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Janusz Czelakowski’s Research on the Theory of Matrices and its Applications in the Seventies and Eighties of the 20th CenturyIn Jacek Malinowski & Rafał Palczewski (eds.), Janusz Czelakowski on Logical Consequence, Springer Verlag. pp. 81-121. 2024.We present, organized by topics, the research of Janusz Czelakowski on the theory of logical matrices as models of propositional logics developed during the seventies and the eighties of the 20th century.
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67Note on algebraic models for relevance logicZeitschrift fur mathematische Logik und Grundlagen der Mathematik 36 (6): 535-540. 1990.
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40On weakening the Deduction Theorem and strengthening Modus PonensMathematical Logic Quarterly 50 (3): 303-324. 2004.This paper studies, with techniques ofAlgebraic Logic, the effects of putting a bound on the cardinality of the set of side formulas in the Deduction Theorem, viewed as a Gentzen-style rule, and of adding additional assumptions inside the formulas present in Modus Ponens, viewed as a Hilbert-style rule. As a result, a denumerable collection of new Gentzen systems and two new sentential logics have been isolated. These logics are weaker than the positive implicative logic. We have determined thei…Read more
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37The Strong Version of a Sentential LogicStudia Logica 105 (4): 703-760. 2017.This paper explores a notion of “the strong version” of a sentential logic S, initially defined in terms of the notion of a Leibniz filter, and shown to coincide with the logic determined by the matrices of S whose filter is the least S-filter in the algebra of the matrix. The paper makes a general study of this notion, which appears to unify under an abstract framework the relationships between many pairs of logics in the literature. The paradigmatic examples are the local and the global conseq…Read more
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52Compatibility operators in abstract algebraic logicJournal of Symbolic Logic 81 (2): 417-462. 2016.This 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 …Read more
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72Algebraic Study of Two Deductive Systems of Relevance LogicNotre Dame Journal of Formal Logic 35 (3): 369-397. 1994.In this paper two deductive systems associated with relevance logic are studied from an algebraic point of view. One is defined by the familiar, Hilbert-style, formalization of R; the other one is a weak version of it, called WR, which appears as the semantic entailment of the Meyer-Routley-Fine semantics, and which has already been suggested by Wójcicki for other reasons. This weaker consequence is first defined indirectly, using R, but we prove that the first one turns out to be an axiomatic e…Read more
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25The Strong Version of a Sentential LogicStudia Logica 105 (4): 703-760. 2017.This paper explores a notion of “the strong version” of a sentential logic S, initially defined in terms of the notion of a Leibniz filter, and shown to coincide with the logic determined by the matrices of S whose filter is the least S-filter in the algebra of the matrix. The paper makes a general study of this notion, which appears to unify under an abstract framework the relationships between many pairs of logics in the literature. The paradigmatic examples are the local and the global conseq…Read more
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20Erratum to J. M. Font, The simplest protoalgebraic logicMathematical Logic Quarterly 60 (1-2): 91-91. 2014.
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66Beyond Rasiowa's Algebraic Approach to Non-classical LogicsStudia Logica 82 (2): 179-209. 2006.This paper reviews the impact of Rasiowa's well-known book on the evolution of algebraic logic during the last thirty or forty years. It starts with some comments on the importance and influence of this book, highlighting some of the reasons for this influence, and some of its key points, mathematically speaking, concerning the general theory of algebraic logic, a theory nowadays called Abstract Algebraic Logic. Then, a consideration of the diverse ways in which these key points can be generaliz…Read more
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13Belnap's four-valued logic and De Morgan latticesLogic Journal of the IGPL 5 (1): 1--29. 1997.This paper contains some contributions to the study of Belnap's four-valued logic from an algebraic point of view. We introduce a finite Hilbert-style axiomatization of this logic, along with its well-known semantical presentation, and a Gentzen calculus that slightly differs from the usual one in that it is closer to Anderson and Belnap's formalization of their “logic of first-degree entailments”. We prove several Completeness Theorems and reduce every formula to an equivalent normal form. The …Read more
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65An abstract algebraic logic approach to tetravalent modal logicsJournal of Symbolic Logic 65 (2): 481-518. 2000.This 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". T…Read more
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80Update to “A Survey of Abstract Algebraic Logic”Studia Logica 91 (1): 125-130. 2009.A definition and some inaccurate cross-references in the paper A Survey of Abstract Algebraic Logic, which might confuse some readers, are clarified and corrected; a short discussion of the main one is included. We also update a dozen of bibliographic references.
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104Taking Degrees of Truth SeriouslyStudia Logica 91 (3): 383-406. 2009.This is a contribution to the discussion on the role of truth degrees in manyvalued logics from the perspective of abstract algebraic logic. It starts with some thoughts on the so-called Suszko’s Thesis (that every logic is two-valued) and on the conception of semantics that underlies it, which includes the truth-preserving notion of consequence. The alternative usage of truth values in order to define logics that preserve degrees of truth is presented and discussed. Some recent works studying t…Read more
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103On the infinite-valued Łukasiewicz logic that preserves degrees of truthArchive for Mathematical Logic 45 (7): 839-868. 2006.Łukasiewicz’s infinite-valued logic is commonly defined as the set of formulas that take the value 1 under all evaluations in the Łukasiewicz algebra on the unit real interval. In the literature a deductive system axiomatized in a Hilbert style was associated to it, and was later shown to be semantically defined from Łukasiewicz algebra by using a “truth-preserving” scheme. This deductive system is algebraizable, non-selfextensional and does not satisfy the deduction theorem. In addition, there …Read more
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72On łukasiewicz's four-valued modal logicStudia Logica 70 (2): 157-182. 2002.ukasiewicz''s four-valued modal logic is surveyed and analyzed, together with ukasiewicz''s motivations to develop it. A faithful interpretation of it in classical (non-modal) two-valued logic is presented, and some consequences are drawn concerning its classification and its algebraic behaviour. Some counter-intuitive aspects of this logic are discussed in the light of the presented results, ukasiewicz''s own texts, and related literature.
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21On Special Implicative FiltersMathematical Logic Quarterly 45 (1): 117-126. 1999.In her well-known book, Rasiowa states without proof that in implicative algebras there is a one-to-one correspondence between kernels of epimorphisms and the so-called special implicative filters, and that in the logic whose algebraic counterpart is the class of implicative algebras the deductive filters coincide with the special implicative filters. We show that neither claim is true, and how to repair the situation by redefining some of the notions involved. We answer other questions concerni…Read more
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54On the Closure Properties of the Class of Full G-models of a Deductive SystemStudia Logica 83 (1-3): 215-278. 2006.In this paper we consider the structure of the class FGModS of full generalized models of a deductive system S from a universal-algebraic point of view, and the structure of the set of all the full generalized models of S on a fixed algebra A from the lattice-theoretical point of view; this set is represented by the lattice FACSs A of all algebraic closed-set systems C on A such that (A, C) ε FGModS. We relate some properties of these structures with tipically logical properties of the sententia…Read more
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39Note on a six-valued extension of three-valued logicJournal of Applied Non-Classical Logics 3 (2): 173-187. 1993.ABSTRACT In this paper we introduce a set of six logical values, arising in the application of three-valued logics to time intervals, find its algebraic structure, and use it to define a six-valued logic. We then prove, by using algebraic properties of the class of De Morgan algebras, that this semantically defined logic can be axiomatized as Belnap's ?useful? four-valued logic. Other directions of research suggested by the construction of this set of six logical values are described
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90M-Sets and the Representation ProblemStudia Logica 103 (1): 21-51. 2015.The “representation problem” in abstract algebraic logic is that of finding necessary and sufficient conditions for a structure, on a well defined abstract framework, to have the following property: that for every structural closure operator on it, every structural embedding of the expanded lattice of its closed sets into that of the closed sets of another structural closure operator on another similar structure is induced by a structural transformer between the base structures. This question ar…Read more
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54Leibniz-linked Pairs of Deductive SystemsStudia Logica 99 (1-3): 171-202. 2011.A pair of deductive systems (S,S’) is Leibniz-linked when S’ is an extension of S and on every algebra there is a map sending each filter of S to a filter of S’ with the same Leibniz congruence. We study this generalization to arbitrary deductive systems of the notion of the strong version of a protoalgebraic deductive system, studied in earlier papers, and of some results recently found for particular non-protoalgebraic deductive systems. The necessary examples and counterexamples found in the …Read more
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50Leibniz filters and the strong version of a protoalgebraic logicArchive for Mathematical Logic 40 (6): 437-465. 2001.A filter of a sentential logic ? is Leibniz when it is the smallest one among all the ?-filters on the same algebra having the same Leibniz congruence. This paper studies these filters and the sentential logic ?+ defined by the class of all ?-matrices whose filter is Leibniz, which is called the strong version of ?, in the context of protoalgebraic logics with theorems. Topics studied include an enhanced Correspondence Theorem, characterizations of the weak algebraizability of ?+ and of the expl…Read more
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32Modality and possibility in some intuitionistic modal logicsNotre Dame Journal of Formal Logic 27 (4): 533-546. 1986.
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2Assertional logics, truth-equational logics, and the hierarchies of abstract algebraic logicIn Janusz Czelakowski (ed.), Don Pigozzi on Abstract Algebraic Logic, Universal Algebra, and Computer Science, Springer Verlag. 2018.We establish some relations between the class of truth-equational logics, the class of assertional logics, other classes in the Leibniz hierarchy, and the classes in the Frege hierarchy. We argue that the class of assertional logics belongs properly in the Leibniz hierarchy. We give two new characterizations of truth-equational logics in terms of their full generalized models, and use them to obtain further results on the internal structure of the Frege hierarchy and on the relations between the…Read more
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On semilattice-based logics with an algebraizable assertional companionReports on Mathematical Logic 109-132. 2011.
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5Addendum to the paper 'Belnap's four-valued logic and De Morgan lattices'Logic Journal of the IGPL 7 (5): 671-672. 1999.
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9On the Sentential Logics Associated with Strongly Nice and Semi-Nice General LogicsLogic Journal of the IGPL 2 (1): 55-76. 1994.