Friday, 29 April 2011

Breckenridge and Magidor on Arbitrary Reference: An apparent counterexample

[This is an early draft of a paper which, since being posted, has grown and changed title. Email me if you would like a copy. - TH 9/4/15]

In an interesting paper forthcoming in Phil. Studies, Breckenridge and Magidor argue for this thesis:

Arbitrary Reference (AR): It is possible to fix the reference of an expression arbitrarily. When we do so, the expression receives its ordinary kind of semantic-value, though we do not and cannot know which value in particular it receives.

Their primary argument in favour of AR is that it can be used to give an attractive account of 'instantial reasoning' such as this (their 'Argument 1'):

(1) There is someone x such that for every person y, x loves y [Premise]
(2) Let John be such a person
(3) For every person y, John loves y [Existential Instantiation on 1]
(4) Let Jane be an arbitrary person
(5) John loves Jane [Universal Instantiation on 3]
(6) There is some person x such that x loves Jane [Existential Generalisation on 5]
(7) But since Jane was an arbitrary person, for every person y there is some person x such that x loves y [Universal Generalisation on 6] 

I will not attempt to rehearse, or even summarize, their arguments, since they state them well and their paper is freely available on Magidor's website. My purpose here is to give an apparent counterexample to the claim that AR can be used to give an attractive account of instantial reasoning.

The following appears to be a logical truth: 

(Unref) If (all unreferred-to objects are white and there is an unreferred-to object), then there is a white object.

(By 'unreferred-to object', I mean an object which is never referred to by anyone or anything.) Here is a quasi-formal argument for (Unref):

(1) All unreferred-to objects are white and there is some unreferred-to object. [Assumption]
(2) All unreferred-to objects are white. [Conjunction Elimination on 1]
(3) There is some unreferred-to object. [Conjunction Elimination on 1]
(4) Let O be such an object.
(5) O is white. [Universal Instantiation on 2]
(6) There is some white object [Existential Generalization on 5]

(Unref) now follows from (1) - (6) by conditional proof.

This seems to be a valid argument. But the theory of instantial reasoning advanced by Breckenridge and Magidor seems to imply that the expression 'O' above refers to an unreferred-to object, which is absurd.

Tristan Haze
The University of Sydney

Breckenridge, Wylie & Magidor, Ofra (forthcoming). 'Arbitrary reference'. Philosophical Studies. 

There is a post about this paper on Ross Cameron's blog here.


  1. Jeff Sanford Russell29 April 2011 at 06:40

    I'm a bit worried about this argument, since "unreferred-to" is the sort of predicate that is liable to produce paradoxes. For instance, the set of unreferred-to natural numbers is non-empty. So let N be the smallest unreferred-to natural number. So N is unreferred-to; but 'N' refers to N, so N is referred-to; contradiction.

    This doesn't show that the argument you gave isn't right, but it does cast some suspicion over it. It would be interesting to see how particular ways of resolving the semantic paradoxes handle this case.

  2. Thanks for the comment!

    The big difference, I think, between the argument to paradox in your comment and my counterexample, is that yours quite clearly does involve an attempt at reference to a particular object. And so one gets a paradox, because it seems there must be a *particular* object answering to that description, but that on the other hand there can't be.

    My idea with "arbitrary reference" (arguably a misnomer) to some unreferred object O is roughly this: such a thing appears to be do-able in valid arguments, without paradox. This seems to me much more clearly true than any specific proposal about the semantics of expressions like 'O'.

    So, here we have a datum that must (I say) be accommodated by any account of the semantics of instantial reasoning. Breckenridge and Magidor's account seems to fail here.

  3. Something I should probably mention, Jeff, since you're worried about 'unreferred-to': there appears to be a similar but distinct problem with arguments involving empty predicates, like 'unicorn' (assuming for the sake of argument that there are no unicorns).

  4. Neat!

    Another class of arguments that seems problematic for that account are certain kinds of proof by contradiction. E.g. to show there is no largest prime you start by assuming, for contradiction, that there is such a prime, p. Then given p is the largest prime you go on to construct a prime larger than p, which is the contradiction.

    The Breckenridge Magidor account doesn't seem to be able to account for the term "p" whose reference was apparently fixed arbitrarily, since if it does refer it doesn't refer to a largest prime. (I assume its still arbitrary even if, had there been a largest prime, there would have been only one thing it could have been -- maybe I should have chosen a different example.)

  5. Thanks Andrew! I agree, this seems to be a problem - perhaps a special case of the one I alluded to in the comment above yours. And it's a good case, because it shows how reasoning involving empty predicates can lead quite directly to truths which are more than just tautologies in a minimal logical sense.

    By the way, I should have said this already: one reason I think the case of unreferred-to objects seems to provide a stronger overall case against the Breckenridge-Magidor account is that there will be valid arguments involving *categorical* instantial reasoning about unreferred-to objects - reasoning not under any assumption. Whereas with "no such object" cases, there may be a reply (and a patch) which crucially involves the fact that assumptions are being made.

    My original counterexample doesn't bring this out, since that argument does use an assumption (because I wanted to make it an argument for a logical truth, not just a logical consequence).

  6. "unreferred to" would also mean undefinable by reference?

    or if this is not so, then merely stating unreferred to _object_ is already a case of performing reference, but without calling it as such on a higher level view.

  7. I'm not sure; could you say something more about what you mean by 'undefinable by reference'? I can cash out the predicate '... is unreferred-to' as '... is an object which never gets referred to (past, present or future)'.

    I think we can talk about such objects without referring to any particular one. In ordinary language, 'refer' and its cognates may reasonably be used for this sort of talking-about, but it isn't reference in the relevant sense of reference to a particular object. Or so I say.

  8. I'm just a novice at this logic business so I apologize for not making myself clear.

    I was highlighting the object part in "unreferred to object." unreferred to is clear enough, but how does it get attached to an object. I think this whole complex is itself a reference already, and it is a performance of reference.

    (3) There is some unreferred-to object. [Conjunction Elimination on 1]
    (4) Let O be such an object.
    (5) O is white. [Universal Instantiation on 2]

    depending on whether you 3 or 4 to be sufficient for "referring", before performing 3 or 4, the object is not referred to. but 3 or 4 instantiates that reference.

    so basically, in classic parlance, referred-to is a second order concept. the referent is flat, but the referential act is not.

  9. 'I'm just a novice at this logic business so I apologize for not making myself clear.'

    No worries - I'm glad for the interest! Some academic types appear to look down their noses at novices or anyone with a non-professional interest in their area, but they tend to be idiots.

    'depending on whether you 3 or 4 to be sufficient for "referring", before performing 3 or 4, the object is not referred to. but 3 or 4 instantiates that reference.'

    I deny this. Again, there's a terminological complication in the fact that 'reference' in ordinary language is used more widely than it is being used here. Here it is being used for designation of particular objects.

    Why do I deny that 'O' is a referring expression? I'm wary of being too theoretical too soon, here. In the first instance, I deny it because it just doesn't seem to be the case. More theoretically, and having looked at Breckenridge and Magidor's proposal, I deny it because their proposal seems incompatible with the evident validity of arguments such as the one under discussion.

    'referred-to is a second order concept. the referent is flat, but the referential act is not.'

    Again, I'm not quite sure what you mean here, or what the relevance is to my argument, but it sounds sort of intriguing, so clarifications are welcome.

    (I say it sounds intriguing because, for reasons not alluded to in the above article, I do feel that reference is no ordinary relation, and that clarifying the manner in which it is peculiar would throw light on the theory of reference.)

  10. Yikes, got something typed out but then the internet ate it. It is probably for the best, I'll try again later.


    hope that works. by "liar's thought" i mean something like "i'm not actually thinking this"

  12. Lots of mistakes i know, but i thought i needed to write what i had down, and then correct it. if i don't do it then my short term memory would just eat everything.

    I think i also made a mistake with calling wittgenstein's showing vs telling (seeing vs representing for the reader/listener) by some other name.

  13. 'Is it correct to presume that when you said
    "Why do I deny that 'O' is a referring expression?", you want O to be simply the white, unreferred to object?'

    No, I don't think so... One thing which may be bothering you is the sentence in the original argument: 'There is some unreferred-to object'. The way I was using this sentence, which is the customary way such sentences are used among logicians, there is no implication that there is just one such object. It just means there is at least one.

    'but how can you say something about O without referring to it, or, how is it that O the expression in that particular place isn't referring?'

    Well, I say here we have to look closer at how expressions like 'O' are actually used. The quoted fragment of yours could be given a referential reading - you may designate some particular object, calling it 'O', and ask how I can say something about it. But in the case of 'O' as I take it to work, such a reading contains a presupposition I want to resist.

    I don't think there *is* some particular object which is the referent of 'O' as used in the original argument, any more than 'Pierre' refers to some particular Frenchman when I say 'Let Pierre be some Frenchman', and go on to make a general argument about Frenchmen. (Breckenridge and Magidor, it should be noted, maintain that even in this case, we have an expression which now designates some particular object, a Frenchman - although we can't possibly know which one.)

  14. The last paragraph there should have begun: 'I don't think there *is* some particular object which is *having something said about it* by means of 'O' as used in the original argument, any more than 'Pierre' can be immediately used to *say something about* some particular Frenchman when I say 'Let Pierre be some Frenchman', with a view to going on to make a general argument about Frenchmen.

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