In logic and mathematics, contraposition, or transposition, refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as § Proof by contrapositive. The contrapositive of a statement has its antecedent and consequent negated and swapped.

Conditional statement ${\displaystyle P\rightarrow Q}$. In formulas: the contrapositive of ${\displaystyle P\rightarrow Q}$ is ${\displaystyle \neg Q\rightarrow \neg P}$.

If P, Then Q. — If not Q, Then not P. "If it is raining, then I wear my coat." — "If I don't wear my coat, then it isn't raining."

The law of contraposition says that a conditional statement is true if, and only if, its contrapositive is true.

Contraposition (${\displaystyle \neg Q\rightarrow \neg P}$) can be compared with three other operations:

Note that if ${\displaystyle P\rightarrow Q}$ is true and one is given that ${\displaystyle Q}$ is false (i.e., ${\displaystyle \neg Q}$), then it can logically be concluded that ${\displaystyle P}$ must be also false (i.e., ${\displaystyle \neg P}$). This is often called the law of contrapositive, or the modus tollens rule of inference.

In the Euler diagram shown, if something is in A, it must be in B as well. So we can interpret "all of A is in B" as:

It is also clear that anything that is not within B (the blue region) cannot be within A, either. This statement, which can be expressed as: