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 D^{n} for D...D with n iterations of D. Let D* be the super-definitely operator: D*p if and only if Dp and D^{2}p and D^{3}p and ....
According to Williamson, in his Vagueness book:
- If D*p, then DD*p, and indeed D*D*p.
- If Dp_{1}&Dp_{2}&..., then D(p_{1}&p_{2}&...).
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.
- Ds_{1} but not D^{2}s_{1}
- D^{2}s_{2} but not D^{3}s_{2}
- D^{3}s_{3} but not D^{4}s_{3}
- ...
- D*s_{0}
Let p be the disjunction s_{1} or s_{2} or .... Assume the very plausible axiom:
- If p is a disjunction, finite or infinite, that has a disjunct q such that D^{n}q, then D^{n}p.
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, s_{0} is maximally definite, but p has less higher level definiteness than s_{0}.
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 D^{n}p) 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.
2 comments:
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.
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.)
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