1. **State the problem:** We need to use the indirect method (proof by contradiction) to derive $\neg q$ from the premises: $$p \to (q \lor r), \quad p \to \neg r, \quad p$$ 2. **Assume the opposite of what we want to prove:** Assume $q$ is true (we want to derive a contradiction). 3. **Use the premises with $p$ true:** Since $p$ is true, from $p \to (q \lor r)$ we get: $$q \lor r$$ From $p \to \neg r$, since $p$ is true, we get: $$\neg r$$ 4. **Analyze the disjunction $q \lor r$ knowing $\neg r$ is true:** Since $q \lor r$ is true and $r$ is false ($\neg r$), it follows that: $$q$$ But this was assumed already in step 2. 5. **Conclusion:** No contradiction arises here by assuming $q$. However, since $p \to \neg r$ and $p$ implies $\neg r$, and knowing from $p \to (q \lor r)$ and $p$ that $q \lor r$ is true, to avoid contradiction, $r$ must be false and so $q$ must be true. This means the assumptions are consistent with $q$ being true. **But** since the problem wants us to prove $\neg q$ indirectly, let's check thoroughly: Re-examine the assumptions for contradiction: - From $p$ and $p \to \neg r$, $r$ is false. - From $p$ and $p \to (q \lor r)$, $q \lor r$ true, but $r$ false, so $q$ true. This contradicts deriving $\neg q$, so either the problem has a typo or requires re-evaluation. If the goal is indeed to derive $\neg q$ using indirect proof, assume $q$ is true, then from the premises one obtains $q$ true consistently. Therefore, indirect proof cannot derive $\neg q$ here; the premises imply $q$ is true under $p$. **Summary:** The premises and $p$ imply $q$ is true, so $\neg q$ cannot be derived indirectly without contradiction. Hence, the solution shows $\neg q$ cannot be derived from the given premises and $p$ by the indirect method.