Monday, June 25, 2012

In-the-limit vagueness

I don't know much about vagueness, so I suspect that this is much more learnedly discussed in the literature. But I am in the midst of unpacking, so I don't have time to look things up.

Let D be the definitely operator. Write Dn for D...D with n iterations of D. Let D* be the super-definitely operator: D*p if and only if Dp and D2p and D3p and ....

According to Williamson, in his Vagueness book:

  1. If D*p, then DD*p, and indeed D*D*p.
Williamson doesn't give the argument, but here's one. Suppose:
  1. If Dp1&Dp2&..., then D(p1&p2&...).
Then (1) follows quite easily. For let pn be Dnp. Then if D*p, we have Dpn for all n since Dpn is just Dn+1p, and by (2) it follows that we definitely have the conjunction of the pn. But the conjunction of the pn is equivalent to D*p. So we have DD*p. Iterating this argument we get DnD*p for all n and hence D*D*p.

The point behind the super-definitely operator is to capture the idea of maximal definiteness. But it doesn't. For given arbitrarily high levels of vagueness one can have a situation structurally similar to the following:

  • People who are 5 1/4 feet tall are short but not definitely short.
  • People who are 5 1/8 feet tall are definitely short but not definitely definitely short.
  • People who are 5 1/16 feet tall are definitely definitely short but not definitely definitely definitely short.
  • ...
  • People who are 5 feet tall are super-definitely short.
Now consider an infinite sequence of people, x1, x2, x3, ..., such that xn has height 5+1/22+n. Let sn be the proposition that xn is short. Suppose x0 is exactly five feet tall and let s0 be the proposition that x0 is short. Then by the above we have:
  • Ds1 but not D2s1
  • D2s2 but not D3s2
  • D3s3 but not D4s3
  • ...
  • D*s0

Let p be the disjunction s1 or s2 or .... Assume the very plausible axiom:

  1. If p is a disjunction, finite or infinite, that has a disjunct q such that Dnq, then Dnp.
Fix any n. Then Dnp. For one of the disjuncts of p is sn and Dnsn by the above. Hence D*p by definition of D*.

If D* captured the idea of maximal definiteness, then p would have to be maximally definite. But I don't think p is maximally definite. Each of the disjuncts in p has some higher level vagueness, and this vagueness does not disappear in the disjunction (in the way it perhaps does in "Sam is bald or not bald"). Intuitively, s0 is maximally definite, but p has less higher level definiteness than s0.

We might say that p suffers from in-the-limit higher level vagueness.

I am also not sure I want to say that DD*p. I grant that each conjunct of D*p (the n conjunct being Dnp) holds definitely. But I am not happy with saying that the whole infinite conjunction holds definitely. Thus I wonder if (2) shouldn't be rejected.


Heath White said...

Wow, that is good!

I think you reach the right interpretation, that D*p as defined does not capture the idea of maximal definiteness (non-vagueness).

I can't see the problem with 2, though.

It seems to me to follow that non-vague predicates cannot be defined in terms of vague predicates and a D operator. That is important.

Alexander R Pruss said...

Your conclusion is really interesting.

(By the way, in my possible worlds book I use the D* type of construction to generate an S4-satisfying necessity operator from a non-S4-satisfying necessity operator. I think a similar argument to the one in this post shows that the resulting operator probably isn't THE absolute necessity operator.)