contraries

Contraries are a relation between two propositions in classical logic in which the two statements cannot both be true, but can both be false. This notion is a key part of the traditional square of opposition in Aristotelian logic. Contraries typically pair universal propositions, such as “All S are P” and “No S are P.” If one of these universal claims is true, the other must be false; but it is possible for both to be false when some S are P and some S are not P.

Contraries are different from contradictories, which are pairs that cannot both be true or both be false (such as “All S are P” and “Some S are not P”). They are also distinct from subcontraries, which pair statements like “Some S are P” and “Some S are not P,” and which cannot both be false.

Examples help clarify: “All swans are white” and “No swans are white” are contraries. They cannot both be true, but they can both be false if there exists at least one white swan and at least one non-white swan. In modern logic, the terminology of contraries is less central, but the concept remains a useful way to describe how certain universal claims relate to one another in classical syllogistic reasoning.