Definition:Universal Quantifier

Definition
The symbol $$\forall$$ is called the universal quantifier.

It expresses the fact that, in a particular universe of discourse, all object have a particular property.

That is:


 * $$\forall x:$$ means: "For all objects $$x$$, it is true that ...".

In the language of set theory, this can be formally defined:
 * $$\forall x \in S: P \left({x}\right) \ \stackrel {\mathbf {def}} {=\!=} \ \left\{{x \in S: P \left({x}\right)}\right\} = S$$

where $$S$$ is some set and $$P \left({x}\right)$$ is a propositional function on $$S$$.

Propositional Expansion
The universal quantifier can be considered as a repeated conjunction.

See the definition of the propositional expansion of $\forall x$.

Variants
Some authors use $$\left({x}\right)$$ to mean $$\forall x$$, but the bespoke symbol is usually preferred as there is then no room for ambiguity.

Some authors use $$\bigwedge$$, which is appropriate when considering the propositional expansion.

Historical Note
The symbol $$\forall$$ was first used by Gerhard Gentzen in Untersuchungen über das logische Schließen (1935: Mathematische Zeitschrift 39).

He invented it in analogy with the existential quantifier symbol $$\exists$$ which he borrowed from Bertrand Russell.

Russell himself used the notation $$\left({x}\right)$$ for for all $$x$$. See his Mathematical Logic as Based on the Theory of Types (1908: American Journal of Mathematics, 30).

Also see

 * Universal statement


 * Existential quantifier
 * Existential statement