The topology of surprise

Abstract

In this paper we present a topological epistemic logic, with modalities for knowledge (modelled as the universal modality), knowability (represented by the topological interior operator), and unknowability of the actual world. The last notion has a non-self-referential reading (modelled by Cantor derivative: the set of limit points of a given set) and a self-referential one (modelled by Cantor's perfect core of a given set: its largest subset without isolated points, where x is isolated iff $\left\{x\right\}$ is open). We completely axiomatize this logic, showing that it is decidable and pspace-complete, and we apply it to the analysis of a famous epistemic puzzle: the Surprise Exam Paradox.