I wish to present a proof that vagueness is impossible. Of course, vagueness is possible; and so there must be something wrong with the proof. But it is far from clear where the error lies and, indeed, all of the assumptions upon which the proof depends are ones that have commonly been accepted. This suggests that we may have to radically alter our current conception of vagueness if we are to make proper sense of what it is.

The present investigation was largely motivated by an interest in what one might call the ‘global’ aspect of vagueness. We may distinguish between the indeterminacy of a predicate in its application to a single case (the local aspect) and in its application to a range of cases (the global aspect). In the first case, it is indeterminate how a predicate, such as a bald, applies in a given case; and, in the second case, it is indeterminate how a predicate applies across a range of cases. Given such a distinction, the question arises as to whether one might understand the indeterminacy of a predicate in its application to a range of cases in terms of its indeterminacy in application to a single case; and considered from this point of view, the result can be seen to show that there is no reasonable way in which this might be done.

But the result can also be seen to arise from an interest in higher-order vagueness. It has often been observed that there is a difficulty in conceiving of indeterminacy in the presence of higher-order vagueness. For it cannot consist in these cases being borderline and those other cases not being borderline, for that would be compatible with there being a sharp line between the borderline cases and the non-borderline cases, contrary to the existence of higher-order vagueness; and, for similar reasons, it cannot be taken to consist in these cases being borderline borderline and those other cases not being borderline borderline or in something else of this sort. But it is hard to pin this difficulty down and, certainly, the failure of one particular attempt to characterize indeterminacy in the presence of higher order vagueness does nothing to establish a failure in principle. Considered from this point of view, the present result can be seen to provide a vindication of those (such as Graff-Fara [2003], Sainsbury [1991], Wright ([1987], [1992])) who have suggested that the existence of higher order vagueness does indeed stand in the way of having a reasonable conception of what indeterminacy might be.

I begin by giving an informal presentation of the result and its proof and I then consider the various responses that might be made to the alleged impossibility. Most of these are found wanting; and my own view, which I hint at rather than argue for, is that it is only by giving up on the notion of single-case indeterminacy, as it is usually conceived, and by modifying the principles of classical logic that one can evade the result and thereby account for the possibility of vagueness. There are two appendices, one providing a formal presentation and proof of the impossibility theorem and the other giving a counter-example to the theorem under a certain relaxation of its assumptions. The mathematics is not difficult but those solely interested in the philosophical implications of the results should be able to get by without it.

The general line of argument goes back to Wright [1987] and further discussion and developments are to be found in Sainsbury[1990, 1991], Wright [1992], Heck [1993], Edgington [1993], GomezTorrente [1997, 2002], Graff-Fara[2002, 2004], and Williamson [1997, 2002]. It would be a nice question to discuss how these various arguments relate to one another and to the argument in this paper. I shall not go into this question, but let me observe that my own approach is in a number of ways more general. It relies, for the most part, on weaker assumptions concerning the underlying logic and the logic of definitely and on weaker constraints concerning the behavior of vague terms; and it also provides a more flexible framework within which to develop arguments of this sort.