The Art of Logic · Part 10 · Propositions · B2–C1
The Square of Opposition
Part 10 of The Art of Logic: if you know that one of the four propositions is true, or false, you often know about the other three. Medieval teachers drew it as a square. Modal verbs of deduction (must be true, can’t be true, might be true), with Philo, Sophia, Marcos and Demetria.
Tonight we draw a square. At its corners, the four letters: A and E at the top, I and O at the bottom.
Because each corner is related to the other three in a different way. Know one corner, and you can reason about the rest.
Contradictories are the diagonals: A and O (All S is P / Some S is not P), and E and I (No S is P / Some S is P). They can never both be true and never both be false: exactly one is true.
Contraries are A and E, along the top: they cannot both be true, but they can both be false. Subcontraries are I and O, along the bottom: they cannot both be false, but they can both be true. Subalterns run down the sides: if A is true, I is true; if E is true, O is true; but not the other way round.
A modern note: logicians since the nineteenth century point out that the traditional square assumes the subject exists. “All unicorns are white” and “No unicorns are white” both seem true if there are no unicorns. Modern logic keeps only the contradictories as always safe. Sister Miriam Joseph teaches the traditional square, which works well whenever the subject really exists.
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