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  1. Priestley Duality for Paraconsistent Nelson’s Logic.Sergei P. Odintsov - 2010 - Studia Logica 96 (1):65-93.
    The variety of N4? -lattices provides an algebraic semantics for the logic N4?, a version of Nelson 's logic combining paraconsistent strong negation and explosive intuitionistic negation. In this paper we construct the Priestley duality for the category of N4?-lattices and their homomorphisms. The obtained duality naturally extends the Priestley duality for Nelson algebras constructed by R. Cignoli and A. Sendlewski.
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  • Belnap-Dunn Semantics for Natural Implicative Expansions of Kleene's Strong Three-Valued Matrix II. Only One Designated Value.Gemma Robles, Francisco Salto & José M. Méndez - 2019 - Journal of Applied Non-Classical Logics 29 (3):307-325.
    ABSTRACTThis paper is a sequel to ‘Belnap-Dunn semantics for natural implicative expansions of Kleene's strong three-valued matrix with two designated values’, where a ‘bivalent’ Belnap-Dunn semant...
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  • Partiality and its Dual in Natural Implicative Expansions of Kleene’s Strong 3-Valued Matrix with Only One Designated Value.Gemma Robles & José M. Méndez - forthcoming - Logic Journal of the IGPL.
    Equivalent overdetermined and underdetermined bivalent Belnap–Dunn type semantics for the logics determined by all natural implicative expansions of Kleene’s strong 3-valued matrix with only one designated value are provided.
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  • Behavioral Algebraization of Logics.Carlos Caleiro, Ricardo Gonçalves & Manuel Martins - 2009 - Studia Logica 91 (1):63-111.
    We introduce and study a new approach to the theory of abstract algebraic logic (AAL) that explores the use of many-sorted behavioral logic in the role traditionally played by unsorted equational logic. Our aim is to extend the range of applicability of AAL toward providing a meaningful algebraic counterpart also to logics with a many-sorted language, and possibly including non-truth-functional connectives. The proposed behavioral approach covers logics which are not algebraizable according to the standard approach, while also bringing a new (...)
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  • The Lattice of Belnapian Modal Logics: Special Extensions and Counterparts.Sergei P. Odintsov & Stanislav O. Speranski - 2016 - Logic and Logical Philosophy 25 (1):3-33.
    Let K be the least normal modal logic and BK its Belnapian version, which enriches K with ‘strong negation’. We carry out a systematic study of the lattice of logics containing BK based on: • introducing the classes of so-called explosive, complete and classical Belnapian modal logics; • assigning to every normal modal logic three special conservative extensions in these classes; • associating with every Belnapian modal logic its explosive, complete and classical counterparts. We investigate the relationships between special extensions (...)
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  • Proofs, Disproofs, and Their Duals.Heinrich Wansing - 2010 - In Lev Beklemishev, Valentin Goranko & Valentin Shehtman (eds.), Advances in Modal Logic, Volume 8. CSLI Publications. pp. 483-505.
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  • On the Embedding of Nelson's Logics.S. P. Odintsov - 2002 - Bulletin of the Section of Logic 31 (4):241-248.
  • On Axiomatizing Shramko-Wansing’s Logic.Sergei P. Odintsov - 2009 - Studia Logica 91 (3):407-428.
    This work treats the problem of axiomatizing the truth and falsity consequence relations, $ \vDash _t $ and $ \vDash _f $, determined via truth and falsity orderings on the trilattice SIXTEEN₃. The approach is based on a representation of SIXTEEN₃ as a twist-structure over the two-element Boolean algebra.
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  • On Axiomatizing Shramko-Wansing’s Logic.Sergei P. Odintsov - 2009 - Studia Logica 91 (3):407 - 428.
    This work treats the problem of axiomatizing the truth and falsity consequence relations, $ \vDash _t $ and $ \vDash _f $, determined via truth and falsity orderings on the trilattice SIXTEEN₃. The approach is based on a representation of SIXTEEN₃ as a twist-structure over the two-element Boolean algebra.
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  • The Logic of Generalized Truth Values and the Logic of Bilattices.Sergei P. Odintsov & Heinrich Wansing - 2015 - Studia Logica 103 (1):91-112.
    This paper sheds light on the relationship between the logic of generalized truth values and the logic of bilattices. It suggests a definite solution to the problem of axiomatizing the truth and falsity consequence relations, \ and \ , considered in a language without implication and determined via the truth and falsity orderings on the trilattice SIXTEEN 3 . The solution is based on the fact that a certain algebra isomorphic to SIXTEEN 3 generates the variety of commutative and distributive (...)
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  • Nelson's Negation on the Base of Weaker Versions of Intuitionistic Negation.Dimiter Vakarelov - 2005 - Studia Logica 80 (2-3):393-430.
    Constructive logic with Nelson negation is an extension of the intuitionistic logic with a special type of negation expressing some features of constructive falsity and refutation by counterexample. In this paper we generalize this logic weakening maximally the underlying intuitionistic negation. The resulting system, called subminimal logic with Nelson negation, is studied by means of a kind of algebras called generalized N-lattices. We show that generalized N-lattices admit representation formalizing the intuitive idea of refutation by means of counterexamples giving in (...)
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  • A First Order Nonmonotonic Extension of Constructive Logic.David Pearce & Agustín Valverde - 2005 - Studia Logica 80 (2-3):321-346.
    Certain extensions of Nelson's constructive logic N with strong negation have recently become important in arti.cial intelligence and nonmonotonic reasoning, since they yield a logical foundation for answer set programming (ASP). In this paper we look at some extensions of Nelson's .rst-order logic as a basis for de.ning nonmonotonic inference relations that underlie the answer set programming semantics. The extensions we consider are those based on 2-element, here-and-there Kripke frames. In particular, we prove completeness for .rst-order here-and-there logics, and their (...)
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  • The Class of Extensions of Nelson's Paraconsistent Logic.Sergei P. Odintsov - 2005 - Studia Logica 80 (2-3):291-320.
    The article is devoted to the systematic study of the lattice εN4⊥ consisting of logics extending N4⊥. The logic N4⊥ is obtained from paraconsistent Nelson logic N4 by adding the new constant ⊥ and axioms ⊥ → p, p → ∼ ⊥. We study interrelations between εN4⊥ and the lattice of superintuitionistic logics. Distinguish in εN4⊥ basic subclasses of explosive logics, normal logics, logics of general form and study how they are relate.
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  • BK-Lattices. Algebraic Semantics for Belnapian Modal Logics.Sergei P. Odintsov & E. I. Latkin - 2012 - Studia Logica 100 (1-2):319-338.
    Earlier algebraic semantics for Belnapian modal logics were defined in terms of twist-structures over modal algebras. In this paper we introduce the class of BK -lattices, show that this class coincides with the abstract closure of the class of twist-structures, and it forms a variety. We prove that the lattice of subvarieties of the variety of BK -lattices is dually isomorphic to the lattice of extensions of Belnapian modal logic BK . Finally, we describe invariants determining a twist-structure over a (...)
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  • Priestley Duality for Paraconsistent Nelson’s Logic.Sergei P. Odintsov - 2010 - Studia Logica 96 (1):65-93.
  • Non-Classical Negation in the Works of Helena Rasiowa and Their Impact on the Theory of Negation.Dimiter Vakarelov - 2006 - Studia Logica 84 (1):105-127.
    The paper is devoted to the contributions of Helena Rasiowa to the theory of non-classical negation. The main results of Rasiowa in this area concerns–constructive logic with strong (Nelson) negation.
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  • Extensions of Priest-da Costa Logic.Thomas Macaulay Ferguson - 2014 - Studia Logica 102 (1):145-174.
    In this paper, we look at applying the techniques from analyzing superintuitionistic logics to extensions of the cointuitionistic Priest-da Costa logic daC (introduced by Graham Priest as “da Costa logic”). The relationship between the superintuitionistic axioms- definable in daC- and extensions of Priest-da Costa logic (sdc-logics) is analyzed and applied to exploring the gap between the maximal si-logic SmL and classical logic in the class of sdc-logics. A sequence of strengthenings of Priest-da Costa logic is examined and employed to pinpoint (...)
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