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:

A conditional statement is always logically equivalent to its contrapositive, not the converse or inverse.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The given statement is a simple conditional of the form, if it is raining then peacocks dance. In logical terms, if we let P stand for it is raining and Q stand for peacocks dance, the statement tells us P implies Q, written as \(P \rightarrow Q\). We must find which option is a statement that is logically guaranteed to be true whenever the original statement is true.

Step 2: Key Formula or Approach:
For any conditional statement \(P \rightarrow Q\), the only logically equivalent statement that is always true alongside it is its contrapositive, \(\text{not } Q \rightarrow \text{not } P\). The converse \(Q \rightarrow P\) and the inverse \(\text{not } P \rightarrow \text{not } Q\) are not logically guaranteed by the original statement, even though they might sound similar.

Step 3: Detailed Explanation:
Option (A), Peacocks dance only when it is raining, actually claims that rain is the only possible reason peacocks ever dance, which is the converse type of claim, \(Q \rightarrow P\), and this is not guaranteed by the original statement, since peacocks might dance for other reasons too.

Option (B), When peacocks dance, it is raining, is exactly the converse, \(Q \rightarrow P\). The original sentence never rules out peacocks dancing for reasons other than rain, so this cannot be concluded with certainty.

Option (C), When peacocks are not dancing, it is not raining, translates to \(\text{not } Q \rightarrow \text{not } P\), which is precisely the contrapositive of the original statement \(P \rightarrow Q\). Since a conditional statement and its contrapositive always have the same truth value, this option must necessarily be true whenever the given sentence is true.

Option (D), When it is not raining, peacocks do not dance, translates to \(\text{not } P \rightarrow \text{not } Q\), which is the inverse of the original statement. The inverse is not logically guaranteed by the original conditional, because peacocks could still dance for some unrelated reason even without rain.

Step 4: Final Answer:
Only the contrapositive form, when peacocks are not dancing it is not raining, is logically guaranteed to be true based solely on the given statement.
\[ \boxed{\text{When peacocks are not dancing, it is not raining.}} \]