Full Idea
Quine is unwilling to suppose second-order logic intelligible. He holds to Mill's account of the referential role of a predicate: it multiply denotes any and all objects to which it applies, and there is no need for a further 'predicative' entity.
Gist of Idea
Quine rejects second-order logic, saying that predicates refer to multiple objects
Source
report of Willard Quine (Philosophy of Logic [1970]) by Harold Hodes - Logicism and Ontological Commits. of Arithmetic p.130
Book Reference
-: 'Journal of Philosophy' [-], p.130
A Reaction
If we assume that 'quantifying over' something is a commitment to its existence, then I think I am with Quine, because you end up with a massive commitment to universals, which I prefer to avoid.