What Is a "Contrapositive Statement"?

A contrapositive statement is a type of logical statement. Logic, a branch of mathematics or sometimes philosophy, dictates that statements can be either true or false. A contrapositive statement can be either true or false, but it must have the same truth value as its original statement.

Form

• If a statement is expressed as "P => Q," or "If P, then Q," a contrapositive statement then takes the form "~Q => ~P," or "If not Q, then not P."

Examples

• An example of a statement is "If John is sweating, then it is warm outside." The contrapositive of this statement would be "If it is not warm outside, then John is not sweating." A contrapositive does not have to be a true statement. It only needs to share a truth value with the original statement.

Uses

• Contrapositive statements are often used in logical mathematical proofs, as they are sometimes easier to prove than the original statement.