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 is the most general (extensional) semantics validating Stalnaker’s epistemic-doxastic axioms [22] for “strong belief”, understood as subjective certainty. We prove two completeness results, and we also give a topological semantics for update (dynamic conditioning), i.e. the operation of revising with “hard information” (modeled by restricting the topology to a subspace). Using this, we show that our setting fits well with the defeasibility analysis of knowledge [18]: topological knowledge coincides with undefeated true belief. Finally, we compare our semantics to the older topological interpretation of belief in terms of Cantor derivative [23].