Full Idea
The defining factor of second-order logic is that, while the domain of its individual variables may be arbitrary, the range of the first-order variables is all the properties of the objects in its domain (or, thinking extensionally, of the sets objects).
Gist of Idea
In second-order logic the higher-order variables range over all the properties of the objects
Source
Stephen Read (Thinking About Logic [1995], Ch.2)
Book Reference
Read,Stephen: 'Thinking About Logic' [OUP 1995], p.47
A Reaction
The key point is that the domain is 'all' of the properties. How many properties does an object have. You need to decide whether you believe in sparse or abundant properties (I vote for very sparse indeed).