-founded semantics of an arbitrary logic program coincides with the 3-valued stable model semantics. The 3-valued stable semantics is closely related to non-monotonic formalisms in AI. Namely, every program P can be translated into a suitable autoepistemic (resp. default) theory P ˆ so that the 3-valued stable semantics of P coincides with the (3-valued) autoepistemic (resp. default) semantics of P ˆ. Similar results hold for circumscription and CWA. Moreover, it can be shown that the 3-valued stable semantics has a natural extension to the class of all disjunctive logic programs and deductive databases. Finally, following upon the recent approach developed by Gelfond and Lifschitz, we extend all of our results to more general logic programs which, in addition to the use of negation as failure, permit the use of classical negation.

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Akama, Seiki, and Yasunori Nagata. "Prior’s Three-Valued Modal Logic Q and its Possible Applications." Journal of Advanced Computational Intelligence and Intelligent Informatics 11, no. 1 (2007): 105–10. http://dx.doi.org/10.20965/jaciii.2007.p0105.

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Prior proposed a three-valued modal logic Q as a “correct” modal logic from his philosophical motivations. Unfortunately, Prior’s Q and many-valued modal logic have been neglected in the tradition of many-valued and modal logic. In this paper, we introduce a version of three-valued Kripke semantics for Q, which aims to establish Prior’s ideas based on possible worlds. We investigate formal properties of Q and prove the completeness theorem of Q. We also compare our approach with others and suggest possible applications.

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Konikowska, Beata. "A two-valued logic for reasoning about different types of consequence in Kleene's three-valued logic." Studia Logica 49, no. 4 (1990): 541–55. http://dx.doi.org/10.1007/bf00370164.

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Fitting, Melvin, and Marion Ben-Jacob. "Stratified, Weak Stratified, and Three-Valued Semantics1." Fundamenta Informaticae 13, no. 1 (1990): 19–33. http://dx.doi.org/10.3233/fi-1990-13104.

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We investigate the relationship between three-valued Kripke/Kleene semantics and stratified semantics for stratifiable logic programs. We first show these are compatible, in the sense that if the three-valued semantics assigns a classical truth value, the stratified approach will assign the same value. Next, the familiar fixed point semantics for pure Horn clause programs gives both smallest and biggest fixed points fundamental roles. We show how to extend this idea to the family of stratifiable logic programs, producing a semantics we call weak stratified. Finally, we show weak stratified semantics coincides exactly with the three-valued approach on stratifiable programs, though the three-valued version is generally applicable, and does not require stratification assumptions.

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Takano, Mitio. "Cut-free systems for three-valued modal logics." Notre Dame Journal of Formal Logic 33, no. 3 (1992): 359–68. http://dx.doi.org/10.1305/ndjfl/1093634401.

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Petrukhin, Ya I. "Natural deduction system for three-valued Heyting’s logic." Moscow University Mathematics Bulletin 72, no. 3 (2017): 133–36. http://dx.doi.org/10.3103/s002713221703007x.

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Barbosa, João, Mário Florido, and Vítor Santos Costa. "A Three-Valued Semantics for Typed Logic Programming." Electronic Proceedings in Theoretical Computer Science 306 (September 19, 2019): 36–51. http://dx.doi.org/10.4204/eptcs.306.10.

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Walker, E. A. "Stone algebras, conditional events, and three valued logic." IEEE Transactions on Systems, Man, and Cybernetics 24, no. 12 (1994): 1699–707. http://dx.doi.org/10.1109/21.328927.

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Marchenkov, Sergey S., and Anatoliy V. Chernyshev. "Basic positively closed classes in three-valued logic." Discrete Mathematics and Applications 28, no. 3 (2018): 157–65. http://dx.doi.org/10.1515/dma-2018-0015.

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Abstract Basic positively closed classes are intersections of positively precomplete classes. We prove that three-valued logic contains exactly 79 basic positively closed classes. Each class is described in terms of endomorphism semigroups.

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Besslich, Ph W., and E. A. Trachtenberg. "Three-valued quasi-linear transformation for logic synthesis." IEE Proceedings - Computers and Digital Techniques 143, no. 6 (1996): 391. http://dx.doi.org/10.1049/ip-cdt:19960466.

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Pynko, Alexej P. "Extensions of Hałkowska-Zajac's three-valued paraconsistent logic." Archive for Mathematical Logic 41, no. 3 (2002): 299–307. http://dx.doi.org/10.1007/s001530100115.

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Rubinson, Claude. "Nulls, three-valued logic, and ambiguity in SQL." ACM SIGMOD Record 36, no. 4 (2007): 13–17. http://dx.doi.org/10.1145/1361348.1361350.

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Stojmenović, Ivan. "On sheffer symmetric functions in three-valued logic." Discrete Applied Mathematics 22, no. 3 (1988): 267–74. http://dx.doi.org/10.1016/0166-218x(88)90099-6.

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Alves, E. H., and J. A. D. Guerzoni. "Extending Montague's system: a three valued intensional logic." Studia Logica 49, no. 1 (1990): 127–32. http://dx.doi.org/10.1007/bf00401558.

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Shan, Jing Yi, Zhi Xiang Yin, Xin Yu Tang, and Jing Jing Tang. "A DNA Computing Model for the AND Gate in Three-Valued Logical Circuit." Applied Mechanics and Materials 610 (August 2014): 764–68. http://dx.doi.org/10.4028/www.scientific.net/amm.610.764.

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Multiple-valued logic is an extended form of Boolean logic. In daily life, people often encounter the problem about the multiple-valued logic. With further study on Boolean logic, multiple-valued logic has been paid more and more attention by researchers. This paper achieves the operation of AND gate in three-valued logic by using the DNA hairpin structure. The experiment makes the DNA hairpin structure as the basic structure, and the molecular beacon as the input signal, and at last judges the logical results according to the intensity of fluorescence and gel electrophoresis. This method has the advantages that it has high sensitivity, good feasibility, and it is easy to observe. In addition, this method reduces the hybrid competition to a certain extent, and it is a new attempt to the research on multiple-valued logic.

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NAISH, LEE, and HARALD SØNDERGAARD. "Truth versus information in logic programming." Theory and Practice of Logic Programming 14, no. 6 (2013): 803–40. http://dx.doi.org/10.1017/s1471068413000069.

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AbstractThe semantics of logic programs was originally described in terms of two-valued logic. Soon, however, it was realised that three-valued logic had some natural advantages, as it provides distinct values not only for truth and falsehood but also for “undefined”. The three-valued semantics proposed by Fitting (Fitting, M. 1985. A Kripke–Kleene semantics for logic programs. Journal of Logic Programming 2, 4, 295–312) and Kunen (Kunen, K. 1987. Negation in logic programming. Journal of Logic Programming 4, 4, 289–308) are closely related to what is computed by a logic program, the third truth value being associated with non-termination. A different three-valued semantics, proposed by Naish, shared much with those of Fitting and Kunen but incorporated allowances for programmer intent, the third truth value being associated with underspecification. Naish used an (apparently) novel “arrow” operator to relate the intended meaning of left and right sides of predicate definitions. In this paper we suggest that the additional truth values of Fitting/Kunen and Naish are best viewed as duals. We use Belnap's four-valued logic (Belnap, N. D. 1977. A useful four-valued logic. In Modern Uses of Multiple-Valued Logic, J. M. Dunn and G. Epstein, Eds. D. Reidel, Dordrecht, Netherlands, 8–37), also used elsewhere by Fitting, to unify the two three-valued approaches. The truth values are arranged in a bilattice, which supports the classical ordering on truth values as well as the “information ordering”. We note that the “arrow” operator of Naish (and our four-valued extension) is essentially the information ordering, whereas the classical arrow denotes the truth ordering. This allows us to shed new light on many aspects of logic programming, including program analysis, type and mode systems, declarative debugging and the relationships between specifications and programs, and successive execution states of a program.

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Cobreros, Pablo, Paul Égré, David Ripley, and Robert van Rooij. "Foreword: Three-valued logics and their applications." Journal of Applied Non-Classical Logics 24, no. 1-2 (2014): 1–11. http://dx.doi.org/10.1080/11663081.2014.909631.

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Gehrke, Mai, Carol Walker, and Elbert Walker. "A Mathematical Setting for Fuzzy Logics." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 05, no. 03 (1997): 223–38. http://dx.doi.org/10.1142/s021848859700021x.

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The setup of a mathematical propositional logic is given in algebraic terms, describing exactly when two choices of truth value algebras give the same logic. The propositional logic obtained when the algebra of truth values is the real numbers in the unit interval equipped with minimum, maximum and -x=1-x for conjunction, disjunction and negation, respectively, is the standard propositional fuzzy logic. This is shown to be the same as three-valued logic. The propositional logic obtained when the algebra of truth values is the set {(a, b)|a≤ b and a,b∈[0,1]} of subintervals of the unit interval with component-wise operations, is propositional interval-valued fuzzy logic. This is shown to be the same as the logic given by a certain four element lattice of truth values. Since both of these logics are equivalent to ones given by finite algebras, it follows that there are finite algorithms for determining when two statements are logically equivalent within either of these logics. On this topic, normal forms are discussed for both of these logics.

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Petrukhin, Yaroslav. "Generalized Correspondence Analysis for Three-Valued Logics." Logica Universalis 12, no. 3-4 (2018): 423–60. http://dx.doi.org/10.1007/s11787-018-0212-9.

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Abdullaev, T. R., and G. U. Juraev. "Application three-valued logic in symmetric block encryption algorithms." Journal of Physics: Conference Series 2131, no. 2 (2021): 022082. http://dx.doi.org/10.1088/1742-6596/2131/2/022082.

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Abstract The issues of limiting the use of binary logic for the further development of science engineering are discussed. The effectiveness of the use of the ternary number system at this stage in the development of information technologies is substantiated and shown. A method is proposed for increasing the informational entropy of plaintext by adding random data using ternary logic in the process of symmetric encryption. To reliably hide the added random data, the first transforming function is proposed to choose gamming with a key.

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