For all symbol (∀)

How to read it

Oscar Levin’s Discrete Mathematics reads ∀ as “for all” or “every”: ∀x (x ≥ 0) asserts that every number is greater than or equal to 0. Whether that is true depends on what x ranges over; Levin notes that this is usually implied rather than stated. forall x: Calgary points out that a quantifier is always followed by a variable, and that the letter chosen makes no difference: ∀x H(x) and ∀y H(y) say the same thing. NIST’s Digital Library of Mathematical Functions lists ∀ as “for every”, and Unicode names it FOR ALL (U+2200), the universal quantifier. Its partner is ∃, “there exists”: Levin gives the rule that “not everything is P”, ¬∀x P(x), is the same as “something is not P”, ∃x ¬P(x).

Also searched as: ∀, for each symbol, for every symbol, universal quantifier, upside down A.

Other ways of writing it

forall x: Calgary notes that some texts have no symbol for the universal quantifier at all: they put the variable in brackets in front of the formula, writing (x)P(x) for ∀x P(x). Others use a large ⋀, the n-ary form of ∧, with the variable underneath or as a subscript.