Definition:Occurrence (Predicate Logic)

Context
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:
 * $$P \left({x, y}\right) \implies \forall u: \left({\exists v: R \left({u, v}\right) \implies Q \left({u, y}\right)}\right)$$