Contraposition

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 ${\displaystyle \neg Q\rightarrow \neg P}$.^[1]

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.^[2]

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

Inversion (the inverse), ${\displaystyle \neg P\rightarrow \neg Q}$

"If it is not raining, then I don't wear my coat." Unlike the contrapositive, the inverse's truth value is not at all dependent on whether or not the original proposition was true, as evidenced here.

Conversion (the converse), ${\displaystyle Q\rightarrow P}$

"If I wear my coat, then it is raining." The converse is actually the contrapositive of the inverse, and so always has the same truth value as the inverse (which as stated earlier does not always share the same truth value as that of the original proposition).

Negation (the logical complement), ${\displaystyle \neg (P\rightarrow Q)}$

"It is not the case that if it is raining then I wear my coat.", or equivalently, "Sometimes, when it is raining, I don't wear my coat." If the negation is true, then the original proposition (and by extension the contrapositive) is false.

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.^[3]