Open formula

An open formula is a formula that contains at least one free variable. 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∈[math]\displaystyle{ \mathbb{N} }[/math]: 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

[math]\displaystyle{ F_{n} = 2^{2^n} + 1, }[/math]

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 fur n = 0,...,4, no larger value of n is known that obtains a true formula, (As of 2023); for example, [math]\displaystyle{ F_5 = 4 \,294 \,967 \,297 = 641 \cdot 6\,700\,417 }[/math] is not a prime. Thus the closed formula ∀n P(F_n) is false.

See also

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

References

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