## Thursday, October 27, 2011

### First- but not second-order vagueness

Consider a view on which there is first-order vagueness but no higher-order vagueness. Thus, it can be vague that Smith is bald, but it can't be vague that it's vague that Smith is bald. Smith is always definitely definitely bald, or definitely vaguely bald, or definitely definitely non-bald. In respect of higher-order vagueness, we just go epistemicist. Call this the "intermediate view".

Why would one do that? Well, there are two main alternatives. One is vagueness all the way up. The other is sharpness all the way down.

Sharpness all the way down—i.e., epistemicism—has trouble with ordinary language intuitions such as that no one becomes bald by the loss of one normal[note 1] hair. The intermediate view says that here we distinguish: you can become vaguely bald by the loss of one normal hair, and you can become definitely bald by the loss of one normal hair, but you can't move from definitely non-bald to definitely bald by the loss of one normal hair. This may not be exactly what the ordinary language intuition holds, but it arguably does it all the justice that it needs to have done. Moreover, it is sometimes just obvious that someone is neither definitely bald nor definitely non-bald. Sharpness all the way down also has difficulties with incompletely introduced terms, such as when I tell you that every square is a squibble and only quadrilaterals are squibbles, but say and think nothing else. Is a non-square rectangle a squibble? Surely the answer is that the sentence, given how "squibble" was introduced, lacks the precision for the answer. But if we have sharpness all the way down, there should be an answer.

Vagueness all the way up also does justice to the intuition that you can't become bald by losing one normal hair, and does so in exactly the same way as the intermediate view does. It also has the advantage of doing justice to the further intuition that one doesn't become vaguely bald by the loss of one hair. But the further intuition is an intuition about much less common language, quite possibly about technical language (while "vague" is an ordinary word, it's not clear that philosophers use it in the technical sense), and the cost of giving it up is much less. The major advantage of the intermediate view over the vagueness all the way up view is that the intermediate view allows an analysis of vague language in a logically classical metalanguage. This is a way of holding on to the intuition that there is something importantly right about classical logic. The intermediate view opens up theoretical possibilities closed by the vagueness all the way up view—it seems to me to be not uncommon to observe about some view of vagueness that "it has trouble with higher-order vagueness". But if there is no higher-order vagueness, then that's not a problem!

The intermediate view allows one to say that all vague predicates are basically like "is a squibble". If we take the intermediate view in this direction, vagueness is simply due to the fact that there is a gap between the precise cases in which the predicate has been specified to hold and the precise cases in which the predicate has been specified not to hold.

Moreover, the intermediate view allows me to make sense of the intuition which I have—but which many do not share—that the world is fundamentally non-vague, that God creates and knows the world with perfect precision. For any vague predicate P can be replaced with a trio of non-vague predicates that carve up the world more precisely—definitely P, vaguely P and definitely not P—and which of the three predicates applies to x fully determines what we should say in regard to x and P. Moreover, I can further suppose that these three predicates are more fundamental than P whenever P is subject to vagueness.

There is a theistic argument for the view here.

Finally, the intermediate view allows a refinement to my theistic story about vagueness. On that theistic story, all predicates are fully sharp, because God has given us language, either by giving us the predicates directly or by giving certain predicate-production rules that result in sharp predicates. There is nothing particularly mysterious on this view about the source of sharpness or about the appearance of vagueness—one can inherit sharp terminology from someone else and the appearance of vagueness is explained by the fact that God didn't tell us what all the sharp boundaries are (why should he? it would be a massive waste of our time to keep track of them).

But that view has two difficulties. The first is that it doesn't at all do justice to the intuition that one doesn't become bald by the loss of one normal hair. The second is that it seems that we can introduce underdetermined terms like "squibble", and it is odd to suppose that God steps in, or has stepped in when setting up the general predicate-production rules, to fill in the gaps in our linguistic stipulations, even when we intended them not to be filled in, as in the squibble case.

The intermediate view helps with the one-hair problem, as already noted, and solves the second. For instead of supposing that the general predicate-production rules that God has enacted for us (in whatever way that happens, whether by enacting the conventions that underlie them or by making the rules implicit in the teleology of our nature—I like the latter version) always yield sharp predicates, we may suppose that they always sharply yield predicates, some of which are vague, but vague in the sharp way that the intermediate view recognizes.

There is, however, another problem with the intermediate view. It seems that by stipulation we can raise the level of vagueness. For instance, suppose I stipulate a predicate P by stipulating what it definitely applies to and what it definitely does not apply to. And suppose that I make use of vague terms in stipulating these. For instance, let's say I stipulate "acceptable employee" as follows. Someone is definitely an acceptable employee if and only if she does her tasks well enough and does not harm the company. Someone is definitely not an acceptable employee if and only if she harms the company. If I can stipulate "squibble" as I did, I should be able to stipulate "acceptable employee" as I did. But my stipulation used vague terms. Suppose Patrick definitely does not harm the company but only vaguely does his tasks well enough. Then Patrick is vaguely definitely an acceptable employee.

But this problem is easily fixed by replacing the intermediate view with a "bounded vagueness" view on which there is a number n such that every predicate has no vagueness above the nth level. The number n may be very large for ordinary languages like English. And since ordinary languages keep on evolving and adding predicates, the number may be continually increasing. But the point is that it's always finite. And as long as it's always finite, we maintain most of the advantages of the intermediate view, while avoiding the above disadvantage. In particular, we can work with a classical metalanguage. And, as a bonus, we can now do some justice to the intuition that one doesn't become definitely bald by the loss of a hair, though there will be some iterated intuition this won't work for—but that iterated intuition will be much less plausible, I think.

We do lose this argument, but I think we can get around the problem there in another way (e.g., by saying vagueness stays at the level of sentences, not the level of propositions, or by saying that the problem is solved by God having the precisified beliefs—perhaps involving a long iterated list of "definitely" and "vaguely" operators—that ground the vague stuff).

Alrenous said...

I learned I need to consider second-order vagueness...or rather, the lack thereof.

I believe in absolute sharpness, just due to the law of identity. Fundamentally, particles must be exactly what they are, and a balding man has exactly as many hairs as he has, made of exactly as many particles as make them up.

Moreover, even linguistic vagueness falls to the no infinities principle.
A vaguely bald person is either probably bald - say 70% - or partly bald. If it has meta-vagueness, then again it's either 70% probably or 70% true. If that goes on forever, it multiplies out to 0% overall. Not even slightly vague. It has to stop or very quickly approach arbitrarily close to stopping.

I think you're working harder here than you have to, though. Sharp communication is expensive, and for baldness it isn't worth it. It's not that baldness per se is sharp or vague, just our words referring to it.

Alexander R Pruss said...

"Fundamentally, particles must be exactly what they are..."

This leaves open the possibility that there is non-fundamental vagueness.

"A vaguely bald person is either probably bald - say 70% - or partly bald. If it has meta-vagueness, then again it's either 70% probably or 70% true. If that goes on forever, it multiplies out to 0% overall."

Vagueness is not probability, so it might not multiply.

In any case, there is no guarantee that a sequence of numbers less than 1 multiplies to zero. For instance the product of the numbers in the following sequence is non-zero: 1, 1-1/2, 1-1/4, 1-1/8, ...

"It's not that baldness per se is sharp or vague, just our words referring to it."

That's a tempting move, but I am afraid it doesn't seem to cut it. Suppose that the word "bald" is vague. So it's vague whether "bald" applies to some individual x. But language is itself a part of the world. So the vagueness about whether "bald" applies to x seems to be a genuine vagueness about the world. You might say that this is just a matter of vagueness in the word "applies". But that just gives us another step in a regress: it is now vague whether "applies" applies to "bald" and x.

Alrenous said...

If higher order vagueness is between us and the fundamental particles, then their precise properties don't exist in any meaningful sense, as we cannot detect them. The higher-level vagueness takes the former ontological place of the fundamental properties.

"Vagueness is not probability, so it might not multiply."

It can either be characterized by a number, or it can't.

I'm open to it not so being, but I would need to know which alternative you propose.

"In any case, there is no guarantee that a sequence of numbers less than 1 multiplies to zero."

It very quickly approaches arbitrarily close to stopping.

At some near, finite point it's [1 - epsilon] and in practice you can ignore epsilon any time it is smaller than your instrument error, because by definition it is indistinguishable from a perfectly precise reality.

"But language is itself a part of the world."

I had not thought of it that way. Still, that works out to exactly my point.

You've limited vagueness to the realm of language. More precisely, there's vague relationships between two things. Neither of the things themselves are vague beyond first order.