Definition:Universal Quantifier

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

It expresses the fact that, in a particular universe of discourse, all objects 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:
 * $\forall x \in S: P \paren x := \set {x \in S: P \paren x} = S$

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:

Suppose our universe of discourse consists of the objects $\mathbf X_1, \mathbf X_2, \mathbf X_3, \ldots$ and so on.

Also see

 * Definition:Universal Statement


 * Definition:Existential Quantifier
 * Definition:Existential Statement