Abstract

Topological semantics have proved to be a very fruitful approach in formal epistemology, two noticeable representatives being the interior semantics and topological evidence models. In this paper, we introduce the concept of _quasi-factive evidence_ as a way to account for untruthful evidence in the interior semantics. This allows us to import concepts from topological evidence models, thereby connecting the two frameworks in spite of their apparent disparities. This approach sheds light on the interpretation of belief in the interior semantics, and gives meaning to concepts that used to be essentially technical: the closure-interior semantics can be interpreted as the condition of existence of a _quasi-factive_ justification, while the extremally disconnected spaces are now characterized as those where the available information is always consistent. But our most important result is the equivalence between the interior-closure-interior semantics and what we call the _strengthening condition_, along with a sound and complete axiomatization. Finally, we build on this strengthening condition to introduce a notion of relative plausibility.