in logic, a theory introduced by the British philosopher Bertrand Russell (Russell, Bertrand) in his Principia Mathematica (1910–13) to deal with logical paradoxes arising from the unrestricted use of predicate (predicate calculus) functions as variables. Arguments of three kinds can be incorporated as variables: (1) In the pure functional calculus of the first order, only individual variables exist. (2) In the second-order calculus, propositional variables are introduced. (3) Higher orders are achieved by allowing predicate functions as variables. The type of a predicate function is determined by the number and type of its arguments. By not allowing predicate functions with arguments of equal or higher type to be used together, contradictions within the system are avoided.