free and bound variables

In the entry first-order language (, I have mentioned the use of variables without mentioning what variables really are. A variable is a symbol that is supposed to range over the universe of discourse. Unlike a constant, it has no fixed value.

There are two ways in which a variable can occur in a formula: free or bound. Informally, a variable is said to occur free in a formula φ if and only if it is not within the “scope” of a quantifierMathworldPlanetmath. For instance, x occurs free in φ if and only if it occurs in it as a symbol, and no subformula of φ is of the form x.ψ. Here the x after the is to be taken literally : it is x and no other symbol.

Variables in Terms

To formally define free (resp. bound) variables in a formula, we start by defining variables occurring in terms, which can be easily done inductively: let t be a term (in a first-order language), then Var(t) is

  • if t is a variable v, then Var(t) is {v}

  • if t is f(t1,,tn), where f is a function symbol of arity n, and each ti is a term, then Var(t) is the union of all the Var(ti).

Free Variables

Now, let φ be a formula. Then the set FV(φ) of free variablesMathworldPlanetmath of φ is now defined inductively as follows:

  • if φ is t1=t2, then FV(φ) is Var(t1)Var(t2),

  • if φ is R(t1,,tn), then FV(φ) is Var(t1)Var(tn)

  • if φ is ¬ψ, then FV(φ) is FV(ψ)

  • if φ is ψσ, then FV(φ) is FV(ψ)FV(σ), and

  • if φ is xψ, then FV(φ) is FV(ψ)-{x}.

If FV(φ), it is customary to write φ as φ(x1,,xn), in order to stress the fact that there are some free variables left in φ, and that those free variables are among x1,,xn. When x1,,xn appear free in φ, then they are considered as place-holders, and it is understood that we will have to supply “values” for them, when we want to determine the truth of φ. If FV(φ)=, then φ is called a sentenceMathworldPlanetmath. Another name for a sentence is a closed formula. A formula that is not closed is said to be open.

Bound Variables

Bound variables in formulas are inductively defined as well: let φ be a formula. Then the set BV(φ) of bound variables of φ

  • if φ is an atomic formula, then BV(φ) is , the empty setMathworldPlanetmath,

  • if φ is ¬ψ, then BV(φ) is BV(ψ)

  • if φ is ψσ, then BV(φ) is BV(ψ)BV(σ), and

  • if φ is xψ, then BV(φ) is BV(ψ){x}.

Thus, a variable x is bound in φ if and only if xψ is a subformula of φ for some formula ψ.

The set of all variables occurring in a formula φ is denoted Var(φ), and is FV(φ)BV(φ).

Note that it is possible for a variable to be both free and bound. In other words, FV(φ) and BV(φ) are not necessarily disjoint. For example, consider the following formula φ of the lenguage {+,,0,1} of ring theory :


Then FV(φ)={x,y} and BV(φ)={x}: the variable x occurs both free and bound. However, the following lemma tells us that we can always avoid this situation :

Lemma 1. It is possible to rename the bound variables without affecting the truth of a formula. In other words, if φ=x(ψ), or x(ψ), and z is a variable not occurring in ψ, then φz(ψ[z/x]), where ψ[z/x] is the formula obtained from ψ by replacing every free occurrence of x by z.

As a result of the lemma above, we see that every formula is logically equivalent to a formula φ such that FV(φ)BV(φ)=.

Title free and bound variables
Canonical name FreeAndBoundVariables
Date of creation 2013-03-22 12:42:57
Last modified on 2013-03-22 12:42:57
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 24
Author CWoo (3771)
Entry type Definition
Classification msc 03C07
Classification msc 03B10
Synonym occur free
Synonym occur bound
Synonym closed
Synonym open
Related topic Substitutability
Defines free variable
Defines bound variable
Defines free occurrence
Defines bound occurrence
Defines occurs free
Defines occurs bound