# Definition:Alphabetic Substitution

## Definition

Let $\mathbf C$ be a WFF of the language of predicate logic $\mathcal L_1$.

Consider the (abbreviated) WFF $Q x: \mathbf C$.

Let $y$ be another variable such that $y$ does not occur in $\mathbf C$.

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

Then to all intents and purposes, the WFFs:

- $Q x: \mathbf C$
- $Q y: \mathbf C'$

will have the same interpretation.

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

This change is called **alphabetic substitution**.

## Also known as

Some sources refer to this process as **alphabetic replacement**.

## Caution

As a consequence of the formal grammar of $\mathcal L_1$, it is essential that only the *free* occurrences of $x$ are replaced.

If this is not adhered to, the statement of the WFF may change, in much the same way as demonstrated on Confusion of Bound Variables.

## Note

In practice, the method of **alphabetic substitution** will be employed mainly to avoid dealing with expressions like:

- $\paren {\exists x: \paren {\forall y: x = y \lor \paren {\exists x: x > y} } }$

where the variable $x$ is bound twice.

It is a formal way of ensuring that such erratic (although well-defined by the parentheses) statements have to be dealt with in practical situations.

## Example

Take the WFF:

- $\map P {x, y} \implies \forall x: \paren {\exists y: \map R {x, y} \implies \map Q {x, y} }$

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:

- $\forall x: \paren {\exists y: \map R {x, y} \implies \map Q {x, y} }$

The scope of the quantifier $\exists$ is:

- $\exists y: \map R {x, y}$

By making the **alphabetic changes** of the bound occurrences of $x$ with $u$, and of $y$ with $v$, we get:

- $\map P {x, y} \implies \forall u: \paren {\exists v: \map R {u, v} \implies \map Q {u, y} }$

## Also see

It can be seen that **alphabetic substitution** is a specific example of a substitution for free occurrences.

## Sources

- 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 1$: Some mathematical language: Variables and quantifiers - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*: $\S 2.3$