W.V.O. Quine, answered from the texts and cited to the page.
The criterion of ontological commitment is that to be is to be the value of a variable — to determine the ontological commitments of a theory, regiment the theory in canonical first-order notation and read off the values that the bound variables must range over for the theory to be true.1234 The criterion is a criterion of regimented theory's commitments, not a doctrine about what exists; it does not tell us which theory to adopt, but once a theory is adopted, the criterion tells us what that theory says exists.
Existence, on this view, is what bound variable quantification ranges over in regimented theory. The criterion is neutral on the metaphysics of what exists — it can be deployed by Platonists and nominalists, realists and instrumentalists — but it disciplines the question of what a theory is committed to in a way that resists ontological evasion through informal paraphrase.5
'Ravens are black' becomes Wx(x is a raven:— - x is black). So I have insisted down the years that to be is to be the value of a variable. More precisely, what one takes there to be are what one admits as values of one's bound variables. The point has been recognized as obvious and trivial, but it has also been deemed unacceptable, even by readers who share my general philosophical outlook.Pursuit of Truth, pp. 35–37
When I inquire into the ontological commitments of a given doctrine or body of theory, I am merely asking what, according to that theory, there is. I might say in passing, though it is no substantial point of disagreement, that Carnap does not much like my terminology here. Now if he had a better use for this fine old word 'ontology,' I should be inclined to cast about for another word for my own meaning.THE PARADOX OF EPIMENIDES, pp. 136–137
The entities presup- posed by statements w'hich use combinators turn out, under such reasoning, to be just the entities that must be reckoned as argu- ments or values of functions in order that the statements in question be true. f See above, pp. 85ff. But it is to the familiar quantifications1 form of discourse that our criterion of ontological commitment primarily and fundamentally applies.From a Logical Point of View, pp. 111–112
To show that some given object is required in theory, what have to show is less a we no more nor than that that object is required, for the truth of the theory, to be among the values over which the bound variables range. Appreciation of this point affords us more than an explication of "a exists," since the existentially quantified identity "(Ax) (x =a)" is one case of existential quantification among 8 J.I first urged the inscrutability of reference with the help, pp. 104–105
Names are,, in fact, altogether immaterial to the ontological issue, for I have shown, in connection with 'Pegasus' and 'pegasiz;e', that names can be converted to descriptions, and Russell has shown that descriptions can be eliminated. > 6 See Esays > II and III. ON WHAT THERE IS Whatever we say with the help of names can be said in a language which shuns names altogether.From a Logical Point of View, pp. 19–20
The criterion of ontological commitment is among the most widely deployed tools in twentieth-century analytic metaphysics. Carnap's 'Empiricism, Semantics, and Ontology' (1950) replied with the linguistic-frameworks distinction between internal and external questions of existence: questions of existence within a chosen linguistic framework are to be settled by the framework's rules; questions of existence external to all frameworks are pseudo-questions. Stephen Yablo's figuralism and Eli Hirsch's quantifier variance (in Quantifier Variance and Realism, 2011) develop more recent challenges. Theodore Sider's Writing the Book of the World (2011) defends a structural realism against quantifier variance. Peter van Inwagen's Existence (2014) systematically defends the Quinean criterion. The position remains central to contemporary metaphysics.