SOLUTION: Use Indirect Proof to solve the following argument (E v F) ⊃ (C • D) (D v G) ⊃ H E v G /H Thank you!

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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
NumberStatementLines UsedReason
1(E v F) -> (C & D)
2(D v G) -> H
3E v G
ConclusionH
4~HAssumption for Indirect Proof
5~(D v G)2,4Modus Tollens
6~D & ~G5De Morgan’s Law
7~D6Simplification
8~G6Simplification
9E3,8Disjunctive Syllogism
10E v F9Addition
11C & D1,10Modus Ponens
12D11Simplification
13D & ~D7,12Conjunction
14H4-13Indirect 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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