contrapositive

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Contraposition

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Etymology tree

Proto-Indo-European *ḱe?

Proto-Indo-European *ḱóm

Proto-Italic *kom

Proto-Italic *kom-

Proto-Indo-European *-teros

Proto-Italic *-teros

Proto-Italic *komterosder.

Proto-Italic *komterād

Latin contrābor.

English contra-

English positive

English contrapositive

contrapositive (plural contrapositives)

From a conditional statement, its inverse, its converse, and its contrapositive are defined as follows:
Proposition: "If P then Q." ( $P\rightarrow Q$ )

Inverse: "If not P then not Q." ( $\neg P\rightarrow \neg Q$ )

Converse: "If Q then P." ( $Q\rightarrow P$ )

Contrapositive: "If not Q then not P." ( $\neg Q\rightarrow \neg P$ )
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.