Belief, knowledge and evidence
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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 `evidence yields belief and knowledge' is satisfied. Our approach relies on previous works of the first author <cit.> who introduced a modal system containing S5-style principles for the reasoning about intutionistic truth (i.e. proof) and, inspired by <cit.>, combined that system with concepts of intuitionistic belief and knowledge. We consider that combined system and replace the constructive concept of proof with a classical notion of evidence. This results in a logic that combines modal system S5 with classical epistemic principles where □φ reads as `φ is evident' in an epistemic sense. Inspired by <cit.>, 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).
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