difference between first order and second order logic
Second-order logic 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.