In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory.
First-order logic only uses discrete variables (eg. the variable x represents a person) whereas second-order logic extends uses variables that range over sets of individuals. For example, the second-order sentence ∀P ∀x (x ∈ P ∨ x ∉ P) says that for every set P of people and every person 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.
Reference:
http://en.wikipedia.org/wiki/Second-order_logic
