Definition:Universal Statement

Definition
A universal statement is one which expresses the fact that all objects (in a particular universe of discourse) have a particular property.

That is, a statement of the form:
 * $\forall x: P \left({x}\right)$

where:
 * $\forall$ is the universal quantifier
 * $P$ is a predicate symbol.

It means:
 * All $x$ (in some given universe of discourse) have the property $P$.

Note that if there exist no $x$ in this particular universe, $\forall x: P \left({x}\right)$ is always true: see vacuous truth.

Also see

 * Existential Statement

Dummy Variable
In the expression $\forall x: P \left({x}\right)$, the symbol $x$ is known as a dummy variable, or bound variable.

Thus, the meaning of $\forall x: P \left({x}\right)$ does not change if $x$ is replaced by another symbol.

That is, $\forall x: P \left({x}\right)$ means the same thing as $\forall y: P \left({y}\right)$ or $\forall \alpha: P \left({\alpha}\right)$. And so on.