Vagueness and Introspection

Version of March 05, 2007. An extended abstract of the paper appeared in the Proceedings of the 2006 Prague Colloquium on "Reasoning about Vagueness and Uncertainty".We compare three strategies to model the notion of vague knowledge in epistemic logic. Williamson's margin for error semantics typically uses non-transitive Kripke structures, but invalidates the principle of positive introspection. On the contrary, Halpern's two-dimensional semantics preserves the introspection principle, but using more complex uncertainty relations that are transitive. We present a modification of the standard epistemic semantics, which validates introspection over one-dimensional non-transitive structures, and study its correspondence with Halpern's approach. While the semantics can be seen as the diagonalization of an explicit two-dimensional semantics, it affords a more intuitive representation of the uncertainty characteristic of vague knowledge. We examine the implications of the semantics concerning higher-order vagueness and the status of the non-transitivity of perceptual indiscriminability. We respond to a potential objection against our approach by giving a dynamic model of the way subjects with inexact knowledge make successive approximations of their margin of error

Reliability, Margin for Error and Self-Knowledge

Forthcoming in D.H. Pritchard & V. Hendricks (eds.), New Waves in Epistemology, (Adelshot: Ashgate Publishing).Margin for error principles play a central role in the epistemology of Timothy Williamson, both in his account of vague knowledge (Williamson 1992, 1994), and in his attack against the luminosity of knowledge (Williamson 2000). The present paper pursues two objectives: the first is an attempt to refine and systematize the modal analysis of the reliability of knowledge given by Williamson, and to delimit the scope of margin for error principles. The second is a criticism of Williamson's thesis that knowledge is not luminous, elaborating on previous work by Dokic & Egre (2004), based on the intuition that knowledge is modular and that a representation of this modularity is needed at the logical level in order to avoid the paradoxical conclusions which result rom Williamson's assumptions

Inexact Knowledge 2.0

Many of our sources of knowledge only afford us knowledge that is inexact. When trying to see how tall something is, or to hear how far away something is, or to remember how long something lasted, we may come to know some facts about the approximate size, distance or duration of the thing in question but we don’t come to know exactly what its size, distance or duration is. In some such situations we also have some pointed knowledge of how inexact our knowledge is. That is, we can knowledgeably pinpoint some exact claims that we do not know. We show that standard models of inexact knowledge leave little or no room for such pointed knowledge. We devise alternative models that are not afflicted by this shortcoming

Margin for error and the transparency of knowledge

Forthcoming in Synthese.In chapter 5 of Knowledge and its Limits, Williamson formulates an argument against the principle (KK) of epistemic transparency, or luminosity of knowledge, namely “that if one knows something, then one knows that one knows it” (Williamson 2000: 115). Principle (KK), which corresponds to axiom schema 4 of propositional modal logic, is also called “positive introspection” and was originally defended by Hintikka in his seminal work on epistemic logic (Hintikka 1962: c. 5, “Knowing that one knows”). Williamson's argument proceeds by reductio: from the description of a situation of approximate knowledge, he shows that a contradiction can be derived on the basis of principle (KK) and additional epistemic principles that he claims are better grounded. One of them is a reflective form of the margin for error principle defended by Williamson in his account of knowledge. We argue that Williamson's reductio rests on the inappropriate identification of distinct forms of knowledge. More specifically, an important distinction between perceptual knowledge and non-perceptual knowledge is wanting in his statement and analysis of the puzzle. The (KK) principle and the margin for error principle can coexist, provided their domain of application is referred to the right sort of knowledge

Inexact knowledge and dynamic introspection

Cases of inexact observations have been used extensively in the recent literature on higher-order evidence and higher-order knowledge. I argue that the received understanding of inexact observations is mistaken. Although it is convenient to assume that such cases can be modeled statically, they should be analyzed as dynamic cases that involve change of knowledge. Consequently, the underlying logic should be dynamic epistemic logic, not its static counterpart. When reasoning about inexact knowledge, it is easy to confuse the initial situation, the observation process, and the result of the observation; I analyze the three separately. This dynamic approach has far reaching implications: Williamson’s influential argument against the KK principle loses its force, and new insights can be gained regarding synchronic and diachronic introspection principles

Not much higher-order vagueness in Williamson’s ’logic of clarity’

This paper deals with higher-order vagueness in Williamson's 'logic of clarity'. Its aim is to prove that for 'fixed margin models' (W,d,α ,[ ]) the notion of higher-order vagueness collapses to second-order vagueness. First, it is shown that fixed margin models can be reformulated in terms of similarity structures (W,~). The relation ~ is assumed to be reflexive and symmetric, but not necessarily transitive. Then, it is shown that the structures (W,~) come along with naturally defined maps h and s that define a Galois connection on the power set PW of W. These maps can be used to define two distinct boundary operators bd and BD on W. The main theorem of the paper states that higher-order vagueness with respect to bd collapses to second-order vagueness. This does not hold for BD, the iterations of which behave in quite an erratic way. In contrast, the operator bd defines a variety of tolerance principles that do not fall prey to the sorites paradox and, moreover, do not always satisfy the principles of positive and negative introspection

The visual presence of determinable properties

Several essays in this volume exploit the idea that in visual experience, and in other forms of consciousness, something is present to consciousness, or phenomenally present to the experiencing subject. This is a venerable idea. Hume, for example, understood conscious experience in terms of the various items ‘present to the mind’. However, it is not obvious how the idea should be understood (‘present to’ is not a standard English modification of the adjective ‘present’) and there are grounds for worrying that there is no good way of making it precise. Here I explore a way of making precise the idea that properties of things, such as their shapes and colours, are present to us in visual experience. I argue that this important idea is coherent, well motivated and empirically plausible, provided that we reject two traditional assumptions: (i) that maximally determinate properties, rather than just determinable properties, are visually present; (ii) that we can tell through introspection exactly which properties are visually present to us