Full Idea
A nominalist will only countenance first-order logic.
Gist of Idea
Nominalists only accept first-order logic
Source
George Molnar (Powers [1998], 12.2.2)
Book Reference
Molnar,George: 'Powers: a study in metaphysics', ed/tr. Mumford,Stephen [OUP 2003], p.210
A Reaction
This is because nominalist will not acknowledge properties as entities to be quantified over. Plural quantification seems to be a strategy for extending first-order logic while retaining nominalist sympathies.