05872589

j

05872589 Lewitzka, Steffen $\in_K$: a non-Fregean logic of explicit knowledge. Stud. Log. 97, No. 2, 233-264 (2011). 2011 Institute of Philosophy and Sociology of the Polish Academy of Sciences, Warsaw; Springer, Dordrecht EN epistemic logic non-Fregean logic logical omniscience explicit knowledge classical abstract logic self-reference epistemic paradox liar paradox

doi:10.1007/s11225-011-9304-8

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.