Page "Second-order logic" Paragraph 1

First-order logic uses only variables that range over individuals ( elements of the domain of discourse ); second-order logic has these variables as well as additional variables that range over sets of individuals.

For example, the second-order sentence says that for every set P of individuals and every individual x, either x is in P or it is not ( this is the principle of bivalence ).

Second-order logic also includes variables quantifying over functions, and other variables as explained in the section Syntax below.

Both first-order and second-order logic use the idea of a domain of discourse ( often called simply the " domain " or the " universe ").

The domain is a set of individual elements which can be quantified over.