With the nine rules of inference and the ten rules of replacement taught in Lessons 13-17 of Intermediate Logic, we can construct a formal proof for any valid propositional argument. But for the benefit of the logic student, I introduce two additional rules in Lessons 18 and 19: the conditional proof, and the reductio ad absurdum. The conditional proof will often simplify a proof, especially one that has a conditional in the conclusion, making the proof shorter or easier to solve. The reductio ad absurdum method usually does not shorten a proof or make it that much easier to solve, but understanding the concept of reductio is beneficial for purposes outside of formal logic, such as understanding proofs in mathematics and apologetics.
Both conditional proof and reductio ad absurdum start with making assumptions. I want to clarify what happens with those assumptions.
To use conditional proof, you start by assuming the antecedent of a conditional. If you then deduce the consequent using that assumption and the other premises, you can conclude the entire conditional using conditional proof. The conditional proof reasoning can be symbolized like this: p → q, ∴ p ⊃ q. In other words, if the assumed proposition p implies another proposition q, we can conclude if p then q.
One misconception that new logic students often make is thinking that the assumption actually “comes from” some previous step in the proof. They think that the assumption must appear somewhere else in order to make it. This is not the case. The assumed antecedent doesn’t come from anywhere; it is quite simply assumed. I tell my students we get the antecedent from our imagination; from Narnia, Middle Earth, Badon Hill. Using conditional proof, you are allowed to assume any antecedent you wish, as long as you use conditional proof properly from that point on.
A similar point can be made about reductio ad absurdum. To use the reductio method, you assume the negation of some proposition you want to conclude. If from that assumption and the other premises you deduce a self-contradiction, you can then conclude the original (un-negated) assumption. The reductio reasoning can be symbolized like this: ~p → (q • ~q), ∴ p. In other words, if assuming the negation of a proposition leads to a self-contradiction, we can conclude the proposition. But again, the assumption for the reductio does not need to appear elsewhere in the proof. We assume it “out of thin air.” Using reductio, you can assume anything you wish. If that assumption gives you a self-contradiction, you can conclude its opposite.