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.

Hi Ross, a quick worry about your answer to the first problem. You criticise the view that ‘n’ doesn’t refer by saying that it’s difficult to see how the inference could be valid in that case. But in order to make your story about supposing for reductio that ‘an arbitrary even prime greater than 2’ refers and then semantically descending, it seems that you’re going to need valid inference without reference at some point. If so there might be a dialectical problem: your main objection to the no reference view is one that your own view ends up having to face too, which leaves the no reference view looking like an overall winner. Do you have any thoughts about how to avoid that kind of move?

I was thinking that in the reductio case no term is being used that lacks reference. Terms are being mentioned that lack reference, and it’s the hypothesis that they do refer that is being tested – but we never disquote until we say that there are no such things: and when we say that, the truth of the claim doesn’t (obviously) require reference. But in the genuine arbitrary reference cases, we’re actually using the term, so reference seems to be required for truth. Is that satisfying?

If reasoning with parameters (i.e., instantial constants/variables; e.g., “the arbitrary n”) is just a notational expedient to quantificational reasoning about then I don’t see the Frege-Geach problem. And isn’t it just an expedient? When I teach my students instantiation and generalization rules I teach them that they need to be able to take off the quantifiers in order to give connectives wide scope and use inference rules, but that parameters are used to keep track of what quantifiers they’re allowed to put back on.BTW, there’s some good discussion in the philosophy of mathematics literature about this topic recently. See Pettigrew’s “Platonism and Aristotelianism in Mathematics”, Shapiro’s “Identity, Indiscernibility, and ante rem Structuralism: The Tale of i and –i” (both in the October 08 Philosophia Mathematica) , or Burgess’s “Putting Structuralism in it’s Place” (available on his website). I’m attracted to Pettigrew’s view that mathematical reasoning involves the employment of permanent parameters.

I don’t mean to place too much weight on the Frege-Geach point. In my paper, I basically assume you’ve got a case of reference, and am asking what’s the best way of accounting for that. I don’t give any argument at all in the paper in favour of the claim that there is reference. Magidor and Breckenridge do in their paper, so let me just refer you to that – I can’t remember if they specifically deal with your suggestion that (if I’m understanding right) these are disguised cases of quantification.

I was wondering whether there was ever a good reason to accept the first rule you proposed?Let a be an arbitrary object. Either Fa or ¬Fa. Fa entails AxFx by the rule. ¬Fa entails Ax¬Fx by the rule. So AxFx or Ax¬Fx reasoning by cases. But this is absurd (no matter what your theory of arbitrary reference is – I only used the rule and classical logic.)Another modification to the rule might be: if you can prove *of* the arbitrary F that it is G, then you can prove that every F is G. In your case you can prove, de dicto, that the arbitrary object, whatever it may be, is referred to by you at t. But there’s no particular object that you can show to be referred to by you at t.

Hi Andrew. I’m assuming the defender of the rule just won’t think it can be used in a sub-proof in the manner you suggest. So, roughly, you can’t conclude from the fact that if assumption A is true the rule lets you prove that p that the rule lets you prove ‘if A then p’. I don’t see that there would be a big problem with not allowing the rule to be used within a sub-proof like this: do you think there is?

Thanks Ross… I could only find an abstract of the paper though. Got a link?Andrew. You said “Fa entails AxFx by the rule.” This looks to me like binding a variable that occurs free in an assumption line. That’s bad for precisely the reason that we don’t want to use CP to screw with scope: e.g., don’t want to get “(x)Fx v (x)~Fx” from “(x)(Fx v ~Fx)” (essentially what you did). That’s precisely the sort of thing I had in mind in my first post. I think that these sort of restrictions support the idea that instantial reasoning is just expedited quantificational reasoning (which seems to me to obviate the need to theorize about arbitrary referents), though this is something I’ll read and think more about.

Hi Jeremy,I think only the abstract is presently available. Think of me as giving a trailer for the forthcoming attraction! I’m not totally against the hidden quantifier view, incidentally – I’m kind of unopinionated about this. But I think the Magidor/Brecekenridge view is cool – and that’s gotta be a guide to truth, or metaphysics is screwed :-)Awesome, my word verification below is ‘grawp’. Give infinite bloggers an infinite amount of time, and their word verifications will create Harry Potter!

Hi Ross. Yeah, I wasn’t saying there’s anything technically wrong with non-classical approaches. I was just wondering why anyone should feel the need to defend themselves against a problem that only arises if you adopt some non-classical logic. (E.g. I don’t feel I’m obliged to defend myself against the analogous argument against contingency, because someone thinks that snow is white entails that snow is necessarily white.)jrshipley”You said “Fa entails AxFx by the rule.” This looks to me like binding a variable that occurs free in an assumption line.”I didn’t make the rule up! (It’s not a variable though, best to think of ‘a’ as a special kind of term for doing arbitrary reference since clearly that rule isn’t licensed with ordinary names.) The restriction you suggest gets you something closer to the “necessitation rule” that is being discussed.

On an unrelated note – I was wondering what the ontic indeterminacy of the most natural meaning actually adds to your theory?Presumably you will say things like “Either Fred is bald or he isn’t, but I don’t know which.” Will you also go on to say “and it’s indeterminate which.”? I’m finding it hard to distinguish this from the plain old ontic indeterminist about vagueness who thinks that indeterminacy necessarily involves ignorance.Similarly, what makes it different from the epistemicist about indeterminacy? Someone who accepts the existence of indeterminacy but thinks it’s indeterminate whether p just means that I have (a certain kind of) ignorance regarding p.

Andrew >>"I didn't make the rule up!"Right. I meant only to suggest that the problem you pointed to seemed to me to support my point. Sorry to belabor that point, but I'm gripped by compulsion… Here's a filled-out argument corresponding roughly to your objection:(1) (x)(Fx v ~Fx)(2) Fx v ~Fx |UI (corresponds to "let x be an arbitrary element of the domain")(3) ___Fx |assumption(4) ___(x)Fx |UG(5) Fx -> (x)Fx |conditional proof(6) ___~Fx |assumption(7) ___(x)~Fx |UG(8) ~Fx -> (x)~Fx |conditional proof(9) (x)Fx v (x)~Fx |hypothetical syllogism (2), (5), (8)I'm, btw, following Hurley (whose text I teach) in having both instantial constants and instantial variables, and in those terms the "x" occurring free on line (2) is a variable (not much depends on this label, as far as I can tell). In Hurley's system lines (4) and (7) are illicit due to restrictions on UG. If you allow UG on variables that occur free in assumption lines then you can mess with scope in all kinds of undesirable ways. The point I'm fixated on is that the restrictions we have on reasoning with instantial constants/variables arise out of the need to (in a sense) preserve scope from the original formulas that have been instantiated; this is my clue that instantial reasoning remains quantified reasoning (as opposed to singular). We just suppress the quantifiers to make things easier notationally, but our restrictions on when we may put the quantifiers back remind us that they never really went away.

hey man,I defended an arbitrary reference account to deal with the sorites in a short paper at Arche in 05. A copy of my paper is still up I think (Google ‘Andrew McGonigal Arche’ – it’s called ‘Vagueness and Context’). Elizabeth’s minutes of the discussion are up too! It’s funny looking back at them. Those were the days, when we were young and free, and hadn’t heard the terrible news about the many minds theory of vagueness…

Awesome! Was I at it? I can’t remember it, but may have internalised it: in any case, I’ll try and get hold of it.

Yeah – and we even have a record of what you asked me!’Ross observed that there seems to be a general problem with truthmaker arguments based on arbitrariness. What could qualify truthmakers as fully non-arbitrary, and why should why expect such constraints to hold on the truth-making relation? Andrew replied that he doesn’t want to hang too much on truthmaker theory per se, but simply use truthmaker talk to highlight a counter-intuitive feature of the view. There seems, at least to him, to be a need for a deeper (non-trivial) explanation of why truths within the theory hold’It was when I was talking to John H about the failure of supervenience that we came up with the many minds thing.Looking back at the minutes, I think that I may have missed one of Crispin’s points. Williamson preserves supervenience by postulating an unknowable (non-semantic) use property. I think that Crispin was suggesting that we could make a similar move here. There is some wholly sharp non-semantic property that explains why x gets selected rather than y, but it is unknowable what it us. If you replace ‘unknowable’ with ‘onticly indeterminate’ you get something similar to your view, it looks like.

Excellent! I’m channelling my supervisor but (as always) with an extra metaphysical flourish!I’m embarrassed not to remember any of this! We’ll have to chat about it when you’re Leeds-based once more.

I've posted an apparent counterexample to the Breckenridge-Magidor account here. Comments welcome.