Euclidean Hierarchy in Modal Logic

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.