Definition:Closed Statement

Definition

Let $P$ be a statement.

$P$ is a closed statement if and only if $P$ contains only bound occurrences of any variables that may appear in it.

That is, such that it contains no free occurrences of variables.

Examples

True Statement

The statement:

$\exists x \in \Z: x^2 + 2 = 11$

is a closed statement that is true, as it is satisfied by the integers $x = 3$ and $x = -3$.

False Statement

The statement:

$\exists x \in \R: x^2 = -1$

is a closed statement that is false, as it is satisfied by no real number $x$.

Also see

• Definition:Open Statement

Sources

• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 1$: Some mathematical language: Variables and quantifiers
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): variable: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): variable: 2.