(Or "second order typed lambda-calculus").
An extension of typed lambda-calculus
allowing functions which take types as parameters.
E.g. the polymorphic
function "twice" may be written:
twice = /\ t . \
(f :: t -> t) . \ (x :: t) . f (f x)
(where "/\" is an upper case Greek lambda and "(v :: T)" is usually written as v with subscript T).
The parameter t will be bound to the type to which twice is applied, e.g.:
takes and returns a function of type Int -> Int.
(Actual type arguments are often written in square brackets [ ]).
Function twice itself has a higher type:
twice :: Delta t . (t -> t) -> (t -> t)
(where Delta is an upper case Greek delta).
Thus /\ introduces an object which is a function of a type and Delta introduces a type which is a function of a type.
Polymorphic lambda-calculus was invented by Jean-Yves Girard in 1971 and independently by John C. Reynolds in 1974.