Which among the following are contradictory propositions : (A) All judges are…

Concept

In traditional (Aristotelian) logic a categorical proposition is fixed by two features: its quantity — universal or particular — and its quality — affirmative or negative. That yields four standard forms: the universal affirmative “All S are P” (type A), the universal negative “No S are P” (type E), the particular affirmative “Some S are P” (type I) and the particular negative “Some S are not P” (type O).

On the square of opposition, two propositions are contradictories when they carry the same subject term and the same predicate term and differ in both quantity and quality. Such a pair can never be true together and can never be false together — exactly one member is true. The two contradictory pairings are therefore A with O, and E with I.

One further fact is needed before comparing statements: E and I convert without change of meaning (“No S are P” is equivalent to “No P are S”, and “Some S are P” to “Some P are S”), whereas A and O do not convert. So a statement written with the terms in the reverse order may still be the same proposition in disguise.

Application

Classify every listed statement first. The letters below are the statement labels printed inside the question stem, not the answer choices.

Now hunt for a pair that shares both terms and differs in both quantity and quality. “All judges are lawyers” is universal and affirmative over the terms judges and lawyers. “Some judges are not lawyers” is particular and negative over exactly those same two terms. Quantity differs, quality differs, the terms agree — that is an A–O pair, so these two statements are contradictories.

The list does contain a second contradictory pair: “No lawyers are judges” converts to “No judges are lawyers” (type E) and stands against “Some judges are lawyers” (type I), an E–I pair. That combination is simply not offered among the answer choices, so it cannot be selected here.

Cross-check

• “All judges are lawyers” with “Some judges are lawyers”: same terms, both affirmative, differing in quantity alone. That is subalternation — both are true together whenever every judge is a lawyer, so they are not contradictories.

• “All judges are lawyers” with “No lawyers are judges” (i.e. “No judges are lawyers”): both universal, differing in quality alone. That is contrariety — they cannot both be true, but both are false when some judges are lawyers and some are not, so they are not contradictories.

• “No lawyers are judges” with “Some lawyers are not judges”: same terms, both negative, differing in quantity alone. Subalternation again — the universal negative entails the particular negative, so the two hold together.

• Truth test on the A–O pair: if “All judges are lawyers” is true, no judge falls outside the lawyers, so “Some judges are not lawyers” must be false. If it is false, at least one judge falls outside the lawyers, so “Some judges are not lawyers” must be true. Exactly one of the two is true in every situation — the defining mark of contradiction.

Result: the contradictory pair among the combinations offered is “All judges are lawyers” together with “Some judges are not lawyers”.