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 logically equivalent only to its contrapositive, not to its converse or inverse.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The sentence "When it is raining, peacocks dance" is a conditional statement of the form "if it is raining, then peacocks dance." In logical terms, if we let \(R\) stand for "it is raining" and \(D\) stand for "peacocks dance," the sentence says \(R \rightarrow D\). We are asked which option is necessarily true, meaning which option is guaranteed by this one sentence alone, without adding extra assumptions.

Step 2: Key Formula or Approach:
A conditional statement \(R \rightarrow D\) is logically equivalent only to its contrapositive, "not \(D\) implies not \(R\)," written \(\lnot D \rightarrow \lnot R\). It is not equivalent to its converse, \(D \rightarrow R\), and it is not equivalent to its inverse, \(\lnot R \rightarrow \lnot D\). Only a statement matching the contrapositive can be called necessarily true.

Step 3: Detailed Explanation:
Option (A), "Peacocks dance only when it is raining," says dancing happens only under rain, which is the converse \(D \rightarrow R\). This is not guaranteed, since peacocks could dance for other reasons too. Option (B), "When peacocks dance, it is raining," is the same converse statement in different words, and is also not guaranteed. Option (D), "When it is not raining, peacocks do not dance," is the inverse \(\lnot R \rightarrow \lnot D\), which again does not follow from the original sentence, since peacocks might still dance without rain. Option (C), "When peacocks are not dancing, it is not raining," says \(\lnot D \rightarrow \lnot R\), which is exactly the contrapositive of the original statement \(R \rightarrow D\), and the contrapositive of a true conditional is always true.

Step 4: Final Answer:
The statement that is necessarily true is option (C), "When peacocks are not dancing, it is not raining."