Euclidean Hierarchy in Modal Logic

For a Euclidean space , let L_n denote the modal logic of chequered subsets of . 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 . 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.