# Unique Readability Theorem of Predicate Calculus

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## Theorem

Each WFF of predicate logic which starts with a left bracket or a negation sign has exactly one main connective.

## Proof

We have the following cases:

- $\mathbf A = \neg \mathbf B$, where $\mathbf B$ is a WFF of length $k$.
- $\mathbf A = \left({\mathbf B \circ \mathbf C}\right)$ where $\circ$ is one of the binary connectives.
- $\mathbf A = p \left({t_1, t_2, \ldots, t_n}\right)$, where $t_1, t_2, \ldots, t_n$ are terms, and $p \in \mathcal P_n$.
- $\mathbf A = ( Q x: \mathbf B )$, where $\mathbf B$ is a WFF of length $k-5$, $Q$ is a quantifier ($\forall$ or $\exists$) and $x$ is a variable.

We deal with these in turn.

Cases 1 and 2 are taken care of by Language of Propositional Logic has Unique Parsability.

Cases 3 and 4 do not start with either a left bracket or a negation sign, so do not have to be investigated.

$\blacksquare$

## Sources

- 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*: $\S 2.2$: Theorem $2.2.4$