Question 1157891: III. Use Indirect Proof to solve the following arguments
(K v L) ⊃ (M • N)
(N v O) ⊃ (P • ~K) / ~K
Thank you!
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
The conclusion is ~K
The opposite of this is ~~K, or more simply just K
The idea is to assume ~~K or K is the case and show how this leads to a contradiction. Note how lines 4 and 12 contradict each other, so this is the key to this indirect proof. The term "indirect proof" is the same as "proof by contradiction".
| Number | Statement | Lines Used | Reason |
| 1 | | (K v L) -> (M & N) | | |
| 2 | | (N v O) -> (P & ~K) | | |
| Conclusion | ~K | | |
| 3 | ~~K | | Assumption for Indirect Proof |
| 4 | K | 3 | Double Negation |
| 5 | K v L | 4 | Addition |
| 6 | M & N | 1,5 | Modus Ponens |
| 7 | N & M | 6 | Commutation |
| 8 | N | 7 | Simplification |
| 9 | N v O | 8 | Addition |
| 10 | P & ~K | 2,9 | Modus Ponens |
| 11 | ~K & P | 10 | Commutation |
| 12 | ~K | 11 | Simplification |
| 13 | K & ~K | 4,12 | Conjunction |
| 14 | | ~K | 3-13 | Indirect Proof |
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