An open formula is a formula that contains at least one free variable. Some educational resources use the term "open sentence",[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 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
studied by Fermat in connection to the primality. The attachement 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 ∀Fn P(Fn) is false.
References and notes
- 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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