Full Idea
The logic of ZF Set Theory is classical first-order predicate logic with identity.
Gist of Idea
The logic of ZF is classical first-order predicate logic with identity
Source
George Boolos (Must We Believe in Set Theory? [1997], p.121)
Book Reference
Boolos,George: 'Logic, Logic and Logic' [Harvard 1999], p.121
A Reaction
This logic seems to be unable to deal with very large cardinals, precisely those that are implied by set theory, so there is some sort of major problem hovering here. Boolos is fairly neutral.