To prove the argument:

Premises:

1. $P \lor Q$
2. $P \rightarrow \neg R$
3. $R$

Conclusion:

$Q$

We will construct a formal proof using the rules of inference, as outlined.

Proof:

1. $P \lor Q$ — Premise
2. $P \rightarrow \neg R$ — Premise
3. $R$ — Premise
4. $\neg \neg R$ — Double Negation Introduction (DNE) on (3)
5. $\neg P$ — Modus Tollens (MT) using (2) and (4)
6. $Q$ — Disjunction Elimination (DS) on (1) and (5)

Explanation:

1. The first step assumes the disjunction $P \lor Q$.
2. The second step introduces the conditional $P \rightarrow \neg R$.
3. The third step introduces the premise $R$.
4. From $R$, we introduce double negation $\neg \neg R$, allowing the use of Modus Tollens.
5. Using Modus Tollens on $P \rightarrow \neg R$ and $\neg \neg R$, we derive $\neg P$.
6. Finally, using Disjunction Elimination (Modus Tollendo Ponens, DS) on $P \lor Q$ and $\neg P$, we conclude $Q$.

Thus, $Q$ is proven.

Do you have any questions or want further details on any step?

Here are five related questions to explore further:

1. How does Modus Tollens work with conditionals in logical proofs?
2. Can $P \lor Q$ be true if both $P$ and $Q$ are false?
3. How does Disjunction Elimination (DS) help simplify logical arguments?
4. What is the importance of double negation in formal logic?
5. Can we reach the same conclusion if the premises were altered slightly?

Tip: Modus Tollens is often useful when the negation of the conclusion helps derive the negation of the premise.