Definition:Occurrence (Predicate Logic)

Definition

Each place where $S$ appears in $\mathbf A$ is called an occurrence of $S$ in $\mathbf A$.

Note that $S$ may consist of a single symbol, but may not be null.

Let $Q$ be an occurrence of a quantifier in $\mathbf A$.

Let $\mathbf B$ be a well-formed part of $\mathbf A$ such that $\mathbf B$ begins (omitting outer parentheses) with $Q x$.

That is, such that $\mathbf B = \paren {Q x: \mathbf C}$ for some WFF $\mathbf C$.

$\mathbf B$ is called the scope of the quantifier $Q$.

Let $Q x$ be an occurrence of a quantifier in $\mathbf A$.

Any occurrence of the variable $x$ in the scope of $Q$ is called a bound occurrence.

An occurrence of a variable $x$ in $\mathbf A$ is said to be a free occurrence if and only if it is not bound.

Consider the (abbreviated) WFF $Q x: \mathbf C$ where $Q$ is a quantifier.

Let $y$ be another variable such that:

$y$ is freely substitutable for $x$ in $\mathbf C$

$y$ does not occur freely in $\mathbf C$.

Let $\mathbf C'$ be the WFF resulting from substituting $y$ for all free occurrences of $x$ in $\mathbf C$.

The change from $Q x: \mathbf C$ to $Q y: \mathbf C'$ is called alphabetic substitution.