Expressivism about Vagueness

According to the theory of vague propositions in chapter 6, there's a more straightfor­ward sense in which vagueness-related uncertainty differs from ordinary uncertainty, at least assuming a certain kind of permissive epistemology.

Our uncertainty about the correct verdict in a court case seems to be in stark contrast with the kinds of vagueness-related uncertainty we have encountered so far. If we both have the evidence that a particular glass is two thirds full, and we additionally had all other relevant precise information about the glass, then I think it would be outright incoherent to have very high credence that the glass is pretty full. It would be similarly irrational to have very high credence that it isn't pretty full. In this case, it seems that the amount to which two people can have opposing opinions about whether the glass is pretty full, assuming they are both knowledgeable about the relevant precise matters, is extremely limited.^[120] The theory of chapter 6 has an explanation for this limitation: there is some particular middling credence that one is supposed to have in the proposition that the glass is pretty full, given that you have all the relevant precise evidence.

The situation here bears a striking resemblance to a certain picture of disagreements about conditional matters.

Expressivism has most famously been applied to moral talk, and this fits the same mould. According to a schematic and simplistic version of this view, when one utters a moral sentence, such as ‘it is good that Alice is happy', one does not represent oneself as believing or knowing a proposition—the proposition that it’s good that Alice is happy—one represents oneself as taking some kind of non-doxastic positive attitude towards Alice being happy. Generally, moral sentences are asserted to express a negative or positive non-doxastic attitude to a non-moral proposition, rather than a doxastic attitude to a moral one.

To illuminate my preferred theory of vagueness it will be instructive to compare it against a view one might call ‘expressivism about vagueness' that is inspired by the expressivist views about conditionals and epistemic modals (and to a lesser extent, moral propositions) outlined above.

To fix ideas let us start with a simple conception of propositions as sets of possible worlds. Taking the lessons of chapter 5 to heart, such a view seems quite unfriendly to the existence of vague propositions, since one can be ignorant about whether Harry is bald even if you know which possible world obtains (i.e. you are not ignorant about any set-of-worlds proposition). Our expressivist thus maintains that vague sentences do not express propositions at all. When one utters a vague sentence, one does not represent oneself as having a high credence in a proposition: one merely represents oneself as having certain patterns of credences among the precise, sets-of-possible- worlds propositions.

To make this idea more rigorous, we may repurpose some of the formalism we introduced in chapter 6. Recall that an evidential role is a function, E, from maximally strong consistent precise propositions (possible worlds in this framework) to real numbers in [0,1]. Let us suppose that each vague sentence for a language gets associated, via the conventional patterns of use for that language, not with a set of possible worlds but with an evidential role. To illustrate, the sentence ‘Harry is bald' in English is associated with a role that maps worlds where Harry has no hairs to 1, worlds where he has lots of hair to O, and worlds where his hair number lies in the border region to credences strictly between 0 and 1. Intuitively, the number E(w) corresponds to the credence you would express by uttering ‘Harry is bald' if you knew you were in world w.7

⁷               There are some analogies to be drawn here between this idea and the theory developed in Horwich [72].

Taking a cue from expressivism about conditionals, the idea is that when one utters a vague sentence, one does not express high credence in a vague proposition; rather one expresses that one has a certain pattern of credences in the precise. In particular, by uttering ‘Harry is bald', associated with evidential role E, one expresses that one's credences are such that the sum £_w E(W)Cr(W) is high: roughly, that one assigns low credences to worlds where E(w) is low, and high credence to worlds where E(w) is high. By analogy, according to the expressivist about conditionals, when one utters an indicative sentence, ‘if P then Q’ one's utterance is proper not if one has high credence in a conditional proposition, but rather if one has a certain distribution of credences over categorical propositions: in particular, if one's credence in P ∧ Q and in P is such that the former divided by the latter is high.

Again, drawing on the parallel view about conditionals, one could introduce the notion of a credence in a vague sentence:

Expressivism about Vagueness: To have a credence of x in a vague sentence S is just for £_w E(w)Cr(w) = x where E is the evidential role associated with S in the language, and Cr represent your distribution of credences over possible worlds.⁸ Note that expressivists about vagueness and conditionals alike can introduce, as we have done above, the technical notion of a credence in a conditional sentence or a credence in a vague sentence.

When I assert the sentence ‘Harry is bald' and my audience takes me to be reliable, they ought to adjust their credences accordingly: they ought to become less confident that Harry has very high hair numbers, and more confident that he has lower hair numbers. IfI had expressed a proposition, they could achieve this simply by conditioning their credences on the proposition I expressed. Since I haven't expressed hairs' or its negation to people either. However, this pattern oflinguistic inclinations obtains because we are ignorant of something whereas the application of ‘bald' is precluded by our linguistic competence alone. Note, however, that this theory relies on a certain kind of‘meaning as use' theory of meaning whereas the present proposal does not. Moreover, it falls afoul of the problems we discussed in chapter 4. The theory can explain why competent English speakers are usually disinclined to utter the sentence ‘Harry is bald' or ‘Harry is not bald' when they know that Harry's hair number is in a certain range, but it doesn't explain why they don't know whether he's bald in those circumstances and this, I take it, is one of the most important jobs for a theory of vagueness.

⁸               To cut down on formalism I have made the simplifying assumption there are a small (i.e.

a proposition they should condition their credences on the evidential role associated with the sentence. Formally, this is achieved by the following equation:⁹

Here W denotes the set of all worlds, and we write Cr(w) for a world w to denote the credence assigned to w’s singleton. I write Cr(P || E) to denote a function that takes a set of worlds and an evidential role to a real number, and reserve Cr(P | Q) for the function that takes two sets of worlds and maps them to the conditional probability of one on the other. We can also see that each precise proposition—a set of worlds— corresponds to a special kind of evidential role: one that assigns either 1 or 0 to each world. If E(w) = 1 or 0 for each world w, then conditioning on E in the above sense is exactly equivalent to ordinary Bayesian conditioning on the set of worlds that E maps to 1.

The notion of conditioning on an evidential role is important for the theory of communication on this view. However, one could also apply this machinery to the view defended in chapter 6 that inexact experiences involve the acquisition of vague evidence. Indeed, it will deliver similar results to the theory of chapter 6 as witnessed by the following fact: for someone who doesn't already have any vague evidence, conditioning on an evidential role, E, has exactly the same effect as conditioning on a vague proposition which has that evidential role. However, because vague propositions are not always probabilistically independent of one another, the theory of chapter 6 will predict more interesting results for those who learn two vague propositions in succession.¹⁰

Expressivism about vagueness suffers some familiar problems. For starters, one might be worried by arguments from the apparent truth of quantified belief reports. I can say things like ‘Alice is pretty sure that Harry is bald, Bob is unsure that Harry is bald, so there's something that Alice is pretty sure about which Bob isn't'. If the straightforward expressivist is correct, then this inference isn't in general good, because Alice and Bob don't really believe anything. Note, however, that we have introduced the notion of a credence (and thus a high credence) in an evidential role, and, thus, it would be easy to cash out this kind of quantification as quantification over

evidential roles instead of quantification over sets of worlds. Since evidential roles can be learnt (we can condition on them), they’re the objects of belief, and they are the meanings of simple vague sentences, it is already looking as though they play a large portion of the proposition role.

This form of expressivism also suffers from a Frege-Geach problem: although we have specified the conditions under which one can assert a simple vague sentence such as ‘Harry is bald’, we haven’t explained how the assertability conditions of complex sentences get determined from their parts. The matter is simple for a negated sentence: if the evidential role of P maps w to E(w), then the evidential role for — P should map w to 1 — E(w), which guarantees that one’s formal ‘credence’ in a negated vague sentence is 1 minus one’s formal credence in the sentence. But it is very hard to say what happens to disjunctions and conjunctions. In the theory of vague propositions of chapter 6, each vague proposition determines a unique evidential role but not conversely: distinct propositions can have the same evidential role. At the extreme, take a vague proposition, P, that has a role that assigns every world the value style='font-size:7.0pt;line-height:122%'>1 (such a proposition must exist by the Principle of Plenitude). Then — P has exactly the same role, but clearly — P is not identical to P. One’s credence in the disjunction Pv—P ought always be 1, while one’s credence in P vP should be one’s credence in P, which is 2. But if the only information the disjuncts contribute to a disjunction are their evidential role, then P v —P and P v P are constructed from the same evidential roles and thus whatever our rule for v is, they must have the same output.

I do not know how to solve the Frege-Geach problem, but we may assume that it involves assigning some kind of technical device—a gizmo¹¹—to each sentence of the language to account for its role in communication.^{[121] [122]} The problems gestured at above show that whatever these gizmos are, they have to be richer than evidential roles: in addition to an operation of negation, there must be operations of disjunction and conjunction that they are closed under. Moreover, we can also define what it is for a gizmo to be true or false using the now familiar disquotational schema: for example, the gizmo corresponding to ‘Harry is bald’ is true if Harry is bald, and false otherwise.

These gizmos, whatever they are, are beginning to look more and more like propositions, for we now know that they are closed under the Boolean operations of conjunction, disjunction, and negation, and there is a straightforward notion of truth that can be applied to them. If we add into the mix that almost all sentences of a natural language are vague, and therefore don’t express sets of worlds, then it is hard to see this view as an alternative to the theory of vague propositions in chapter 6: it is the gizmos that are playing the proposition role, not sets of worlds, and the gizmos have most of the distinctive features of the propositions of chapter 6.13

Indeed, some moral expressivists are willing to make exactly this concession.^{[123] [124]} For these expressivists, propositions—conceived merely as occupiers of the proposition role—are easy to come by. However, such theorists will typically also want to maintain that there is an important distinction between propositions to be made: some propos­itions correspond to the kinds of propositions we originally represented by sets of worlds, and others do not—the latter are, in some sense, ‘metaphysically lightweight’.

Unfortunately, spelling out this notion of being ‘metaphysically lightweight' has proved notoriously difficult, and it has long been an important challenge for such theories to offer some precise articulation of this idea. Indeed, without this distinction, the view effectively collapses into realism (see Dorr [33], Dreier [36], Field [50]). Since the problem of spelling out the difference between the metaphysically lightweight propositions bears a strong parallel with the problem of making sense of the notion of there being no fact of the matter, it will be worth taking a look at how some existing expressivists have attempted to articulate similar distinctions.

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