A Principle of Plenitude for Vague Propositions

According to linguistic theories of vagueness, vague propositions have a derivative status. Usually there are no vague propositions or there are and they are parasitic on the vocabulary of the language.

An alternative to the modal way of individuating propositions, in which they are identified with sets of possible worlds, would be to individuate them by their attitudinal roles and to identify them with sets of maximally specific epistemic (bouletic/doxastic/etc.) possibilities. How should we express the idea that propos­itions exist independently of language and thought?

I propose the following Principle of Plenitude for propositions:

For every possible evidential role in thought, there is some proposition with that evidential role.

Intuitively the evidential role of a proposition is a profile of the strength of evidential support that that proposition receives from each maximally specific precise descrip­tion of the world. We may state the Principle of Plenitude rigorously as follows. Let P denote the set of maximally strong consistent precise propositions: the set of propositions that are (i) precise and (ii) entail any other precise proposition that they are consistent with. Then, an evidential role will be represented by a function E : P → [0,1], telling you what your ur-priors should look like conditional on each maximally strong consistent precise proposition:

The Principle of Plenitude.

As an example, the Principle of Plenitude entails that there is a proposition p such that conditional on the proposition that Harry has N hairs, everyone’s ur-prior in p ought to be 0.798 (set E(w) = 0.798 whenever w entails that Harry has N hairs). The Principle of Plenitude entails the existence of an abundance of such propositions. Note that while the evidential roles generate propositions, they do not individuate them. For example, by Plenitude there is a proposition, p, such that one is required to have credence 2 in it conditional on any maximally specific precise proposition. It follows by probability theory that —p has the same evidential role as p, yet these propositions are always distinct in a Boolean algebra.

Roughly, the Principle of Plenitude guarantees that we have the following kind of picture: the space of epistemic possibilities will be divided up into non-overlapping cells, representing the maximally strong consistent precise propositions. Furthermore if for each cell I pick a proportion of that cell that I want to ‘fill, plenitude guarantees there will be some vague proposition that fills each cell to that proportion. The notion of filling a cell by some proportion simply means having that probability conditional on the cell according to every ur-prior. (Notice that a proposition generated by the Principle of Plenitude is such that every ur-prior agrees about what proportion that proposition takes up of each cell, even if the ur-priors disagree about the size of the cells themselves. This is an important feature, although I shall defer its discussion until chapter 8.)

Although plenitude will be enough for most purposes, an even stronger principle is motivated by the picture I have been sketching.

Let P' be any subpartition of P and let E a function from P' into [0,1]. Then there is some proposition, p, such that for every coherent ur-prior Pr and every proposition x ∈ P', Pr(p | x) = E(x).

This principle allows us to generate vague propositions from evidential relationships to other vague propositions that have already been generated.^[99] In appendix 18.2 it is shown that both principles are consistent.

It is tempting to ask why one ought to have a particular credence, 0.798 say, in the proposition that Harry is bald given one knows the relevant facts about his head. This is simply not a question with a reasonable answer on this way of individuating propositions. Part of what it is to be the proposition that Harry is bald is to have the specific epistemic profile it in fact has, which includes this conceptual requirement. It is like asking, on the modal way of individuating propositions, why the proposition that Harry is bald corresponds to one set of worlds and not another.

However, the thought which I take it this question is trying to latch on to might be better expressed by the question: why does the sentence ‘Harry is bald' denote a proposition with one set of conceptual requirements rather than another? This is presumably a question of metasemantics and a question whose answer is vague. The appearance of precision is really an illusion. A similar problem arises for the modal way of individuating propositions. One might ask why the proposition that Harry is bald contains worlds where Harry has 784 hairs but not worlds where he has 785 hairs.

Another issue that might initially seem problematic for the Principle of Plenitude, is the existence of higher-order vagueness. Given that it is vague which propositions are precise, it's going to be vague which partition of logical space into non-overlapping cells corresponds to the partition of logical space into maximally strong consistent precise propositions.^[100] Note, however, that this is not inconsistent with the Principle of Plenitude as stated, or even the claim that the Principle of Plenitude is determinate. The determinacy of plenitude guarantees that determinately, for any evidential role on the partition of maximally strong consistent precise propositions, there is some proposition satisfying that role. There is vagueness concerning which partition to use, and thus vagueness concerning which functions are evidential roles. But so long as whatever the partition and evidential roles are, there are vague propositions and ur-priors which accord with that role conditional on each member of the partition, the Principle of Plenitude will come out determinate. What one shouldn't expect, however, is that a proposition have its role determinately, or that it be completely determinate which class of probability functions are the rational ur-priors. A propos­ition p might have probability 2 conditional on each cell, but there might be things that are not determinately not cells (i.e. not determinately not members of the partition of maximally strong consistent precise propositions) such that p does not have probability 2 conditional on these ‘would be' cells: p has the role of being 2 on each cell, but not determinately so.

Having developed a plenitudinous theory of vague propositions, we are now in a position to see how to give a theory of inexact evidence based on vague propositions. Recall the example of the tree seen from a distance. Assuming that my credences are initially uniform over the possible heights less than 1000cm, it seemed in that case that after the evidence is in my credences ought to conform to the curve displayed in Figure 6.4. What proposition would have the effect that updating this way would cause this change in credence? Here the Principle of Plenitude kicks in. According to the principle there will be a proposition that has the desired effect on my credences. The curve depicted can be represented by a function, E, mapping each possible exact height of the tree to a number. Suppose for simplicity we identify the cells of our partition with the partition of propositions saying that the tree is exactly «cm tall (this simplification is harmless), then it follows that E is effectively an evidential role of the kind we were looking for: something which maps each member of this partition to a number between 0 and 1. The Principle of Plenitude entails there is a proposition, p, with evidential role E. In particular, if Cr is a coherent prior and e is the agent's total evidence, so that Cr(∙ | e) is my credence before the observing and is thus uniform over the partition p_n for n ≤ 1000, then conditioning on p will result in a curve like that in Figure 6.3: Cr(p_n | e ∧ p) = E(p_n).

6.4