Question 1179864: I. Use an ordinary proof (not conditional or indirect) to solve the following arguments.
1)
1. I v (N • F)
2. I ⊃ F /F
2)
1. P ⊃ ~M
2. C ⊃ M
3. ~L v C
4. (~P ⊃ ~E) • (~E ⊃ ~C)
5. P v ~P /~L
3)
1. O ⊃ (Q • N)
2. (N Ú E) ⊃ S / O ⊃ S
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down each argument step-by-step using ordinary proofs:
**1) Argument:**
1. I v (N • F)
2. I ⊃ F / F
**Proof:**
3. ~I ∨ F (2, Implication)
4. I ∨ (N • F) (1, Copy)
5. ~I ∨ F (3, Copy)
6. F ∨ ~I (5, Commutation)
7. F ∨ (N • F) (4, Resolution, 6)
8. F ∨ (N • F) (7, Copy)
9. F ∨ N (8, Distribution)
10. F (9, Simplification)
**Therefore, F is proven.**
**2) Argument:**
1. P ⊃ ~M
2. C ⊃ M
3. ~L v C
4. (~P ⊃ ~E) • (~E ⊃ ~C)
5. P v ~P / ~L
**Proof:**
6. ~P ∨ ~M (1, Implication)
7. ~C ∨ M (2, Implication)
8. ~P ⊃ ~E (4, Simplification)
9. ~E ⊃ ~C (4, Simplification)
10. P ∨ ~P (5, Copy)
11. P (10, Tautology)
12. ~M (6, 11, Modus Ponens)
13. ~C (7, 12, Modus Tollens)
14. ~L (3, 13, Disjunctive Syllogism)
**Therefore, ~L is proven.**
**3) Argument:**
1. O ⊃ (Q • N)
2. (N ∨ E) ⊃ S / O ⊃ S
**Proof:**
3. ~O ∨ (Q • N) (1, Implication)
4. N ∨ E ⊃ S (2, Copy)
5. ~O ∨ Q (3, Simplification)
6. ~O ∨ N (3, Simplification)
7. N ∨ E (6, Addition)
8. S (4, 7, Modus Ponens)
9. ~O ∨ S (6, 8, Hypothetical Syllogism)
10. O ⊃ S (9, Implication)
**Therefore, O ⊃ S is proven.**
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