# Language of Propositional Logic has Unique Parsability/Lemma

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## Lemma

Let $\mathcal L_0$ be the language of propositional logic.

Let $\mathbf A$ be a WFF.

Suppose that $\mathbf A = \left({B \circ C}\right) = \left({D * E}\right)$.

Then $\mathbf B = \mathbf D$, ${\circ} = {*}$, and $\mathbf C = \mathbf E$.

## Proof

The WFFs $\mathbf B$ and $\mathbf D$ are strings which both start in the same place, right after the first left bracket in $\mathbf A$.

By Initial Part of WFF of PropLog is not WFF, neither $\mathbf B$ nor $\mathbf D$ can be an initial part of the other.

Therefore $\mathbf B = \mathbf D$.

It follows that $* = \circ$ and $\mathbf C = \mathbf E$.

Hence the result.

$\blacksquare$

## Sources

- 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): $\S 1.4$: Main Connective: Theorem $1.4.2$