1. **Stating the problem:** We are asked to prove the validity of the logical statements Modus Ponens, Modus Tollens, and Syllogism using truth tables and negation. 2. **Recall the definitions:** - Modus Ponens: From $p \to q$ and $p$, conclude $q$. - Modus Tollens: From $p \to q$ and $\neg q$, conclude $\neg p$. - Syllogism: From $p \to q$ and $q \to r$, conclude $p \to r$. 3. **Construct truth tables:** Let’s define columns for $p$, $q$, $r$, $p \to q$, $q \to r$, and the conclusions. | $p$ | $q$ | $r$ | $p \to q$ | $q \to r$ | Modus Ponens ($q$) | Modus Tollens ($\neg p$) | Syllogism ($p \to r$) | |-----|-----|-----|-----------|-----------|--------------------|-------------------------|-----------------------| | T | T | T | T | T | T | F | T | | T | T | F | T | F | T | F | F | | T | F | T | F | T | F | T | T | | T | F | F | F | T | F | T | F | | F | T | T | T | T | T | F | T | | F | T | F | T | F | T | F | F | | F | F | T | T | T | F | T | T | | F | F | F | T | T | F | T | F | 4. **Interpretation:** - Modus Ponens is valid because whenever $p \to q$ and $p$ are true, $q$ is true. - Modus Tollens is valid because whenever $p \to q$ is true and $q$ is false, $p$ is false. - Syllogism is valid because whenever $p \to q$ and $q \to r$ are true, $p \to r$ is true. 5. **Negation:** Negating these tautologies results in contradictions, confirming their validity. 6. **Summary:** All three statements are tautologies and can be used as valid inference rules. --- **Tugas 2:** - Tautology example: $p \lor \neg p$ (Law of excluded middle) - Contradiction example: $p \land \neg p$ (Contradiction) These are fundamental logical truths and falsehoods respectively.