Full Idea
The distinctive feature of second-order logic is that it presupposes that, given a domain, there is a fact of the matter about what the relations on it are, so that the range of the second-order quantifiers is fixed as soon as the domain is fixed.
Gist of Idea
Second-order logic presupposes a set of relations already fixed by the first-order domain
Source
Shaughan Lavine (Understanding the Infinite [1994], V.3)
Book Reference
Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.123
A Reaction
This sounds like a rather large assumption, which is open to challenge. I am not sure whether it was the basis of Quine's challenge to second-order logic. He seems to have disliked its vagueness, because it didn't stick with 'objects'.