Belief, knowledge and evidence

We present a logical system that combines the well-known classical epistemic concepts of belief and knowledge with a concept of evidence such that the intuitive principle textit`evidence yields belief and knowledge' is satisfied. Our approach relies on previous works of the first author citelewjlc2, lewigpl, lewapal who introduced a modal system containing S5-style principles for the reasoning about intutionistic truth (i.e. textitproof) and, inspired by citeartpro, combined that system with concepts of textitintuitionistic belief and knowledge. We consider that combined system and replace the constructive concept of textitproof with a classical notion of textitevidence. This results in a logic that combines modal system S5 with classical epistemic principles where squarevarphi reads as `varphi is evident' in an epistemic sense. Inspired by citelewapal, and in contrast to the usual possible worlds semantics found in the literature, we propose here a relational, frame-based semantics where belief and knowledge are not modeled via accessibility relations but directly as sets of propositions (sets of sets of worlds).