Topological Approaches to Epistemic Logic

Sophia Knight (Uppsala University) and Aybüke Özgün (University of Amsterdam)

Epistemic logic is an umbrella term for a species of modal logics whose main objects of study are knowledge and belief. As a field of study, epistemic logic uses modal logic and mathematical tools to formalize, clarify and solve the questions that drive (formal) epistemology, and its applications extend not only to philosophy, but also to theoretical computer science, artificial intelligence and economics (for a survey, see Handbook of Epistemic Logic). Hintikka is considered the founding father of modern epistemic logic. In his book Knowledge and Belief: An Introduction to the Logic of the Two Notions (1962)—inspired by insights in (von Wright, 1951)—Hintikka formalizes knowledge and belief as basic modal operators, denoted by K and B, respectively, and interprets them using standard possible worlds semantics based on (relational) Kripke structures. Ever since—as Kripke semantics provides a natural and relatively easy way of modelling epistemic logics—it has been one of the prominent and most commonly used semantic structures in epistemic logic, and research in this area has widely advanced based on the formal ground of Kripke semantics.

However, standard Kripke semantics possesses some features that make the notions of knowledge and belief it implements too strong—leading to the problem of logical omniscience—and, more importantly for this lecture, is lacking the ingredients that make it possible to talk about the nature and grounds of acquired knowledge and belief. However, we not only seek an easy way to model knowledge and belief, but also study the emergence, usage, and transformation of evidence as an inseparable component of a rational and idealized agent’s justified belief and knowledge.

For this purpose, topological spaces are proven to be rich mathematical objects to formalize the aforementioned epistemic notions, and, in turn, evidence-based information dynamics: while providing a deeper insight into the evidence-based interpretation of knowledge and belief, topological semantics also generalizes the standard relational semantics of epistemic logic.

This course will cover several variants of topological semantics for epistemic/doxastic logics including the interior-based topological semantics á la McKinsey (1941) and McKinsey & Tarski (1944) as well as the so-called subset space semantics introduced by Moss & Parikh (1992). We focus mainly on the recent developments in the field and the conceptual arguments behind using topological semantics. It is not going to be a very technical course. We in particular will not go through difficult meta-logical results in detail, however, provide the relevant sources for the interested students. Please find the detailed schedule, course materials, and reading list for each lecture here.

Course Materials