# Definition:Universal Statement

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## 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:

- $\forall x: P \paren x$

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

### Bound Variable

In the universal statement:

- $\forall x: P \paren x$

the symbol $x$ is a **bound variable**.

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

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

## 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