The McKinsey-Tarski Theorem for Topological Evidence Logics

Abstract

We prove an analogue of the McKinsey and Tarski theorem for the recently introduced dense-interior semantics of topological evidence logics. In particular, we show that in this semantics the modal logic \(\mathsf {S4.2}\) is sound and complete for any dense-in-itself metrizable space. As a result \(\mathsf {S4.2}\) is complete with respect to the real line \(\mathbb {R}\), the rational line \(\mathbb {Q}\), the Baire space \(\mathfrak {B}\), the Cantor space \(\mathfrak {C}\), etc. We also show that an extension of this logic with the universal modality is sound and complete for any idempotent dense-in-itself metrizable space, obtaining as a result that this logic is sound and complete with respect to \(\mathbb {Q}\), \(\mathfrak {B}\), \(\mathfrak {C}\), etc.