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

Arbitrary Example

Consider the statement expressed symbolically as:

$\exists x: \exists y: x \text { is the son of $y$}$

This is a closed statement, as both the variables $x$ and $y$ appears as bound variables.

Whether it is true or false depends upon the ranges of $x$ and $y$.

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

• Results about closed statements can be found here.

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.