1. **Problem Statement:** We need to state and prove the rule of disjunction syllogism, which is a valid argument form in logic. 2. **Rule of Disjunction Syllogism:** If we have a disjunction $A \lor B$ (meaning "$A$ or $B$"), and we know that $\neg A$ (not $A$) is true, then we can conclude $B$. 3. **Formal Proof Setup:** Given: - $A \lor B$ (Premise 1) - $\neg A$ (Premise 2) To Prove: - $B$ 4. **Proof Steps:** - From $A \lor B$, we know at least one of $A$ or $B$ is true. - From $\neg A$, we know $A$ is false. - Since $A$ is false, and $A \lor B$ is true, $B$ must be true. 5. **Symbolic Proof Using Natural Deduction:** \begin{align*} 1.&\quad A \lor B & \text{Premise} \\ 2.&\quad \neg A & \text{Premise} \\ 3.&\quad \text{Assume } A & \text{Assumption for indirect proof} \\ 4.&\quad \bot & \text{From } 2 \text{ and } 3 \text{ (contradiction)} \\ 5.&\quad B & \text{From } 1, 3, 4 \text{ by disjunction elimination} \end{align*} 6. **Explanation:** The disjunction $A \lor B$ means at least one of $A$ or $B$ is true. If $A$ is false ($\neg A$), then the only way for the disjunction to hold is if $B$ is true. This logical reasoning is the basis of the disjunction syllogism rule. **Final conclusion:** $$B$$