Grammar
In chapter 1, we drew a distinction between people based on how much money they had. Not the distinction between being rich and not rich, but a slightly more demanding one—that of being clearly rich as opposed to not clearly rich.
The most natural way to express this distinction in English is to prefix, as I just have done, an adverb such as ‘clearly’ or ‘definitely’ to a verb or verb-phrase, as in ‘clearly rich’, ‘clearly richer than’, and so on. These expressions fall under the same grammatical class as many other words of interest to logicians such as ‘not’ and ‘possibly’; compare ‘Harry is clearly bald’ to ‘Harry is not bald’, and ‘Harry is possibly bald’. These adverbial expressions play an important role in the way that many philosophers informally talk about vagueness.It is common among logicians to regiment adverbial modification in formal contexts using operators—things which result in a sentence when prefixed to a sentence— so that ‘Harry is not bald’ becomes the slightly more awkward ‘it’s not the case that Harry is bald’ and ‘Harry is possibly bald’ becomes ‘it’s possible that Harry is bald’. For our purposes this regimentation is innocuous and I shall frequently adopt it in what follows. The operator locutions can further be divided according to whether they take the complementizer ‘that’ (‘it’s possible that’) or ‘whether’ (‘it’s contingent whether’). The most common ways of drawing the distinction I have just alluded to all use expressions that fall into one of these classes.
Adverbs: Harry is definitely/clearly/determinately/as a matter of fact/borderline/indeter- minately tall.
‘that’ operators: It’s clear/determinate/definite/a fact that Harry is tall.
size=1 color=black face="Times New Roman">‘wh’ operators: It’s borderline/it’s indeterminate/it’s vague/there’s no fact of the matter/it’s unclear [whether/when/how Harry became tall]/[who/where/what the size of/which the oldest tall person is].
A common way of representing these operators in logical notation is to use ∆p for the operators expressed in the second class (e.g.
‘it’s determinate thatp,) and Vp for the operators in the third class (e.g. ‘it’s indeterminate whether p’). It is typically assumed that the latter can be defined in terms of the former by writing —∆p Λ — Δ—p: it is indeterminate whether p iff neither p nor its negation is determinate. Given a similar assumption, the former can be defined from the latter by p Λ —Vp: it is determinate that p iff p is true and isn't indeterminate. I shall adopt this notation and the two assumptions throughout the book.One notable thing about expressions in these classes that distinguish them from, say, verbs is that adverbs and operators can iterate. One can combine different adverbs with other adverbs, such as with ‘clearly not bald' and ‘not clearly bald', as well as iterating them, such as in ‘not not bald’ and ‘clearly clearly bald’. Analogous remarks apply to operators.
This way of speaking does not obviously relate to the metalinguistic attributions of vagueness that the linguistic theorist is prone to make. If I say that it is not the case that Harry is bald, for example, it is not at all obvious I have said anything about the English sentence ‘Harry is bald’ More generally, attaching an operator to a sentence does not involve saying something about the sentence that it precedes; it would be analogous to suggesting that appending ‘runs’ to ‘Alice’ results in a sentence that says something about the name ‘Alice’—the resulting sentence is not about names at all, it is about the person Alice.[41] This suggests, in particular, that when I say ‘it is clear that Harry is bald’ I have not said anything about the English sentence ‘Harry is bald’. I have said something about Harry and the status of his hairline; this way of speaking seems, therefore, to be fundamentally unsuited to a linguistic theory of vagueness.
If we want to talk about the sentence rather than whatever that sentence is about it is useful to have some canonical way of referring to that sentence; the use of quotation marks serves this purpose well.
A linguistic theorist must reject the thought that the adverbial and operator expressions listed above play a central role in the theory of vagueness; she might therefore prefer to theorize with a predicate—something which attaches to a name to form a sentence such as ‘S is definitely true’ (see McGee’s [102]; many other linguistic theorists use this locution as well). She may then combine this with the relevant quotation names of sentences to draw distinctions that at least seem to correspond to the distinction we introduced earlier, thus saying, for example, ‘the sentence “Patrick Stewart is bald” is definitely true’ instead of ‘Patrick Stewart is definitely bald’.One thing to note about these particular locutions is that they do not completely do away with the adverbial notion of clarity. In the sentence ‘S is clearly true’, ‘clearly’ appears as an adverb and the verb it is modifying is a linguistic truth predicate. We can regiment this in terms of operators with the sentence ‘it’s clear that S is true’; as we mentioned already, this does not appear to be about the sentence ‘S is true’ but what that sentence itself is about. In this case the sentence happens to be yet another sentence, S, but in general uses of the adverb ‘clearly’ need not be so. Thus, while the distinction between clear and unclear truth is perfectly acceptable to the adverbialist, it is not obvious whether linguistic theorists ought to be using the locutions ‘definite truth’ and ‘clear truth’ to state their theories; if anything is linguistic in this example it is the truth predicate and not the adverb ‘clearly’ which is modifying it. Better, then, that the linguistic theorist adopt some primitive simple predicate, ‘is vague’ or ‘is definite’, to do the work that she wants; it will be this linguistic notion of vagueness that this theorist will be attempting to analyse.
Another structural difference between the preferred terminology of the linguistic theorist and that of the adverbialist comes out with the point about iteration.
When one applies an operator to a sentence one gets a sentence back, so that an operator can be grammatically applied again and again. By contrast, predicates take something of one grammatical type as input—a singular term—and output something of a different type—a sentence. Predicates thus cannot be iterated—one cannot, for example, say ‘Harry is tall tall’—so we must find some other way to paraphrase apparently coherent talk involving iterations of the above adverbs and operators. While one can say, for example, that the sentence S is vaguely vague, the word ‘vaguely’ here is an adverb which modifies the predicate ‘vague’ but does not predicate anything of a sentence.[42] The linguistic theorist might capture something like the original thought by instead saying S' is vague, where S' is a name for the sentence ‘S is vague’ (for example, S' could be the quotation name “S is vague”). Something similar to adverbial iteration is achieved by ascribing vagueness to a sentence, ‘S is vague’, which in turn ascribes vagueness to another S.4.2