A non-standard Semantics for Inexact Knowledge with Introspection

Forthcoming in S. Artemov and R. Parikh, Proceedings of the ESSLLI 2006 Workshop on Rationality and Knowledge.Standard Kripke models are inadequate to model situations of inexact knowledge with introspection, since positive and negative introspection force the relation of epistemic indiscernibility to be transitive and euclidian. Correlatively, Williamson's margin for error semantics for inexact knowledge invalidates axioms 4 and 5. We state a non-standard semantics for modal logic which is shown to be complete for K45, without constraining the accessibility relation to be transitive or euclidian. The semantics corresponds to a system of modular knowledge, in which iterated modalities and simple modalities are not on a par. We show how the semantics helps to solve Williamson's luminosity paradox, and argue that it corresponds to an integrated model of perceptual and introspective knowledge that is psychologically more plausible than the one defended by Williamson. A generalized version of the semantics is formulated, in which modalities are iteration-sensitive up to degree n and insensitive beyond n. The multi-agent version of the semantics yields a resource-sensitive logic with implications for the representation of common knowledge in situations of bounded rationality