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.