Definition:Substitution for Free Occurrence

Definition
Let $\mathbf C$ be a WFF of predicate calculus.

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

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

Let $\phi \left({y_1, \ldots, y_n}\right)$ be freely substitutable for $x$ in $\mathbf C$.

We denote:


 * $\mathbf C \left({x \,//\, \phi \left({y_1, \ldots, y_n}\right)}\right)$

for the result of replacing all free occurrences of $x$ in $\mathbf C$ by $\phi \left({y_1, \ldots, y_n}\right)$.

This is referred to as the substitution of $\phi \left({y_1, \ldots, y_n}\right)$ for free occurrence of $x$.

Example
Let $\mathbf C$ be the WFF:
 * $R \left({x}\right) \lor \left({Q \left({x}\right) \implies \exists x: P \left({x, z}\right)}\right)$.

Then $\mathbf C \left({x \,//\, u}\right)$ is the WFF:
 * $R \left({u}\right) \lor \left({Q \left({u}\right) \implies \exists x: P \left({x, z}\right)}\right)$.

Note that the second and third occurrences of $x$ in $\mathbf C$ are not free but bound occurrences of $x$.

Also known as
Some sources use the notation $\mathbf C \left({x \gets y}\right)$ for $\mathbf C \left({x \,//\, y}\right)$.

The symbol $\gets$ can be referred to as gets, thus $\mathbf C \left({x \gets y}\right)$ is sometimes voiced as $x$ gets $y$ in $\mathbf C$.

Also see

 * Confusion of Bound Variables, showing that it is essential that $\phi \left({y_1, \ldots, y_n}\right)$ is freely substitutable for $x$