Universal quantification

Lore & Background

Universal quantification is a logical constant that asserts a predicate is satisfied by every member of a domain of discourse. It is typically denoted by the symbol ∀, a turned 'A' in a sans-serif font, first used in this way by Gerhard Gentzen in 1935, by analogy with Giuseppe Peano's turned E (∃) for existential quantification and later use by Bertrand Russell. The universal quantifier is encoded as U+2200 in Unicode and as \forall in LaTeX.

Reader's Guide

Universal quantification is fundamental to formal logic and mathematics, allowing precise statements about all elements of a set. For example, 'For all natural numbers n, 2·n = n + n' is a single statement using universal quantification, more rigorous than an infinite conjunction. The truth of a universal statement depends on the domain of discourse; a single counterexample makes it false. Negation of a universal quantifier yields an existential quantifier with a negated predicate: ¬∀x P(x) is equivalent to ∃x ¬P(x). The universal quantifier moves unchanged across logical connectives like ∧, ∨, →, and ↚ when the other operand is unaffected. Its notation and properties are essential for formalizing mathematical reasoning and distinguishing from existential quantification.

Did You Know?

• The universal quantifier symbol ∀ was first used by Gerhard Gentzen in 1935.
• Universal quantification is distinct from existential quantification, which asserts a property holds for at least one member of the domain.
• The negation of a universally quantified statement is equivalent to an existentially quantified negation: ¬∀x P(x) ≡ ∃x ¬P(x).
• The universal quantifier is encoded as U+2200 ∀ FOR ALL in Unicode.