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.

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

$F_{n}=2^{2^{n}}+1,$

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 also

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

Bibliography

• 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