Let $F,G$ be two distinct atomic formulae. Show that no formula that can be formed from $F,G$ using just the conditional $\to$ is logically equivalent to $\neg F \vee \neg G$. We are then told that we can find a property not had by $\neg F \vee \neg G$ and use induction to show that this property is had by any formula formed from $F,G$ using just $\to$.
Wikipedia defines logically equivalent as so: two statements are equivalent if they have the same truth value in every model
Hence, I would imagine I could just use the standard model of arithmetic, and fine the property that was asked for. However, there is definitely more to this, but I don't know how to do it because I have not ever worked on any induction on complexity problems. There is only a few relevant examples in Boolos and Jeffrey's book, but they all seem very vague to me. Could someone please explain this process to me and show me how one would solve this problem? Thank you.
Edit: This is how my text defines induction
"Base Step: Atomic formulas have the property. Induction Step: If a more complex formula is formed by applying a logical operator to a simpler formula or formulas, then, assuming (as induction hypothesis) that the simpler formula or formulas have the property, so does the more complex formula. The induction step will usually be divided into cases, according as the operator is ∼ or & or ∨ or ∀ or ∃."
I believe the idea is to use induction on the number of times the operator $\to$ appears in the formula.
As a hint: what happens when both $F, G$ are True?