Question 1208946: Prove that the argument is valid using the method of natural deduction.
1. Q
2. (R horseshoe Q) horseshoe (P dot tilde P) // S
Answer by textot(100)   (Show Source):
You can put this solution on YOUR website! Certainly, let's prove the validity of the argument using the method of natural deduction.
**Argument:**
1. Q
2. (R horseshoe Q) horseshoe (P dot tilde P) // S
**To Prove:** S
**Method:** Natural Deduction
**Proof:**
1. Q (Premise)
2. Assume R (Assumption)
3. Q (Reiteration 1)
4. R horseshoe Q (Conditional Introduction 2, 3)
5. (R horseshoe Q) horseshoe (P dot tilde P) (Premise)
6. P dot tilde P (Modus Ponens 4, 5)
7. P (Simplification 6)
8. tilde P (Simplification 6)
9. P dot tilde P (Conjunction Introduction 7, 8)
10. (R horseshoe Q) horseshoe (P dot tilde P) (Reiteration 5)
11. tilde (R horseshoe Q) (Modus Tollens 9, 10)
12. tilde R (Conditional Negation 2, 11)
13. R horseshoe tilde R (Conditional Introduction 2, 12)
14. S (Explosion 13)
**Explanation:**
1. **Premise:** The first statement of the argument is given as a premise.
2. **Assumption:** We temporarily assume R to be true.
3. **Reiteration:** We reiterate the premise Q.
4. **Conditional Introduction:** From the assumption R and the reiterated Q, we infer R horseshoe Q.
5. **Premise:** The second statement of the argument is given as a premise.
6. **Modus Ponens:** Applying Modus Ponens to lines 4 and 5, we infer P dot tilde P.
7. **Simplification:** We simplify P dot tilde P to obtain P.
8. **Simplification:** We simplify P dot tilde P to obtain tilde P.
9. **Conjunction Introduction:** We combine P and tilde P to obtain P dot tilde P.
10. **Reiteration:** We reiterate the second premise.
11. **Modus Tollens:** Applying Modus Tollens to lines 9 and 10, we infer tilde (R horseshoe Q).
12. **Conditional Negation:** From the assumption R and the derived tilde (R horseshoe Q), we infer tilde R using Conditional Negation.
13. **Conditional Introduction:** From the assumption R and the derived tilde R, we infer R horseshoe tilde R.
14. **Explosion:** Since R horseshoe tilde R is a contradiction, we can infer any statement, including S, using the rule of Explosion.
**Conclusion:**
We have successfully proven that the argument is valid using the method of natural deduction.
**Note:**
* "horseshoe" represents the material conditional (→).
* "tilde" represents negation (¬).
* "dot" represents conjunction (∧).
Let me know if you'd like to explore other proof methods or have any further questions!
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