This paper propounds a systematic examination of the link between the Knower Paradox and provability interpretations of modal logic. The aim of the paper is threefold: to give a streamlined presentation of the Knower Paradox and related results; to clarify the notion of a syntactical treatment of modalities; finally, to discuss the kind of solution that modal provability logic provides to the Paradox. I discuss the respective strength of different versions of the Knower Paradox, both in the framework of first-order (...) arithmetic and in that of modal logic with fixed point operators. It is shown that the notion of a syntactical treatment of modalities is ambiguous between a self-referential treatment and a metalinguistic treatment of modalities, and that these two notions are independent. I survey and compare the provability interpretations of modality respectively given by Skyrms, B. (1978, The Journal of Philosophy 75: 368–387) Anderson, C.A. (1983, The Journal of Philosophy 80: 338–355) and Solovay, R. (1976, Israel Journal of Mathematics 25: 287–304). I examine how these interpretations enable us to bypass the limitations imposed by the Knower Paradox while preserving the laws of classical logic, each time by appeal to a distinct form of hierarchy. (shrink)

Instability occurs when the very fact of choosing one particular possible option rather than another affects the expected values of those possible options. In decision theory: An act is stable iff given that it is actually performed, its expected utility is maximal. When there is no stable choice available, the resulting instability can seem to pose a dilemma of practical rationality. A structurally very similar kind of instability, which occurs in cases of anti-expertise, can likewise seem to create dilemmas of (...) epistemic rationality. One possible line of response to such cases of instability, suggested by both Jeffrey (1983) and Sorensen (1987), is to insist that a rational agent can simply refuse to accept that such instability applies to herself in the first place. According to this line of thought it can be rational for a subject to discount even very strong empirical evidence that the anti-expertise condition obtains. I present a new variety of anti-expertise condition where no particular empirical stage-setting is required, since the subject can deduce a priori that an anti-expertise condition obtains. This kind of anti-expertise case is therefore not amenable to the line of response that Jeffrey and Sorensen recommend. (shrink)

This paper is intended to show how epistemologists can profit from the study of ways in which 'know' is vague. Topics include the kk thesis, Incorrigibility of sense data, A resemblance between infinity and vagueness, Common knowledge, Naive holism, Question-Begging, Epistemic universalizability, The prediction paradox, The completability of epistemology, And harman's social knowledge cases.

Epistemologists have profited from studies of the ways in which ‘know’ is ambiguous. We can also profit by studying the ways in which ‘know’ is vague. After classifying sources of vagueness for ‘know,’ I spend the second section examining theories of vagueness. With the exception of the theory that vague predicates are incoherent, which I try to refute, we need not take a stand on a particular theory to show that the vagueness of knowledge has substantive epistemological implications. The third (...) section supports the thesis through a survey of ways in which appeals to vagueness can be applied to the field's issues. First, I show how higher order vagueness is evidence against the KK thesis, the incorrigibility of sense data, and the completability of epistemology. An interesting resemblance between infinity and vagueness provides the point of departure for the next type of application. For this resemblance suggests a new way of handling apparently infinite belief structures. The approach is illustrated with an analysis of common knowledge. The third type of application concerns the ways in which disguised sorites creep into our thinking about knowledge. In addition to brief illustrations concerning naive holism, question begging, and an objection to incorrigibilism, I provide more detailed illustrations involving Jonathan Adler's sceptical appeal to epistemic universalizability, the prediction paradox, and William Lycan's objection to Gilbert Harman's social knowledge cases. I conclude with some general remarks on the lessons to be learned from the vagueness of knowledge. (shrink)