Vagueness and Introspection

Intransitivity and introspection. One central and debated aspect of the notion of inexact knowledge concerns the non-transitivity of the relation of indiscriminability and how it should be represented. On the epistemic account of vagueness put forward by Williamson, the intransitivity of the relation of indiscriminability is presented as the main source for vagueness ([5]: 237). In [4] and in the appendix to [5], Williamson formulates a fixed margin for error semantics for propositional modal logic in which the relation of epistemic uncertainty, based on a metric between worlds, is thus reflexive and symmetric, but non-transitive and non-euclidian. An important consequence of the semantics is that it invalidates the principles of positive introspection (if I know p, then I know that I know p) as well as negative introspection (if I don’t know p, then I know that I don’t know p). In [1], we argued against Williamson that models of inexact knowledge that preserve the introspection principles can sometimes be desirable, and we presented a non-standard epistemic semantics for the notion of inexact knowledge, in which non-transitive and non-euclidian Kripke models can nevertheless validate positive as well as negative introspection. In [2], Halpern also argued against Williamson that an adequate model of vague knowledge need not invalidate the introspection principles, but following a different route. Instead of taking intransitivity as a primitive, and proving that the introspection principles can be preserved for a logic with one epistemic operator, as we did in [1], Halpern proposes a bimodal account of inexact knowledge that preserves the introspection principles, and he shows that there is a way to derive intransitivity. For Halpern, the intransitivity of vague knowledge is more characteristic of our reports on what we perceive than about our actual perception. Despite these differences, one can establish a precise correspondence between Halpern’s semantics and the semantics presented in [1]. The object of this paper is to spell out the details of this correspondence, and thus to compare two strategies in order to keep together introspection and nontransitivity. Like Halpern, but contra Williamson, we think it does make sense to preserve the introspection principles within a logic of inexact knowledge; unlike Halpern, but in agreement with Williamson, we are ready to see non-transitivity as a property of perceptual knowledge proper.