To solve the problem, we need to analyze the given statements and implications step by step.
### Step-by-Step Solution:
1. **Understanding the Given Implication**:
We have the statement \( P \implies (Q \implies (R \implies P)) \). This means that if \( P \) is true, then \( Q \) must be true, and if \( Q \) is true, then \( R \) must be true, which ultimately leads back to \( P \).
2. **Analyzing the Second Statement**:
We are also given that \( \neg S \implies R \). This means if \( S \) is false, then \( R \) must be true.
3. **Using the Implications**:
From the first statement, we can infer that if \( P \) is true, then \( Q \) and \( R \) must also be true.
4. **Contraposition of the Second Statement**:
From \( \neg S \implies R \), we can use the contrapositive which states that \( \neg R \implies S \). This means if \( R \) is false, then \( S \) must be true.
5. **Combining the Implications**:
Now, since \( R \) must be true whenever \( \neg S \) is true (from our second statement), we can conclude that if \( S \) is false, then \( R \) is true.
6. **Conclusion on \( Q \)**:
Since \( P \implies Q \implies R \implies P \), if \( R \) is true, then \( Q \) must also be true. Thus, we can conclude that \( \neg S \implies Q \) must also hold true.
7. **Final Result**:
Therefore, we can conclude that if \( \neg S \) is true, then \( Q \) must also be true. Hence, the correct answer among the options provided is that \( \neg Q \implies S \).
### Final Answer:
The correct option is \( \neg Q \implies S \).
