Field’s Theory

Let us begin by outlining a fairly pared-down presentation of Field’s theory. This version is governed by the following two axioms, where Q represents a rational credence function mapping propositions into the interval [0,1]:

F2.

The first axiom quite clearly bears a close resemblance to the additivity law from classical probability theory, except that the usual sum is being used to calculate the value of ΔA V ΔB rather than the usual disjunction A V B (the standard additivity law states that Q(AvB) = Q(A)+Q(B)+Q(A∧B)). The second axiom effectively amounts to the requirement that rational degrees of belief respect logical laws, where these laws are understood broadly enough to include the laws governing the determinacy operator.

The above theory is in fact strictly weaker than Field’s theory. According to Field, Q must additionally respect the logic of determinacy S4, which controversially states that if something is determinate, it’s determinately determinate.^[CIII] However, the theory can be developed perfectly consistently without making this assumption. None of the problems I shall raise here will depend on this unnecessarily controversial assumption, so I shall simply put it to one side for now.

Two important consequences of these axioms are the following, which we already stated informally in the last section:

Iteration follows from the first axiom by setting A equal B and then applying some fairly trivial logic using F2.

It is worth stressing, however, that the principle Rejection, even in conjunction with the axiom F2, does not imply Iteration. In fact, although there is an entailment in one direction, Iteration and Rejection correspond to two very different ideas that ought to be kept theoretically separate.

Rejection allows for a particularly striking characterization of vagueness-related uncertainty. The standard probability calculus only allows you to be in one of two states regarding p: (i) be certain whether p: either having a credence of 1 in p and 0 in its negation, or vice versa, or (ii) be uncertain whether p: both p and its negation get a credence strictly between 1 and 0. However, in Field's calculus there is a third option: one assigns both p and its negation a credence of 0. Your credences are not intermediate, as they would be if you were uncertain, nor do they assign 1 to p or its negation, as they would if you were certain whether p. Whereas the epistemicist, for example, would treat uncertainty about whether Harry is bald just as she would treat uncertainty about, say, the mass of the moon, someone adopting Field's calculus can say something distinctive about the former case: vagueness does not merely amount to a kind of uncertainty, for one's credences in p aren't intermediate between 1 and 0. Yet this does not commit us to being certain about p either, since we do not assign 1 to p or its negation.

Iteration entails something much stronger: it entails that anyone who is certain that A is borderline at the second order or higher, must also assign A and its negation credence 0. To illustrate, let us suppose that I am certain that A is second-order borderline: i.e. that Q(VVA) = 1. It follows that my credence in A and —A must both be 0: if my credence in A were non-zero, then I would assign some credence to A being determinately determinate (applying Iteration twice to my non-zero credence in A), and this is simply impossible if I'm certain that A is second-order indeterminate.

It is worth stressing that the thought behind these two ideas is quite different. The first tells us that we shouldn't be uncertain—in the sense of having middling credences—about the borderline. The second idea is that we should extend this state to the things we believe to be higher-order borderline. It turns out that neither of these theses is particularly friendly to the theory we developed in chapter 6.

7.1