Substitutional Quantification and Supervaluations

(Cross-posted with Matters of Subtsance.)

Let U be the (universal) substitutional quantifier: its truth-conditions are

"UxF(x)" is true iff, for every name n, "F(n)" is true.

(Normal quotes are doing double-duty as quasi-quotes here.)

Peter van Inwagen has an argument that we can't understand substitutional quantification. It goes like this:

(1) We can't understand a sentence unless we can specify what proposition it expresses.
(2) The only proposition we know of with the right truth-conditions to be expressed by "UxF(x)" is the proposition that, for every name n, "F(n)" is true. (Call this proposition "UU".)
(3) Friends of substitutional quantification say that UU is not what is expressed by "UxF(x)".
(4) There are no other candidates to be the proposition expressed by "UxF(x)".
(5) So if friends of substitutional quantification are right, we can't understand "UxF(x)".

I want to respond to this argument, but I don't know whether my response rejects premise (1) or (4). So I'll outline the basic idea, and then maybe someone can help me know which premise I'm rejecting.

Suppose some sort of supervaluationism is the right treatment of vagueness, and set aside higher order vagueness. Then a sentence like "Fido is red" doesn't express a proposition simpliciter; rather, it expresses a proposition relative to every precisification of "red".

(Since we can understand "Fido is red", this alone might be enough to lead us to deny (1). But it's not clear how this denial gives us any positive reason to think we should be able to understand substitutional quantification. I want to aim higher. So let's press on.)

The truth-conditions for this sentence with the determinacy operator are:

"Det(Fido is red)" is true iff "Fido is red" is true on every precisification of "red".

Now, we can think about precisifications in a number of ways. One of them is an explicitly semantic way: the precisifications of a term are the precise meanings it can have. But another is a bit more syntactic, relating more precise terms to less. If we have semantic precisifications, we can easily define syntactic ones as follows: T is a syntactic precisification of T* iff T's semantic value is a semantic precisification of T*. If we don't have semantic precisifications, we might take the syntactic ones as primitive, or we might be able to define them some other way (maybe by appealing to metalinguistic predicates like "admits of borderline cases" and some others).

If we have the syntactic understanding of precisification, then we have the truth-conditions

"Det(Fido is red)" is true iff "Fido is R" is true for every term R that is a precisification of "red",

which look remarkably similar to the ones we had for the substitutional quantifier.

So here's my basic idea: think of "x" as a maximally vague name --- a name such that every precise name is a (syntactic) precisification of it. Then think of "U" as a determinacy operator. This gives us essentially the truth-conditions we want.

How does van Inwagen's argument look now, with this understanding of the substitutional quantifiers? That depends, I think, on what we say about the proposition expressed by "Det(Fido is red)". I think there are very good reasons to think that this sentence does not express the proposition that "Fido is R" is true for every term R that is a precisification of red. (One very good reason is that it won't embed right at all --- it might be necessary, say, that Det(Fido is red), even though it certainly isn't necessary that "red" is even a word, much less that it has precisifications. And these thoughts extend to the truth-conditions that go via semantic precisifications, too.) But are we in any position at all to specify a proposition it expresses?

Here I don't know what to say, and this is why I don't know which premise I reject in van Inwagen's argument. On the one hand, maybe we have some recipe for specifying a proposition expressed by "Det(Fido is red)". If so, then we can use the same recipe to specify one expressed by "UxF(x)", and I deny premise (4). Maybe we think "Det(Fido is red)" expresses the conjunction of all the propositions expressed by "Fido is R", where R is a (syntactic) precisification of "red", for instance. If so, then we can say that "UxF(x)" expresses the conjunction of all propositions expressed by sentences of the form "F(a)" for some name "a".

On the other hand, maybe we can't specify any proposition expressed by "Det(Fido is red)". (Maybe we dislike the conjunction proposal for both the "Det" and "U" cases because we think it misses out on the "that's-all"-ish nature of the quantifications involved in the truth-conditions.) Nonetheless, I think it's entirely clear that we understand "Det(Fido is red)". And I also think (but I haven't argued for it) that one way we can come to understand a vague term by learning a recipe for figuring out what its precisifications are, so we can understand what the "x" in "UxF(x)" is doing. But in this case, "UxF(x)" is essentially just "Det F(x)"

There's a lot of details I've left out --- stuff about variable-binding, the viability of the syntactic characterization of precisifications, how to think of modally embedded substitutional quantifications, and so on. But setting these techy details aside, I'm wondering what the right thing to say about the argument is. Or, more to the point, I'm wondering what we should deny when we run a parody argument for our inability to understand the sentence "Det(Fido is red)".

Thoughts, anyone?