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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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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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The aim of this paper is to show that every topological space gives rise to a wealth of topological models of the modal logic S4.1. The construction of these models is based on the fact that every space defines a Boolean closure algebra (to be called a McKinsey algebra) that neatly reflects the structure of the modal system S4.1. It is shown that the class of topological models based on McKinsey algebras contains a canonical model that can be used to (...) |
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We prove that each countable rooted K4 -frame is a d-morphic image of a subspace of the space $\mathbb{Q}$ of rational numbers. From this we derive that each modal logic over K4 axiomatizable by variable-free formulas is the d-logic of a subspace of $\mathbb{Q}$ . It follows that subspaces of $\mathbb{Q}$ give rise to continuum many d-logics over K4 , continuum many of which are neither finitely axiomatizable nor decidable. In addition, we exhibit several families of modal logics finitely axiomatizable (...) |
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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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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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This book illustrates the program of Logical-Informational Dynamics. Rational agents exploit the information available in the world in delicate ways, adopt a wide range of epistemic attitudes, and in that process, constantly change the world itself. Logical-Informational Dynamics is about logical systems putting such activities at center stage, focusing on the events by which we acquire information and change attitudes. Its contributions show many current logics of information and change at work, often in multi-agent settings where social behavior is essential, (...) |
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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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The simplest bimodal combination of unimodal logics \ and \ is their fusion, \, axiomatized by the theorems of \ for \ and of \ for \, and the rules of modus ponens, necessitation for \ and for \, and substitution. Shehtman introduced the frame product \, as the logic of the products of certain Kripke frames: these logics are two-dimensional as well as bimodal. Van Benthem, Bezhanishvili, ten Cate and Sarenac transposed Shehtman’s idea to the topological semantics and introduced (...) |
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This chapter presents a new semantics for inductive empirical knowledge. The epistemic agent is represented concretely as a learner who processes new inputs through time and who forms new beliefs from those inputs by means of a concrete, computable learning program. The agent’s belief state is represented hyper-intensionally as a set of time-indexed sentences. Knowledge is interpreted as avoidance of error in the limit and as having converged to true belief from the present time onward. Familiar topics are re-examined within (...) |
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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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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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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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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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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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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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In topological modal logic, it is well known that the Cantor derivative is more expressive than the topological closure, and the ‘elsewhere’, or ‘difference’, operator is more expressive than the ‘somewhere’ operator. In 2014, Kudinov and Shehtman asked whether the combination of closure and elsewhere becomes strictly more expressive when adding the Cantor derivative. In this paper we give an affirmative answer: in fact, the Cantor derivative alone can define properties of topological spaces not expressible with closure and elsewhere. To (...) |
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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 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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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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We show that if we interpret modal diamond as the derived set operator of a topological space, then the modal logic of Stone spaces isK4and the modal logic of weakly scattered Stone spaces isK4G. As a corollary, we obtain thatK4is also the modal logic of compact Hausdorff spaces andK4Gis the modal logic of weakly scattered compact Hausdorff spaces. |
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