The theory of vague propositions outlined in chapter 6 appealed freely to the orthodox Bayesian theory of credences and learning, which assumes, among other things, that the correct theory of rational credence is one governed by the classical axioms of probability theory.
I assumed this theory when I argued that one's credences after conditioning on a vague proposition conform to the kinds of smooth curves depicted in Figure 6.4.
class=a7 style='text-indent:18.0pt'>However, the classical theory of probability is not entirely uncontroversial, especially when it is applied to vague propositions. For one thing, it takes sides on the question of what kind of doxastic attitude we should take towards P when we know it is borderline. This question turns out to be surprisingly central to the study of vagueness, and if one accepts the probability calculus, the answer is that one should be uncertain. That is, one should have the same credal attitude concerning Harry's baldness as one should have about the outcome of a coin flip, for example. For according to the probability calculus, the only alternative to having middling degrees of belief about a borderline proposition is to either assign a credence of 1 to that proposition, or to its negation—and this seems to be absurd: I shouldn't be certain, for example, that any number is the last small number.This dichotomy, however, has been challenged in recent years by Hartry Field [53], who has given a precise formulation of the idea that we shouldn't be uncertain about the borderline.[CII] Crucially, this idea doesn't collapse into the view that one should be certain about the locations of cutoff points, because it relaxes the probabilistic axioms governing rational degrees of belief.
Field's theory is one of the most influential alternatives to the standard probabilistic picture. However, it also serves as a good springboard for a discussion of the present theory of vagueness-related uncertainty, since it articulates in a particularly precise way, positions on two of the most central questions concerning this issue—it tells us what kind of attitude we ought to have toward a proposition we know to be borderline, and it tells us how this attitude ought to be extended to propositions we know to be higher-order borderline.
This is summarized by the following two principles, which I'll state informally for now:Iteration: The credal attitude one ought to have towards A when one is certain that A is higher-order borderline is the same as the attitude one ought to have when one is certain that A is borderline at the first order.
Rejection: The credal attitude one ought to have towards A when one is certain that A is borderline is simply that of fully rejecting A and fully rejecting —A: one’s credence in both A and —A must be 0.
Although Field’s theory of credences will be a useful springboard for this discussion, many of the points I shall be considering here generalize to other questions involving attitudes like knowledge and assertion.
7.1