In logic, contraposition is a law, which says that a conditional statement is logically equivalent to its contrapositive. The contrapositive of the statement has its antecedent and consequent inverted and flipped. 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:
- Inversion (the inverse): "If something is not a bat, then it is not a mammal." 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. The inverse here is clearly not true.
- Conversion (the converse): "If something is a mammal, then it is a bat." The converse is actually the contrapositive of the inverse and so always has the same truth value as the inverse, which is not necessarily the same as that of the original proposition.
- Negation: "There exists a bat that is not a mammal. " If the negation is true, the original proposition (and by extension the contrapositive) is untrue. Here, of course, the negation is untrue.
Read more about Contraposition: Simple Proof Using Venn Diagrams, Formal Definition, Simple Proof By Contradiction, More Rigorous Proof of The Equivalence of Contrapositives, Comparisons, Application