Relatively Unrestricted Quantification

There are four broad grounds upon which the intelligibility of quantification over absolutely everything has been questioned—one based upon the existence of semantic indeterminacy, another on the relativity of ontology to a conceptual scheme, a third upon the necessity of sortal restriction, and the last upon the possibility of indefinite extendibility. The argument from semantic indeterminacy derives from general philosophical considerations concerning our understanding of language. For the Skolem–Lowenheim Theorem appears to show that an understanding of quantification over absolutely everything (assuming a suitably infinite domain) is semantically indistinguishable from the understanding of quantification over something less than absolutely everything; the same first-order sentences are true and even the same first-order conditions will be satisfied by objects from the narrower domain. From this it is then argued that the two kinds of understanding are indistinguishable tout court and that nothing could count as having the one kind of understanding as opposed to the other. The second two arguments reject the bare idea of an object as unintelligible, one taking it to require supplementation by reference to a conceptual scheme and the other taking it to require supplementation by reference to a sort. Thus we cannot properly make sense of quantification over mere objects, but only over objects of such and such a conceptual scheme or of such and such a sort. The final argument, from indefinite extendibility, rejects the idea of a completed totality. For if we take ourselves to be quantifying over all objects, or even over all sets, then the reasoning of Russell’s paradox can be exploited to demonstrate the possibility of quantifying over a more inclusive domain. The intelligibility of absolutely unrestricted quantification, which should be free from such incompleteness, must therefore be rejected. The ways in which these arguments attempt to the undermine the intelligibility of absolutely unrestricted quantification are very different; and each calls for extensive discussion in its own right. However, my primary concern in the present paper is with the issue of indefinite extendibility; and I shall only touch upon the other arguments in so far as they bear upon this particular issue. I myself am not persuaded by