Hybrid Logics for Spacetimes along the Causal Ladder

Abstract

We study how Modal and Hybrid Logic can be used to describe the causal structure of spacetime. Motivated by Malament’s Theorem, which shows that the causal order alone can recover the Lorentzian manifold structure up to conformal factor, we introduce a Hybrid Language capable of expressing spacetime notions such as future/past cones and chronological/causal diamonds. In terms of this language, the modalities F, P, @, and U are definable. Within this language we give axioms for characterising several levels of the Causal Ladder – non-total viciousness, chronology, causality, and distinguishability. The broader aim is to lay groundwork toward complete logics for the spacetimes classified by the Causal Ladder though this larger goal is left for future work