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:
 * $\forall x \in S: P \left({x}\right) := \left\{{x \in S: P \left({x}\right)}\right\} = S$

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