Vagueness and Precision

Since indices are permuted within a cell in such a way that no index gets sent out of its cell by a symmetry, it follows that any proposition that is either a cell, or a union of cells will be left alone by each symmetry.

Precision: A proposition p is precise if and only if σp = p for every symmetry σ.

A vague proposition is defined as a proposition that is not precise. Note that this definition meets the challenge, raised in chapter 12, of giving a direct account of precision that does not go via the modal characterization.

If we want a definition of determinacy and borderlineness we can get that from the notion of precision as in chapter 12; a proposition is determinate iff it is entailed by the strongest true precise proposition. There is also a direct definition in terms of symmetries:

Determinacy: It's determinate that p if and only if every proposition that p is mapped to under a symmetry is true.

A borderline proposition is defined as a proposition which is neither determinate nor has a determinate negation. This is equivalent to saying that borderline propositions are those that are both mapped to a truth and to a falsehood by some symmetry.

Our abstract analysis of precision in terms of being fixed by every symmetry entails some desirable structural features. For example, we can infer that the precise propositions form a complete Boolean algebra: negations and arbitrary conjunctions and disjunctions of precise propositions are also precise.

If we go beyond the abstract definition and invoke the particular notion of sym­metry we have been using, the Rational Supervenience principle and Indifference principle (but not Plenitude), likewise fall out of our definition.name="_ftnref173" title="">^[173] If i and j are indices that belong to the same maximally strong consistent precise proposition, then there must be some symmetry that maps i to j. Since symmetries preserve utilities this means u(i) = u(j). This guarantees Indifference, which amounts to the claim that indices in the same cells have the same utilities. Showing the rational supervenience thesis is a bit more involved, but it is basically shown by proving that if the conditional probability of a proposition on a maximally strong consistent precise proposition is bounded by two rational numbers according to one prior, it is bounded by those numbers according to every prior.⁹

It thus follows that we can provide a fairly compact ‘axiomatization of the main principles defended in this book:

Symmetry. A proposition is precise if and only if it is fixed by every symmetry.

Plenitude. For any function from the maximally specific precise propositions to [0,1], E, there is a proposition, p, such that Pr(p | w) = E(w) for every maximally strong consistent precise proposition w and conceptually coherent ur-prior Pr.

Let me now turn to some points of clarification in our analysis. The first point to observe is that while there are symmetries in the space of conceptually coherent priors and values, these may not represent symmetries in the credences and values of informed people.

Despite the fact that symmetries can be broken amongst the credences of people whose evidence is vague, it is still natural to think that the space of rational values and credences of informed people are closed under a symmetry in an extended sense. If V is the value function of a possible rational person, so is the function V(σ(p)), which assigns p the value of its image under the symmetry σ. Clearly when V is a value for a prior credence function, these two value functions are identical. But even when V corresponds to the values of an informed person, symmetries do take you outside the set of possible rational values.

Another point that requires some clarification is the role that the preservation of rational desires is playing in our notion of symmetry. What would happen if we just worked with a notion of symmetry that preserves initial credences?¹¹ Evidently, the Indifference principle would no longer be a consequence of our definitions, so the principle is needed for my specific project. But would a definition purely in terms of credences give an extensionally adequate characterization of precision?

In section 13.2, we effectively showed that, provided we accepted a richness condition on the space of coherent priors, one can characterize the set of precise propositions as those which are fixed by all symmetries that preserve all coherent initial credences.

For a function to be a rational prior it is certainly necessary that it be probabilis­tically and conceptually coherent. But is this sufficient? Perhaps priors ought also to satisfy the principal principle, support sensible inductive hypotheses, respect the principle of indifference, and so on. Once these further constraints are taken into account, one might think that there are propositions that all initial priors must agree on. Indeed, at the extreme end of the spectrum there are some who think that there is exactly one prior which it is rational to adopt (see Carnap [25] and, more recently, Williamson [160]). On such views, the richness condition on rational priors fails quite dramatically.

This conception of a rational prior might be thought to pose a problem for my abstract analysis of precision. Let me focus on one potential example like this. Suppose that a principle of indifference regarding purely haecceitistic differences is in operation. Imagine, for example, a world exactly like our own except that two people have swapped qualitative roles: Bob has led a life qualitatively indiscernible from the life Bill actually has led, and conversely Bill has led a life qualitatively exactly like the one Bob has led. You might think that any rational prior should assign just as much confidence to the switched possibility as to actuality. This, in some sense, substantiates the idea that it is hard to distinguish between such possibilities without possessing any de re evidence. More generally, any permutation of individuals will naturally induce a permutation on qualitatively identical worlds—a general principle of haecceitistic indifference would require that these permutations preserve rational prior credence. Thus, although the proposition that Bob is exactly 175cm tall seems to be precise, assuming the haecceitistic indifference principle, there's a symmetry preserving rational prior credence that maps it to the distinct proposition that Bill is exactly 175cm tall.

The account I have developed states things in terms of credences that are concep­tually coherent, not in terms of a generic notion of rationality. Conceptual coherence is a fairly weak constraint: a prior which makes no purely conceptual confusions may still be irrational in the wider sense. There is nothing conceptually incoherent about a prior that supports strange inductive inferences, for example, but many would not count such a prior completely rational. Similarly, while it may, in some sense, be unreasonable to have priors that find it more likely that Bob has a certain qualitative role than that Bill does, there is nothing conceptually incoherent about this belief. It is not, for example, like having priors that assign the proposition that Harry is both poor and is a billionaire a high probability.

Provided we are clear about the distinction between rationality in toto and con­ceptual coherence, then it may indeed be possible to extensionally capture the notion of precision without invoking desires. However, I would imagine that there will be some who, despite my attempts at elucidation, find the distinction between a rational prior and a merely conceptually coherent one too obscure to be bearing the burden of explicating this important philosophical notion alone. For those wishing to theorize only with the notion of rationality in toto, it is more important that the constraint that symmetries preserve desires be included. It seems quite evident that I could coherently have haecceitist cares: that I could care about what happens to Bob but not about Bill, for example. This is even more striking when it comes to caring about oneself—surely Bob needn't be completely indifferent between what happens to Bob and what happens to Bill.

Letmealsopointoutthatevenifthe definitionofsymmetryinterms ofpreserving a certain class of priors is extensionally adequate for characterizing vagueness, it might not do the job of a good explanatory theory. For example, if the principle that one should not care intrinsically about the vague is true, it calls out for an explanation. Presumably the explanation ought to have something to do with vagueness. An abstract analysis of vagueness purely in terms of coherent credences does not provide any such explanation, yet a theory that invokes bouletic notions could provide such an explanation. For example, according to my theory, in order for something to play the role of a particular vague proposition, your bouletic attitudes toward it ought to be aligned in certain ways.

Let me end this discussion by considering the question of whether our abstract analysis in terms of symmetries can act as a reduction of vagueness to more basic notions. In this section I showed how one could start off with nothing but a certain class of priors and utilities, and from these characterize the class of precise propos­itions. From the notion of a conceptually coherent prior and utility, we introduced a class of symmetries that preserved values according to every prior and utility, and from this I characterized a precise proposition as something that is fixed by every symmetry. Could this be considered a reduction of the notion of precision to the normative notion of being conceptually coherent?

There are reasons to resist this further reductive claim. In particular, it is hard to get a handle on the notion of a conceptually coherent prior without already having the concepts of vagueness and precision at our disposal. A crucial distinction that came up in our earlier discussion was the difference between being rational in toto, and merely being conceptually coherent.

To distinguish mere conceptual coherence from general rationality we noted that the former satisfied a richness condition: roughly, although one can have pretty much any opinion or preference about the precise matters without committing a conceptual confusion, not all such opinions are rational in the wider sense. Formally, for any probability function over precise matters there is a conceptually coherent prior over all matters that agrees with it; this may not hold when restricted to rational priors. Similarly, for any utility over the cells, there's a conceptually coherent utility that agrees with it and is constant within each cell. Explaining what a conceptually coherent prior or utility is to someone by appeal to these richness conditions would require them to already possess the concept of precision and vagueness.

If, like me, you think that completely reductive analyses are rare, the above conclu­sion is hardly surprising; the value of abstract analyses is of an entirely different nature altogether. What, then, have we gained from our analysis if not a reduction? At least one important result is to widen the circle of concepts that vagueness and precision are related to. Even if you don't have the concept of a coherent prior at your disposal, we have still succeeded in related vagueness and precision to overall rationality: we have discovered that it is simply irrational, for example, to care about the vague, even if we can't get the converse claim without invoking notions that presuppose the concept of vagueness. Note also that such analyses deliver important structural features of the target concepts. By analogy, analyses of counterfactuals in terms of similarity are not usually proposed as reductive analyses, reducing counterfactuals to an antecedently understood notion of similarity; rather similarity is usually taken to be a back-formation from our judgements about counterfactuals (see Lewis [90] and Stalnaker [140]). The value of these analyses is that they predict the validity and invalidity of inference patterns that have puzzled earlier philosophers thinking about counterfactuals, and subsume them under a more general theory.¹²

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