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# Euclidean Hierarchy in Modal Logic

Johan van Benthem, Guram Bezhanishvili and Mai Gehrke
Studia Logica: An International Journal for Symbolic Logic
Vol. 75, No. 3 (Dec., 2003), pp. 327-344
Stable URL: http://www.jstor.org/stable/20016563
Page Count: 18
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## Abstract

For a Euclidean space ${\Bbb R}^{n}$, let $L_{n}$ denote the modal logic of chequered subsets of ${\Bbb R}^{n}$. 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 $L_{\infty}$ of chequered subsets of ${\Bbb R}^{\infty}$. As a result, we obtain that $L_{\infty}$ is also a logic over Grz, and that $L_{\infty}$ 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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