The Vagueness of the Existential Quantifier

I’m in the middle of assembling a two-piece table. The top and the base didn’t compose anything when I started, and I’m now at a point at which they’re a borderline case of composing something. Consequently, it’s indeterminate whether there is anything in addition to the top and the base. Consequently, it’s indeterminate what there is. Q.E.D..

A lot of people think it can’t be indeterminate what there is. (What part of ‘Q.E.D.’ don’t they understand??) Ted Sider argues that, if it were indeterminate what there is, then some numerical sentence -- like ‘∃x∃y(Cx & Cy & x≠y & ∀z(Cz → (x=z v y=z)))’, which says that there are exactly two concrete objects -- would have to contain some vague expression. But numerical sentences don’t have any vague expressions. 

Here’s the argument that ‘∃’ isn’t vague (see p.128 in Sider’s book; he focuses on the universal quantifier).

    (S1) In order for ‘∃’ to be vague, ‘∃’ has to have multiple precisifications
    (S2) In order for ‘∃’ to have multiple precisifications, ‘∃’ has to have multiple candidate extensions.
    (S3) ‘∃’ doesn’t have multiple candidate extensions
    (C) So, ‘∃’  isn’t vague

Let’s grant S1 and S2 (though I’m happy to hear worries about these). Sider’s argument for S3 runs as follows: In order for there to be multiple candidate extensions for ‘∃’, there’d have to be something, x, that’s in one but not the other. But whichever extension doesn’t contain x isn’t even a candidate for being an extension of ‘∃’, since a constraint on any candidate precisification of ‘∃’ is that it ranges over everything.

Here’s a way of resisting the argument, which occurred to me after re-reading Katherine Hawley’s very nice paper “Vagueness and Existence”. (It’s not entirely clear to me whether this strategy for blocking the argument is just a straightforward application of what she already says in §5, or whether the strategy requires some substantial further assumptions.) Sider’s argument for S3 works just fine if we suppose that the candidate semantic value of ‘∃’ has to be a first-order property. But suppose (as some think, right?) that ‘∃’ expresses a second-order property. In that case, the candidate extensions of ‘∃’ are going to be sets of properties. Indeed, they'll be the same sets of properties that are candidates for being the extension of ‘has at least one instance’. Because the top and the base are a borderline case of composing something, the property being composed of the top and the base will be a borderline case of having an instance. (NB. It does *not* follow that there is an object that's a borderline case of being an instance of this property.)

We can now specify the different candidate extensions of ‘∃’: one includes the property of being composed of the top and the base, and the other doesn’t. And it certainly doesn’t follow from the fact that the latter extension doesn’t include this property that the corresponding precisification doesn’t range over everything. That’s because, on this understanding of the semantic value of ‘∃’, the extension of ‘∃’ doesn't consist of the things ‘∃’ ranges over, but rather the properties that have instances. So the argument fails if we take quantifiers to express second-order properties.

So ‘∃’ does have different precisifications on this picture., which differ with respect to whether they map the property of being composed of the top and the base onto T or F. A further question (which I’m having trouble thinking clearly about) is whether this opens the door to existential vagueness without ontic vagueness. After all, we can blame the vagueness of ‘∃’ on linguistic indecision with respect to the aforementioned candidate precisifications of ‘∃’. But perhaps ontic vagueness sneaks in in some other way.

Comments

[Bryan Pickel]

Hi Dan,

I haven't read the paper, I think I agree with you that Sider's argument can be avoided if one "intensionalizes" the semantic value of the quantifiers. Nonetheless, I don't think that this sort of strategy should be tied to the success of a Frege-style 2nd-level property account of quantification.

2nd level property accounts of the existential quantifier have trouble delivering accounts of iterated quantification as occurs in 'ExEyRxy'. It looks like this sentence is processed by feeding the semantic value of 'Rxy' into 'Ey' and then feeding the result into 'Ex'. The trouble is that semantic value of 'Rxy' is either interpreted as being of sentence type, in which case the semantic value of the quantifier is going to be something which takes things of sentence type and yields another sentence, so not a second level property. Or, it is going to be a 2-place relation.

In the latter case, 'Ey' can't be interpreted as designating a property of a monadic property. But then we'd have to go in for type shifting of the existential quantifier, since the semantic value of 'Ex' is going to have to accept the semantic value of 'EyFxy' as its argument. So, the trouble is your quantifiers will have to type shift a whole lot. Type shifting can be avoided if one complicates the syntax so that there's a level of semantic processing between 'Fxy' which is of sentence type and 'Ey'. This is somewhat standard in the semantics of natural language. The trouble is that these kinds of accounts still tend to require a domain semantics, which is, I take it, what you're trying to get away from.

Anyhow, even if one takes the quantifiers as sentential operators there's nothing in itself wrong with making them intensional, even if one interprets them a sentential operators rather than properties of properties.

[Dan Korman]

Thanks Bryan, this is helpful. Is this trouble with iterated quantification a well-known problem for Fregean accounts?

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