Arbitrary Reference

I posted a while back toying with a view of vagueness whereby there was a sharp cut-off in any sorites series as a result of there always being a unique most meaning among the candidate meanings (i.e. those that fit equally well with usage) for any vague expression; since naturalness is a reference magnet – and since it is ex hypothesi not trumped by usage – this is the meaning we will in fact mean, thus determining that the cut-off is where it is. (I further toyed with the idea that it is ontically indeterminate which meaning is the unique most natural, thus yielding the conclusion that it’s determinate that there’s a sharp cut-off in the sorites series but that it’s ontically indeterminate where it is – but forget about this complication for now.)

I’ve also been thinking about this with respect to arbitrary reference. What’s going on when we reason as follows? Let n be an arbitrary multiple of 4. n is a multiple of 2, all multiples of 2 are even, so every multiple of 4 is even. In particular, what, if anything, is referred to by ‘n’ throughout? Maybe it doesn’t refer; but then it’s hard to see how the sentences could be truth-apt, and we get a kind of Frege-Geach problem. Maybe it refers to a special kind of entity: the arbitrary multiple of 4; but that’s kind of weird. Ofra Magidor and Wylie Breckenridge have a really interesting paper where they argue that n actually refers to some particular multiple of 4 – we just cannot know which one. But in virtue of what do I refer to this particular multiple of 4 rather than some other? In virtue of nothing, they say: this is a brute fact. The semantic facts, on their view, are not fixed by the non-semantic facts: all the non-semantic facts could have been just the same but you have referred to some other multiple of 4 by ‘n’. I don’t like brute semantic facts, but I like a lot about their account, so I am quite attracted to extending the above account of vagueness to cases of arbitrary reference: ‘n’ refers to the most natural arbitrary multiple of 4. (Psst! – and it’s ontically indeterminate what this is. But again, forget this just now.)

There are two problems, one of which is encountered by both Magidor and Breckenridge and myself, the other of which might be thought to tell in favour of Magidor and Breckenridge’s view over my variant. I’d appreciate any thoughts on what I have to say about these.

First the common problem. Any view that takes us to genuinely refer to an F when we aim to refer to an arbitrary F has to have something to say about the case where there can be no Fs. For example, suppose we reason as follows. Let n be an arbitrary even prime greater than 2. n is (because it’s even) divisible by 2. So n is divisible by a number other than itself or 1. So n is not prime. Reductio: there is no such n. This chain of reasoning is perfectly good; but it’s obviously hopeless to take ‘n’ to refer to any even prime greater than 2, precisely because there are no such things. (I guess we could go Meinongian, and claim that there are such things, and ‘n’ refers to one, but that n doesn’t exist. But let’s not.) So what’s going on in this case? This is a case where those who postulate special entities as the referents in the cases of arbitrary reference – the arbitrary F – are at an advantage over those who take us to refer to an F; for if the arbitrary even prime greater than 2 isn’t really an even prime greater than 2, there can be no objection to its existence on these grounds. But of course, such views face other problems: such as, if the arbitrary F isn’t an F, what is it? I think we should treat cases like this as not really being cases of arbitrary reference after all. Despite their surface similarity to such cases, these cases, I suggest, are really reductios on the hypothesis that we have a case of genuine reference. So when we say ‘Let n be an arbitrary even prime greater than 2’, I suggest we are really supposing for reductio the hypothesis that ‘an arbitrary even prime greater than 2’ refers. Then, of course, we need some principle that lets us semantically descend, and conclude that there are no even primes greater than 2 if that expression cannot refer.

Now to the other problem. While I might not know what the most natural F is when I refer to an arbitrary F, there are some things I do know. I do know, for example, that if I refer to an arbitrary property I do not refer to grue, because grue is less natural than green. So when I say ‘Let F be an arbitrary property’, I can conclude that F is not identical to grue. But can’t I then conclude that all properties are not identical to grue, for isn’t one of the rules we’re trying to capture the one that says that if x is an arbitrary F and x is G then all Fs are G? But this rule would then take us wrong, for it’s not true that all properties are not identical to grue, for grue is identical to grue.

If this is a problem for my view, however, there is as much of a problem with Magidor and Breckenridge’s view. Indeed, any view that takes you to refer in a case of arbitrary reference has such a problem, including views that take you to refer to a special kind of entity (the arbitrary F), for the above rule would tell you to infer that all the Fs have the property of having being referred to by you when you said ‘Let n be an arbitrary F’. If I, at time t, say ‘Let n be an arbitrary number’ then, if ‘n’ refers – no matter what it refers to, or how the reference fact is determined – then n has the property having been referred to by me at t. If we follow the rule that tells us to infer that all Fs are G if the arbitrary F is G, it follows that all numbers were referred to by me at t. This is false: either I referred to a particular number, or to a special entity that is the arbitrary number, but I certainly didn’t refer to each number.

So anyone who takes cases of arbitrary reference to really be cases of reference can’t admit that rule in full generality. But views which take us to refer to an F (rather than to a special entity, the arbitrary F) when we say ‘Let a be an arbitrary F’ obviously needed to restrict this rule in any case. Suppose I say ‘Let n be an arbitrary multiple of 4’. We want to be able to reason as follows: n is even, hence every multiple of 4 is even. But suppose, as a matter of fact (putting aside why this is the case), ‘n’ refers, arbitrarily, to 28. 28 is a multiple of 14. So can’t we now conclude, mistakenly, that all multiples of 4 are multiples of 14? The rule had better be restricted so that we cannot so infer. Magidor and Breckenridge respond to this problem by modifying the rule to say that we can only conclude that every number is F if we can prove that the arbitrary number n is F. Because you can’t know that n is 28, you can’t prove that n is a multiple of 14, and hence you can’t conclude that all multiples of 4 are multiples of 14.

I think Magidor and Breckenridge are basically right to restrict the rule so that the properties we can conclude that all Fs have aren’t the ones that n has if n was our arbitrary F but rather just those ones that we can prove that n has from a certain basis. But the basis can’t be the properties we know that n has: for while that would deal with the problem immediately above, since we can’t know that n, our arbitrary multiple of 4, is a multiple of 14, even if it is, this won’t deal with the prior problem, since we can know that n was referred to at t when I said at t ‘Let n be an arbitrary multiple of 4’. I think instead we should restrict the rule as follows: if a is an arbitrary F, then if you can prove that a is G from facts that are true solely in virtue of a being an F (i.e. excluding those facts that are true in virtue of a being the particular F that it is), conclude that all Fs are G. 28 isn’t a multiple of 14 in virtue of being a multiple of 4, it’s a multiple of 14 in virtue of being that particular multiple of 4, but it is even in virtue of being a multiple of 4, and that’s why we conclude that all multiples of 4 are even but why we can’t conclude that they’re all multiples of 14. Nor was 28 the referent of ‘n’ solely in virtue of being a multiple of 14: on my view, it is true in virtue of being the most natural multiple of 14; on Magidor and Breckenridge’s view it is not true in virtue of anything. Either way, the move to ‘all multiples of 14 were referred to by ‘n’ at t’ is blocked.

This also lets me respond to what would otherwise have been an advantage of Magidor and Breckenridge’s approach over my own (I owe the objection to Ofra). Suppose we say ‘let n be an arbitrary number and let m be an arbitrary number’? If the reference facts are just brutely settled, they might be brutely settled so that ‘n’ and ‘m’ co-refer and they might not be. Either way, we can’t prove either that n is identical to m or that n is distinct from m, so we can’t ever conclude that arbitrary Fs a and b are identical (unless we can prove that there’s only one) or that they are distinct: and of course, that’s exactly as it should be. But the worry is that I can know that n=m because I know that ‘n’ and ‘m’ co-refer: they must both refer to the most natural number.

But once the rule isn’t restricted to the properties we can prove n has from the basis of facts we know about n but rather, as it has to be to deal with the reference problem, to the properties we can prove n has on the basis of facts that hold solely in virtue of n being a number, this problem dissolves. n is not identical to m, if it is, solely in virtue of being a number. It is in virtue of n being the particular number that it is, i.e. m, that it is identical to m. Likewise if n is in fact distinct from m, this is true in virtue of n being the particular number that it is - one other than m. With this restriction on the rule – and let me re-emphasise that any account that takes us to refer in cases of arbitrary reference must place some such restriction – I think there will be no unwelcome consequences to my approach. (At least, no additional unwelcome consequences over the brute facts view!) And the advantage is that, at the price of accepting these facts about naturalness, we avoid both brute semantic facts and the postulation of weird entities like the arbitrary number.