While composing the following question, I had an "ah-ha" moment. I still want to post the question along with my answer to show what I have learned. Any comments or additional answers will be greatly appreciated.
Recently I encountered a theorem in An Introcution to Mathematical Logic and Type Theory: To Truth through Proof by Peter B. Andrews where the author states that it can be used in inductive proofs without using integers. The theorem is stated as
1000 Principle of Induction on the Construction of a Wff. Let $\mathscr{R}$ be a property of formulas, and let $\mathscr{R}(\mathbf{A})$ mean that $\mathbf{A}$ has property $\mathscr{R}$. Suppose
(1) $\mathscr{R}(\mathbf{q})$ for each propositional variable $\mathbf{q}$.
(2) Whenever $\mathscr{R}(\mathbf{A})$, then $\mathscr{R}(\mathord{\sim} \mathbf{A})$.
(3) Whenever $\mathscr{R}(\mathbf{A})$ and $\mathscr{R}(\mathbf{B})$, then $\mathscr{R}([\mathbf{A} \lor \mathbf{B}])$.
Then every wff has property $\mathscr{R}$.
I am quite familiar with mathematical induction on integers. Mine typically take the following form:
Blah, blah, blah. Therefore, the statement holds for $n=1$.
Now assume that the statement is true for some integer $n$. Now consider the statement for $n+1$. Yadda, yadda, yadda (eventually transforming the statement for $n+1$ into a statement involving $n$). Therefore, the statement holds for $n+1$.
I'm also comfortable with using strong induction:
Blah, blah, blah. Therefore the statement holds for $n=1$.
Now assume that the statement is true for all integers $k$ such that $1 \le k \le n$ for some integer $n$. Now consider the statement for $n+1$. Yadda, yadda, yadda (eventually transforming the statement for $n+1$ into a statement involving integers $k$ between $1$ and $n$). Therefore, the statement holds for $n+1$.
One example where I would use this is in graph theory to prove a statement about trees. I proceed by induction on the number of vertices in a tree. This maps the object of interest (a tree) to the integers.
When I tried to apply the theorem about wffs, I got stuck how to proceed. How do I apply this theorem to a proof?