Full Idea
First-order predicate calculus is an extensional logic, while quantified modal logic is intensional (which has grave problems of interpretation, according to Quine).
Clarification
'Extensions' concern items in the world; 'intensions' concern internal meaning
Gist of Idea
First-order predicate calculus is extensional logic, but quantified modal logic is intensional (hence dubious)
Source
Joseph Melia (Modality [2003], Ch.3)
Book Reference
Melia,Joseph: 'Modality' [Acumen 2003], p.63
A Reaction
The battle is over ontology. Quine wants the ontology to stick with the values of the variables (i.e. the items in the real world that are quantified over in the extension). The rival view arises from attempts to explain necessity and counterfactuals.