September 14, 2007

The indispensability of identity conditions

Sometimes metaphysicians can only be dealt with by doing some metaphysics. So, as much as it pains me, this is a post on metaphysics.

Philosophers who fancy themselves `Quinean meta-ontologists' believe in a simple condition for existence claims: something of type K exists if and only if statements of the form [there is something such that it is a K] are true. That is, if and only if

(Ex)(Kx)

For example, to say that cats exist, represent [x is a cat] as Cx, and then assert that

(Ex)(Cx)

It's not immediately obvious how to translate `something exists' into the formal idiom. A typical strategy is to use self-identity, so that `something exists' is taken to be formalised as

(Ex)(x=x)


(NB Such a strategy is, I think, incompatible with the claim that existence is captured entirely by the existential quantifier. If that were so, then one wouldn't need identity to say that something exists. But that's not the main argument I'm making in this post.)

All cats are self-identical. Or, at least, the identity of cats (ie, whether one cat is or is not the same as `another' cat) is not considered deeply controversial. How do we say that all cats are self-identical?

(x)(Cx -> x=x)

Note that, from the claims that (1) cats exist and (2) all cats are self-identical, one easily concludes that something exists. Here's the derivation in all its glory, if you're curious:

(Ex)(Cx)
(x)(Cx -> x=x)
Ca
Ca -> a=a
a=a
---------------
(Ex)(x=x)


Some `Quinean meta-ontologists' believe in propositions. That is, they believe that propositions exist. If [x is a proposition] is symbolised as Px, then they believe that

(Ex)(Px)

Now, this is a departure from Quine. Quine very firmly believed that propositions did not exist. His argument was that there were no clear identity conditions for propositions -- it is generally impossible to tell whether one propositions was the same as `another'. Self-identity of propositions is, therefore, controversial.

The `Quinean meta-ontologists' who depart from Quine in this way claim that they do not need identity conditions. Some have claimed that they do not need identity conditions at all, for any type of thing, while others seem to commit to the far more modest claim that they do not need identity conditions for propositions. I take this to mean that they do not assert or assume that

(x)(Px -> x=x)


I claim (or at least I suggest as likely to be true, pending proof) that, from these two assumptions (or, more precisely, one assumption and one lack of assumption), one cannot conclude that something exists. That is, from (Ex)(Px) and the ordinary rules of first-order logic *without identity*, and without using (x)(Px -> x=x), one cannot derive (Ex)(x=x). In short, from the assumption `there is something such that it is a proposition', one cannot conclude that `there is something'.

It may be objected that I have crippled the `Quinean meta-ontologist' by forbidding the use of the logic of the identity relation =. But the `Quinean meta-ontologist' has done it to herself, by abjuring identity conditions for propositions. If there are no identity conditions for propositions, then neither of these two claims, nor any equivalent to them, can be assumed:

(x)(x=x)
(x)(Px -> x=x)

The first of these captures, as an axiom, the standard rule of =-introduction used when identity is added to first-order logic. The second is equivalent to it under the assumption (Ex)(Px). There are therefore both unacceptable.

And I claim (or at least suggest as likely to be true, pending proof) that at least one of these, or something equivalent to them, must be used to derive the desired conclusion. I thereby conclude that identity conditions for propositions (and, indeed, every other type of entity one would countenance) are indispensable to make good on the programme of `Quinean meta-ontology'.

The challenge to the `Quinean meta-ontologist' who believes in propositions is to give the required proof.

4 comments:

Andrew Bailey said...

Quine says `no entity without identity conditions.' The Deniers disagree. And I don't think you've done the Deniers justice--your criticism isn't sensitive to an important distinction any Denier worth her salt would make. Doing philosophy without identity conditions is one thing. Doing philosophy without identity is another.

The Denier can freely abstain from giving identity conditions for propositions while nonetheless employing an identity relation in her logic. You've simply assumed that to abjure identity conditions is to abjure the logic of the identity relation. But why assume this without argument?

Two other points. First, it's never been clear what Quine actually wanted when asking for identity conditions for Fs. The clearest statement of the Quinean slogan I've found is something like this: don't posit the existence of Fs without a non-circular analysis of the open sentence `x is the same F as y.' But analyzing such open sentences without circularity (even for such well-behaved entities as sets) is notoriously difficult. That sets don't submit to this charge (but there apparently are such things) is good reason, I think, to jettison the requirement.

Second, Chisholm answered Quine's specific charge long ago--he gave identity conditions for properties that just might do the trick. F is the same property as G iff for all x, Fx iff Gx and for any subject S, S apprehends (I forget Chisholm's technical vocabulary here--but it was an epistemic relation) F iff S apprehends G. Combine this with the standard Chisholmian take on propositions as a special sort of property to answer Quine's charge. Coextensivity along two dimensions suffices (and is necessary) for identity.

So the believer in propositions has options.

Noumena said...

While the issue of identity conditions is still up for debate, the Proposition-Believer (PB) cannot assert any sentence of the form [x is the same proposition as y] without begging the question against Quine. That's just the debate between Quine and the PB: Quine says that, without identity conditions for propositions, [x is the same proposition as y] is nonsense (or nearly so), and hence the PB must provide identity conditions as a necessary (but probably not sufficient) condition on her sentences making sense (or any significant amount of sense). He writes, for example, that `little sense has been made of the term [`proposition', but I believe this applies more generally to any term for Quine] until we have before us some standard of when to speak of propositions as identical and when as distinct' (WO 200).

Now, I don't see how to formalise [x is the same proposition as y] using the standard identity relation except as [x=y]. Perhaps the PB might want identity-with-respect-to-being-propositions, or some such, but this wouldn't be the standard identity relation. So, when we formalise, the sentences that the PB must abjure from if she does not give identity conditions just are the sentences involving the logic of the identity relation.

Chisholm's definition might do the trick. Quine would object that the entities and relations Chisholm is giving here are just as dubious and in need of identity conditions themselves, but that's another debate. One worry that occurs to me, however, is that it seems to make the identity of propositions -- necessary things -- dependent upon the mental abilities of subjects -- contingent beings (excluding a necessarily-existent divine subject for the moment). At worlds with no subjects at all, for example, all true propositions are identified. It would just be a contingent fact that `2+2=4' and `All bachelors are unmarried' aren't identical. You could start talking about all possible subjects, but Al would complain (just like he did on Friday -- though, did you leave town before then?).

Finally, I don't see the circularity in

x=y <-> (z)(z€x <-> z€y)

(where € is my easier-to-type surrogate for the epsilon of set-inclusion). One side has =, and the other side doesn't.

Andrew Bailey said...

"The Proposition-Believer (PB) cannot assert any sentence of the form [x is the same proposition as y] without begging the question against Quine."

False. One can use a sentence of that form provided one has given a non-circular analysis of it. If Quine isn't asking for such an analysis, I really have no idea what the demand for identity conditions is supposed to come to.

"Now, I don't see how to formalise [x is the same proposition as y] using the standard identity relation except as [x=y]. Perhaps the PB might want identity-with-respect-to-being-propositions, or some such, but this wouldn't be the standard identity relation. So, when we formalise, the sentences that the PB must abjure from if she does not give identity conditions just are the sentences involving the logic of the identity relation."

Numerical identities are easy to translate into first order logic. But sortal-relative identities can be translated too. The standard translation of `x is the same proposition as y' is: `x=y and x is a proposition and y is a proposition.'

The sort-relativity disappeared in this translation. Perhaps that's a cost. And if you think it is, it's easy to see what motivates the introduction of relative identity predicates by Geach and van Inwagen. Incidentally, one can, so far as I can tell, help oneself to these resources without saying the crazy things Geach says (eg, that *all* identity is sortal relative and that there's no most general sortal).

Noumena said...

"One can use a sentence of that form provided one has given a non-circular analysis of it. If Quine isn't asking for such an analysis, I really have no idea what the demand for identity conditions is supposed to come to."

Agreed: Quine is demanding the PB give a non-circular analysis of sentences of the form [x is the same proposition as y] before she goes around asserting them. For the PB to concede to this demand, fail to do so, and yet still go around asserting such sentences is begging the question.

The subordinate clause `While the issue of identity conditions is still up for debate' is a critical qualifier. The sentence you put in quotes is still true when the argument is over the legitimacy of asserting sentences of the form [x is the same proposition as y]. But it's not what I said. And I'm not sure why you think the sentence I said is false.