WFFs of PropLog of Length 1

Theorem

The only WFFs of propositional logic of length $1$ are:

• The letters of the formal grammar of propositional logic $\LL_0$
• The tautology symbol $\top$
• The contradiction symbol $\bot$.

Proof

We refer to the rules of formation.

From $\mathbf W: \T \F$, $\top$ and $\bot$ (both of length 1) are WFFs.

From $\mathbf W: \PP_0$, all elements of $\PP_0$ (all of length 1) are WFFs.

Every other rule of formation of the formal grammar of propositional logic consists of an existing WFF in addition to at least one other primitive symbol.

Hence the result.

$\blacksquare$

Sources

• 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.3$: Induction on Length of Wffs