Contraposition

In logic, contraposition is an inference that says that a conditional statement is logically equivalent to its contrapositive. The contrapositive of the statement has its antecedent and consequent inverted and flipped: the contrapositive of ${\displaystyle P\rightarrow Q}$ is thus ${\displaystyle \neg Q\rightarrow \neg P}$. For instance, the proposition "All bats are mammals" can be restated as the conditional "If something is a bat, then it is a mammal". Now, the law says that statement is identical to the contrapositive "If something is not a mammal, then it is not a bat."

The contrapositive can be compared with three other relationships between conditional statements:

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

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