Categorical Logic · Relationships

The Traditional
Square of Opposition

The square maps the logical relationships between the four standard form categorical propositions. Knowing these relationships lets you immediately draw inferences from any single proposition.

A
All S are P
E
No S are P
I
Some S are P
O
Some S are not P
Contraries
Cannot both be true; can both be false
A true → E false
but both can be false
Subcontraries
Cannot both be false; can both be true
I false → O true
but both can be true
Subalternation
Truth flows downward; falsity flows upward
A true → I true
Subalternation
Truth flows downward; falsity flows upward
E true → O true
Contradictories
Opposite truth values — always one true, one false
A↔O  ·  E↔I
Contradictories
A ↔ O  ·  E ↔ I
Contradictory propositions always have opposite truth values. If one is true, the other must be false — and vice versa. There is no middle ground.
"All swans are white" (A) is false, so "Some swans are not white" (O) must be true.
Contraries
A ↔ E  (top)
Contrary propositions cannot both be true at the same time, but they can both be false. If A is true, E must be false — but E being false doesn't make A true.
"All cats are pets" and "No cats are pets" can't both be true — but both are false (some cats are, some aren't).
Subcontraries
I ↔ O  (bottom)
Subcontrary propositions cannot both be false at the same time, but they can both be true. At least one must be true.
"Some birds can fly" and "Some birds cannot fly" are both true — and that's perfectly fine.
Subalternation
A → I  ·  E → O  (sides)
Truth flows down: if the universal is true, the particular must be true. Falsity flows up: if the particular is false, the universal must be false.
If "All dogs are mammals" (A) is true, then "Some dogs are mammals" (I) must also be true.