Euclidean Hierarchy in Modal Logic
Johan van Benthem, Guram Bezhanishvili, Mai Gehrke
Abstract:
For an Euclidean space $\mathbb{R}^n$, let $L_n$ denote the modal
logic of chequered subsets of $\mathbb{R}^n$. For every $n\geq 1$, 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 $\mathbb{R}^\infty$. As a result, we obtain that $L_\infty$
is also a logic over Grz, and that $L_\infty$ has the finite model
property.