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:

2. If Dp₁&Dp₂&..., then D(p₁&p₂&...).

Then (1) follows quite easily. For let p_n be Dⁿp. Then if D*p, we have Dp_n for all n since Dp_n is just Dⁿ⁺¹p, and by (2) it follows that we definitely have the conjunction of the p_n. But the conjunction of the p_n 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 x_n has height 5+1/2²⁺ⁿ. Let s_n be the proposition that x_n 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:

3. 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 s_n and Dⁿs_n 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.