Question 1157893: Use Indirect Proof to solve the following argument
(E v F) ⊃ (C • D)
(D v G) ⊃ H
E v G /H
Thank you!
Found 2 solutions by jim_thompson5910, Edwin McCravy:
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
The conclusion is H, so the opposite of this is ~H
We'll use a proof by contradiction
The idea is to assume ~H is the case, and show that it leads to a contradiction. This will then mean the opposite of ~H, which is H, must be the true case instead.
If ~H is the case, then we can use modus tollens on premise 2 to get ~(D v G) which turns into ~D & ~G after using De Morgan's Law.
After using simplification, ~D & ~G turns into ~G. Use ~G with E v G, and you'll get E. Use the disjunctive syllogism rule here.
Now that we have E, use the addition rule to get E v F. Next up, apply modus ponens on premise 1 to get C & D which can be simplified to D.
Now go back to ~D & ~G. We can also simplify this to ~D. But this contradicts the D we got earlier in the last paragraph. This concludes the proof by contradiction and confirms that the argument as presented, with the conclusion H, is a valid argument.
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Here is a derivation table which may be one format your teacher wants you to use
| Number | Statement | Lines Used | Reason |
| 1 | | (E v F) -> (C & D) | | |
| 2 | | (D v G) -> H | | |
| 3 | | E v G | | |
| Conclusion | H | | |
| 4 | ~H | | Assumption for Indirect Proof |
| 5 | ~(D v G) | 2,4 | Modus Tollens |
| 6 | ~D & ~G | 5 | De Morgan’s Law |
| 7 | ~D | 6 | Simplification |
| 8 | ~G | 6 | Simplification |
| 9 | E | 3,8 | Disjunctive Syllogism |
| 10 | E v F | 9 | Addition |
| 11 | C & D | 1,10 | Modus Ponens |
| 12 | D | 11 | Simplification |
| 13 | D & ~D | 7,12 | Conjunction |
| 14 | | H | 4-13 | Indirect Proof |
Note how lines 7 and 12 contradict each other.
Line 14 is the opposite of line 4.
Answer by Edwin McCravy(20054)  (Show Source):
You can put this solution on YOUR website!
1. (E v F) ⊃ (C • D)
2. (D v G) ⊃ H
3. E v G /H
| 4. ~H Assumption for Indirect Proof
| 5. ~(D v G) 2,4, Modus Tollens
| 6. ~D • ~G 5, DeMorgan's Law
| 7. ~D 6, Simplification
| 8. ~D v ~C 7, Addition
| 9. ~(D • C) 8, DeMorgan's Law
|10. ~(C • D) 9, Commutation
|11. ~(E v F) 1,10, Modus Tollens
|12. ~E • ~F 11, DeMorgan's Law
|13. ~E 12, Simplification
|14. G 3,13 Disjunctive Syllogism
|15. ~G • ~D 6, Commutation
|16. ~G 15, Simplification
|17. G • ~G 14,16, Conjunction
18. H Lines 4-17 Indirect Proof
Edwin
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