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
For example, to say that cats exist, represent [x is a cat] as Cx, and then assert that
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
(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:
(x)(Cx -> x=x)
Ca -> a=a
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
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)(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.