Question

‘When it is raining, peacocks dance.’
Based only on this sentence, which one of the following options is necessarily
true?

• A. Peacocks dance only when it is raining.
• B. When peacocks dance, it is raining.
• C. When peacocks are not dancing, it is not raining.
• D. When it is not raining, peacocks do not dance.

Hint:

Remember that a statement if R then D is logically equivalent only to its contrapositive if not D then not R, not to its converse or inverse.

Answer 1

The Correct Option is C

A common trap in this kind of question is confusing a conditional statement with its converse or inverse. The sentence only tells us that rain guarantees peacock dancing, so rain is a sufficient condition for dancing, not that dancing only happens because of rain, and not that no rain guarantees no dancing. So any option that reverses the direction, dancing implies raining, or negates both sides in the same direction, no rain implies no dancing, cannot be logically guaranteed. The only statement that must be true, using the rule that \(P \to Q\) always implies \(not\ Q \to not\ P\), is the one saying that if peacocks are not dancing, it cannot be raining. That is exactly option (C), so it is the necessarily true statement.