1. **State the problem:** We need to check the validity of the argument given about the party, neighbors, and police contact. 2. **Identify the propositions:** - Let $P$ = "The party was shut down." - Let $L$ = "There was loud music." - Let $N$ = "Neighbors contacted the police." 3. **Translate the argument into logical form:** - Premise 1: Neighbors did not contact police, so $\neg N$. - Premise 2: Volume was not excessive, so $\neg L$. - Premise 3: If absence of loud music justifies shutdown, then neighbors contacted police. Formally, if $\neg L \to P$, then $N$. - Premise 4: Neighbors did not contact police, $\neg N$. 4. **Analyze the conditional:** The statement "if the absence of loud music were sufficient to justify shutdown, then neighbors contacted police" can be written as: $$ (\neg L \to P) \to N $$ Given $\neg N$, by contrapositive: $$ \neg N \to \neg (\neg L \to P) $$ Since $\neg N$ is true, it follows that: $$ \neg (\neg L \to P) $$ 5. **Evaluate $\neg (\neg L \to P)$:** Recall that $A \to B$ is logically equivalent to $\neg A \lor B$. So, $$ \neg L \to P \equiv L \lor P $$ Therefore, $$ \neg (\neg L \to P) \equiv \neg (L \lor P) \equiv \neg L \land \neg P $$ 6. **Given $\neg L$ (no loud music), from above we get:** $$ \neg L \land \neg P $$ This means: - No loud music ($\neg L$) - Party was not shut down ($\neg P$) 7. **Conclusion:** The argument concludes that the party was not shut down, which matches $\neg P$. **Therefore, the argument is valid.** **Summary:** - Neighbors did not contact police ($\neg N$) - Volume was not excessive ($\neg L$) - If absence of loud music justified shutdown, neighbors would have contacted police ($(\neg L \to P) \to N$) - Since neighbors did not contact police, the absence of loud music does not justify shutdown ($\neg (\neg L \to P)$) - Given no loud music, the party was not shut down ($\neg P$) Hence, the logical reasoning is consistent and valid.