Vagueness and Preferences

Maximize and Averaging leave V fairly unconstrained. Indeed, for someone who is not in a position to make very much true, someone in a coma perhaps, Maximize imposes hardly any constraints at all.

It is natural to think that there is another concept we can relate V to that would further constrain an agent's value function.

Suppositional Preference: The agent considers things to be better for her on the supposition of A than on the supposition of B if and only if V (A) > V (B). She considers things to be at least as good iff V (A) ≥ V (B).

Indeed, provided your suppositional preferences satisfy some natural structural con­straints (to be discussed shortly), one can show that it is always possible to rep­resent your suppositional preferences using a value function V, to some degree of uniqueness, relative to some probability function, also determined to some degree of uniqueness. The function V is, strictly speaking, a convenient fiction, but the fact that it's always possible to represent rational preferences in the above way entitles us to use it.

If we include suppositional preferences among the attitudes used to rationalize action, then one's attitudes towards vague propositions are not irrelevant.

Let us begin by noting that the supposition of a vague proposition can clearly change the order of your preferences. There are things that would be great to buy, on the supposition that you're rich, but, since they would bankrupt you otherwise, would not be a good idea to buy. Since the proposition that you're rich is vague, the truth of this vague proposition is relevant to your decision—if you were to learn it, you should behave differently.

It is far from obvious, however, that this shows that our attitudes towards the vague are not practically indispensable. There are closely related precise propositions that would produce this same reordering on their supposition, such as the proposition that you have more than a certain amount of money. What we want, then, is a vague proposition whose supposition induces a reordering of the precise propositions that could not be obtained by the supposition of any precise proposition.

Speaking purely formally, such examples are quite simple to construct. Imagine that there are two equiprobable maximally strong consistent precise propositions A and B whose values are V (A) = 2 and V (B) = 1. Conditional on any precise proposition the order of A and B will remain the same, with A beating B, provided they are ranked (if the precise proposition being supposed is inconsistent with a proposition, that proposition won't be ranked). Suppose furthermore that A is partitioned into two equiprobable vague propositions, Ai and A₂ with V(A1style='font-style:italic'>) = 0 and V(A₂) = 4. B V Ai is a vague proposition, and on its supposition the ordering of A and B is reversed, since clearly V(A ∧ (B V A₁)) = 0 whilst V(B ∧ (B V A₁)) = V(B) = 1.

This is one type of formal counterexample to the thesis that attitudes towards the vague are practically irrelevant. Note, however, that in this type of example one had to care intrinsically about the vague. A₁ and A₂ had different values, even though they entailed the exact same precise propositions—they both entail A, and since A is a maximally strong consistent precise proposition, they have the same precise consequences. The difference in value has nothing to do with the value of some precise matter. Clearly there's nothing wrong with valuing vague things—one can care about being rich, or bald, or whatever—but these cases normally come with valuing some precise underlying parameter such as money in dollars, or hair number. The case being described here is one where, even once one has supposed all of the precise facts to be a certain way (i.e. one has supposed A), one still has preferences for the vague proposition A2 over A1.

Whether this kind of preference is rational is something we'll return to later. For now let me describe another class of examples that do not depend on this feature. In this case, consider three equiprobable maximally strong consistent precise proposi­tions A, B, and C, with values 1, 2, and 5 respectively and suppose that these values remain the same for any proposition that entails A, B, or C (for example if X entails C then V(X) = V(C) = 5; this ensures that no one cares intrinsically about the vague). Now suppose that the propositions you are in a position to make true are B, A V B, and A V C.^[142]

I claim that there is no precise proposition such that upon learning it you would rank A V B below A V C, and A V C below B.

is in some particular precise range. Your evidence after looking won't be anything like, say, the proposition that the jar is between 65% and 75% full; it is more likely that your total evidence regarding the jar will be a vague proposition. Perhaps your evidence after looking is that the jar is pretty full. Being pretty full is perfectly compatible with the jar being in the larger of the ranges, although when it's in the high 60s, it's borderline whether the glass is pretty full. After learning that the glass is pretty full, I can't rule out the possibility that the percentage is in the 60s, but I become much less confident in it.

The case I have described above instantiates the formal example I gave earlier. My preference ranking after seeing that the jar is pretty full will plausibly be to rank option (2) below (1) and option (1) below (3) which seems to be distinctive to this piece of vague information.

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