<P> 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 . </P> <P> First - order logic quantifies only variables that range over individuals (elements of the domain of discourse); second - order logic, in addition, also quantifies over relations . For example, the second - order sentence ∀ P ∀ x (x ∈ P ∨ x ∉ P) (\ displaystyle \ forall P \, \ forall x (x \ in P \ lor x \ notin P)) says that for every unary relation (or 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 quantification over sets, functions, and other variables as explained in the section Syntax and fragments . 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 over which individual elements may be quantified . </P> <P> The syntax of second - order logic tells which expressions are well formed formulas . In addition to the syntax of first - order logic, second - order logic includes many new sorts (sometimes called types) of variables . These are: </P>

Difference between first order and second order logic
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