Definition:Freely Substitutable

Definition

Let $\mathbf C$ be a WFF of predicate logic.

Let $x$ be a variable in $\mathbf C$.

Let $\phi \left({y_1, \ldots, y_n}\right)$ be a term in which the variables $y_1, \ldots y_n$ occur.

Then $\phi \left({y_1, \ldots, y_n}\right)$ is freely substitutable for $x$ in $\mathbf C$ if and only if no free occurrence of $x$ occurs in a well-formed part of $\mathbf C$ which is of the form:

$( Q y_i: \mathbf B )$

where $Q$ is a quantifier and $\mathbf B$ is a WFF.

Also known as

One often sees free for as a convenient abbreviation for freely substitutable for.

Also see

• Confusion of Bound Variables, showing what goes wrong when substituting a variable that is not free for another.

Sources

• 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability: $\S 2.3$
• 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\mathrm{II}.8$ Further Semantic Notions: Definition $\mathrm{II.8.9}$