We are tasked with proving that $R \rightarrow S$ is a valid conclusion from the premises:

1. $\neg P \vee Q$ (Premise 1)
2. $\neg Q \vee R$ (Premise 2)
3. $R \rightarrow S$ (Premise 3)

Step-by-Step Proof

Let's break down the logic step by step.

1. Premise: $\neg P \vee Q$

This is equivalent to saying, "If $P$ is false, then $Q$ must be true."

2. Premise: $\neg Q \vee R$

This means, "If $Q$ is false, then $R$ must be true."

3. Premise: $R \rightarrow S$

This means, "If $R$ is true, then $S$ must be true."

We need to show that $R \rightarrow S$ is a valid conclusion from the first two premises. The conclusion essentially states that if $R$ is true, then $S$ will follow.

Step-by-Step Proof by Contradiction

We can assume $R$ is true and show that $S$ must follow using the premises.

Assume $R$ is true:

• From Premise 3, if $R$ is true, then $S$ must be true (direct conclusion from $R \rightarrow S$).

If $R$ is false:

We want to show what happens if $R$ were false.

1. From Premise 2: $\neg Q \vee R$, if $R$ is false, then $\neg Q$ must be true, meaning $Q$ is false.
2. If $Q$ is false, we can go back to Premise 1: $\neg P \vee Q$. Since $Q$ is false, $\neg P$ must be true, meaning $P$ is false.

At this point, we have $P$ false, $Q$ false, and $R$ false, which contradicts the premise that $R \rightarrow S$.

Therefore, $R$ must be true.

Once we have shown that $R$ is true, $R \rightarrow S$ directly leads to the conclusion that $S$ is true, which proves that $R \rightarrow S$ is a valid conclusion.

Conclusion

Thus, we have shown that $R \rightarrow S$ logically follows from the premises $\neg P \vee Q$, $\neg Q \vee R$, and $R \rightarrow S$. Therefore, $R \rightarrow S$ is a valid conclusion.

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Would you like any further clarification on any part of the proof? Here are some related questions to explore:

1. What is the significance of proof by contradiction in logic?
2. How does disjunction ($\vee$) interact with implications?
3. Can you explain how the truth table for $R \rightarrow S$ looks?
4. What happens if $P$ is true in this scenario?
5. How can these logical principles apply to real-world situations?

Tip: When proving logical conclusions, looking for contradictions can be a powerful technique to show the necessity of a statement.