Journal articles: 'Algebraic logic'

1

Rota, Gian-Carlo. "Algebraic logic." Advances in Mathematics 61, no. 2 (August 1986): 184. http://dx.doi.org/10.1016/0001-8708(86)90075-7.

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Yang, Eunsuk. "Basic Core Fuzzy Logics and Algebraic Routley–Meyer-Style Semantics." Axioms 10, no. 4 (October 25, 2021): 273. http://dx.doi.org/10.3390/axioms10040273.

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Recently, algebraic Routley–Meyer-style semantics was introduced for basic substructural logics. This paper extends it to fuzzy logics. First, we recall the basic substructural core fuzzy logic MIAL (Mianorm logic) and its axiomatic extensions, together with their algebraic semantics. Next, we introduce two kinds of ternary relational semantics, called here linear Urquhart-style and Fine-style Routley–Meyer semantics, for them as algebraic Routley–Meyer-style semantics.

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3

De Araujo Feitosa, Hércules, Mariana Matulovic, and Ana Claudia de J. Golzio. "A basic epistemic logic and its algebraic model." INTERMATHS 4, no. 2 (December 30, 2023): 28–37. http://dx.doi.org/10.22481/intermaths.v4i2.14133.

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In this paper we propose an algebraic model for a modal epistemic logic. Although it is known the existence of algebraic models for modal logics, considering that there are so many different modal logics, so it is not usual to give an algebraic model for each such system. The basic epistemic logic used in the paper is bimodal and we can show that the epistemic algebra introduced in the paper is an adequate model for it.

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Höfner, Peter, and Bernhard Möller. "Algebraic Neighbourhood Logic." Journal of Logic and Algebraic Programming 76, no. 1 (May 2008): 35–59. http://dx.doi.org/10.1016/j.jlap.2007.10.004.

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Dang, H. H., P. Höfner, and B. Möller. "Algebraic separation logic." Journal of Logic and Algebraic Programming 80, no. 6 (August 2011): 221–47. http://dx.doi.org/10.1016/j.jlap.2011.04.003.

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Maddux, Roger D. "Finitary Algebraic Logic." Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 35, no. 4 (1989): 321–32. http://dx.doi.org/10.1002/malq.19890350405.

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7

Hsiang, Jieh, and Anita Wasilewska. "Automating Algebraic Proofs in Algebraic Logic." Fundamenta Informaticae 28, no. 1,2 (1996): 129–40. http://dx.doi.org/10.3233/fi-1996-281208.

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8

van Alten, C. J. "The finite model property for knotted extensions of propositional linear logic." Journal of Symbolic Logic 70, no. 1 (March 2005): 84–98. http://dx.doi.org/10.2178/jsl/1107298511.

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AbstractThe logics considered here are the propositional Linear Logic and propositional Intuitionistic Linear Logic extended by a knotted structural rule: . It is proved that the class of algebraic models for such a logic has the finite embeddability property, meaning that every finite partial subalgebra of an algebra in the class can be embedded into a finite full algebra in the class. It follows that each such logic has the finite model property with respect to its algebraic semantics and hence that the logic is decidable.

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9

Font, Josep Maria, and Miquel Rius. "An abstract algebraic logic approach to tetravalent modal logics." Journal of Symbolic Logic 65, no. 2 (June 2000): 481–518. http://dx.doi.org/10.2307/2586552.

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AbstractThis paper contains a joint study of two sentential logics that combine a many-valued character, namely tetravalence, with a modal character; one of them is normal and the other one quasinormal. The method is to study their algebraic counterparts and their abstract models with the tools of Abstract Algebraic Logic, and particularly with those of Brown and Suszko's theory of abstract logics as recently developed by Font and Jansana in their “A General Algebraic Semantics for Sentential Logics”. The logics studied here arise from the algebraic and lattice-theoretical properties we review of Tetravalent Modal Algebras, a class of algebras studied mainly by Loureiro, and also by Figallo. Landini and Ziliani, at the suggestion of the late Antonio Monteiro.

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10

ALBUQUERQUE, HUGO, JOSEP MARIA FONT, and RAMON JANSANA. "COMPATIBILITY OPERATORS IN ABSTRACT ALGEBRAIC LOGIC." Journal of Symbolic Logic 81, no. 2 (June 2016): 417–62. http://dx.doi.org/10.1017/jsl.2015.39.

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AbstractThis paper presents a unified framework that explains and extends the already successful applications of the Leibniz operator, the Suszko operator, and the Tarski operator in recent developments in abstract algebraic logic. To this end, we refine Czelakowski’s notion of an S-compatibility operator, and introduce the notion of coherent family of S-compatibility operators, for a sentential logic S. The notion of coherence is a restricted property of commutativity with inverse images by surjective homomorphisms, which is satisfied by both the Leibniz and the Suszko operators. We generalize several constructions and results already existing for the mentioned operators; in particular, the well-known classes of algebras associated with a logic through each of them, and the notions of full generalized model of a logic and a special kind of S-filters (which generalizes the less-known notion of Leibniz filter). We obtain a General Correspondence Theorem, extending the well-known one from the theory of protoalgebraic logics to arbitrary logics and to more general operators, and strengthening its formulation. We apply the general results to the Leibniz and the Suszko operators, and obtain several characterizations of the main classes of logics in the Leibniz hierarchy by the form of their full generalized models, by old and new properties of the Leibniz operator, and by the behaviour of the Suszko operator. Some of these characterizations complete or extend known ones, for some classes in the hierarchy, thus offering an integrated approach to the Leibniz hierarchy that uncovers some new, nice symmetries.

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11

Sen, Jayanta, and M. K. Chakraborty. "Linear Logic and Lukasiewicz ℵ0- Valued Logic: A Logico-Algebraic Study." Journal of Applied Non-Classical Logics 11, no. 3-4 (January 2001): 313–29. http://dx.doi.org/10.3166/jancl.11.313-329.

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12

Voutsadakis, George. "Categorical Abstract Algebraic Logic: Referential Algebraic Semantics." Studia Logica 101, no. 4 (June 28, 2013): 849–99. http://dx.doi.org/10.1007/s11225-013-9500-9.

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13

Dai, Jianhua. "Generalized Rough Logics with Rough Algebraic Semantics." International Journal of Cognitive Informatics and Natural Intelligence 4, no. 2 (April 2010): 35–49. http://dx.doi.org/10.4018/jcini.2010040103.

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The collection of the rough set pairs <lower approximation, upper approximation> of an approximation (U, R) can be made into a Stone algebra by defining two binary operators and one unary operator on the pairs. By introducing a more unary operator, one can get a regular double Stone algebra to describe the rough set pairs of an approximation space. Sequent calculi corresponding to the rough algebras, including rough Stone algebras, Stone algebras, rough double Stone algebras, and regular double Stone algebras are proposed in this paper. The sequent calculi are called rough Stone logic (RSL), Stone logic (SL), rough double Stone logic (RDSL), and double Stone Logic (DSL). The languages, axioms and rules are presented. The soundness and completeness of the logics are proved.

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14

Sain, I. "Finite schematizable algebraic logic." Logic Journal of IGPL 5, no. 5 (September 1, 1997): 699–751. http://dx.doi.org/10.1093/jigpal/5.5.699.

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15

Bryant, B. "Algebraic and logic programming." Information and Software Technology 33, no. 6 (July 1991): 463. http://dx.doi.org/10.1016/0950-5849(91)90085-p.

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16

Maddux, Roger D. "Finitary algebraic logic II." Mathematical Logic Quarterly 39, no. 1 (1993): 566–69. http://dx.doi.org/10.1002/malq.19930390159.

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17

Metcalfe, George, and Franco Montagna. "Substructural fuzzy logics." Journal of Symbolic Logic 72, no. 3 (September 2007): 834–64. http://dx.doi.org/10.2178/jsl/1191333844.

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AbstractSubstructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval [0, 1]. In this paper, we introduce Uninorm logic UL as Multiplicative additive intuitionistic linear logic MAILL extended with the prelinearity axiom ((A → B) ∧ t) V ((B → A)∧ t). Axiomatic extensions of UL include known fuzzy logics such as Monoidal t-norm logic MIX and Gödel logic G, and new weakening-free logics. Algebraic semantics for these logics are provided by subvarieties of (representable) pointed bounded commutative residuated lattices. Gentzen systems admitting cut-elimination are given in the framework of hypersequents. Completeness with respect to algebras with lattice reduct [0, 1] is established for UL and several extensions using a two-part strategy. First, completeness is proved for the logic extended with Takeuti and Titani's density rule. A syntactic elimination of the rule is then given using a hypersequent calculus. As an algebraic corollary, it follows that certain varieties of residuated lattices are generated by their members with lattice reduct [0, 1].

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18

Yang, Eunsuk. "Fixpointed Idempotent Uninorm (Based) Logics." Mathematics 7, no. 1 (January 20, 2019): 107. http://dx.doi.org/10.3390/math7010107.

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Idempotent uninorms are simply defined by fixpointed negations. These uninorms, called here fixpointed idempotent uninorms, have been extensively studied because of their simplicity, whereas logics characterizing such uninorms have not. Recently, fixpointed uninorm mingle logic (fUML) was introduced, and its standard completeness, i.e., completeness on real unit interval [ 0 , 1 ] , was proved by Baldi and Ciabattoni. However, their proof is not algebraic and does not shed any light on the algebraic feature by which an idempotent uninorm is characterized, using operations defined by a fixpointed negation. To shed a light on this feature, this paper algebraically investigates logics based on fixpointed idempotent uninorms. First, several such logics are introduced as axiomatic extensions of uninorm mingle logic (UML). The algebraic structures corresponding to the systems are then defined, and the results of the associated algebraic completeness are provided. Next, standard completeness is established for the systems using an Esteva–Godo-style approach for proving standard completeness.

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19

Zhang, Xiaohong, Xiangyu Ma, and Xuejiao Wang. "Filters in Strong BI-Algebras and Residuated Pseudo-SBI-Algebras." Mathematics 8, no. 9 (September 4, 2020): 1513. http://dx.doi.org/10.3390/math8091513.

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The concept of basic implication algebra (BI-algebra) has been proposed to describe general non-classical implicative logics (such as associative or non-associative fuzzy logic, commutative or non-commutative fuzzy logic, quantum logic). However, this algebra structure does not have enough characteristics to describe residual implications in depth, so we propose a new concept of strong BI-algebra, which is exactly the algebraic abstraction of fuzzy implication with pseudo-exchange principle (PEP). Furthermore, in order to describe the characteristics of the algebraic structure corresponding to the non-commutative fuzzy logics, we extend strong BI-algebra to the non-commutative case, and propose the concept of pseudo-strong BI (SBI)-algebra, which is the common extension of quantum B-algebras, pseudo-BCK/BCI-algebras and other algebraic structures. We establish the filter theory and quotient structure of pseudo-SBI- algebras. Moreover, based on prequantales, semi-uninorms, t-norms and their residual implications, we introduce the concept of residual pseudo-SBI-algebra, which is a common extension of (non-commutative) residual lattices, non-associative residual lattices, and also a special kind of residual partially-ordered groupoids. Finally, we investigate the filters and quotient algebraic structures of residuated pseudo-SBI-algebras, and obtain a unity frame of filter theory for various algebraic systems.

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20

Platzer, A. "Differential-algebraic Dynamic Logic for Differential-algebraic Programs." Journal of Logic and Computation 20, no. 1 (November 18, 2008): 309–52. http://dx.doi.org/10.1093/logcom/exn070.

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21

Voutsadakis, George. "Categorical Abstract Algebraic Logic: (ℐ,N)-Algebraic Systems." Applied Categorical Structures 13, no. 3 (June 2005): 265–80. http://dx.doi.org/10.1007/s10485-005-5797-5.

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22

Voutsadakis, George. "Categorical Abstract Algebraic Logic: Partially Ordered Algebraic Systems." Applied Categorical Structures 14, no. 1 (February 2006): 81–98. http://dx.doi.org/10.1007/s10485-005-9006-3.

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23

Hirani, Anil, and V. S. Subrahmanian. "Algebraic Foundations of Logic Programming, I: The Distributive Lattice of Logic Programs." Fundamenta Informaticae 13, no. 3 (July 1, 1990): 317–32. http://dx.doi.org/10.3233/fi-1990-13306.

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Given a logic program P, the operator TP associated with P is closely related to the intended meaning of P. Given a first order language L that is generated by finitely many non-logical symbols, our aim is to study the algebraic properties of the set {TP|P is a general logic program in language L} with certain operators on it. For the operators defined in this paper the resulting algebraic structure is a bounded distributive lattice. Our study extends (to the case of general logic programs), the work of Mancarella and Pedreschi who initiated a study of the algebraic properties of the space of pure logic programs. We study the algebraic properties of this set and identify the ideals and zero divisors. In addition, we prove that our algebra satisfies various non-extensibility conditions.

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24

Rounds, William C., and Guo-Qiang Zhang. "Clausal Logic and Logic Programming in Algebraic Domains." Information and Computation 171, no. 2 (December 2001): 183–200. http://dx.doi.org/10.1006/inco.2001.3073.

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25

Kuznetsov, Stepan. "Action Logic is Undecidable." ACM Transactions on Computational Logic 22, no. 2 (May 15, 2021): 1–26. http://dx.doi.org/10.1145/3445810.

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Action logic is the algebraic logic (inequational theory) of residuated Kleene lattices. One of the operations of this logic is the Kleene star, which is axiomatized by an induction scheme. For a stronger system that uses an -rule instead (infinitary action logic), Buszkowski and Palka (2007) proved -completeness (thus, undecidability). Decidability of action logic itself was an open question, raised by Kozen in 1994. In this article, we show that it is undecidable, more precisely, -complete. We also prove the same undecidability results for all recursively enumerable logics between action logic and infinitary action logic, for fragments of these logics with only one of the two lattice (additive) connectives, and for action logic extended with the law of distributivity.

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26

Ahmed, Tarek. "Amalgamation in universal algebraic logic." Studia Scientiarum Mathematicarum Hungarica 49, no. 1 (March 1, 2012): 26–43. http://dx.doi.org/10.1556/sscmath.2011.1184.

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Let (Kα: α ≧ ω) be a system of varieties definable by schemas. We characterize the amalgamation base, strong amalgamation base, and super amalgamation base of the class \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $S\mathfrak{N}\mathfrak{r}_\alpha $ \end{document}Kα+ω at this abstract level.

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27

Bancerek, Grzegorz. "Algebraic Approach to Algorithmic Logic." Formalized Mathematics 22, no. 3 (September 1, 2014): 225–55. http://dx.doi.org/10.2478/forma-2014-0025.

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Summary We introduce algorithmic logic - an algebraic approach according to [25]. It is done in three stages: propositional calculus, quantifier calculus with equality, and finally proper algorithmic logic. For each stage appropriate signature and theory are defined. Propositional calculus and quantifier calculus with equality are explored according to [24]. A language is introduced with language signature including free variables, substitution, and equality. Algorithmic logic requires a bialgebra structure which is an extension of language signature and program algebra. While-if algebra of generator set and algebraic signature is bialgebra with appropriate properties and is used as basic type of algebraic logic.

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Banerjee, Mohua, and Mihir K. Chakraborty. "Rough Sets Through Algebraic Logic." Fundamenta Informaticae 28, no. 3,4 (1996): 211–21. http://dx.doi.org/10.3233/fi-1996-283401.

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29

Hirsch, Robin, and Ian Hodkinson. "Complete representations in algebraic logic." Journal of Symbolic Logic 62, no. 3 (September 1997): 816–47. http://dx.doi.org/10.2307/2275574.

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AbstractA boolean algebra is shown to be completely representable if and only if it is atomic, whereas it is shown that neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any fixed dimension (at least 3) are elementary.

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30

Goldblatt, R. "Algebraic polymodal logic: a survey." Logic Journal of IGPL 8, no. 4 (July 1, 2000): 393–450. http://dx.doi.org/10.1093/jigpal/8.4.393.

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31

Sayed-Ahmed, Tarek. "Independence Results in Algebraic Logic." Logic Journal of the IGPL 14, no. 1 (January 1, 2006): 87–96. http://dx.doi.org/10.1093/jigpal/jzi058.

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Svozil, K. "Quantum Logic in Algebraic Approach." Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 32, no. 1 (March 2001): 113–15. http://dx.doi.org/10.1016/s1355-2198(00)00005-8.

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33

Beall, J. "Algebraic Methods in Philosophical Logic." Australasian Journal of Philosophy 81, no. 3 (September 2003): 442–44. http://dx.doi.org/10.1080/713659690.

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34

Andréka, H., M. Ferenczi, I. Németi, and Gy Serény. "Algebraic logic conference, Budapest, 1988." Journal of Symbolic Logic 54, no. 2 (June 1989): 686. http://dx.doi.org/10.1017/s0022481200027493.

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35

Turunen, Esko. "Algebraic structures in fuzzy logic." Fuzzy Sets and Systems 52, no. 2 (December 1992): 181–88. http://dx.doi.org/10.1016/0165-0114(92)90048-9.

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36

Venema, Y. "Algebraic and Coalgebraic Logic Corner." Journal of Logic and Computation 19, no. 2 (August 14, 2008): 303. http://dx.doi.org/10.1093/logcom/exn098.

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37

Raftery, James G. "Inconsistency lemmas in algebraic logic." Mathematical Logic Quarterly 59, no. 6 (November 2013): 393–406. http://dx.doi.org/10.1002/malq.201200020.

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38

Moss, Lawrence S., and Satish R. Thatte. "Modal logic and algebraic specifications." Theoretical Computer Science 111, no. 1-2 (April 1993): 191–210. http://dx.doi.org/10.1016/0304-3975(93)90187-x.

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39

Jacobs, Bart, and Jorik Mandemaker. "Coreflections in Algebraic Quantum Logic." Foundations of Physics 42, no. 7 (May 10, 2012): 932–58. http://dx.doi.org/10.1007/s10701-012-9654-8.

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40

Fu, Jun, Jinzhao Wu, and Hongyan Tan. "A Deductive Approach towards Reasoning about Algebraic Transition Systems." Mathematical Problems in Engineering 2015 (2015): 1–12. http://dx.doi.org/10.1155/2015/607013.

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Algebraic transition systems are extended from labeled transition systems by allowing transitions labeled by algebraic equations for modeling more complex systems in detail. We present a deductive approach for specifying and verifying algebraic transition systems. We modify the standard dynamic logic by introducing algebraic equations into modalities. Algebraic transition systems are embedded in modalities of logic formulas which specify properties of algebraic transition systems. The semantics of modalities and formulas is defined with solutions of algebraic equations. A proof system for this logic is constructed to verify properties of algebraic transition systems. The proof system combines with inference rules decision procedures on the theory of polynomial ideals to reduce a proof-search problem to an algebraic computation problem. The proof system proves to be sound but inherently incomplete. Finally, a typical example illustrates that reasoning about algebraic transition systems with our approach is feasible.

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41

Ahmed, Tarek Sayed. "Algebraic Logic, Where Does it Stand Today?" Bulletin of Symbolic Logic 11, no. 4 (December 2005): 465–516. http://dx.doi.org/10.2178/bsl/1130335206.

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AbstractThis is a survey article on algebraic logic. It gives a historical background leading up to a modern perspective. Central problems in algebraic logic (like the representation problem) are discussed in connection to other branches of logic, like modal logic, proof theory, model-theoretic forcing, finite combinatorics, and Gödel's incompleteness results. We focus on cylindric algebras. Relation algebras and polyadic algebras are mostly covered only insofar as they relate to cylindric algebras, and even there we have not told the whole story. We relate the algebraic notion of neat embeddings (a notion special to cylindric algebras) to the metalogical ones of provability, interpolation and omitting types in variants of first logic. Another novelty that occurs here is relating the algebraic notion of atom-canonicity for a class of boolean algebras with operators to the metalogical one of omitting types for the corresponding logic. A hitherto unpublished application of algebraic logic to omitting types of first order logic is given. Proofs are included when they serve to illustrate certain concepts. Several open problems are posed. We have tried as much as possible to avoid exploring territory already explored in the survey articles of Monk [93] and Németi [97] in the subject.

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Rivieccio, Umberto, Paulo Maia, and Achim Jung. "Non-involutive twist-structures." Logic Journal of the IGPL 28, no. 5 (November 29, 2018): 973–99. http://dx.doi.org/10.1093/jigpal/jzy070.

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

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43

Castiglioni, JosÉ Luis, Víctor FernÁndez, Héctor Federico Mallea, and HernÁn Javier San MartÍn. "On subreducts of subresiduated lattices and some related logics." Journal of Logic and Computation 34, no. 5 (June 20, 2023): 856–86. http://dx.doi.org/10.1093/logcom/exad042.

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Abstract Subresiduated lattices were introduced during the decade of 1970 by Epstein and Horn as an algebraic counterpart of some logics with strong implication previously studied by Lewy and Hacking. These logics are examples of subintuitionistic logics, i.e. logics in the language of intuitionistic logic that are defined semantically by using Kripke models, in the same way as intuitionistic logic is defined, but without requiring of the models some of the properties required in the intuitionistic case. Also in relation with the study of subintuitionistic logics, Celani and Jansana get these algebras as the elements of a subvariety of that of weak Heyting algebras. Here, we study both the implicative and the implicative-infimum subreducts of subresiduated lattices. Besides, we propose a calculus whose equivalent algebraic semantics is given by these classes of algebras. Several expansions of these calculi are also studied together with some interesting properties of them.

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Карниэлли, У. "Formal polynomials, heuristics and proofs in logic." Logical Investigations 16 (April 7, 2010): 280–94. http://dx.doi.org/10.21146/2074-1472-2010-16-0-280-294.

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This note surveys some previous results on the role of formal polynomials as a representation method for logical derivation in classical and non-classical logics, emphasizing many-valued logics, paraconsistent logics and modal logics. It also discusses the potentialities of formal polynomials as heuristic devices in logic and for expressing certain meta-logical properties, as well as pointing to some promising generalizations towards algebraic geometry.

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45

Voutsadakis, G. "Categorical Abstract Algebraic Logic: Categorical Algebraization of Equational Logic." Logic Journal of IGPL 12, no. 4 (July 1, 2004): 313–33. http://dx.doi.org/10.1093/jigpal/12.4.313.

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46

Hirsch, R., I. Hodkinson, and A. Kurucz. "On modal logics betweenK × K × KandS5 × S5 × S5." Journal of Symbolic Logic 67, no. 1 (March 2002): 221–34. http://dx.doi.org/10.2178/jsl/1190150040.

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AbstractWe prove that everyn-modal logic betweenKnandS5nis undecidable, whenever n ≥ 3. We also show that each of these logics is non-finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov–Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a reduction of the (undecidable) representation problem of finite relation algebras.

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47

Spitters, Bas. "Constructive algebraic integration theory." Annals of Pure and Applied Logic 137, no. 1-3 (January 2006): 380–90. http://dx.doi.org/10.1016/j.apal.2005.05.031.

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48

Banaschewski, Bernhard. "ALGEBRAIC CLOSURE WITHOUT CHOICE." Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 38, no. 1 (1992): 383–85. http://dx.doi.org/10.1002/malq.19920380136.

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49

CINTULA, PETR, ROSTISLAV HORČÍK, and CARLES NOGUERA. "NONASSOCIATIVE SUBSTRUCTURAL LOGICS AND THEIR SEMILINEAR EXTENSIONS: AXIOMATIZATION AND COMPLETENESS PROPERTIES." Review of Symbolic Logic 6, no. 3 (May 15, 2013): 394–423. http://dx.doi.org/10.1017/s1755020313000099.

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AbstractSubstructural logics extending the full Lambek calculus FL have largely benefited from a systematical algebraic approach based on the study of their algebraic counterparts: residuated lattices. Recently, a nonassociative generalization of FL (which we call SL) has been studied by Galatos and Ono as the logic of lattice-ordered residuated unital groupoids. This paper is based on an alternative Hilbert-style presentation for SL which is almost MP-based. This presentation is then used to obtain, in a uniform way applicable to most (both associative and nonassociative) substructural logics, a form of local deduction theorem, description of filter generation, and proper forms of generalized disjunctions. A special stress is put on semilinear substructural logics (i.e., logics complete with respect to linearly ordered algebras). Axiomatizations of the weakest semilinear logic over SL and other prominent substructural logics are provided and their completeness with respect to chains defined over the real unit interval is proved.

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Lávička, Tomáš, and Carles Noguera. "A New Hierarchy of Infinitary Logics in Abstract Algebraic Logic." Studia Logica 105, no. 3 (December 23, 2016): 521–51. http://dx.doi.org/10.1007/s11225-016-9699-3.

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Journal articles on the topic 'Algebraic logic'

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