Question 1179695: INSTRUCTIONS: Use natural deduction to derive the conclusion in each problem.
Prove this using natural deduction.
NOTE: Use * for dot, v for wedge, ~ for tilde, = for triple bar (or copy and paste ≡), and > for horseshoe (or copy and paste ⊃ )
1. M ⊃ (∼B ⊃ J)
2. B ⊃ (~M * ~M)
3. ∼J / ~M
Answer by RBryant(14)   (Show Source):
You can put this solution on YOUR website!
Here is a proof using the Introduction and Elimination Rules of Natural Deduction:
1. M ⊃ (∼B ⊃ J)
2. B ⊃ (~M * ~M)
3. ∼J / ∴ ~M
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4. M Assumption
5. ~B ⊃ J 1,4 ⊃E
6. B Assumption
7. ~M & ~M 2,6 &I
8. ~M 7 &E
9. M & ~M 4,8 &I
10. ~B 6-9 ~I
11. B & ~B 6,10 &I
12. ~M 4-11 ~I
QED
HERE is a Proof using Copi Rules:
1. M ⊃ (∼B ⊃ J)
2. B ⊃ (~M * ~M)
3. ∼J / ∴ ~M
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4. B ⊃ (~M & ~M) 2 Tautology
5. ~(~B ⊃ J) ⊃ ~M 4 Transposition
6. (~B ⊃ J) v ~M 5 Material Implication
7. ~ ~B v J v ~M 6 Material Implication
8. B v J v ~M 7 Double Negation
9. (B v ~M) v J 8 Association
10. B v ~M 9 Disjunctive Syllogism
11. ~B ⊃ ~M 10 Material Implication
12. M ⊃ B 11 Transposition
13. M ⊃ ~M 4,12 Hypothetical Syllogism
14. ~M v ~M 13 Material Implication
15. ~M 14 Tautology
QED
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