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:

$\forall x: \map P x$

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, $\forall x: \map P x$ is always true: see vacuous truth.

Bound Variable

In the universal statement:

$\forall x: \map P x$

the symbol $x$ is a bound variable.

Thus, the meaning of $\forall x: \map P x$ does not change if $x$ is replaced by another symbol.

That is, $\forall x: \map P x$ means the same thing as $\forall y: \map P y$ or $\forall \alpha: \map P \alpha$.

And so on.

Also known as

A universal statement can also be referred to as a universal sentence, or more wordily, a sentence of a universal character.

Also see

• Definition:Existential Statement

• Definition:Bound Variable

Sources

• 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) ... (previous) ... (next): $\S 1.3$: Universal and Existential Sentences
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 3$: Statements and conditions; quantifiers