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  1. Neighborhood Semantics for Modal Logic.Eric Pacuit - 2017 - Cham, Switzerland: Springer.
    This book offers a state-of-the-art introduction to the basic techniques and results of neighborhood semantics for modal logic. In addition to presenting the relevant technical background, it highlights both the pitfalls and potential uses of neighborhood models – an interesting class of mathematical structures that were originally introduced to provide a semantics for weak systems of modal logic. In addition, the book discusses a broad range of topics, including standard modal logic results ; bisimulations for neighborhood models and other model-theoretic (...)
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  • Strong Completeness of Modal Logics Over 0-Dimensional Metric Spaces.Robert Goldblatt & Ian Hodkinson - 2020 - Review of Symbolic Logic 13 (3):611-632.
    We prove strong completeness results for some modal logics with the universal modality, with respect to their topological semantics over 0-dimensional dense-in-themselves metric spaces. We also use failure of compactness to show that, for some languages and spaces, no standard modal deductive system is strongly complete.
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  • The mckinsey–tarski theorem for locally compact ordered spaces.Guram Bezhanishvili, Nick Bezhanishvili, Joel Lucero-Bryan & Jan van Mill - 2021 - Bulletin of Symbolic Logic 27 (2):187-211.
    We prove that the modal logic of a crowded locally compact generalized ordered space is $\textsf {S4}$. This provides a version of the McKinsey–Tarski theorem for generalized ordered spaces. We then utilize this theorem to axiomatize the modal logic of an arbitrary locally compact generalized ordered space.
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  • Tree-like constructions in topology and modal logic.G. Bezhanishvili, N. Bezhanishvili, J. Lucero-Bryan & J. Van Mill - 2020 - Archive for Mathematical Logic 60 (3):265-299.
    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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  • 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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  • 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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  • Krull dimension in modal logic.Guram Bezhanishvili, Nick Bezhanishvili, Joel Lucero-Bryan & Jan van Mill - 2017 - Journal of Symbolic Logic 82 (4):1356-1386.
    We develop the theory of Krull dimension forS4-algebras and Heyting algebras. This leads to the concept of modal Krull dimension for topological spaces. We compare modal Krull dimension to other well-known dimension functions, and show that it can detect differences between topological spaces that Krull dimension is unable to detect. We prove that for aT1-space to have a finite modal Krull dimension can be described by an appropriate generalization of the well-known concept of a nodec space. This, in turn, can (...)
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  • A New Proof of the McKinsey–Tarski Theorem.G. Bezhanishvili, N. Bezhanishvili, J. Lucero-Bryan & J. van Mill - 2018 - Studia Logica 106 (6):1291-1311.
    It is a landmark theorem of McKinsey and Tarski that if we interpret modal diamond as closure, then \ is the logic of any dense-in-itself metrizable space. The McKinsey–Tarski Theorem relies heavily on a metric that gives rise to the topology. We give a new and more topological proof of the theorem, utilizing Bing’s Metrization Theorem.
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