Relevant Epistemic Logic and Knowledge as Belief Based on Correct Evidence

Abstract

Relevant logics avoid some problematic principles of classical and intuitionistic logic, such as explosion – everything follows from a contradiction – or the paradoxes of implication – for example, that every implication follows from its consequent [7,12]. Relevant epistemic logics are epistemic logics based on relevant logics [1,3,8,13,14,15,16,17,19,20,21,22]. The motivation for studying them is that they provide a much finer-grained model of epistemic attitudes than that provided by classical epistemic logic. Relevant epistemic logics avoid many problematic closure principles rooted in classical logic – for example, a belief base may be inconsistent but non-trivial, or it may contain a consequent of an implication without containing the implication itself. In addition, informational interpretations of relevant logics [6,7,9,11] suggest that relevant logics provide a natural framework for epistemic logic. Existing relevant epistemic logics model variants of evidence-based belief. In this paper we focus on the interplay between evidence, belief and knowledge. This fills a gap in the literature and complements recent extensions of relevant logics with knowledge-like operators [22]. In particular, we present a reductionist framework in which belief and knowledge are based on various kinds of evidence. Our main technical result is a representation theorem for a certain class of distributive lattices with operators. This theorem entails a completeness result for a large family of reductionist relevant epistemic logics of knowledge, evidence and belief.