Full Idea
Henkin proved that there is no first-order treatment of branching quantifiers, which do not seem to involve any idea that is fundamentally different from ordinary quantification.
Gist of Idea
A limitation of first-order logic is that it cannot handle branching quantifiers
Source
Ian Hacking (What is Logic? [1979], §13)
Book Reference
'A Philosophical Companion to First-Order Logic', ed/tr. Hughes,R.I.G. [Hackett 1993], p.247
A Reaction
See Hacking for an example of branching quantifiers. Hacking is impressed by this as a real limitation of the first-order logic which he generally favours.