---
tag: 04T1CT7212
kind: statement
role: theorem
name: Contrapositive
aliases: [contraposition, law-of-contraposition]
msc: [03B05]
uses: [0ZNX6DQ3EJ, 0XR7A7RSXB, 06GYGS41JK]
sources:
  - url: https://en.wikipedia.org/wiki/Contraposition
    locator: lead section
---

## Statement

Let $p$ and $q$ be [[propositions|0ZGC7ZESYN]].
The [[implication|0ZNX6DQ3EJ]] $p \to q$ is [[logically equivalent|06GYGS41JK]] 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|0BB5TPGQ1N]].
Do not confuse the contrapositive with the [[converse|0RVYE75E2M]] $q \to p$.

## 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.