Definition:Alphabetic Substitution

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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