Full Idea
To get any advantage from moving to second-order logic, we need to assign to second-order variables a role different from merely ranging over collections made up of things the first-order variables range over.
Gist of Idea
Second-order variables need to range over more than collections of first-order objects
Source
Vann McGee (Logical Consequence [2014], 7)
Book Reference
'Bloomsbury Companion to Philosophical Logic', ed/tr. Horsten,L/Pettigrew,R [Bloomsbury 2014], p.47
A Reaction
Thus it is exciting if they range over genuine properties, but not so exciting if you merely characterise those properties as sets of first-order objects. This idea leads into a discussion of plural quantification.
Related Idea
Idea 18763 Basic variables in second-order logic are taken to range over subsets of the individuals [Anderson,CA]