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  1. The D-Logic of the Rational Numbers: A Fruitful Construction.Joel Lucero-Bryan - 2011 - Studia Logica 97 (2):265-295.
    We present a geometric construction that yields completeness results for modal logics including K4, KD4, GL and GL n with respect to certain subspaces of the rational numbers. These completeness results are extended to the bimodal case with the universal modality.
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  • Tree-like constructions in topology and modal logic.G. Bezhanishvili, N. Bezhanishvili, J. Lucero-Bryan & J. Van Mill - forthcoming - Archive for Mathematical Logic:1-35.
    Within ZFC, we develop a general technique to topologize trees that provides a uniform approach to topological completeness results in modal logic with respect to zero-dimensional Hausdorff spaces. Embeddings of these spaces into well-known extremally disconnected spaces then gives new completeness results for logics extending S4.2.
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  • The Tangled Derivative Logic of the Real Line and Zero-Dimensional Space.Robert Goldblatt & Ian Hodkinson - 2016 - In Lev Beklemishev, Stéphane Demri & András Máté (eds.), Advances in Modal Logic, Volume 11. CSLI Publications. pp. 342-361.
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  • Positive Provability Logic for Uniform Reflection Principles.Lev Beklemishev - 2014 - Annals of Pure and Applied Logic 165 (1):82-105.
    We deal with the fragment of modal logic consisting of implications of formulas built up from the variables and the constant ‘true’ by conjunction and diamonds only. The weaker language allows one to interpret the diamonds as the uniform reflection schemata in arithmetic, possibly of unrestricted logical complexity. We formulate an arithmetically complete calculus with modalities labeled by natural numbers and ω, where ω corresponds to the full uniform reflection schema, whereas n<ω corresponds to its restriction to arithmetical Πn+1-formulas. This (...)
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  • A Sound and Complete Axiomatization for Dynamic Topological Logic.David Fernández-Duque - 2012 - Journal of Symbolic Logic 77 (3):947-969.
    Dynamic Topological Logic (DFH) is a multimodal system for reasoning about dynamical systems. It is defined semantically and, as such, most of the work done in the field has been model-theoretic. In particular, the problem of finding a complete axiomatization for the full language of DFH over the class of all dynamical systems has proven to be quite elusive. Here we propose to enrich the language to include a polyadic topological modality, originally introduced by Dawar and Otto in a different (...)
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  • Completeness and Definability of a Modal Logic Interpreted Over Iterated Strict Partial Orders.Philippe Baldiani & Levan Uridia - 2012 - In Thomas Bolander, Torben Braüner, Silvio Ghilardi & Lawrence Moss (eds.), Advances in Modal Logic, Volume 9. CSLI Publications. pp. 71-88.
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  • Subspaces of $${\Mathbb{Q}}$$ Whose D-Logics Do Not Have the FMP.Guram Bezhanishvili & Joel Lucero-Bryan - 2012 - Archive for Mathematical Logic 51 (5-6):661-670.
    We show that subspaces of the space ${\mathbb{Q}}$ of rational numbers give rise to uncountably many d-logics over K4 without the finite model property.
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  • The Polytopologies of Transfinite Provability Logic.David Fernández-Duque - 2014 - Archive for Mathematical Logic 53 (3-4):385-431.
    Provability logics are modal or polymodal systems designed for modeling the behavior of Gödel’s provability predicate and its natural extensions. If Λ is any ordinal, the Gödel-Löb calculus GLPΛ contains one modality [λ] for each λ < Λ, representing provability predicates of increasing strength. GLPω has no non-trivial Kripke frames, but it is sound and complete for its topological semantics, as was shown by Icard for the variable-free fragment and more recently by Beklemishev and Gabelaia for the full logic. In (...)
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  • Foreword.Daniele Mundici - 1998 - Studia Logica 61 (1):1-1.
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  • A Canonical Topological Model for Extensions of K4.Christopher Steinsvold - 2010 - Studia Logica 94 (3):433 - 441.
    Interpreting the diamond of modal logic as the derivative, we present a topological canonical model for extensions of K4 and show completeness for various logics. We also show that if a logic is topologically canonical, then it is relationally canonical.
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  • Foreword.Lev Beklemishev, Guram Bezhanishvili, Daniele Mundici & Yde Venema - 2012 - Studia Logica 100 (1-2):1-7.
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  • Matching Topological and Frame Products of Modal Logics.Philip Kremer - 2016 - Studia Logica 104 (3):487-502.
    The simplest combination of unimodal logics \ into a bimodal logic is their fusion, \, axiomatized by the theorems of \. Shehtman introduced combinations that are not only bimodal, but two-dimensional: he defined 2-d Cartesian products of 1-d Kripke frames, using these Cartesian products to define the frame product \. Van Benthem, Bezhanishvili, ten Cate and Sarenac generalized Shehtman’s idea and introduced the topological product \, using Cartesian products of topological spaces rather than of Kripke frames. Frame products have been (...)
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  • Topological Completeness of the Provability Logic GLP.Lev Beklemishev & David Gabelaia - 2013 - Annals of Pure and Applied Logic 164 (12):1201-1223.
    Provability logic GLP is well-known to be incomplete w.r.t. Kripke semantics. A natural topological semantics of GLP interprets modalities as derivative operators of a polytopological space. Such spaces are called GLP-spaces whenever they satisfy all the axioms of GLP. We develop some constructions to build nontrivial GLP-spaces and show that GLP is complete w.r.t. the class of all GLP-spaces.
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  • Modal Languages for Topology: Expressivity and Definability.Balder ten Cate, David Gabelaia & Dmitry Sustretov - 2009 - Annals of Pure and Applied Logic 159 (1-2):146-170.
    In this paper we study the expressive power and definability for modal languages interpreted on topological spaces. We provide topological analogues of the van Benthem characterization theorem and the Goldblatt–Thomason definability theorem in terms of the well-established first-order topological language.
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  • Spatial Logic of Tangled Closure Operators and Modal Mu-Calculus.Robert Goldblatt & Ian Hodkinson - 2017 - Annals of Pure and Applied Logic 168 (5):1032-1090.
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  • The Modal Logic of Stone Spaces: Diamond as Derivative: Modal Logic of Stone Spaces.Guram Bezhanishvili - 2010 - Review of Symbolic Logic 3 (1):26-40.
    We show that if we interpret modal diamond as the derived set operator of a topological space, then the modal logic of Stone spaces is _K4_ and the modal logic of weakly scattered Stone spaces is _K4G_. As a corollary, we obtain that _K4_ is also the modal logic of compact Hausdorff spaces and _K4G_ is the modal logic of weakly scattered compact Hausdorff spaces.
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  • On Neighbourhood Product of Some Horn Axiomatizable Logics.Andrey Kudinov - 2018 - Logic Journal of the IGPL 26 (3):316-338.
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  • 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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  • 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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  • Scattered and Hereditarily Irresolvable Spaces in Modal Logic.Guram Bezhanishvili & Patrick J. Morandi - 2010 - Archive for Mathematical Logic 49 (3):343-365.
    When we interpret modal ◊ as the limit point operator of a topological space, the Gödel-Löb modal system GL defines the class Scat of scattered spaces. We give a partition of Scat into α-slices S α , where α ranges over all ordinals. This provides topological completeness and definability results for extensions of GL. In particular, we axiomatize the modal logic of each ordinal α, thus obtaining a simple proof of the Abashidze–Blass theorem. On the other hand, when we interpret (...)
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  • 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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