There are two ways that one might try to defend the quantifier account, neither very satisfactory. First, one might simply restrict the predicates of the language of the theory to those that express intrinsic properties (and relations). Perhaps one could argue that the theories Quine had primarily in mind, theories of mathematics and natural science, obey this restriction already with respect to their primitive predicates (though this has been contested even for physics, and is not plausible for the special sciences).  But this would be a rather severe restriction; and there is no evidence that Quine had intended the criterion to be restricted in this way. Moreover, it would needlessly inject controversies over which properties and relations are intrinsic into the application of the criterion, threatening both its foundational and polemical roles.