contradictories

Contradictories are pairs of propositions that cannot both be true and cannot both be false. In classical two-valued logic this means one proposition is the negation of the other; the truth of one guarantees the falsity of the other. In Aristotelian logic the term is used for the cross-cutting pairs in the square of opposition: All S are P (A) contradicts Some S are not P (O), and No S are P (E) contradicts Some S are P (I).

Examples illustrate the idea: All swans are white (A) and Some swans are not white (O) are contradictories. No swans are white (E) and Some swans are white (I) are contradictories as well. In each pair, at least one is true and the other false in standard logic.

Relation to other oppositions differs in type. Contraries, such as All S are P and No S are P, cannot both be true but can both be false. Subcontraries, such as Some S are P and Some S are not P, cannot both be false but can both be true. Contradictories form the cross-cutting links in the square: each member of a contradictory pair excludes the truth of the other and vice versa.

In modern logic the concept generalizes to p and not p, reflecting the law of non-contradiction. In some non-classical logics, such as paraconsistent systems, both a statement and its negation may be treated as true without collapsing the system; in those contexts the strict classical notion of contradictories is treated with appropriate nuances.