forall

ForAll, commonly written with the symbol ∀, is the universal quantifier in formal logic. It expresses that a given property holds for every element of a specified domain. In mathematics and computer science, ∀ is used to state axioms, definitions, and theorems in a concise, general form. The expression ∀x P(x) reads as “for all x, P(x).”

Semantics and scope are central to its use. In first-order logic, ∀x P(x) asserts that P(x) is true for every x in the domain of discourse, with x bound by the quantifier and the formula following the quantifier determining its scope. The domain can be any set relevant to the problem, and in many logics the domain may be empty; in standard first-order logic, ∀x P(x) is typically considered true when the domain is empty (vacuous truth).

Examples illustrate its application. Examples include ∀n ∈ N, n + 1 > n, which states that every natural number’s successor is greater than the number itself, and ∀x ∈ R, x2 ≥ 0, which holds for all real numbers x. These statements convey universal properties about entire sets of objects.

Relations to other quantifiers are foundational. The universal quantifier is dual to the existential quantifier ∃, with logical equivalences such as ¬∃x P(x) ≡ ∀x ¬P(x) and ¬∀x P(x) ≡ ∃x ¬P(x). In practice, ∀ is used across proofs, definitions, and specifications, including in programming languages or formal specification tools where “forall” or “forAll” denotes the same general idea, sometimes with language-specific syntax or semantics.

Variations include higher-order and intuitionistic logics, where the behavior and admissible inferences of ∀ can differ from classical, especially regarding negation and domain assumptions.