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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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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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We introduce the concept of a connected logic (over S4) and show that each connected logic with the finite model property is the logic of a subalgebra of the closure algebra of all subsets of the real line R, thus generalizing the McKinsey-Tarski theorem. As a consequence, we obtain that each intermediate logic with the finite model property is the logic of a subalgebra of the Heyting algebra of all open subsets of R. |
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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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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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We study hybrid logics in topological semantics. We prove that hybrid logics of separation axioms are complete with respect to certain classes of finite topological models. This characterisation allows us to obtain several further results. We prove that aforementioned logics are decidable and PSPACE-complete, the logics of T 1 and T 2 coincide, the logic of T 1 is complete with respect to two concrete structures: the Cantor space and the rational numbers. |
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Let ${\beta(\mathbb{N})}$ denote the Stone–Čech compactification of the set ${\mathbb{N}}$ of natural numbers (with the discrete topology), and let ${\mathbb{N}^\ast}$ denote the remainder ${\beta(\mathbb{N})-\mathbb{N}}$ . We show that, interpreting modal diamond as the closure in a topological space, the modal logic of ${\mathbb{N}^\ast}$ is S4 and that the modal logic of ${\beta(\mathbb{N})}$ is S4.1.2. |
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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 (...) |
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In the topological semantics for modal logic, S4 is well-known to be complete for the rational line, for the real line, and for Cantor space: these are special cases of S4’s completeness for any dense-in-itself metric space. The construction used to prove completeness can be slightly amended to show that S4 is not only complete, but also strongly complete, for the rational line. But no similarly easy amendment is available for the real line or for Cantor space and the question (...) |
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We prove completeness of the propositional modal logic S 4 for the measure algebra based on the Lebesgue-measurable subsets of the unit interval, [0, 1]. In recent talks, Dana Scott introduced a new measure-based semantics for the standard propositional modal language with Boolean connectives and necessity and possibility operators, and . Propositional modal formulae are assigned to Lebesgue-measurable subsets of the real interval [0, 1], modulo sets of measure zero. Equivalence classes of Lebesgue-measurable subsets form a measure algebra, , and (...) |
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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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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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We prove completeness of preferential conditional logic with respect to convexity over finite sets of points in the Euclidean plane. A conditional is defined to be true in a finite set of points if all extreme points of the set interpreting the antecedent satisfy the consequent. Equivalently, a conditional is true if the antecedent is contained in the convex hull of the points that satisfy both the antecedent and consequent. Our result is then that every consistent formula without nested conditionals (...) |
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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 (...) |
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