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  1.  7
    Stable Canonical Rules.Guram Bezhanishvili, Nick Bezhanishvili & Rosalie Iemhoff - 2016 - Journal of Symbolic Logic 81 (1):284-315.
  2.  27
    A Bimodal Perspective on Possibility Semantics.Johan van Benthem, Nick Bezhanishvili & Wesley H. Holliday - 2017 - Journal of Logic and Computation 27 (5):1353–1389.
    In this article, we develop a bimodal perspective on possibility semantics, a framework allowing partiality of states that provides an alternative modelling for classical propositional and modal logics. In particular, we define a full and faithful translation of the basic modal logic K over possibility models into a bimodal logic of partial functions over partial orders, and we show how to modulate this analysis by varying across logics and model classes that have independent topological motivations. This relates the two realms (...)
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  3.  17
    Krull Dimension in Modal Logic.Guram Bezhanishvili, Nick Bezhanishvili, Joel Lucero-Bryan & Jan van Mill - 2017 - Journal of Symbolic Logic 82 (4):1356-1386.
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  4.  10
    Stable Modal Logics.Guram Bezhanishvili, Nick Bezhanishvili & Julia Ilin - 2018 - Review of Symbolic Logic 11 (3):436-469.
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  5.  10
    Locally Finite Reducts of Heyting Algebras and Canonical Formulas.Guram Bezhanishvili & Nick Bezhanishvili - 2017 - Notre Dame Journal of Formal Logic 58 (1):21-45.
    The variety of Heyting algebras has two well-behaved locally finite reducts, the variety of bounded distributive lattices and the variety of implicative semilattices. The variety of bounded distributive lattices is generated by the →-free reducts of Heyting algebras, while the variety of implicative semilattices is generated by the ∨-free reducts. Each of these reducts gives rise to canonical formulas that generalize Jankov formulas and provide an axiomatization of all superintuitionistic logics. The ∨-free reducts of Heyting algebras give rise to the (...)
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  6.  12
    The Bounded Proof Property Via Step Algebras and Step Frames.Nick Bezhanishvili & Silvio Ghilardi - 2014 - Annals of Pure and Applied Logic 165 (12):1832-1863.
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  7.  19
    Algebraic and Topological Semantics for Inquisitive Logic Via Choice-Free Duality.Nick Bezhanishvili, Gianluca Grilletti & Wesley H. Holliday - 2019 - In Rosalie Iemhoff, Michael Moortgat & Ruy de Queiroz (eds.), Logic, Language, Information, and Computation. WoLLIC 2019. Lecture Notes in Computer Science, Vol. 11541. Springer. pp. 35-52.
    We introduce new algebraic and topological semantics for inquisitive logic. The algebraic semantics is based on special Heyting algebras, which we call inquisitive algebras, with propositional valuations ranging over only the ¬¬-fixpoints of the algebra. We show how inquisitive algebras arise from Boolean algebras: for a given Boolean algebra B, we define its inquisitive extension H(B) and prove that H(B) is the unique inquisitive algebra having B as its algebra of ¬¬-fixpoints. We also show that inquisitive algebras determine Medvedev’s logic (...)
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  8.  10
    Cofinal Stable Logics.Guram Bezhanishvili, Nick Bezhanishvili & Julia Ilin - 2016 - Studia Logica 104 (6):1287-1317.
    We generalize the \}\)-canonical formulas to \}\)-canonical rules, and prove that each intuitionistic multi-conclusion consequence relation is axiomatizable by \}\)-canonical rules. This yields a convenient characterization of stable superintuitionistic logics. The \}\)-canonical formulas are analogues of the \}\)-canonical formulas, which are the algebraic counterpart of Zakharyaschev’s canonical formulas for superintuitionistic logics. Consequently, stable si-logics are analogues of subframe si-logics. We introduce cofinal stable intuitionistic multi-conclusion consequence relations and cofinal stable si-logics, thus answering the question of what the analogues of cofinal (...)
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  9.  11
    On Modal Logics Arising From Scattered Locally Compact Hausdorff Spaces.Guram Bezhanishvili, Nick Bezhanishvili, Joel Lucero-Bryan & Jan van Mill - 2019 - Annals of Pure and Applied Logic 170 (5):558-577.
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  10.  14
    Stable Formulas in Intuitionistic Logic.Nick Bezhanishvili & Dick de Jongh - 2018 - Notre Dame Journal of Formal Logic 59 (3):307-324.
    In 1995 Visser, van Benthem, de Jongh, and Renardel de Lavalette introduced NNIL-formulas, showing that these are exactly the formulas preserved under taking submodels of Kripke models. In this article we show that NNIL-formulas are up to frame equivalence the formulas preserved under taking subframes of frames, that NNIL-formulas are subframe formulas, and that subframe logics can be axiomatized by NNIL-formulas. We also define a new syntactic class of ONNILLI-formulas. We show that these are the formulas preserved in monotonic images (...)
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  11.  24
    Frame Based Formulas for Intermediate Logics.Nick Bezhanishvili - 2008 - Studia Logica 90 (2):139-159.
    In this paper we define the notion of frame based formulas. We show that the well-known examples of formulas arising from a finite frame, such as the Jankov-de Jongh formulas, subframe formulas and cofinal subframe formulas, are all particular cases of the frame based formulas. We give a criterion for an intermediate logic to be axiomatizable by frame based formulas and use this criterion to obtain a simple proof that every locally tabular intermediate logic is axiomatizable by Jankov-de Jongh formulas. (...)
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  12.  53
    Canonical Formulas for Wk4.Guram Bezhanishvili & Nick Bezhanishvili - 2012 - Review of Symbolic Logic 5 (4):731-762.
    We generalize the theory of canonical formulas for K4, the logic of transitive frames, to wK4, the logic of weakly transitive frames. Our main result establishes that each logic over wK4 is axiomatizable by canonical formulas, thus generalizing Zakharyaschev’s theorem for logics over K4. The key new ingredients include the concepts of transitive and strongly cofinal subframes of weakly transitive spaces. This yields, along with the standard notions of subframe and cofinal subframe logics, the new notions of transitive subframe and (...)
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  13.  19
    An Algebraic Approach to Canonical Formulas: Modal Case.Guram Bezhanishvili & Nick Bezhanishvili - 2011 - Studia Logica 99 (1-3):93-125.
    We introduce relativized modal algebra homomorphisms and show that the category of modal algebras and relativized modal algebra homomorphisms is dually equivalent to the category of modal spaces and partial continuous p-morphisms, thus extending the standard duality between the category of modal algebras and modal algebra homomorphisms and the category of modal spaces and continuous p-morphisms. In the transitive case, this yields an algebraic characterization of Zakharyaschev’s subreductions, cofinal subreductions, dense subreductions, and the closed domain condition. As a consequence, we (...)
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  14. All Proper Normal Extensions of S5-Square Have the Polynomial Size Model Property.Nick Bezhanishvili & Maarten Marx - 2003 - Studia Logica 73 (3):367 - 382.
    We show that every proper normal extension of the bi-modal system S5 2 has the poly-size model property. In fact, to every proper normal extension L of S5 2 corresponds a natural number b(L) - the bound of L. For every L, there exists a polynomial P(·) of degree b(L) + 1 such that every L-consistent formula is satisfiable on an L-frame whose universe is bounded by P(||), where || denotes the number of subformulas of . It is shown that (...)
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  15.  85
    Sahlqvist Correspondence for Modal Mu-Calculus.Johan van Benthem, Nick Bezhanishvili & Ian Hodkinson - 2012 - Studia Logica 100 (1-2):31-60.
    We define analogues of modal Sahlqvist formulas for the modal mu-calculus, and prove a correspondence theorem for them.
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  16.  19
    Instantial Neighbourhood Logic.Johan van Benthem, Nick Bezhanishvili, Sebastian Enqvist & Junhua Yu - 2017 - Review of Symbolic Logic 10 (1):116-144.
    This paper explores a new language of neighbourhood structures where existential information can be given about what kind of worlds occur in a neighbourhood of a current world. The resulting system of ‘instantial neighbourhood logic’ INL has a nontrivial mix of features from relational semantics and from neighbourhood semantics. We explore some basic model-theoretic behavior, including a matching notion of bisimulation, and give a complete axiom system for which we prove completeness by a new normal form technique. In addition, we (...)
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  17.  28
    Choice-Free Stone Duality.Nick Bezhanishvili & Wesley H. Holliday - 2020 - Journal of Symbolic Logic 85 (1):109-148.
    The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean (...)
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  18.  24
    Sahlqvist Correspondence for Modal Mu-Calculus.Johan Benthem, Nick Bezhanishvili & Ian Hodkinson - 2012 - Studia Logica 100 (1-2):31-60.
    We define analogues of modal Sahlqvist formulas for the modal mu-calculus, and prove a correspondence theorem for them.
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  19.  12
    All Proper Normal Extensions of S5-Square Have the Polynomial Size Model Property.Nick Bezhanishvili & Maarten Marx - 2003 - Studia Logica 73 (3):367-382.
    We show that every proper normal extension of the bi-modal system S5⁲ has the poly-size model property. In fact, to every proper normal extension L of S5⁲ corresponds a natural number b-the bound of L. For every L, there exists a polynomial P of degree b + 1 such that every L-consistent formula φ is satisfiable on an L-frame whose universe is bounded by P, where |φ| denotes the number of subformulas of φ. It is shown that this bound is (...)
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  20.  7
    Multiple-Conclusion Rules, Hypersequents Syntax and Step Frames.Nick Bezhanishvili & Silvio Ghilardi - 2014 - In Rajeev Goré, Barteld Kooi & Agi Kurucz (eds.), Advances in Modal Logic, Volume 10. CSLI Publications. pp. 54-73.
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  21.  11
    A New Game Equivalence, its Logic and Algebra.Sebastian Enqvist, Nick Bezhanishvili & Johan Benthem - 2019 - Journal of Philosophical Logic 48 (4):649-684.
    We present a new notion of game equivalence that captures basic powers of interacting players. We provide a representation theorem, a complete logic, and a new game algebra for basic powers. In doing so, we establish connections with imperfect information games and epistemic logic. We also identify some new open problems concerning logic and games.
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  22.  19
    Extendible Formulas in Two Variables in Intuitionistic Logic.Nick Bezhanishvili & Dick de Jongh - 2012 - Studia Logica 100 (1-2):61-89.
    We give alternative characterizations of exact, extendible and projective formulas in intuitionistic propositional calculus IPC in terms of n-universal models. From these characterizations we derive a new syntactic description of all extendible formulas of IPC in two variables. For the formulas in two variables we also give an alternative proof of Ghilardi’s theorem that every extendible formula is projective.
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  23.  30
    A Topological Approach to Full Belief.Alexandru Baltag, Nick Bezhanishvili, Aybüke Özgün & Sonja Smets - 2019 - Journal of Philosophical Logic 48 (2):205-244.
    Stalnaker, 169–199 2006) introduced a combined epistemic-doxastic logic that can formally express a strong concept of belief, a concept of belief as ‘subjective certainty’. In this paper, we provide a topological semantics for belief, in particular, for Stalnaker’s notion of belief defined as ‘epistemic possibility of knowledge’, in terms of the closure of the interior operator on extremally disconnected spaces. This semantics extends the standard topological interpretation of knowledge with a new topological semantics for belief. We prove that the belief (...)
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  24.  15
    Extendible Formulas in Two Variables in Intuitionistic Logic.Nick Bezhanishvili & Dick Jongh - 2012 - Studia Logica 100 (1-2):61-89.
    We give alternative characterizations of exact, extendible and projective formulas in intuitionistic propositional calculus IPC in terms of n -universal models. From these characterizations we derive a new syntactic description of all extendible formulas of IPC in two variables. For the formulas in two variables we also give an alternative proof of Ghilardi’s theorem that every extendible formula is projective.
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  25.  25
    All Normal Extensions of S5-Squared Are Finitely Axiomatizable.Nick Bezhanishvili & Ian Hodkinson - 2004 - Studia Logica 78 (3):443-457.
    We prove that every normal extension of the bi-modal system S52 is finitely axiomatizable and that every proper normal extension has NP-complete satisfiability problem.
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  26.  17
    A New Game Equivalence, its Logic and Algebra.Johan van Benthem, Nick Bezhanishvili & Sebastian Enqvist - 2019 - Journal of Philosophical Logic 48 (4):649-684.
    We present a new notion of game equivalence that captures basic powers of interacting players. We provide a representation theorem, a complete logic, and a new game algebra for basic powers. In doing so, we establish connections with imperfect information games and epistemic logic. We also identify some new open problems concerning logic and games.
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  27.  26
    Pseudomonadic Algebras as Algebraic Models of Doxastic Modal Logic.Nick Bezhanishvili - 2002 - Mathematical Logic Quarterly 48 (4):624-636.
    We generalize the notion of a monadic algebra to that of a pseudomonadic algebra. In the same way as monadic algebras serve as algebraic models of epistemic modal system S5, pseudomonadic algebras serve as algebraic models of doxastic modal system KD45. The main results of the paper are: Characterization of subdirectly irreducible and simple pseudomonadic algebras, as well as Tokarz's proper filter algebras; Ordertopological representation of pseudomonadic algebras; Complete description of the lattice of subvarieties of the variety of pseudomonadic algebras.
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  28.  17
    Tarski's Theorem on Intuitionistic Logic, for Polyhedra.Nick Bezhanishvili, Vincenzo Marra, Daniel McNeill & Andrea Pedrini - 2018 - Annals of Pure and Applied Logic 169 (5):373-391.
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  29.  5
    A Propositional Dynamic Logic for Instantial Neighborhood Semantics.Sebastian Enqvist, Nick Bezhanishvili & Johan Benthem - 2019 - Studia Logica 107 (4):719-751.
    We propose a new perspective on logics of computation by combining instantial neighborhood logic $$\mathsf {INL}$$ INL with bisimulation safe operations adapted from $$\mathsf {PDL}$$ PDL. $$\mathsf {INL}$$ INL is a recent modal logic, based on an extended neighborhood semantics which permits quantification over individual neighborhoods plus their contents. This system has a natural interpretation as a logic of computation in open systems. Motivated by this interpretation, we show that a number of familiar program constructors can be adapted to instantial (...)
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  30.  12
    A Propositional Dynamic Logic for Instantial Neighborhood Semantics.Johan van Benthem, Nick Bezhanishvili & Sebastian Enqvist - 2019 - Studia Logica 107 (4):719-751.
    We propose a new perspective on logics of computation by combining instantial neighborhood logic \ with bisimulation safe operations adapted from \. \ is a recent modal logic, based on an extended neighborhood semantics which permits quantification over individual neighborhoods plus their contents. This system has a natural interpretation as a logic of computation in open systems. Motivated by this interpretation, we show that a number of familiar program constructors can be adapted to instantial neighborhood semantics to preserve invariance for (...)
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  31.  7
    Tychonoff Hed-Spaces and Zemanian Extensions of S4.3.Guram Bezhanishvili, Nick Bezhanishvili, Joel Lucero-Bryan & Jan van Mill - 2018 - Review of Symbolic Logic 11 (1):115-132.
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  32.  7
    Admissible Bases Via Stable Canonical Rules.Nick Bezhanishvili, David Gabelaia, Silvio Ghilardi & Mamuka Jibladze - 2016 - Studia Logica 104 (2):317-341.
    We establish the dichotomy property for stable canonical multi-conclusion rules for IPC, K4, and S4. This yields an alternative proof of existence of explicit bases of admissible rules for these logics.
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  33.  9
    The Kuznetsov-Gerčiu and Rieger-Nishimura Logics.Guram Bezhanishvili, Nick Bezhanishvili & Dick de Jongh - 2008 - Logic and Logical Philosophy 17 (1-2):73-110.
    We give a systematic method of constructing extensions of the Kuznetsov-Gerčiu logic KG without the finite model property (fmp for short), and show that there are continuum many such. We also introduce a new technique of gluing of cyclic intuitionistic descriptive frames and give a new simple proof of Gerčiu’s result [9, 8] that all extensions of the Rieger-Nishimura logic RN have the fmp. Moreover, we show that each extension of RN has the poly-size model property, thus improving on [9]. (...)
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