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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 zerodimensional Hausdorff spaces. Embeddings of these spaces into wellknown extremally disconnected spaces then gives new completeness results for logics extending S4.2. 



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+1formulas. This (...) 

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 modeltheoretic. 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 (...) 



We show that subspaces of the space ${\mathbb{Q}}$ of rational numbers give rise to uncountably many dlogics over K4 without the finite model property. 

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 GödelLöb calculus GLPΛ contains one modality [λ] for each λ < Λ, representing provability predicates of increasing strength. GLPω has no nontrivial Kripke frames, but it is sound and complete for its topological semantics, as was shown by Icard for the variablefree fragment and more recently by Beklemishev and Gabelaia for the full logic. In (...) 



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. 



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 twodimensional: he defined 2d Cartesian products of 1d 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 (...) 

Provability logic GLP is wellknown 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 GLPspaces whenever they satisfy all the axioms of GLP. We develop some constructions to build nontrivial GLPspaces and show that GLP is complete w.r.t. the class of all GLPspaces. 

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 wellestablished firstorder topological language. 



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. 





Stalnaker, 169–199 2006) introduced a combined epistemicdoxastic 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 (...) 

When we interpret modal ◊ as the limit point operator of a topological space, the GödelLöb modal system GL defines the class Scat of scattered spaces. We give a partition of Scat into αslices 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 (...) 

