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 $\mathit{\text{‘evidence}}\mathit{\text{yields}}\mathit{\text{belief}}\mathit{\text{and}}\mathit{\text{knowledge’}}$ is satisfied. Our approach relies on previous works of the first author \cite{lewjlc2, lewigpl, lewapal} who introduced a modal system containing $S5$-style principles for the reasoning about intutionistic truth (i.e. $\mathit{\text{proof}}$) and, inspired by \cite{artpro}, combined that system with concepts of $\mathit{\text{intuitionistic}}$ belief and knowledge. We consider that combined system and replace the constructive concept of $\mathit{\text{proof}}$ with a classical notion of $\mathit{\text{evidence}}$. This results in a logic that combines modal system $S5$ with classical epistemic principles where $\mathrm{\square }\phi$ reads as `$\phi$ is evident’ in an epistemic sense. Inspired by \cite{lewapal}, 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).