Universal quantification

Universal quantification

Type	Quantifier
Field	Mathematical logic
Statement	${\displaystyle \forall xP(x)}$ is true when ${\displaystyle P(x)}$ is true for all values of ${\displaystyle x}$.
Symbolic statement	${\displaystyle \forall xP(x)}$

In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any", "for all", or "for any". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable.

It is usually denoted by the turned A (∀) logical operator symbol, which, when used together with a predicate variable, is called a universal quantifier ("∀x", "∀(x)", or sometimes by "(x)" alone). Universal quantification is distinct from existential quantification ("there exists"), which only asserts that the property or relation holds for at least one member of the domain.

Quantification in general is covered in the article on quantification (logic). The universal quantifier is encoded as U+2200 ∀ FOR ALL in Unicode, and as \forall in LaTeX and related formula editors.