Definition
The square of opposition is a diagram developed in the Aristotelian and medieval logical tradition that displays the four basic categorical propositions and the logical relations among them. The four forms, labeled A, E, I, and O, are: A — universal affirmative, "all S are P"; E — universal negative, "no S are P"; I — particular affirmative, "some S are P"; and O — particular negative, "some S are not P." Arranged as the four corners of a square, the propositions stand in four relations: contradiction (across diagonals), contrariety (along the top), subcontrariety (along the bottom), and subalternation (down each side).
The square encodes the inferential structure of categorical logic in a single compact image. From the truth or falsity of any one corner, the square lets you read off what follows about the other three — what must be true, what must be false, and what is left undetermined.
Why it matters
How it works
Read clockwise from the top-left, the four corners are A (all S are P), E (no S are P), O (some S are not P), and I (some S are P). The diagonals — A with O, and E with I — are contradictories: if one is true the other is false, and vice versa. The top edge connects A with E: these are contraries, which cannot both be true at once but can both be false (consider "all swans are white" and "no swans are white" — both are false in a world with mixed swans). The bottom edge connects I with O: these are subcontraries, which cannot both be false but can both be true. The vertical sides connect A to I and E to O: these are subalternation relations, where the truth of the universal entails the truth of the particular below it but not the reverse.
Subalternation depends on what is called existential import — the assumption that "all S are P" implies that there is at least one S. Aristotle and the medieval logicians took this for granted, since they were typically reasoning about non-empty categories like humans, swans, and triangles. Modern predicate logic, following Frege and Russell, dropped the assumption: "all S are P" becomes a conditional statement that is vacuously true when no S exists. Under that modern reading, subalternation fails — the universal no longer entails the particular — and the classical square collapses to just the contradictory relations on the diagonals. The square is therefore both a piece of working logic and a historical record of where modern logic departed from classical assumptions.