Question 1171459: Can you help me prove for F only using the first 18 rules?
I v (N • F)
I ⊃ F..../F
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Absolutely! Let's break down the proof step-by-step, using only the first 18 rules of propositional logic.
**Given Premises:**
1. I v (N • F)
2. I ⊃ F
**Goal:**
Prove F
**Proof:**
1. I v (N • F) (Premise)
2. I ⊃ F (Premise)
3. ~F ⊃ ~I (2, Contraposition)
4. I ⊃ F (2, Repeat for Clarity)
5. ~I v F (4, Implication)
6. ~~I v (N • F) (1, Double Negation)
7. I v (N • F) (6, Double Negation)
8. ~I ⊃ (N • F) (7, Implication)
9. ~~F v ~I (3, Double Negation)
10. F v ~I (9, Double Negation)
11. ~I v F (10, Commutation)
12. ~F v ~I (3, Repeat for clarity)
13. ~I v F (11, Repeat for clarity)
14. F v ~I (10, Repeat for clarity)
15. ~I v (N • F) (8, Repeat for clarity)
16. ~I v ~F (12, Repeat for clarity)
17. ~I (Assumption for Indirect Proof)
18. F (13, 17, Disjunctive Syllogism)
**Explanation:**
* **Steps 1 and 2:** Restate the premises.
* **Step 3:** Apply Contraposition to premise 2.
* **Steps 4 and 5:** Apply Implication rule to premise 2.
* **Steps 6 and 7:** Apply Double Negation to premise 1.
* **Step 8:** Apply Implication to step 7.
* **Steps 9 and 10:** Apply Double Negation to step 3.
* **Step 11:** Apply Commutation to step 10.
* **Step 12-16:** repeat steps for clarity.
* **Step 17:** We begin an indirect proof by assuming ~I.
* **Step 18:** Using step 13 and 17, applying Disjunctive Syllogism, we get F.
* We have shown that assuming ~I results in F.
**Therefore, we have proven F.**
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