Archive for Mathematical Logic 49 (4):519-527 (2010)

Authors
Philip Kremer
University of Toronto at Scarborough
Abstract
Let ${{\mathcal L}^{\square\circ}}$ be a propositional language with standard Boolean connectives plus two modalities: an S4-ish topological modality □ and a temporal modality ◦, understood as ‘next’. We extend the topological semantic for S4 to a semantics for the language ${{\mathcal L}^{\square\circ}}$ by interpreting ${{\mathcal L}^{\square\circ}}$ in dynamic topological systems, i.e., ordered pairs 〈X, f〉, where X is a topological space and f is a continuous function on X. Artemov, Davoren and Nerode have axiomatized a logic S4C, and have shown that S4C is sound and complete for this semantics. S4C is also complete for continuous functions on Cantor space (Mints and Zhang, Kremer), and on the real plane (Fernández Duque); but incomplete for continuous functions on the real line (Kremer and Mints, Slavnov). Here we show that S4C is complete for continuous functions on the rational numbers
Keywords Modal logic  Temporal logic  Topology
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DOI 10.1007/s00153-010-0185-8
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References found in this work BETA

The Algebra of Topology.J. C. C. Mckinsey & Alfred Tarski - 1944 - Annals of Mathematics, Second Series 45:141-191.
Dynamic Topological Logic.Philip Kremer & Grigori Mints - 2005 - Annals of Pure and Applied Logic 131 (1-3):133-158.

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Citations of this work BETA

Dynamic Topological Logic Interpreted Over Minimal Systems.David Fernández-Duque - 2011 - Journal of Philosophical Logic 40 (6):767-804.
Dynamic Topological Logic of Metric Spaces.David Fernández-Duque - 2012 - Journal of Symbolic Logic 77 (1):308-328.

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Dynamic Topological Logic Interpreted Over Minimal Systems.David Fernández-Duque - 2011 - Journal of Philosophical Logic 40 (6):767-804.
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