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\(\in_K\): a non-Fregean logic of explicit knowledge. (English) Zbl 1231.03016
Summary: We present a new logic-based approach to reasoning about knowledge which is independent of a possible worlds semantics. \(\in_K\) (Epsilon-\(K\)) is a non-Fregean logic whose models consist of propositional universes with subsets for true, false and known propositions. Knowledge is, in general, not closed under rules of inference; the only valid epistemic principles are the knowledge axiom \(K_{i \varphi} \rightarrow \varphi \) and some minimal conditions concerning common knowledge in a group. Knowledge is explicit and all forms of the logical omniscience problem are avoided. Various stronger epistemic properties such as positive and/or negative introspection, the \(K\)-axiom, closure under logical connectives, etc. can be restored by imposing additional semantic constraints. This yields corresponding sublogics for which we present sound and complete axiomatizations. As a useful tool for general model constructions we study abstract versions of some 3-valued logics in which we interpret truth as knowledge. We establish a connection between \(\in_K\) and the well-known syntactic approach to explicit knowledge proving a result concerning equi-expressiveness. Furthermore, we discuss some self-referential epistemic statements, such as the knower paradox, as relaxations of variants of the liar paradox and show how these epistemic “paradoxes” can be solved in \(\in_K\). Every specific \(\in_K\)-logic is defined as a certain extension of some underlying classical abstract logic.

03B42 Logics of knowledge and belief (including belief change)
03A05 Philosophical and critical aspects of logic and foundations
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