Question 1178947: 1. ~(U v R)
2. (~R v N) ⊃ (P • H)
3. Q ⊃ ~H /~Q
Answer by CPhill(1987)   (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 ~Q from the given premises.
**1. Premises:**
1. ~(U v R)
2. (~R v N) ⊃ (P • H)
3. Q ⊃ ~H
**2. Derivations:**
4. ~U • ~R (De Morgan's Law on premise 1)
5. ~R (Simplification on line 4)
6. ~R v N (Addition on line 5)
7. P • H (Modus Ponens on lines 2 and 6)
8. H (Simplification on line 7)
9. ~Q (Modus Tollens on lines 3 and 8)
**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 addition on line 5. If A is true, then (A v B) is also true, regardless of the truth value of B.
* **Step 7:** We used Modus Ponens on lines 2 and 6. Modus Ponens states that if (A ⊃ B) is true and A is true, then B is true.
* **Step 8:** We used simplification on line 7, the same as step 5.
* **Step 9:** We used Modus Tollens on lines 3 and 8. Modus Tollens states that if (A ⊃ B) is true and ~B is true, then ~A is true.
**Conclusion:**
We were able to derive ~Q from the given premises. Therefore, the argument is **valid**.
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