Topology and Justified True Belief: A Baseless, Evidence-Free (and Pointless) Approach

Topological spaces and their interior operators provide semantics for modal logic S4. This semantics has recently attracted an epistemic interpretation as a model of epistemic logic, on the basis of the idea that the generating of a topology from a “base” can be interpreted as the justifying of knowledge with evidence. This is called the evidence-based approach. This paper proposes an alternative approach that may be called evidence-free, in the sense that a topology or knowledge is generated from something other than a base or evidence. (We also use the algebraic, “pointless” formalism of topology, so as to more perspicuously present the structural essence of the correspondence between semantics and logic.) The new semantics provides a logic that describes the relationship between belief and a notion of justified true belief.