Full Idea
The contribution of the quantifier to the truth conditions of sentences of which it is a part cannot be adequately explained if it is treated as other than a second-level predicate (for instance, if it is viewed as name).
Gist of Idea
A quantifier is a second-level predicate (which explains how it contributes to truth-conditions)
Source
report of Gottlob Frege (Begriffsschrift [1879]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2
Book Reference
George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.23
A Reaction
They suggest that this makes it something like a 'property of properties'. With this account it becomes plausible to think of numbers as quantifiers (since they do, after all, specify quantities).