contrapositive

== English == === Etymology === From contra- + positive. === Noun === contrapositive (plural contrapositives) (logic) The inverse of the converse of a given logical implication. ==== Usage notes ==== From a conditional statement, its inverse, its converse, and its contrapositive are defined as follows: Proposition: "If P then Q." ( P → Q {\displaystyle P\rightarrow Q} ) Inverse: "If not P then not Q." ( ¬ P → ¬ Q {\displaystyle \neg P\rightarrow \neg Q} ) Converse: "If Q then P." ( Q → P {\displaystyle Q\rightarrow P} ) Contrapositive: "If not Q then not P." ( ¬ Q → ¬ P {\displaystyle \neg Q\rightarrow \neg P} ) If a conditional statement is true then its contrapositive is, too. Thus, if the statement "If I'm Roman, then I can speak Latin" is true, then it logically follows that the statement "If I can't speak Latin, then I'm not Roman" must also be true. In classical logic (more generally, logics employing the law of excluded middle), a conditional statement is true if and only if its contrapositive is true. In this framework, if the statement "If I can't speak Latin, then I'm not Roman" is true, then the statement "If I'm Roman, then I can speak Latin" must also be true. This only applies to the contrapositive, while the inverse and the converse do not depend on the original statement. ==== Derived terms ==== ==== Translations ==== ==== See also ==== converse inverse