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The modal logic of continuous functions on cantor space

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Abstract

Let \(\mathcal{L}\) be a propositional language with standard Boolean connectives plus two modalities: an S4-ish topological modality \(\square\) and a temporal modality \(\bigcirc\), understood as ‘next’. We extend the topological semantic for S4 to a semantics for the language \(\mathcal{L}\) by interpreting \(\mathcal{L}\) in dynamic topological systems, i.e. ordered pairs \(\langle X, f\rangle\) , 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. Zhang and Mints have shown that S4C is complete relative to a particular topological space, Cantor space. The current paper produces an alternate proof of the Zhang-Mints result.

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  1. Department of Philosophy, University of Toronto, Toronto, Canada

    Philip Kremer

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  1. Philip Kremer
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Correspondence to Philip Kremer.

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Kremer, P. The modal logic of continuous functions on cantor space. Arch. Math. Logic 45, 1021–1032 (2006). https://doi.org/10.1007/s00153-006-0024-0

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