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An **open formula** is a formula that contains at least one free variable.^{[citation needed]}

An open formula does not have a truth value assigned to it, in contrast with a closed formula which constitutes a proposition and thus can have a truth value like *true* or *false*. An open formula can be transformed into a closed formula by applying a quantifier for each free variable. This transformation is called capture of the free variables to make them bound variables.

For example, when reasoning about natural numbers, the formula "*x*+2 > *y*" is open, since it contains the free variables *x* and *y*. In contrast, the formula "∃*y* ∀*x*: *x*+2 > *y*" is closed, and has truth value *true*.

Open formulas are often used in rigorous mathematical definitions of properties, like

- "
*x*is an aunt of*y*if, for some person*z*,*z*is a parent of*y*, and*x*is a sister of*z*"

(with free variables *x*, *y*, and bound variable *z*) defining the notion of "aunt" in terms of "parent" and "sister".
Another, more formal example, which defines the property of being a prime number, is

- "
*P*(*x*) if ∀*m*,*n*∈:*m*>1 ∧*n*>1 →*x*≠*m*⋅*n*",

(with free variable *x* and bound variables *m*,*n*).

An example of a closed formula with truth value *false* involves the sequence of Fermat numbers

studied by Fermat in connection to the primality. The attachment of the predicate letter P (*is prime*) to each number from the Fermat sequence gives a set of closed formulae. While they are true for *n* = 0,...,4, no larger value of *n* is known that obtains a true formula, as of 2023^{[update]}; for example, is not a prime. Thus the closed formula ∀*n* *P*(*F*_{n}) is false.

- Wolfgang Rautenberg (2008),
*Einführung in die Mathematische Logik*(in German) (3. ed.), Wiesbaden: Vieweg+Teubner, ISBN 978-3-8348-0578-2 - H.-P. Tuschik, H. Wolter (2002),
*Mathematische Logik – kurzgefaßt*(in German), Heidelberg: Spektrum, Akad. Verlag, ISBN 3-8274-1387-7