## The Vagueness of “Vague”

May 23, 2008

Back over here I was trying to argue that the determinately operator, $\Delta$, could not be a logical constant because it is vague. In the course of my pitch for this I had to appeal to an argument due to Sorensen for the vagueness of “vague”. Define:

• x is k-small iff x < k or x is small

Since being 0-small is equivalent to being small, 0-small is vague. Since n := Rayo’s number isn’t small, being n-small is equivalent to being less than n, which is precise, so n-small isn’t vague. Since the predicates ‘k-small’ for k < n form a sorites of “is vague”, “is vague” must be vague.

In the comments Robbie pressed me on this last step. It seems the existence of a Sorites isn’t always sufficient for a predicate to be vague. Especially when the terms in the sorites sequence are themselves vague (as in this case.) Anyway, I thought I’d repost some of the discussion here for a wider audience (and also give me a chance to get clearer on what’s going on.) Robbies example was as follows:

“Suppose we have a paradigm electron, Sparky, and a paradigm non-electron, Robbie. Let “item number n” be introduced as the x satisfying the following reference-fixing description: either x is Sparky, and n is small; or x is Robbie, and n is not small. Now the first few terms (where x is determinately small) will determinately refer to Sparky, and the last few (where x is determinately not small) will determinately refer to Robbie. But “item number n” for borderline-small n will be indeterminate in reference between the two.

Now consider the collection of claims: (0) “item 0 is an electron”; (1) “item 1 is an electron”…. (n) “item n is an electron”. This’ll display the characteristics of a forced-march sorites, I guess, and we could turn it into a sorites paradox without too much trouble. Does it show “electron” is vague? Not at all—intuitively the reason we get a sorites is because of the vagueness in the referential terms.”

This left me a bit worried. Not just because I couldn’t establish the conclusion I wanted, but because it seems we don’t have any sufficient condition under which we can conclude that any expression is vague.

That said, I don’t think Sorites are completely useless in testing whether predicates are vague. Since the counterexamples only work when the terms in the Sorites sequence are vague, the existence of a Sorites for F should show at least the following disjunctive claim:

• Either (i) the terms in the Sorites are vague, or (ii) F is vague.

And I think we can get a more useful version for testing for vagueness from this, namely:

• If there is a Sorites for F, and all the terms in the Sorites sequence are precise, the F is vague.

With that in place, I was wondering if I could run a better argument for the vagueness of “vague”.

Let “item number k” refer (determinately) to the predicate “is k-small”. Now we can run a Sorites as follows: (i) item number 0 is vague (ii) if item number k is vague, so is item number k+1, (iii) item number n isn’t vague, where n := Rayo’s number.

The idea is that we have reformulated Sorensens argument so that the terms of the Sorites sequence are completely precise. With our new principle for testing vagueness in place, this should be sufficient to establish its vagueness.

Two things to note. Nothing depends on whether you think it is predicates like “is small” that are vague, or the semantic values of these things that are vague. On the first view a predicates reference is indeterminate between many precise extensions, on the second predicates have exactly one reference, a vague semantic value (a function from precisifications to extensions.) The argument could be run either way as far as I can see.

Secondly, if we can succeed in determinately referring to things at all, the names “item number k”, for each k, must be among those cases. Now I don’t have much of an argument for this yet, but I just can’t see how indeterminacy could enter into the picture there at all. We are talking about abstract entities (strings of letters, or semantic values dependingly) and it seems to me that if we can refer to these kinds of things at all we can refer to them precisely.

Anyway, that’s that – I’m still not 100% sure this will do the job. One worry is that if we go for the second option above (it’s semantic values, not syntactic entities, that are vague) then it isn’t completely clear that determinate reference has been achieved. For suppose the semantic values of predicates are functions from admissible precisifications to extensions. Then, in the presence of higher order vagueness, it will be vague which precisifications are admissible – and thus it will be vague what the domain of these functions are. In other words, it might still be vague what the semantic value of “is k-small” is, and my “item number k” trick won’t do the job. (It’s interesting, I think, that this problem doesn’t arise for the view that it is predicates that are vague.)