An assertion is either an affirmation (κατάφασις) or a negation (ἀπόφασις). In chapter 6 Aristotle defines the two in the fewest words.
Κατάφασις δέ ἐστιν ἀπόφανσίς τινος κατά τινος. Ἀπόφασις δέ ἐστιν ἀπόφανσίς τινος ἀπό τινος.
An affirmation is the assertion of something about something. A negation is the assertion of something away from something. (ch. 6, 17a; Greek quotations throughout, Japanese and English are ToL working translations.)
When an affirmation and a negation form a pair over the same subject and the same predicate “in the same way (μὴ ὁμωνύμως),” that pair is a contradiction (ἀντίφασις). A contradictory pair has exactly one true and one false — and Aristotle adds fine conditions on the match, to block sophistical wordplay.
Universal and particular — the four corners
In chapter 7 Aristotle uses terms like “wise” and “man” and combines them with the quantity of the subject (all / some): universal (καθόλου, predicated of many) or individual (καθ᾿ ἕκαστον, a particular one, like Callias).
In medieval terms, the four corners relate like this:
- Contrary: universal affirmative A and universal negative E. Cannot both be true, but can both be false (when some men are wise and some are not).
- Contradictory: the diagonals. A and particular negative O; E and particular affirmative I. Exactly one of each pair is true.
- Subcontrary: particular affirmative I and particular negative O. Cannot both be false.
- Subalternation: the sides. If A is true, so is I; if E is true, so is O.
The “sentence with no quantifier”
Aristotle also notes that a sentence with no quantifier (“all,” “some”) — “a man is wise” — is awkward to place. It is close to “some man is wise,” but is not universal. He moves on without giving this “indefinite (ἀδιόριστος)” sentence a fixed spot. Later logic puts the ambiguous form outside the square.
The diagram came much later
Aristotle himself only states this in prose; he draws no square. The diagram was built by the commentators of late antiquity. Apuleius (2nd c.) first gave a square figure, and Boethius (6th c.) named the four corners. In the Middle Ages the vowels A, E, I, O settled as the corner labels (from Latin affirmo “I affirm,” giving A and I; nego “I deny,” giving E and O). The “square of opposition” has sat on the first page of the logic textbook ever since.
In modern logic, part of it collapses
Modern logic (Frege, Russell) did not take the square over as-is. The problem is existential import. “All unicorns are wise” counts as true even if no unicorn exists (vacuously true). Then “if A is true, so is I” (subalternation) fails — “some unicorn is wise” is false. For the same reason contrariety and subcontrariety collapse too. What remains is only the diagonal: contradiction. So in modern predicate logic the square shrinks to “the diagonal of contradiction.”
Still, the square has not vanished. It is used wherever term logic is taught, and in the twentieth century Blanché and Sesmat extended it to a hexagon and octagon of opposition, adding “possible / impossible” and the like. Aristotle’s short chapter has been drawn, as the diagram at the entrance to logic, for more than two thousand years.