Question 1179563: I could really use some help. Thank You
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. N ≡ F
2. ~F v ~N
3. D ⊃ N /~(F v D)
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1. M ⊃ (∼B ⊃ J)
2. B ⊃ (~M * ~M)
3. ∼J / ~M
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1. ~X ⊃ ~~O
2. ~X ⊃ A
3. ~(O * A) / X
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! **Problem 1:**
1. N ≡ F (Given)
2. ~F v ~N (Given)
3. D ⊃ N (Given)
4. | F v D (Assumption for Indirect Proof)
5. | | F (Assumption for v Elimination)
6. | | N (1, 5, ≡ Elimination)
7. | | N v ~D (6, v Introduction)
8. | | ~D (2, 7, v Elimination)
9. | | F ⊃ ~D (5, 8, ⊃ Introduction)
10. | | D (Assumption for v Elimination)
11. | | N (3, 10, ⊃ Elimination)
12. | | N v ~D (11, v Introduction)
13. | | ~D (2, 12, v Elimination)
14. | | D ⊃ ~D (10, 13, ⊃ Introduction)
15. | | ~D v ~D (4, 9, 14, v Elimination)
16. | | ~D (15, Tautology)
17. | ~(F v D) (4, 16, ⊃ Introduction)
**Problem 2:**
1. M ⊃ (∼B ⊃ J) (Given)
2. B ⊃ (~M * ~M) (Given)
3. ∼J (Given)
4. | M (Assumption for Indirect Proof)
5. | | ∼B (Assumption for Indirect Proof)
6. | | ∼B ⊃ J (1, 4, ⊃ Elimination)
7. | | J (5, 6, ⊃ Elimination)
8. | | J * ~J (3, 7, * Introduction)
9. | B (5, 8, ~ Introduction)
10. | ~M * ~M (2, 9, ⊃ Elimination)
11. | ~M (10, * Elimination)
12. | M * ~M (4, 11, * Introduction)
13. ~M (4, 12, ~ Introduction)
**Problem 3:**
1. ~X ⊃ ~~O (Given)
2. ~X ⊃ A (Given)
3. ~(O * A) (Given)
4. | ~X (Assumption for Indirect Proof)
5. | ~~O (1, 4, ⊃ Elimination)
6. | O (5, ~~ Elimination)
7. | A (2, 4, ⊃ Elimination)
8. | O * A (6, 7, * Introduction)
9. | (O * A) * ~(O * A) (3, 8, * Introduction)
10. X (4, 9, ~ Introduction)
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