Question 1178946: 1. ~(K v F)
2. ~F ⊃ (K v C)
3. (G v C) ⊃ ~H /~(K v H)
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's analyze this logical argument to determine its validity. We'll use a proof by deduction to see if we can derive ~(K v H) from the given premises.
**1. Premises:**
1. ~(K v F)
2. ~F ⊃ (K v C)
3. (G v C) ⊃ ~H
**2. Derivations:**
4. ~K • ~F (De Morgan's Law on premise 1)
5. ~F (Simplification on line 4)
6. K v C (Modus Ponens on lines 2 and 5)
7. ~K (Simplification on line 4)
8. C (Disjunctive Syllogism on lines 6 and 7)
9. G v C (Addition on line 8)
10. ~H (Modus Ponens on lines 3 and 9)
11. ~K • ~H (Conjunction on lines 7 and 10)
12. ~(K v H) (De Morgan's Law on line 11)
**Explanation of Steps:**
* **Step 4:** We applied De Morgan's Law to premise 1. De Morgan's Law states that ~(A v B) is equivalent to ~A • ~B.
* **Step 5:** We used simplification on line 4. If a conjunction (A • B) is true, then both A and B are true individually.
* **Step 6:** We used Modus Ponens on lines 2 and 5. Modus Ponens states that if (A ⊃ B) is true and A is true, then B is true.
* **Step 7:** We used simplification on line 4, just as in step 5.
* **Step 8:** We used Disjunctive Syllogism on lines 6 and 7. Disjunctive Syllogism states that if (A v B) is true and ~A is true, then B is true.
* **Step 9:** We used addition on line 8. If A is true, then (A v B) is also true, regardless of the truth value of B.
* **Step 10:** We used Modus Ponens on lines 3 and 9. Modus Ponens states that if (A ⊃ B) is true and A is true, then B is true.
* **Step 11:** We used conjunction on lines 7 and 10. If A and B are both true, then (A • B) is true.
* **Step 12:** We applied De Morgan's Law to line 11. De Morgan's Law states that ~A • ~B is equivalent to ~(A v B).
**Conclusion:**
We were able to derive ~(K v H) from the given premises. Therefore, the argument is **valid**.
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