Logical Reasoning: Peacocks Dancing

The question asks what is necessarily true based on the statement: "When it is raining, peacocks dance."

This is a conditional statement. Let:

• P = "It is raining"
• Q = "Peacocks dance"

The original statement can be written in logical notation as: $P \rightarrow Q$ (If P, then Q).

This means that rain is a sufficient condition for peacocks to dance. However, it does not say it's the only condition (necessary condition).

Analyzing the Options

Let's examine each option based on the premise $P \rightarrow Q$:

• Option 1 & 2: "Peacocks dance only when it is raining." / "When peacocks dance, it is raining."
  • This translates to $Q \rightarrow P$ (If Q, then P).
  • This is the converse of the original statement. The converse is not necessarily true. Peacocks might dance for other reasons besides rain.
• Option 3: "When peacocks are not dancing, it is not raining."
  • This translates to $\neg Q \rightarrow \neg P$ (If not Q, then not P).
  • This is the contrapositive of the original statement ($P \rightarrow Q$).
  • The contrapositive is logically equivalent to the original statement. Therefore, if $P \rightarrow Q$ is true, $\neg Q \rightarrow \neg P$ must also be true.
  • This statement is necessarily true.
• Option 4: "When it is not raining, peacocks do not dance."
  • This translates to $\neg P \rightarrow \neg Q$ (If not P, then not Q).
  • This is the inverse of the original statement. The inverse is not necessarily true. The original statement doesn't say what happens when it is *not* raining.

Conclusion

Based on the logical equivalence between a statement and its contrapositive, the statement "When peacocks are not dancing, it is not raining" ($\neg Q \rightarrow \neg P$) is necessarily true if the original statement "When it is raining, peacocks dance" ($P \rightarrow Q$) is true.