Abstract
For a Euclidean space \(\mathbb{R}^n \), let L n denote the modal logic of chequered subsets of \(\mathbb{R}^n \). For every n ≥ 1, we characterize L n using the more familiar Kripke semantics, thus implying that each L n is a tabular logic over the well-known modal system Grz of Grzegorczyk. We show that the logics L n form a decreasing chain converging to the logic L ∞ of chequered subsets of \(\mathbb{R}^\infty \). As a result, we obtain that L ∞ is also a logic over Grz, and that L ∞ has the finite model property. We conclude the paper by extending our results to the modal language enriched with the universal modality.
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van Benthem, J., Bezhanishvili, G. & Gehrke, M. Euclidean Hierarchy in Modal Logic. Studia Logica 75, 327–344 (2003). https://doi.org/10.1023/B:STUD.0000009564.00287.16
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DOI: https://doi.org/10.1023/B:STUD.0000009564.00287.16