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  1. Advances in Modal Logic, Vol. 13.Nicola Olivetti & Rineke Verbrugge (eds.) - 2020 - College Publications.
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  • Axiomatizing the lexicographic products of modal logics with linear temporal logics.Philippe Balbiani & David Fernández-Duque - 2016 - In Lev Beklemishev, Stéphane Demri & András Máté (eds.), Advances in Modal Logic, Volume 11. CSLI Publications. pp. 78-96.
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  • The omega-rule interpretation of transfinite provability logic.David Fernández-Duque & Joost J. Joosten - 2018 - Annals of Pure and Applied Logic 169 (4):333-371.
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  • Provability and Interpretability Logics with Restricted Realizations.Thomas F. Icard & Joost J. Joosten - 2012 - Notre Dame Journal of Formal Logic 53 (2):133-154.
    The provability logic of a theory $T$ is the set of modal formulas, which under any arithmetical realization are provable in $T$. We slightly modify this notion by requiring the arithmetical realizations to come from a specified set $\Gamma$. We make an analogous modification for interpretability logics. We first study provability logics with restricted realizations and show that for various natural candidates of $T$ and restriction set $\Gamma$, the result is the logic of linear frames. However, for the theory Primitive (...)
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  • A topological completeness theorem for transfinite provability logic.Juan P. Aguilera - 2023 - Archive for Mathematical Logic 62 (5):751-788.
    We prove a topological completeness theorem for the modal logic $$\textsf{GLP}$$ GLP containing operators $$\{\langle \xi \rangle :\xi \in \textsf{Ord}\}$$ { ⟨ ξ ⟩ : ξ ∈ Ord } intended to capture a wellordered sequence of consistency operators increasing in strength. More specifically, we prove that, given a tall-enough scattered space X, any sentence $$\phi $$ ϕ consistent with $$\textsf{GLP}$$ GLP can be satisfied on a polytopological space based on finitely many Icard topologies constructed over X and corresponding to the (...)
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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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  • On Provability Logics with Linearly Ordered Modalities.Lev D. Beklemishev, David Fernández-Duque & Joost J. Joosten - 2014 - Studia Logica 102 (3):541-566.
    We introduce the logics GLP Λ, a generalization of Japaridze’s polymodal provability logic GLP ω where Λ is any linearly ordered set representing a hierarchy of provability operators of increasing strength. We shall provide a reduction of these logics to GLP ω yielding among other things a finitary proof of the normal form theorem for the variable-free fragment of GLP Λ and the decidability of GLP Λ for recursive orderings Λ. Further, we give a restricted axiomatization of the variable-free fragment (...)
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  • Provability logic.Rineke Verbrugge - 2008 - Stanford Encyclopedia of Philosophy.
    -/- Provability logic is a modal logic that is used to investigate what arithmetical theories can express in a restricted language about their provability predicates. The logic has been inspired by developments in meta-mathematics such as Gödel’s incompleteness theorems of 1931 and Löb’s theorem of 1953. As a modal logic, provability logic has been studied since the early seventies, and has had important applications in the foundations of mathematics. -/- From a philosophical point of view, provability logic is interesting because (...)
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  • Leo Esakia on Duality in Modal and Intuitionistic Logics.Guram Bezhanishvili (ed.) - 2014 - Dordrecht, Netherland: Springer.
    This volume is dedicated to Leo Esakia's contributions to the theory of modal and intuitionistic systems. Consisting of 10 chapters, written by leading experts, this volume discusses Esakia’s original contributions and consequent developments that have helped to shape duality theory for modal and intuitionistic logics and to utilize it to obtain some major results in the area. Beginning with a chapter which explores Esakia duality for S4-algebras, the volume goes on to explore Esakia duality for Heyting algebras and its generalizations (...)
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  • (1 other version)PSPACE-decidability of Japaridze's polymodal logic.Ilya Shapirovsky - 1998 - In Marcus Kracht, Maarten de Rijke, Heinrich Wansing & Michael Zakharyaschev (eds.), Advances in Modal Logic. CSLI Publications. pp. 289-304.
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  • Nested sequents for provability logic GLP: FIG. 1.Daniyar Shamkanov - 2015 - Logic Journal of the IGPL 23 (5):789-815.
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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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  • 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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  • A many-sorted variant of Japaridze’s polymodal provability logic.Gerald Berger, Lev D. Beklemishev & Hans Tompits - 2018 - Logic Journal of the IGPL 26 (5):505-538.
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