Question 1208946
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!