Can One Explain Propositional Borderlineness in Terms of Sentential Borderlineness?

I have so far been characterizing the contrast between linguistic and non-linguistic theories of vagueness as being concerned with which terms will play a more basic role in the explanations of the vagueness-related phenomena: an adverbial or operator locution such as ‘it’s vague whether’ or the metalinguistic predicate ‘S is borderline in L relative to parameters...

As with many philosophical disputes in which different sides adopt different basic ideologies, it is often desirable to be able to explain the vocabulary of one theory in terms of the other. I have suggested that the notion of a sentence being borderline in language L relative to parameters p can be explained in terms of propositional bor­derlineness: it is just for the sentence in question to express a borderline proposition in L relative to parameters p. As I mentioned earlier, some linguistic theorists accept both the distinction between sentences and the distinction between propositions. It is crucial, then, that they be able to offer some kind of explanation of the latter notion in their preferred vocabulary.

But even if a linguistic theorist does not accept a non-trivial notion of propositional borderlineness, it is prudent for her to be interested in such paraphrases. Indeed, given all the technicalities to follow, it is tempting for the linguistic theorist to adopt the adverbialist way of talking for the sake of convenience, but to insist that when it gets down to it, this way of speaking is really an innocuous shorthand for something else.

The practice of using the operator formalism as shorthand for something linguistic, however, is not innocuous. Indeed, this practice can lead one to think that options open to the non-linguistic theory are also open to the linguistic theory when, in fact, they are not. We will encounter two specific examples of this, relating to the problem of quantifying in (in section 4.4) and Montague’s paradox in (section 4.6). However, it is instructive to first see how some natural concrete paraphrasing strategies deliver the wrong results.

Consider a typical example of a statement we might make using an operator:

(H) It’s borderline whether Harry is bald.

As a first stab: it is natural to paraphrase (H), by picking a language, L (along with other parameters), and sentence, S, such that the claim that S is borderline in L (at parameters p) is a reasonable paraphrase of the claim expressed by (H); one of the things this paraphrase must do is play the same role that (H) does within a theory of vagueness. An example of the paraphrase strategy, therefore, would be to choose English as our language, and the sentence ‘Harry is bald’ as our sentence. Our paraphrase of (H) is thus that the sentence ‘Harry is bald’ is used by English speakers in whatever way is required to make it a vague sentence of English.

This is effectively the strategy Quine adopts in [114] for paraphrasing modal sentences in which only closed formulae appear in the scope of intensional operators.

An obvious objection is that it is not equivalent to (H): English speakers could have used ‘Harry is bald’ in a precise way, to mean that 1+1=2 say, yet it would still have been borderline whether Harry is bald provided he has the same number of hairs he in fact has. The equally obvious reply, which we have already pre-empted, is that we should instead paraphrase (H) with the claim that ‘Harry is bald’ is a vague sentence of English as it is used in the actual world.

But even this seems inadequate: a non-English speaker may not know anything about how English is actually used, but still know that the number of hairs Harry has falls within the borderline region for baldness. It seems, in this case, that one can know whether Harry is borderline bald without knowing whether the sentence ‘Harry is bald’ is vague as it is actually used in English. Conversely, someone who doesn’t speak English may have it on good authority that the sentence ‘Harry is bald’ is vague in English as it is actually used, but have no idea whether Harry has 0 hairs, 1,000,000, or a borderline number, since she might not know what ‘Harry is bald’ actually means in English. Although these are only hyperintensional differences they are important: it is natural to think that the notion of propositional borderlineness is the notion that regulates our epistemic and doxastic attitudes for both English and non-English speakers. It is irrational, for example, to believe that Harry is bald but not determinately so; it is not irrational, if you were rationally mistaken about how English is used, to believe that Harry is bald but that ‘Harry is bald' is not definite as it is actually used by English speakers.

A more pedestrian worry is the dependence of the paraphrase on the choice of sentence and language: the claim that ‘Harry is bald' is borderline in English as it is actually used, and the claim that ‘Harry ist kahl' is borderline in German as it is actually used are two completely different paraphrases of (H). One could, for all we know, be true while the other false—they state completely different facts about different sentences in different languages.

Perhaps we could do better with language independent paraphrases:

E. There is some sentence, S, in some language, L, whose linguistic community actually uses S (i) in such a way that it says that Harry is bald and (ii) in whatever way it takes to make S vague in L.

U. Every sentence, S, in any language, L, whose linguistic community actually (i) uses S in such a way that it says that Harry is bald (ii) uses S in whatever way it takes to make S vague in L.

These paraphrases still mention languages and sentences, unlike (H), but perhaps they do better with regard to our first problem of language dependence.

Unfortunately, according to some linguistic theories (i) and (ii) are incompatible: the vagueness of a sentence relative to its use in a linguistic community precludes that use determining any proposition as being uniquely expressed by S in L.

However, I think the most problematic feature of this strategy is that it's either inadequate or it assumes the distinction between propositions we are trying to dispense with. Consider the following two possibilities.

The proposition that electrons are positively charged is expressed in some language by a vague sentence.

The proposition that Harry is bald is expressed in some language by a precise sentence.

If the first claim was metaphysically possible then E.

It should be noted straight off the bat that coarse-grained theories of propositions, such as the view that propositions are sets of worlds, are already committed to both these possibilities. For example, for some N, the proposition that Harry is bald is necessarily equivalent, and thus identical to the proposition that Harry has N hairs.^[45] It follows that the precise sentence ‘Harry has less than N hairs' expresses the proposition that Harry is bald, demonstrating the second possibility. An analogous argument can be made for the first possibility. The paraphrases E. and U. are therefore simply not adequate on a coarse-grained theory; such a theorist would have to resort to a language dependent paraphrase.

Fine-grained theories (theories that distinguish the proposition that Harry is bald from the proposition that Harry has n hairs, for each n) do not have this problem. It follows that if the paraphrases are adequate neither statement is possible: if E, is adequate the proposition that electrons are positively charged couldn't possibly be expressed by a vague sentence in some language, and if U, is adequate the proposition that Harry is bald couldn't possibly be expressed by a precise sentence.

But facts like this call out for explanation! What is so special about the proposition that Harry is bald that means it can't even possibly be expressed by a precise sentence? Yet it seems hard to see how the difference between propositions which could be expressed by a precise sentence and those which couldn't, could be explained without appealing to the propositional notion of vagueness that we are attempting to eliminate.

To put it another way, while we know that the proposition that Harry is bald is expressed in English by the vague sentence ‘Harry is bald' and in German by the vague sentence ‘Harry ist kahl’, what is it about the proposition that Harry is bald that prevents it being expressed by a precise sentence? Couldn't there be a language with a completely precise predicate, such that there is no unclarity about when it applies in that language and when it doesn't apply, which is such that it applies in L only when the object is bald, and doesn't apply otherwise? How are we supposed to distinguish the proposition that Harry is bald from a proposition that can be expressed by a precise sentence, like the proposition that Harry has less than 2,000 hairs, if not by employing the distinction between vague and precise propositions we are trying to eliminate? It thus seems that if we are to explain why the two possibilities raised above cannot arise, we will need something similar to the operator way of talking to distinguish vague from precise propositions.

4.4