# polymorphic lambda-calculus

(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.:

```	twice Int

```
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.

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Nearby terms: polyadic predicate logic « Polya George « polymorphic « polymorphic lambda-calculus » polymorphism » polynomial » polynomial-time