1. Problem: Determine the validity of each argument. (a) Given: $p \to q$, $q$. Conclusion: $p$. - This is an example of affirming the consequent, which is invalid. - Just because $q$ is true does not guarantee $p$ is true. (b) Given: $p \lor q$, $p$. Conclusion: $q$. - This is invalid because $p$ being true does not imply $q$ is true. (c) Given: $p \to q$, $\neg p$. Conclusion: $q$. - This is invalid; from $\neg p$ we cannot conclude $q$. (d) Given: $p \to q$, $r \to q$. No conclusion stated, so no validity to check. (e) Given: $p \to q$, $q$. Conclusion: $p$. - Same as (a), invalid (affirming the consequent). (f) Given: $p \to q$, $q \lor r$, conclusion: $r \to \neg p$. - This is a more complex argument; validity depends on further proof. 2. Theorem 12: Law of Contrapositive (Negative Inference) - Given: $p \to q$, $\neg q$. - To prove: $\neg p$. - Proof: If $p$ were true, then $q$ would be true by $p \to q$. - But $\neg q$ is given, so $p$ must be false. - Therefore, $\neg p$. 3. Theorem 13: Law of Disjunctive Inference - Given: $p \lor q$, $\neg p$. - To prove: $q$. - Since $p$ is false and $p \lor q$ is true, $q$ must be true. 4. Theorem 14: Law of Equivalence Inference - Given: $p \leftrightarrow q$, $p$. - To prove: $q$. - Since $p$ and $q$ are equivalent, $p$ true implies $q$ true. 5. Theorem 15: Law of Detachment - Given: $p$, $p \to q$. - To prove: $q$. - Since $p$ is true and $p$ implies $q$, $q$ must be true. Final answers: - (a), (b), (c), (e) are invalid arguments. - Theorems 12, 13, 14, 15 are valid inference rules.