A Note on the Modal and Temporal Logics for N-Dimensional Spacetime

Abstract

We generalize an observation made by Goldblatt in "Diodorean modality in Minkowski spacetime" by proving that each $n$-dimensional integral spacetime frame equipped with Robb's irreflexive `after' relation determines a unique temporal logic. Our main result is that, unlike $n$-dimensional spacetime where, as Goldblatt has shown, the Diodorean modal logic is the same for each frame $(\mathbb{R}^{n}, \leq)$, in the case of $n$-dimensional integral spacetime, the frame $(\mathbb{Z}^{n},\leq)$ determines a unique Diodorean modal logic.