Step 1: Understanding the Concept:
Read the sentence as a simple rule: rain forces dancing. Write it as $R\to D$, where $R$ is "it rains" and $D$ is "peacocks dance." We only know this one rule is true, nothing else about peacocks or rain.
Step 2: Key Formula or Approach:
Think of the cases for $R$ and $D$ that keep the rule $R\to D$ true. The rule is broken only in the case where it rains and peacocks do not dance, $R$ true and $D$ false. The three other cases, true-true, false-true, and false-false, all keep the rule true. Any option must hold in all three of these surviving cases to be called necessarily true.
Step 3: Detailed Explanation:
Take options (A) and (B), which both claim dancing forces rain, $D\to R$. In the surviving case where it does not rain but the peacocks dance anyway, false-true, this claim breaks, so it is not guaranteed. Take option (D), which claims no rain forces no dancing, $\lnot R\to\lnot D$. That same false-true case, no rain yet peacocks dancing, breaks this claim too. Now take option (C), which claims no dancing forces no rain, $\lnot D\to\lnot R$. Check it against every surviving case: true-true has $D$ true so the claim's condition does not apply, false-true again has $D$ true so it does not apply, and false-false has $D$ false and $R$ false, so the claim holds. In every surviving case, option (C) never fails.
Step 4: Final Answer:
Only option (C), "When peacocks are not dancing, it is not raining," survives every case allowed by the original rule, so it is the necessarily true statement.