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.
Could you give a real world example of where the examples in lesson 18 number 7. U/:. W > W and number 8. X/ :. Y > X would be used or explain them a bit? Thank you.
Hi Felicity,
Thanks for the great question! 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!” 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.”
But 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,
Jim
My students and I had a couple of questions about one of the problems in Lesson 17. In #5, it seems that the antecedent assumed is from Step 1, since the antecedent in the conclusion is negated. But based on the end of the proof, I’m guessing the antecedent is assumed from the conclusion. Where is it explained that the assumption can be non-negated if it appears negated originally?
Also, why is the CP identified as Step 7, when the conclusion isn’t fully deduced until after the Implication?
After debating all of this, my students wanted me to ask if there was a Rule of Replacement assumed at the beginning, to give us the conditional, which was then deduced at Step 7. And if this is the case, they’re asking how can we just opt to replace something without “showing your work” like in math? If we can just do that, why not just assume the Double Negation and skip that step too? 🙂
Or maybe we have this all wrong, but I promised I’d ask.