# Two Strange Proofs

Mr. Nance,

Could you give real-world examples of the arguments to prove in Intermediate Logic Lesson 18, number 7) U / ∴ W ⊃ W, and number 8) X / ∴ Y ⊃ X, showing how they would be used, or explain them a bit? Thank you.

Thanks for the great question! These two arguments are unusual, so I am not surprised that you are asking about them.

A real-world example for #7 might be Esther 4:16, “I will go to the king which is against the law; if I perish, then I perish!” This argument form basically shows that any proposition implies a tautology.

An example for #8 could be, “God created all things. So even if evolution can be used to explain some fossils, it’s still true that God created all things.” The form of this argument shows that if a proposition is given, any other proposition implies it.

To be honest, my purposes for including those two problems were: 1) to show how very strange the conditional proof is, and 2) to show how this method can be used to simplify otherwise difficult proofs.

Blessings!

# Conditional Proof Assumption

With the nine rules of inference and the ten rules of replacement taught in Lessons 13-17 of Intermediate Logic, any valid propositional argument can be proven. But for the benefit of the logic student, I introduce an additional rule in Lesson 18: the conditional proof. 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. Conditional proof starts with making an assumption. I want to clarify what happens with that assumption.

To use conditional proof, you start by assuming the antecedent of a conditional. If by using that assumption along with the other premises you are able to deduce the consequent, you can conclude the entire conditional using conditional proof. More briefly, if an assumed proposition p implies the proposition q, we can conclude if p then q.

One misconception 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. With conditional proof, you are allowed to assume any antecedent you wish, as long as you use conditional proof correctly from that point on.