Ineradicable?

It's a very weird---fuzzy, elusive thought that if vagueness is eliminable by the application of a (single?) operator, then vagueness is not a deep phenomenon. But there's definitely something to it. Heck and MacFarlane comment on the difficulty:

``Since both the semantic assumptions and the formal principle [S5 analogue] imply that vagueness is eradicable, the assumption of either begs the question against the Indefinitist [ie, the Metaphysicalist.]

...[But] the problem with these [foregoing] remarks is that the claim that vagueness is ineradicable, as it stands, is rather imprecise. The claim can not be that no operator can eliminate vagueness, as the trivial falsum operator would surely do that. The though, rather, is that, while it is almost essential to such views that there are operators which strengthen vague statements, which make them rather less vague (e.g. `Definitely') , there can be no such operator which eliminates vagueness." (Heck, 284)

The problem with the falsum operator is surely that it obliterates all distinctions in the language, making every sentence (and its negation) false!

For the vagueness of tall, there is presumably another description of any tall object which could completely eliminate the vagueness...but this is not an addition of an operator to the original sentence, but rather

a different sentence altogether.

...I'm not convinced that the higher-order sorites poses a serious worry even for standard degree theories. The predicate ``satisfies `tall' to degree 1" is sufficiently theoretical that it's not clear why we should accept a sorites premise formulated with it. Perhaps that is why the objection is usually run using a sentential operator D (for `Definitely'), stipulated to have the following semantics:

[[Ds]] = 1 if [[s]] = 1

We do have a strong inclination to accept a sorites premise for ``definitely tall." But it's not clear that the ordinary meaning of ``definitely" matches that of D as defined above. More plausibly, ``definitely p" means something like ``p is true enough, by a good margin, for present purposes," or ``p has degree 1 by a good margin," and on this understanding we should expect ``definitely p" to take non-extremal degrees, since ``enough" and ``good margin" are vague. If that is right, then a degree theory can say exactly the same thing about a sorites with ``definitely tall" as it says with a sorites about ``tall." (MacFarlane, 31-32)