Full Idea
The natural understanding of first-order logic is that in writing down first-order schemata we are implicitly asserting their validity, that is, making second-order assertions. ...Thus even quantification theory involves reference to classes.
Gist of Idea
Asserting first-order validity implicitly involves second-order reference to classes
Source
Hilary Putnam (Philosophy of Logic [1971], Ch.3)
Book Reference
Putnam,Hilary: 'Philosophy of Logic' [Routledge 1972], p.32
A Reaction
If, as a nominalist, you totally rejected classes, presumably you would get by in first-order logic somehow. To say 'there are no classes so there is no logical validity' sounds bonkers.
Related Idea
Idea 18951 For scientific purposes there is a precise concept of 'true-in-L', using set theory [Putnam]