The effect of a quantifier $\text{Q}$ on an occurrence of a variable lying within its scope. For example, with respect to a variable $x$ and formula

• • $\left[\forall x\phi \left(x,y\right)\right]\wedge \psi \left(x\right)$

the first instance of the variable $x$ is said to be bound by the quantifier $\forall x$. The instance of the variable $y$ is not bound because it is a different variable than that within the quantifier and the second instance of the variable $x$ is not bound because it is outside of the scope of $\forall x$. If a variable is not bound by any quantifier, it is said to be free.

In the case of generalized quantifiers, this becomes more subtle, as more than one variable may be bound by a quantifier. For example, one may define a quantifier $\text{M}$ corresponding to the natural language “as many… as”. An appropriate treatment of this quantifier would accept two formulae—each corresponding to one of the properties whose instances are being compared—and bind one variable in each. For example, one might expect the truth condition:

• • $\text{M}xy(\phi (x),\psi (y))$ is true in a model if and only if the cardinality of the set of elements satisfying $\phi \left(x\right)$ is greater than or equal to the cardinality of the set of elements satisfying $\psi \left(y\right)$.

In this case, when $\text{M}xy$ is applied to a pair of formulae $〈\phi ,\psi 〉$, the quantifier binds all occurrences of $x$ appearing in $\phi$ and all occurrences of $y$ appearing in $\psi$.