In this paper the propositional logic LTop is introduced, as an extension of classical propositional logic by adding a paraconsistent negation. This logic has a very natural interpretation in terms of topological models. The logic LTop is nothing more than an alternative presentation of modal logic S4, but in the language of a paraconsistent logic. Moreover, LTop is a logic of formal inconsistency in which the consistency and inconsistency operators have a nice topological interpretation. This constitutes a new proof of (...) S4 as being "the logic of topological spaces", but now under the perspective of paraconsistency. (shrink)

It is a celebrated result of McKinsey and Tarski [28] thatS4is the logic of the closure algebraΧ+over any dense-in-itself separable metrizable space. In particular,S4is the logic of the closure algebra over the realsR, the rationalsQ, or the Cantor spaceC. By [5], each logic aboveS4that has the finite model property is the logic of a subalgebra ofQ+, as well as the logic of a subalgebra ofC+. This is no longer true forR, and the main result of [5] states that each connected (...) logic aboveS4with the finite model property is the logic of a subalgebra of the closure algebraR+.In this paper we extend these results to all logics aboveS4. Namely, for a normal modal logicL, we prove that the following conditions are equivalent: (i)Lis aboveS4, (ii)Lis the logic of a subalgebra ofQ+, (iii)Lis the logic of a subalgebra ofC+. We introduce the concept of a well-connected logic aboveS4and prove that the following conditions are equivalent: (i)Lis a well-connected logic, (ii)Lis the logic of a subalgebra of the closure algebra$\xi _2^ + $over the infinite binary tree, (iii)Lis the logic of a subalgebra of the closure algebra${\bf{L}}_2^ + $over the infinite binary tree with limits equipped with the Scott topology. Finally, we prove that a logicLaboveS4is connected iffLis the logic of a subalgebra ofR+, and transfer our results to the setting of intermediate logics.Proving these general completeness results requires new tools. We introduce the countable general frame property (CGFP) and prove that each normal modal logic has the CGFP. We introduce general topological semantics forS4, which generalizes topological semantics the same way general frame semantics generalizes Kripke semantics. We prove that the categories of descriptive frames forS4and descriptive spaces are isomorphic. It follows that every logic aboveS4is complete with respect to the corresponding class of descriptive spaces. We provide several ways of realizing the infinite binary tree with limits, and prove that when equipped with the Scott topology, it is an interior image of bothCandR. Finally, we introduce gluing of general spaces and prove that the space obtained by appropriate gluing involving certain quotients ofL2is an interior image ofR. (shrink)

This paper brings together Dana Scottʼs measure-based semantics for the propositional modal logic S4, and recent work in Dynamic Topological Logic. In a series of recent talks, Scott showed that the language of S4 can be interpreted in the Lebesgue measure algebra, M, or algebra of Borel subsets of the real interval, [0,1], modulo sets of measure zero. Conjunctions, disjunctions and negations are interpreted via the Boolean structure of the algebra, and we add an interior operator on M that interprets (...) the □-modality. In this paper we show how to extend this measure-based semantics to the bimodal logic S4C. S4C is interpreted in ‘dynamic topological systems,’ or topological spaces together with a continuous function acting on the space. We extend Scottʼs measure based semantics to this bimodal logic by defining a class of operators on the algebra M, which we call O-operators and which take the place of continuous functions in the topological semantics for S4C. The main result of the paper is that S4C is complete for the Lebesgue measure algebra. A strengthening of this result, also proved here, is that there is a single measure-based model in which all non-theorems of S4C are refuted. (shrink)

The Baire algebra of a topological space X is the quotient of the algebra of all subsets of X modulo the meager sets. We show that this Boolean algebra can be endowed with a natural closure operator, resulting in a closure algebra which we denote $\mathbf {Baire}(X)$. We identify the modal logic of such algebras to be the well-known system $\mathsf {S5}$, and prove soundness and strong completeness for the cases where X is crowded and either completely metrizable and continuum-sized (...) or locally compact Hausdorff. We also show that every extension of $\mathsf {S5}$ is the modal logic of a subalgebra of $\mathbf {Baire}(X)$, and that soundness and strong completeness also holds in the language with the universal modality. (shrink)

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 (...) context. We then provide a sound axiomatization for DFH over this extended language, and prove that it is complete. The polyadic modality is used in an essential way in our proof. (shrink)

We study the measure semantics for propositional modal logics, in which formulas are interpreted in theLebesgue measure algebra${\cal M}$, or algebra of Borel subsets of the real interval [0,1] modulo sets of measure zero. It was shown in Lando (2012) and Fernández-Duque (2010) that the propositional modal logicS4 is complete for the Lebesgue measure algebra. The main result of the present paper is that every logicL aboveS4 is complete for some subalgebra of${\cal M}$. Indeed, there is a single model over (...) a subalgebra of${\cal M}$in which all nontheorems ofLare refuted. This work builds on recent work by Bezhanishvili, Gabelaia, & Lucero-Bryan (2015) on the topological semantics for logics aboveS4. In Bezhanishviliet al., (2015), it is shown that there are logics above that arenotthe logic of any subalgebra of the interior algebra over the real line,${\cal B}$(ℝ), but that every logic above is the logic of some subalgebra of the interior algebra over the rationals,${\cal B}$(ℚ), and the interior algebra over Cantor space,${\cal B}\left( {\cal C} \right)$. (shrink)