The Epistemology of Vagueness
Once one has acknowledged that there are vague contents, questions about their role in thought and practical reasoning quickly become urgent.
Indeed, answering such questions will pave the way for a positive account of propositional vagueness.For example, one might wonder whether one could directly come to learn a vague proposition—whether by perception or some other means—or if we only ever know vague contents by inferring them from precise things we've learned directly.[36] In chapter 6, I argue that the totality of an agent's evidence at any given time is almost always vague. Our investigation builds on the similarities between the effect of learning a vague proposition and the effects of information obtained by imperfect perceptual faculties. Indeed, by studying this relationship we are able to provide the beginnings of a positive account of propositional vagueness.
Central to this positive account is a Principle of Plenitude governing the existence of vague propositions. The principle says, roughly, that for any potential effect an inexact experience might have on one's credences about the precise, there is a vague proposition such that learning it has that effect. The ‘effects' of learning a vague proposition can be likened to what's known as a Jeffrey conditioning on a certain partition of precise propositions, the coefficients of which determine what we call an evidential role. In these terms, the Principle of Plenitude can be informally put as saying that:
Principle of Plenitude. For every evidential role, there is some vague proposition occupying that role.
(The precise formulation of this principle will, of course, have to wait until chapter 6, as will the proper explanation of key formal concepts such as Jeffrey conditioning, and the notion of an evidential role.)
Once one has acknowledged that our information is often vague, it is natural to wonder whether this information could ever be useful.
Could vague information, for example, play an essential role in your decision making—could it effect an outcome of your practical reasoning that couldn't have been arrived at by someone with the same attitudes towards the precise who lacked that information? In chapter 9, I argue that, in deciding what to do, you must sometimes take into account your beliefs and desires (and attitudes more generally) about vague matters. In that chapter we consider, for example, the possibility of agents who agree about the precise, act rationally, but behave differently due to their opinions about the vague.It is worth re-emphasizing at this juncture the importance of the ideology of propositional vagueness to the formulation of these questions. If, for example, all propositions were precise there would be no vague propositions to be uncertain about, to be learnt, to desire, to decide to make true, and so on.
The above aspects of the view I am going to advocate for suggest that attitudes with vague contents play a fairly rich role in thought. It is natural to wonder, then, how autonomous our opinions about the vague are. Are they independent of our beliefs in the precise, similar to the way that opinions about the weather and the economy are independent (there are no a priori constraints connecting our opinions about these subject matters)—or is there a tighter rational connection between them? Indeed, one might wonder whether one of these mental states reduces to the other: is having a credence in a vague proposition just a matter of having certain credences in precise propositions?
This issue relates indirectly to the issue of whether vague propositions carve out real factual distinctions. In a parallel debate, philosophers sceptical of genuine conditional facts—expressivists about conditionals—have argued that to have a credence in a conditional is just have a certain distribution of credences in non-conditional propositions; similarly philosophers sceptical of moral facts have suggested that having a doxastic attitude towards a moral proposition is just a matter of having certain sorts of bouletic attitudes towards non-moral propositions.
Assuming that evidence can be vague, it is natural to think that opinions about the vague are autonomous to a certain degree: one could imagine two people with exactly the same precise evidence (and the same priors) but different vague evidence. Such people could in principle exhibit a rational difference of opinion about the vague without a difference of opinion about the precise.
In chapter 8, I argue that there is nonetheless a sense in which vague beliefs are not autonomous, vindicating one aspect of the expressivist picture: for rational ur-priors—probability functions representing the a priori opinions of a completely uninformed person who is not subject to any conceptual confusions—the probabilities of the vague are completely fixed by the probabilities they assign to the precise. That is, I argue that:Rational Supervenience. Any two ur-priors that agree about the precise propositions agree about every proposition.
A similar set of questions arises concerning the autonomy of our desires about vague matters. If there are vague propositions, it will presumably be possible to have desires with those propositions as contents. It is extremely natural to wonder whether caring about some vague matter amounts to distributing your cares about the precise in a certain way, or whether fixing which precise things you care about determines the vague things you care about. That is: is it possible to rationally care intrinsically about the vague? In chapter 10, I argue that there's a sense in which it is not possible to be rational and care intrinsically about the vague. Putting it roughly, I argue:
Indifference. Every rational set of preferences is indifferent between any two vague propositions that settle all precise matters in the same way.
Here a vague proposition settles a precise one if it entails it or its negation, and two propositions settle a third in the same way if they both entail that proposition or both entail its negation.
Putting these three theses about the epistemology of vagueness together puts us in a position to give an account of what a precise proposition is squarely in terms of their role in thought. More specifically, the notion of a precise proposition can be defined in terms of the set of rational prior probability and utility functions.
For example, Indifference says that a maximally strong consistent precise proposition will always have the feature that any two propositions entailing it receive the same value, according to any of the permissible utility functions. Assuming that the collection of rational utilities is sufficiently rich it can be shown that the maximally strong consistent precise propositions are exactly the propositions with this feature. Once you have identified the maximally strong consistent precise propositions, you have identified the precise propositions (their disjunctions). A similar such characterization can be given in terms of priors, appealing instead to Rational Supervenience (see chapter 13).Of course, this characterization of precision in terms of rational utilities and priors is a bit ham-fisted. In chapter 13, I show that the notion of a precise proposition can be defined succinctly in terms of the notion of a symmetry of a set of probability functions and utility functions. A symmetry is a Boolean automorphism of the space of propositions—a one-to-one function that preserves the Boolean operations—that also preserves the values and probabilities of every proposition. A precise proposition turns out to be exactly a proposition that is mapped to itself by every automorphism that preserves the values and probabilities of every proposition:
Symmetry. A proposition is precise if and only if it is fixed by every symmetry.
A similar characterization of determinacy is possible as well: a proposition p is determinate if every proposition p is mapped to by a symmetry is true. (Again we must wait until chapter 13 for a proper explanation of these definitions.)
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