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System F, also known as the (Girard–Reynolds) polymorphic lambda calculus or the second-order lambda calculus, is a typed lambda calculus that differs from the simply typed lambda calculus by the introduction of a mechanism of universal quantification over types. System F thus formalizes the notion of parametric polymorphism in programming languages, and forms a theoretical basis for languages such as Haskell and ML. System F was discovered independently by logician Jean-Yves Girard (1972) and computer scientist John C. Reynolds (1974). Whereas simply typed lambda calculus has variables ranging over functions, 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 judgment ⊢ Λ α . λ x α . x : ∀ α . α → α {\displaystyle \vdash \Lambda \alpha .\lambda x^{\alpha }.x:\forall \alpha .\alpha \to \alpha } where α {\displaystyle \alpha } is a type variable. The upper-case Λ {\displaystyle \Lambda } is traditionally used to denote type-level functions, as opposed to the lower-case λ {\displaystyle \lambda } which is used for value-level functions. (The superscripted α {\displaystyle \alpha } means that the bound x is of type α {\displaystyle \alpha } ; the expression after the colon is the type of the lambda expression preceding it.) As a term rewriting system, System F is strongly normalizing. However, type inference in System F (without explicit type annotations) is undecidable. Under the Curry–Howard isomorphism, System F corresponds to the fragment of second-order intuitionistic logic that uses only universal quantification. System F can be seen as part of the lambda cube, together with even more expressive typed lambda calculi, including those with dependent types. According to Girard, the "F" in System F was picked by chance.

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