Invalid.

Here's the proof for why this argument is invalid:

Premises:

1. $(p \land q) \to r$ (Premise 1)
2. $(r \lor s) \to p$ (Premise 2)
3. $r \land \neg p$ (Premise 3)

Conclusion to Prove:

$s \lor q$

Analysis:

Step 1: Analyzing Premise 3

From $r \land \neg p$, we know that:

• $r$ is true.
• $p$ is false.

Step 2: Implications from Premise 2

Consider $(r \lor s) \to p$. Since $p$ is false (from Step 1), the only way for $(r \lor s) \to p$ to hold true is if $r \lor s$ is false. Therefore:

• $r \lor s$ must be false.
• Since $r$ is true (from Step 1), it follows that $s$ must be false to make $r \lor s$ false.

Thus, $s$ is false.

Step 3: Implications from Premise 1

Now consider $(p \land q) \to r$. Since $r$ is true (from Step 1), this tells us nothing about $p \land q$, because the implication is satisfied when the consequent $r$ is true, regardless of whether $p \land q$ is true or false.

Step 4: Final Conclusion

Given that $s$ is false (from Step 2), for $s \lor q$ to be true, $q$ must be true. However, none of the premises give us any information about the truth value of $q$. It could be either true or false.

Therefore, the argument does not guarantee the truth of $s \lor q$. Since the conclusion $s \lor q$ cannot be definitively derived from the premises, the argument is invalid.

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Would you like further details or clarifications? Here are some related questions:

1. How would the truth table for this argument look?
2. Can you prove the argument is valid using natural deduction?
3. What happens if $p$ is true in Premise 3?
4. Can we modify the premises to make the argument valid?
5. How can we represent this argument symbolically in terms of a proof system?

Tip: Always verify the truth of each proposition step by step to see if the conclusion follows logically from the premises.