What do we mean by bound and unbound variable? I have difficulty to understand their meanings, Could anyone help me with that?


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This is the formal definition.

Consider any variable $x$ and a formula $A$.

We say that $x$ occurs free in $A$ when:

1) If $A$ is atomic, $x$ occurs free in $A$ iff $x$ occurs in $A$ (i.e. it is a symbol of $A$).

2) $x$ occurs free in $(¬ A)$ iff $x$ occurs free in $A$.

3) $x$ occurs free in $(A \to B)$ iff $x$ occurs free in $A$ or in $B$ (and the same for the other binary connectives)

4) $x$ occurs free in $∀ y A$ α iff $x$ occurs free in $A$ and $x \ne y$.

If a variable $x$ is a symbol of the formula $A$ but does not occur free in $A$, then it is said to occur bound.

What does a statement with a free variable, like e.g. $x=1$, mean?

The details of the semantics of the language dictate "how to read" a formula with free variables.

Usually, we use a "context" [technically called: variable assignment function], i.e. a way to assign a "temporary meaning" to the free variables.

We can compare a free variable to a pronoun of natural language.

To assert "$x$ is red" is the same as "it is red": its meaning depends on what the context assigns to "it".

In the same way, we can read $x=1$ as follows: "it is equal to one", that is true when we assign to "it" the number one as reference.

  • $\begingroup$ I think the distinguishing feature of "free" variables is that they represent a potential dependency between statements, rather than the presence of quantifiers. Some logics (I think PRA is one) have neither quantifiers nor free variables. I realize it is harder to define though, so any definition of "free variable" in the grammatical sense should probably have a caveat "in FOL". $\endgroup$ – DanielV Sep 26 '17 at 22:30

I typically rely on Mauro for answers to my questions, but I wanted to offer the following:

If you have the formula $(\forall x)(Px \land Ry) \lor Qx$, where capital letters are relations and small case letters are individual variables, then the bound variables is the first x, and the free variables are y and the second x.

In other words...

If $\mathfrak U$ is a wff in which there occurs a certain variable in one or more places, but always in a free form, another wff is obtained by placing the existential or the universal quantifier for this variable before $\mathfrak U$, after $\mathfrak U$ has been enclosed by brackets. The occurrence of the variable is said to be bound in the new formula. $\mathfrak U$ is called the scope of the quantifier.

Ackermann, Solvable Cases of the Decision Problem, 1954

wff=well formed formula.

Note that relations and functions may also be free or bound. In the example above all of the relations are free. Binding such variables is the topic of second-order logic.


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