Vagueness, uncertainty and degrees of clarity

The focus of the paper is on the logic of clarity and the problem of higherorder vagueness. We first examine the consequences of the notion of intransitivity of indiscriminability for higher-order vagueness, and compare different theories of vagueness understood as inexact or imprecise knowledge, namely Williamson’s margin for error semantics, Halpern’s twodimensional semantics, and the system we call centered semantics. We then propose a semantics of degrees of clarity, inspired from the signal detection theory model, and outline a view of higher-order vagueness in which the notions of subjective clarity and unclarity are handled asymmetrically at higher orders, namely such that the clarity of clarity is compatible with the unclarity of unclarity. 1 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 ([16]: 237). In [15] and in the Appendix to [16], 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.1 An important consequence of Williamson’s 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 an earlier paper [1], we argued against Williamson that models of inexact knowledge that preserve the introspection principles may sometimes be desirable, and we presented an alternative epistemic semantics for the notion of inexact 1A relation R is transitive if xRy and yRz imply xRz for every x, y and z. A relation is euclidian if xRy and xRz imply yRz for every x, y, z.