${\in_K}$: a Non-Fregean Logic of Explicit Knowledge

Abstract

We present a new logic-based approach to the reasoning about knowledge which is independent of 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 φ → φ 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.