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.

See also

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