The Topology of Belief, Belief Revision and Defeasible Knowledge

We present a new topological semantics for doxastic logic, in which the belief modality is interpreted as the closure of the interior operator. We show that this semantics validates Stalnaker’s epistemicdoxastic axioms [23], and indeed it is the most general (extensional) semantics validating them. We prove, among other things, that in this semantics the doxastic logic KD45 is sound and complete with respect to the class of all extremally disconnected topological spaces. We also give a topological semantics for conditional belief and show its connection to the operation of updating with “hard information” (modeled by restricting the topology to a subspace). We show that our topological notions fit well with the defeasibility analysis of knowledge: topological knowledge coincides with undefeated true belief. We compare our semantics to the older topological interpretation of belief in terms of Cantor derivative (Steinsvold 2006), arguing in favor of our new semantics.