Definition:Occurrence (Predicate Logic)

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

Occurrence
Let $S$ be a string in the alphabet of predicate calculus.

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

See Occurrence (Formal Systems).

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

Scope
Let $x$ be a variable of predicate calculus.

Let $Q$ be a quantifier, i.e. either $\forall$ or $\exists$, such that $Q x$ occurs in $\mathbf A$.

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

(Then $\mathbf B = Q x: \mathbf C$ for some WFF $\mathbf C$.)

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

The scope of a given quantifier is unique, from Quantifier has Unique Scope.

Bound Occurrence
Every occurrence of $x$ in $\mathbf B = Q x: \mathbf C$ is called a bound occurrence of $x$ in $\mathbf A$.

A variable which occurs as a bound occurrence is called a bound variable or dummy variable.

Alphabetic Replacement
The meaning of the expression $\mathbf B = Q x: \mathbf C$ does not change if $x$ is replaced by another symbol.

That is, $\mathbf B = Q x: \mathbf C$ means the same thing as $\mathbf B = Q y: \mathbf C$ or $\mathbf B = Q \alpha: \mathbf C$. And so on.

Thus we may change the bound occurrences of any variable for another variable symbol.

This change is called alphabetic change of a bound occurrence.

Free Occurrence
Any occurrence of $x$ in $\mathbf A$ which is not a bound occurrence is called a free occurrence of $x$ in $\mathbf A$.

A variable which occurs as a free occurrence is called a free variable.

Example
Take the WFF:
 * $P \left({x, y}\right) \implies \forall x: \left({\exists y: R \left({x, y}\right) \implies Q \left({x, y}\right)}\right)$

The first occurrence of $x$ is free.

The other three occurrences of $x$ are bound.

The first and last occurrences of $y$ are free.

The second and third occurrences of $y$ are bound.

The scope of the quantifier $\forall$ is:
 * $\forall x: \left({\exists y: R \left({x, y}\right) \implies Q \left({x, y}\right)}\right)$

The scope of the quantifier $\exists$ is:
 * $\exists y: R \left({x, y}\right)$

By making the alphabetic changes of the bound occurrences of $x$ with $u$, and of $y$ with $v$, we get:
 * $P \left({x, y}\right) \implies \forall u: \left({\exists v: R \left({u, v}\right) \implies Q \left({u, y}\right)}\right)$