16 found
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  1.  24
    Functional Completeness and Axiomatizability within Belnap's Four-Valued Logic and its Expansions.Alexej P. Pynko - 1999 - Journal of Applied Non-Classical Logics 9 (1):61-105.
    In this paper we study 12 four-valued logics arisen from Belnap's truth and/or knowledge four-valued lattices, with or without constants, by adding one or both or none of two new non-regular operations—classical negation and natural implication. We prove that the secondary connectives of the bilattice four-valued logic with bilattice constants are exactly the regular four-valued operations. Moreover, we prove that its expansion by any non-regular connective (such as, e.g., classical negation or natural implication) is strictly functionally complete. Further, finding axiomatizations (...)
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  2.  34
    Definitional equivalence and algebraizability of generalized logical systems.Alexej P. Pynko - 1999 - Annals of Pure and Applied Logic 98 (1-3):1-68.
    In this paper we define and study a generalized notion of a logical system that covers on an equal formal basis sentential, equational and sequential systems. We develop a general theory of equivalence between generalized logics that provides, first, a conception of algebraizable logic , second, a formal concept of equivalence between sequential systems and, third, a notion of equivalence between sentential and sequential systems. We also use our theory of equivalence for developing a general algebraic approach to conjunctive non-pseudo-axiomatic (...)
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  3.  41
    Gentzen's cut-free calculus versus the logic of paradox.Alexej P. Pynko - 2010 - Bulletin of the Section of Logic 39 (1/2):35-42.
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  4.  36
    Characterizing Belnap's Logic via De Morgan's Laws.Alexej P. Pynko - 1995 - Mathematical Logic Quarterly 41 (4):442-454.
    The aim of this paper is technically to study Belnap's four-valued sentential logic . First, we obtain a Gentzen-style axiomatization of this logic that contains no structural rules while all they are still admissible in the Gentzen system what is proved with using some algebraic tools. Further, the mentioned logic is proved to be the least closure operator on the set of {Λ, V, ⌝}-formulas satisfying Tarski's conditions for classical conjunction and disjunction together with De Morgan's laws for negation. It (...)
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  5.  39
    On Priest's logic of paradox.Alexej P. Pynko - 1995 - Journal of Applied Non-Classical Logics 5 (2):219-225.
    The present paper concerns a technical study of PRIEST'S logic of paradox [Pri 79], We prove that this logic has no proper paraconsistent strengthening. It is also proved that the mentioned logic is the largest paraconsistent one satisfaying TARSKI'S conditions for the classical conjunction and disjunction together with DE MORGAN'S laws for negation. Finally, we obtain for the logic of paradox an algebraic completeness result related to Kleene lattices.
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  6.  36
    Algebraic study of Sette's maximal paraconsistent logic.Alexej P. Pynko - 1995 - Studia Logica 54 (1):89 - 128.
    The aim of this paper is to study the paraconsistent deductive systemP 1 within the context of Algebraic Logic. It is well known due to Lewin, Mikenberg and Schwarse thatP 1 is algebraizable in the sense of Blok and Pigozzi, the quasivariety generated by Sette's three-element algebraS being the unique quasivariety semantics forP 1. In the present paper we prove that the mentioned quasivariety is not a variety by showing that the variety generated byS is not equivalent to any algebraizable (...)
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  7.  11
    A relative interpolation theorem for infinitary universal Horn logic and its applications.Alexej P. Pynko - 2006 - Archive for Mathematical Logic 45 (3):267-305.
    In this paper we deal with infinitary universal Horn logic both with and without equality. First, we obtain a relative Lyndon-style interpolation theorem. Using this result, we prove a non-standard preservation theorem which contains, as a particular case, a Lyndon-style theorem on surjective homomorphisms in its Makkai-style formulation. Another consequence of the preservation theorem is a theorem on bimorphisms, which, in particular, provides a tool for immediate obtaining characterizations of infinitary universal Horn classes without equality from those with equality. From (...)
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  8.  52
    Distributive-lattice semantics of sequent calculi with structural rules.Alexej P. Pynko - 2009 - Logica Universalis 3 (1):59-94.
    The goal of the paper is to develop a universal semantic approach to derivable rules of propositional multiple-conclusion sequent calculi with structural rules, which explicitly involve not only atomic formulas, treated as metavariables for formulas, but also formula set variables, upon the basis of the conception of model introduced in :27–37, 2001). One of the main results of the paper is that any regular sequent calculus with structural rules has such class of sequent models that a rule is derivable in (...)
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  9.  11
    Extensions of Hałkowska–Zajac's three-valued paraconsistent logic.Alexej P. Pynko - 2002 - Archive for Mathematical Logic 41 (3):299-307.
    As it was proved in [4, Sect. 3], the poset of extensions of the propositional logic defined by a class of logical matrices with equationally-definable set of distinguished values is a retract, under a Galois connection, of the poset of subprevarieties of the prevariety generated by the class of the underlying algebras of the defining matrices. In the present paper we apply this general result to the three-valued paraconsistent logic proposed by Hałkowska–Zajac [2]. Studying corresponding prevarieties, we prove that extensions (...)
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  10.  14
    Regular bilattices.Alexej P. Pynko - 2000 - Journal of Applied Non-Classical Logics 10 (1):93-111.
    ABSTRACT A bilattice is said to be regular provided its truth conjunction and disjunction are monotonic with respect to its knowledge ordering. The principal result of this paper is that the following properties of a bilattice B are equivalent: 1. B is regular; 2. the truth conjunction and disjunction of B are definable through the rest of the operations and constants of B; 3. B is isomorphic to a bilattice of the form L 1 · L 2 where L 1 (...)
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  11.  23
    A cut-free Gentzen calculus with subformula property for first-degree entailments in lc.Alexej P. Pynko - 2003 - Bulletin of the Section of Logic 32 (3):137-146.
  12.  10
    Four-valued expansions of Dunn-Belnap's logic (I): Basic characterizations.Alexej P. Pynko - 2020 - Bulletin of the Section of Logic 49 (4):401-437.
    Basic results of the paper are that any four-valued expansion L4 of Dunn-Belnap's logic DB4 is de_ned by a unique conjunctive matrix ℳ4 with exactly two distinguished values over an expansion.
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  13.  40
    Many-place sequent calculi for finitely-valued logics.Alexej P. Pynko - 2010 - Logica Universalis 4 (1):41-66.
    In this paper, we study multiplicative extensions of propositional many-place sequent calculi for finitely-valued logics arising from those introduced in Sect. 5 of Pynko (J Multiple-Valued Logic Soft Comput 10:339–362, 2004) through their translation by means of singularity determinants for logics and restriction of the original many-place sequent language. Our generalized approach, first of all, covers, on a uniform formal basis, both the one developed in Sect. 5 of Pynko (J Multiple-Valued Logic Soft Comput 10:339–362, 2004) for singular finitely-valued logics (...)
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  14.  8
    Minimal Sequent Calculi for Łukasiewicz’s Finitely-Valued Logics.Alexej P. Pynko - 2015 - Bulletin of the Section of Logic 44 (3/4):149-153.
    The primary objective of this paper, which is an addendum to the author’s [8], is to apply the general study of the latter to Łukasiewicz’s n-valued logics [4]. The paper provides an analytical expression of a 2(n−1)-place sequent calculus (in the sense of [10, 9]) with the cut-elimination property and a strong completeness with respect to the logic involved which is most compact among similar calculi in the sense of a complexity of systems of premises of introduction rules. This together (...)
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  15.  11
    Subquasivarieties of implicative locally-finite quasivarieties.Alexej P. Pynko - 2010 - Mathematical Logic Quarterly 56 (6):643-658.
  16.  8
    Erratum to “Definitional equivalence and algebraizability of generalized logical systems” Annals of Pure and Applied Logic 98 (1999) 1–68. [REVIEW]Alexej P. Pynko - 2000 - Annals of Pure and Applied Logic 102 (3):283-284.
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