Contrapositive

Statement

Let $p$ and $q$ be propositions. The implication $p \to q$ is logically equivalent to its contrapositive $\neg q \to \neg p$. $$p \to q \equiv \neg q \to \neg p$$

Remarks

The direction from $p \to q$ to $\neg q \to \neg p$ is valid in intuitionistic logic. The other direction needs classical logic. This equivalence is the basis of proof by contrapositive. Do not confuse the contrapositive with the converse $q \to p$.

Explanations

L1 human

"If it rains, the street is wet" says the same as "if the street is not wet, it does not rain". Both sentences are false in exactly one case: rain and a dry street.