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I was hoping that I could get some help with an indirect proof we are doing in my logic class. It is an introductory class, and this was a homework question that I couldn't get. This isn't course-work that is graded, it is just practice for our exam that is tomorrow! I asked it in class today and the professor worked on it for about 45 minutes but he couldn't get it. He and I are both sure that it is something very simple that we aren't catching.

Here it is: (I will use = to show equivalence, and > to show implication).

1. (R*S) = (G*H)

2. R > S

3. H > G

Conclusion: R=H

The point is to use Reductio ad Absurdum (RAA) to solve this. We can use the 18 Rules of Inference (and Replacement), and also conditional proofs.

Here is one pathway I took that did not steer me towards the answer:

4. ~(R=H) by AP (first thing for the RAA)

5. [(R*S) > (G*H) * [(G*H) > (R*S)] by Equiv 1

6. (R*S) > (G*H) by Simp 5

7. (G*H) > (R*S) by Simp 5

8. ~[(R*H) v (~R*~H)] by Equiv 4

9. ~(R * H) * ~(~R * ~H) by DM 8

10. (~Rv~H) * ~(~R*~H) by DM 9

11. (~R v ~H) by Simp 10

12. ~(~R*~H) by simp 10

13. (RvH) by DM 12

14. (SvG) by CD(2,3,13)

Perhaps there is something to the constructive delimma that is set up on line 14? Also, it seems that I will have to at some point use a conditional proof or something to assume a sentence letter, or something, because I don't see how I can show the contradiction (we do the A*~A on the same line thing) without having some sentence letter to prove another?

I dunno, this should be really easy because this is a basic logic book (A Concise Introduction to Logic by Hurley) and a basic class, and the rest of the proofs are very easy in this section. But this one .. We couldn't get it. Does anyone have a suggestion?

Thank you!

AYCHAMO

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# Help with an indirect proof (RAA)

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