Inexact Knowledge with Introspection

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 euclidean. Correlatively, Williamson’s margin for error semantics for inexact knowledge invalidates axioms 4 and 5. We present a new semantics for modal logic which is shown to be complete forK45, without constraining the accessibility relation to be transitive or euclidean. 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. We formulate a generalized version of the semantics, called token semantics, in which modalities are iteration-sensitive up to degree n and insensitive beyond n. The multi-agent version of the semantics yields a resourcesensitive logic with implications for the representation of common knowledge in situations of bounded rationality. 1 Inexact knowledge and introspection Standard modal models for knowledge are commonly S5 models in which the epistemic accessibility relation is an equivalence relation, namely a relation that is reflexive, symmetric and transitive. From an axiomatic point of view, reflexivity corresponds to the fact that knowledge is veridical, symmetry to the idea that if something is true, one knows one will not exclude it, and transitivity to the idea that knowledge is positively introspective, that is the property that whenever I know some proposition, I know that I know it. S5 models can also be described as reflexive models that are euclidean, which also makes them symmetric and transitive. Euclideanness corresponds to the property of negative introspection, namely to the property that whenever I don’t know, I know that I don’t know. S5 models are commonly used to represent situations of social knowledge, for instance in game theory, due to their well-known correspondence with partitional models of information (Osborne & Rubinstein 1994). An important feature of these models is the fact that they represent a notion of precise or exact knowledge in the following sense: whenever an agent fails to discriminate between two worlds or situations w and w, any other situation which he fails to discriminate from w is also a situation which he fails to discriminate from w, and vice versa. In other words, even though one’s knowledge is not necessarily as fine-grained as it should be, it is at least clear cut, since