Restricted Precisification

Something I've wondered for a while is whether supervaluationists are right to think of precisifications as ways of making *every* term in our language precise. Instead, mightn't it be helpful (at least sometimes) to talk about precisifications-for-a-term, i.e. ways of making *that term* precise? Here are two prima facie reasons to think so.

1. Robbie Williams shows in a forthcoming paper that on the usual supervaluationist story there is a problematic interaction between two kinds of vagueness: how-tall-must-it-be-to-be-a-mountain vagueness and problem-of-the-many vagueness. Suppose we resolve the problem of the many by saying that on each precisification just one of the Kilimanjaro candidates is a mountain. Then there is nothing which is a mountain on every precisification. But that undermines something supervaluationists (and the rest of us) typically want to say about how-tall vagueness, namely that some things are definitely mountains.
We could deal with this if we said that there are some things (Kilimanjaro-candidates) which are definitely mountains where what this means is that they are mountains on every acceptable way of making precise how tall something has to be to be a mountain, but which are not definitely mountains where what this means is that they are mountains on every way of specifying precisely which of the Kilimanjaro-candidates counts as a mountain.

2. Precisifying everything at once makes it always false to say of a vague term that it is definitely vague. (No term is vague on every complete precisification, because no term is vague on *any* complete precisification.) But we can truly assert that a term is definitely vague if what it means is that *on all precifications of 'vague'* the term in question counts as vague.

If precisification worked this way, 'Definitely ...' would function differently on different occasions of use. It would always have the same truth-conditions as 'On all precisifications of the salient term(s) ...', but which term(s) are salient will change. Presumably context would help us determine which term(s) are salient on any particular occasion.