Take one pair of terms, say birds and swimmers, and write all four forms with them. Those four statements are not independent: the truth of one limits the truth of the others. Since late antiquity, logicians have drawn those limits as a square.
- Corners: universals on top (A left, E right), particulars below (I under A, O under E). Affirmatives on the left, negatives on the right.
- Contradictories (the diagonals, A–O and E–I): exactly one is true.
- Contraries (the top, A–E): not both true.
- Subcontraries (the bottom, I–O): not both false.
- Subalterns (the sides, A–I and E–O): truth flows down, falsity flows up.
Key idea
The square names four relations among A, E, I and O with the same terms: contradictories, contraries, subcontraries and subalterns. This traditional square assumes the subject class has members.
Why it works
The layout follows quantity and quality. Moving across a side changes quality (affirmative to negative); moving up or down changes quantity (universal to particular); a diagonal changes both. A statement and its full opposite, different in both ways, can never agree, which is why the diagonals are the strongest relation. Later lessons show why each side behaves as it does.
Worked example
Worked example: Naming a relation
How are "No birds are swimmers" and "Some birds are swimmers" related?
- The first is E (top right). The second is I (bottom left).
- Top right to bottom left is a diagonal.
Answer: contradictories
The names come from the old Latin textbooks: contraries are opposed at the top, subcontraries sit "under" them at the bottom, and a subaltern is the statement "below another" on the same side.
Common mistake
Placing the corners in the wrong spots, so every relation comes out wrong. Remember: universals on top, A left and E right; I sits under A and O under E (each particular under the universal with the same quality). Check: the two left corners both affirm; the two right corners both deny.