Open formula

An open formula is a formula that contains at least one free variable. Some educational resources use the term "open sentence", 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 x₁, x₂, x₃.... 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.

Mathematical notations with free and bound variables

$\sum _{i=1}^{n}a_{i}$	(Sum with a finite number of terms)	$i$ is bound, $n$ and $a$ are free
$\int _{a}^{b}f(x)\,\mathrm {d} x$	(Definite integral)	$x$ is bound, $a$, $b$ and $f$ are free
$\lim _{n\to \infty }a_{n}$	(Limit of a sequence for infinite sequences)	$n$ is bound, $a$ is free
$\lim _{x\to x_{0}}f(x)$	(Limit of a function for a Function at the value $x_{0}$)	$x$ is bound, $x_{0}$ and $f$ are free

See also

• First-order logic
• Higher-order logic
• Quantifier (logic)
• Predicate (mathematical logic)

Bibliography

• Wolfgang Rautenberg (2008), [springerlink.com Einführung in die Mathematische Logik] (in German) (3. ed.), Wiesbaden: Vieweg+Teubner, ISBN 978-3-8348-0578-2 {{citation}}: Check |url= value (help)
• H.-P. Tuschik, H. Wolter (2002), Mathematische Logik – kurzgefaßt (in German), Heidelberg: Spektrum, Akad. Verlag, ISBN 3-8274-1387-7