Induction on Formulas (Induction on the number of connectives in a formula) I am working through the textbook of Richard Hodel on an Introduction to Mathematical Logic. I came accross this theorem 1 on page 52 (picture attached below) and I am stuck at trying to figure out this proof. It does not seem right to me, because the proof is referencing it's own conclusion to prove a statement.
The first such instance occurs in the statement "a formula with no connectives is a propositional variable, and each propositional variable has property Q by (1)"
The second such instance occurs in the statement "Now apply (3) to conclude that ..."
How is it possible the theorem is referencing it's own conclusions to prove the it?
I can't imagine a mistake in this textbook would go unnoticed. I feel that perhaps I am not understanding something correctly about the proof.


The proof of the theorem is allowed to depend on $(1)$, $(2)$, and $(3)$ because they aren't conclusions of the theorem. They're premises. To clarify, here's a restatement of Theorem $1$:

If $Q$ is a property of formulas, and if we assume that $(1)$, $(2)$, and $(3)$ are all true, then it necessarily follows that every formula of propositional logic also has property $Q$.

