Quantifier has Unique Scope

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Theorem

Let $\mathbf A$ be a WFF of predicate logic.

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


Then there exists a unique well-formed part of $\mathbf A$ which (omitting outermost parentheses) begins with that occurrence $Q$.


This unique well-formed part of $\mathbf A$ is called the scope of the occurrence of $Q$.


Proof

Existence

First, from the rules of formation of predicate logic, we have that whenever a quantifier is included in a WFF, it appears in the form:

$( Q x: \mathbf B )$

where $\mathbf B$ is itself a WFF.

Hence it is clear that $( Q x: \mathbf B )$ is a well-formed part of $\mathbf A$ which begins with $Q$.

$\Box$


Uniqueness

Now we prove that this well-formed part is unique.

Suppose $\mathbf B$ and $\mathbf C$ are both well-formed parts of $\mathbf A$ which begin with the given occurrence of $Q$.

Since $\mathbf B$ and $\mathbf C$ both begin with the same $Q$, neither one can be the initial part of the other, by Initial Part of WFF of Predicate Logic is not WFF.

Therefore, $\mathbf B$ and $\mathbf C$ are necessarily the same.

Hence the result.

$\blacksquare$


Sources