If two propositions having the same subject and predicate terms are such that…

Concept — the square of opposition

Traditional (Aristotelian) logic recognises four categorical forms built from one subject term S and one predicate term P: A (All S are P), E (No S is P), I (Some S are P) and O (Some S are not P). Two propositions sharing the same subject and predicate terms stand in a contradictory relation when one is the strict denial of the other: they differ in both quantity (universal versus particular) and in quality (affirmative versus negative), so they can never be true together and never false together.

Applying the concept to this statement

The stem fixes two conditions: the two propositions share the same subject and predicate terms, and one asserts exactly what the other rules out. Denying "All S are P" leaves "Some S are not P", and denying "No S is P" leaves "Some S are P". So the denial pairs are A with O and E with I, and in each pair the truth of one member forces the falsity of the other and the falsity of one forces the truth of the other. That two-way exclusion is precisely the contradictory relation.

Cross-check against the other three relations

A strict denial has to rule out both possibilities at once — the pair may not be true together and may not be false together. Reading the table, the contradictory row is the row that closes off both columns. A contrary pair still allows both members to be false (when some S are P and some are not); a sub-contrary pair still allows both members to be true in that same mixed case; and sub-alternation is a one-way inference — truth carries down from the universal to the particular and falsity carries up — rather than a denial at all.

Result

Two propositions with the same subject and predicate terms, one of which denies the other, stand in the contradictory relation.