Immediate Inference and the Square of Opposition
The square of opposition
Immediate inference draws a conclusion from one categorical proposition without a second premise. The traditional square relates A, E, I, and O with the same S and P:
- Contradictories: A–O and E–I differ in truth value; one is true exactly when the other is false.
- Contraries: A and E cannot both be true, but may both be false.
- Subcontraries: I and O cannot both be false, but may both be true.
- Subalternation: A implies I; E implies O, on the traditional assumption that the subject class exists.
Operations
Obversion changes quality and replaces the predicate with its complement; it is valid for all four forms. Conversion swaps S and P: valid for E and I, not for A or O. Contraposition replaces S with complement of P and P with complement of S; valid for A and O.
Worked example
“All citizens are voters” (A). Its contradictory is “Some citizens are not voters” (O). If the A statement is true, the O statement must be false. Its converse, “All voters are citizens,” does not follow: the original says every citizen is a voter, not that every voter is a citizen.
Traps that cost marks
- Illicit conversion of an A or O proposition.
- Treating contraries as contradictories: both may be false.
- Forgetting existential import when using subalternation.
- Obverting incorrectly by failing to complement the predicate.
- Assuming the square works when terms are not used univocally.