Free Variables and Bound Variables

Free Variables And Bound Variables

In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation that specifies places in an expression where substitution may take place. The idea is related to a placeholder (a symbol that will later be replaced by some literal string), or a wildcard character that stands for an unspecified symbol.

In computer programming, the term free variable refers to variables used in a function that are not local variables nor parameters of that function. The term non-local variable is often a synonym in this context.

A bound variable is a variable that was previously free, but has been bound to a specific value or set of values. For example, the variable x becomes a bound variable when we write:

'For all x, (x + 1)2 = x2 + 2x + 1.'

or

'There exists x such that x2 = 2.'

In either of these propositions, it does not matter logically whether we use x or some other letter. However, it could be confusing to use the same letter again elsewhere in some compound proposition. That is, free variables become bound, and then in a sense retire from being available as stand-in values for other values in the creation of formulae.

The term "dummy variable" is also sometimes used for a bound variable (more often in general mathematics than in computer science), but that use creates an ambiguity with the definition of dummy variables in regression analysis.