We struggled to see the exact import of this paper. Cian worried that ‘ontological commitment’ was a philosophical technical term, and that even a really good account of it would still not tell us too much about what really exists. Perhaps the motivation is that Rayo wants to emphasize that the characterization of ontological commitment can be kept apart from Quine’s criterion. Quine’s criterion (to be is to be the value of a variable) has perhaps come to seem constitutive of ontological commitment for some philosophers, which leaves no room for non-Quinean accounts of the ontology of (say) mathematics.
I wondered about an attempt at explaining demand-talk in terms of necessitation. The obvious account, that the truth of P demands that the world contains F iff necessarily(p → Fs exist), ends up saying that asserting any true proposition commits us to the existence of all necessary existents. So we can try ruling necessary existents out by fiat. Say that the truth of p demands of the world that it contain Fs . Maybe this is equivalent to saying that a) necessarily(p → Fs exist) and b) not necessarily p. The problem with that (supposing numbers are necessary existents), it would come out false that the truth of <there are numbers> demands of the world that it contain numbers, and hence that asserting ‘there are numbers’ commits you to numbers. Indeed, it would never be the case that asserting the existence of necessary existents commits you to those necessary existents. I’m not sure what Rayo would think about this. On the one hand, he does want to say that true propositions of mathematics have trivial truth-conditions and hence we can assert them without being committed to numbers. On the other hand, this account of demand-talk would beg the question in favour of the falsity of Quine’s criterion (at least a version of the criterion which is generalized to the language of mathematics) and Rayo seems to be unwilling to build either the truth or falsity of Quine’s criterion into the notion of ontological commitment. So I guess if we want to remain neutral on Quine’s criterion (at least as far as characterising the notion of ontological commitment is concerned) then demand-talk is going to have to remain primitive and unanalysed. But if we take a nominalist line which rejects Quine’s criterion, then we can potentially give a straightforward account of ontological commitment in terms of necessitation.
Rayo suggests that the ‘demands of the world that it contain Fs’ should be generalized to ‘demands of the totality of everything there is that it contain Fs’ if you are a modal realist. If so, I don’t see why we shouldn’t use this latter formulation in general, since we don’t want an account of ontological commitment to beg any questions about modal ontology.
We thought that the ‘extrinsic property worry’ for Quine’s criterion was badly characterised by the extrinsic/intrinsic distinction. Not only, as Rayo admits, do not all extrinsic properties cause trouble, but some intrinsic properties like ‘is composite’ also cause trouble. Plausibly, a thing being composite demands of the world that the world contain parts, but ‘for some x, x is composite’ doesn’t need to have parts among the values of its variables.