Given a tautology of the form (α → β), where α and β are propositions, I want to prove that there exists another proposition γ such that (α → γ) and (γ → β) are both tautologies, and that γ can only use propositional variables that are either in both α and β, or in neither of them (so can't be in only one of them).
I approach this problem by first noticing that all truth assignments can be broken into 4 cases based on the truth table for logical implication. Since (α → β) is a tautology, we can eliminate the case where α is T and β is F as that's impossible to occur. The other case we don't need to consider is where α is F and β is T, since in this case any proposition can be γ.
Left with the two cases, I need to find γ that will be true when both α and β are true, and false when both are false. I noticed that for an arbitrary truth assignment, there must be a set of variables that are true and are in both alpha and beta, and another set of variables all false and in both alpha and beta. I tried to use this observation but soon realized that different truth assignments can be very different yet giving same results, so I don't think this observation would help. I need some suggestions on how I should proceed.