Quantifying In

Another technical issue that must be surmounted by a linguistic theorist is related to what Quine calls the third grade of modal involvement: how a linguistic theorist should say what we would normally say by quantifying into the scope of an operator or adverb.

∃xDef ‘Fx’

Supposing here that ‘Def’ represents our linguistic predicate ascribing definiteness to sentences, then this paraphrase suffers from two problems. Firstly, ‘Fx’ is not a sentence; it is an open formula and our primitive was introduced so as to apply truthfully only to sentences. Secondly, even if a sensible notion of definiteness could be introduced for open formulae, the above quantification is vacuous, since x does not appear free in ⁱDefⁱFx^, ’; it only appears free in the sentence that is mentioned by that sentence.

size=1 color=black face=Cambria>Before one dismisses this as a technical side issue, note that one of the crucial concepts in the study of vagueness is the notion of a predicate being vague or having borderline cases. Adverbialists often introduce a parallel notion for properties: a prop­erty, F, has borderline cases if there is something which is borderline F. Similarly, a property is said to be vague if it’s possible that there is something which is borderline F. This involves exactly the kind of quantification into the scope of an operator that is problematic for the linguistic theorist to emulate.^[46]

A natural way to make sense of the above formula is to invoke a substitutional understanding of the quantifiers.

Provided we restrict ourselves to proper, and hence rigid, names the substitutional analysis is not subject to straightforward counterexamples involving modal contexts. If we were to extend that proposal to quantification into contexts involving definite­ness, however, the results would prove much less satisfactory. Recall that a property, F, has borderline cases iff for some x it’s borderline whether x is F. The linguistic version of this analysis, understood in terms of substitutional quantification, would count the predicate ‘is 29,000ft’ as having borderline cases. Since the sentence ‘Mt Everest is 29,000ft is a borderline sentence, it follows that there’s a proper (rigid) name, a, for which the sentence ‘a is 29,000ft is a borderline sentence (namely a = ‘Mt Everest’). But this seems wrong: ‘is 29,000ft is as good a candidate as any for being a precise predicate.

Indeed, sceptics of modal logic, such as Kneale and Quine, took similar positions to suggest that modal properties only belong to individuals relative to some way of describing them.^{[47] [48]} The analogous move here would be to say that things cannot be borderline or determinately tall in themselves. Rather, a given person, Alice, is borderline tall relative to some names, and determinately tall relative to others.

A much less revisionary move would be to introduce further ideology. Rather than theorizing with a sentential notion of vagueness, one could start instead with a notion that relates an object to a predicate when it is a borderline case of that predicate relative to the relevant parameters: ‘x is a borderline case of predicate F at parameters p^,. Note that this move does not allow us to simply dispense with the old sentential notion of borderlineness. We must have that too, or we would not be able to draw the sentential distinction any more. We cannot straightforwardly define sentential borderlineness in terms of predicate borderlineness: ‘Mt Everest is 29,000ft' is borderline, but we cannot state this in terms of the borderline cases of the predicate ‘is 29,000ft', since there are none. If we are to go this route, then, we will need separate accounts of both predicate and sentential borderlineness.¹¹

Not only that, the above primitive is only good for monadic predicates. We will need a four-place relation to make sense of cases where it's borderline whether two objects stand in a relation: ‘x and y, in that order, are borderline cases of the relation R relative to parameters p’. And we will need a five-place relation to ascribe borderline cases to ternary relations, and so on. (Of course, rather than multiply primitives, one could talk about monadic predicates of tuples.)

Most of the issues raised above arise for plural predicates and plural quantification. An adverbialist can say things like the following: it's borderline whether Tom, Dick, and Harry have enough hair between them to make a hairball, so there are some people such that it's borderline whether they have enough hair between them to make a hairball. A similar theory would have to be constructed for the linguistic theorist to make sense of this sentence.

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