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:
- If D*p, then DD*p, and indeed D*D*p.
- If Dp1&Dp2&..., then D(p1&p2&...).
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.
- Ds1 but not D2s1
- D2s2 but not D3s2
- D3s3 but not D4s3
Let p be the disjunction s1 or s2 or .... Assume the very plausible axiom:
- If p is a disjunction, finite or infinite, that has a disjunct q such that Dnq, then Dnp.
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.