Parametric polymorphism in the context of Type theory

System F (also polymorphic lambda calculus or second-order lambda calculus) is a typed lambda calculus that introduces, to simply typed lambda calculus, a mechanism of universal quantification over types. System F formalizes parametric polymorphism in programming languages, thus forming a theoretical basis for languages such as Haskell and ML. It was discovered independently by logician Jean-Yves Girard (1972) and computer scientist John C. Reynolds.

Whereas simply typed lambda calculus has variables ranging over terms, and binders for them, System F additionally has variables ranging over types, and binders for them. As an example, the fact that the identity function can have any type of the form A → A would be formalized in System F as the statement

The simply typed lambda calculus (⁠${\displaystyle \lambda ^{\to }}$⁠), a formof type theory, is a typed interpretation of the lambda calculus with only one type constructor (⁠${\displaystyle \to }$⁠) that builds function types. It is the canonical and simplest example of a typed lambda calculus. The simply typed lambda calculus was originally introduced by Alonzo Church in 1940 as an attempt to avoid paradoxical use of the untyped lambda calculus.

The term simple type is also used to refer to extensions of the simply typed lambda calculus with constructs such as products, coproducts or natural numbers (System T) or even full recursion (like PCF). In contrast, systems that introduce polymorphic types (like System F) or dependent types (like the Logical Framework) are not considered simply typed. The simple types, except for full recursion, are still considered simple because the Church encodings of such structures can be done using only ${\displaystyle \to }$ and suitable type variables, while polymorphism and dependency cannot.