Euclidean Hierarchy in Modal Logic

Studia Logica 75 (3):327 - 344 (2003)
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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