# Open formula

An **open formula** is a formula that contains at least one free variable.^{[citation needed]} Some educational resources use the term "**open sentence**",^{[1]}^{[unreliable source?]} but this use conflicts with the definition of "sentence" as a formula that does not contain any free variables.

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 quantifiers or specifying of the domain of discourse of individuals for each free variable denoted x, y, z....or x_{1}, x_{2}, x_{3}.... This transformation is called capture of the free variables to make them bound variables, bound to a domain of individual constants.

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*.

An example of 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 false closed formulae when the rank *n* of the Fermat number is greater than 4. Thus the closed formula ∀*n* *P*(*F*_{n}) is false.

## See alsoEdit

## References and notesEdit

## BibliographyEdit

- 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

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