Full Idea
Under its now standard principal interpretation, the monadic predicate variables in second-order logic range over subsets of the domain on individuals.
Gist of Idea
Basic variables in second-order logic are taken to range over subsets of the individuals
Source
C. Anthony Anderson (Identity and Existence in Logic [2014], 1.5)
Book Reference
'Bloomsbury Companion to Philosophical Logic', ed/tr. Horsten,L/Pettigrew,R [Bloomsbury 2014], p.59
A Reaction
This is an interpretation in which properties are just sets of things, which is fine if you are a logician, but not if you want to talk about anything important. Still, we must play the game. Boolos introduced plural quantification at this point.
Related Idea
Idea 18761 Second-order variables need to range over more than collections of first-order objects [McGee]