Question 1206425: Premise:
1.
F
Conclusion:
(G ⊃ H) ∨ (~G ⊃ J)
Use either indirect proof or conditional proof (or both) and the eighteen rules of inference to derive the conclusion of the following symbolized argument.
Found 2 solutions by Edwin McCravy, mccravyedwin:
Answer by Edwin McCravy(20060)   (Show Source):
You can put this solution on YOUR website!
I am not sure what symbol you use for conjunction (AND), maybe a dot,
but I will use &.
Premise:
1. F
Conclusion:
(G ⊃ H) ∨ (~G ⊃ J)
| 2. ~[(G ⊃ H) ∨ (~G ⊃ J)] Assumption for Indirect Proof
| 3. ~(G ⊃ H) & ~(~G ⊃ J) 2, DeMorgan's Law
| 4. ~(~G V H) & ~(~~G ∨ J) 3, Material Implication (twice)
| 5. ~(~G V H) & ~(G ∨ J) 4, Double Negation
| 6. (~~G & ~H) & (~G & ~J) 5, DeMorgan's Law (twice)
| 7. (G & ~H) & (~G & ~J) 6, Double Negation
| 8. (G & ~H) & [~G & ~J] 7, Changing () to [] for clarity
| 9. [(G & ~H) & ~G] & ~J 8, Association
|10. [G & (~H & ~G)] & ~J 9, Association
|11. [G & (~G & ~H)] & ~J 10, Commutation
|12. [(G & ~G) & ~H] & ~J 11, Association
|13. (G & ~G) & [~H & ~J] 12, Association
|14. G & ~G 13, Simplification
15. F lines 2-14 for Indirect Proof
Comment: This is a case where the conclusion is a tautology, and since a
tautology is always true, then the conclusion is always true. So regardless
of what we are given as premises (in this case only F), the conclusion will
always be true.
Edwin
Answer by mccravyedwin(408)  (Show Source):
You can put this solution on YOUR website!
This student has informed me that she is not allowed to use material
implication as I did in the above.
(P ⊃ Q) <=> (~P ∨ Q)
Why teachers don't just prove this early on is beyond me. It makes
proofs a lot simpler.
They wouldn't even need to prove equivalence, they could just prove
(P ⊃ Q) ⊃ (~P ∨ Q), and that would make lots of proofs easier. It's
easy to prove indirectly, anyway.
First we prove
premise
1. P ⊃ Q conclusion ~P ∨ Q
|2. ~(~P ∨ Q) Assumption for Indirect Proof
|3. ~~P & ~Q 2, DeMorgan's law
|4. P & ~Q 3, Double negation
|5. P 4, Simplification
|6. Q 1,5, Modus Ponens
|7. ~Q & P 4, commutation
|8. ~Q 7, Simplification
|9. Q & ~Q 6,8, Conjunction
~P ∨ Q lines 2-9 Indirect proof.
Therefore (P ⊃ Q) ⊃ (~P ∨ Q)
Now we reverse the conclusion and premise:
premise
1. ~P ∨ Q conclusion P ⊃ Q
|2. P Assumption for Conditional Proof
|3. ~~P 2, Double negation
|4. Q 1,3, Disjunctive syllogism
5. P ⊃ Q lines 2-4 Conditional proof.
Therefore (~P ∨ Q) ⊃ (P ⊃ Q)
So we have proved
[(P ⊃ Q) ⊃ (~P ∨ Q)] & [(~P ∨ Q) ⊃ (P ⊃ Q)]
thus they are equivalent. But all we need is the first part,
(P ⊃ Q) ⊃ (~P ∨ Q)
Edwin
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