If the statement-“No squares are circles” is given as true, which of the…

Concept — For one fixed subject–predicate pair the four categorical forms are A, the universal affirmative “All S are P”; E, the universal negative “No S are P”; I, the particular affirmative “Some S are P”; and O, the particular negative “Some S are not P”. On the traditional square of opposition, A and E are contraries, so they can never both be true; A pairs with O and E pairs with I as contradictories, so each of those pairs always carries opposite truth values; and every universal implies its own particular subaltern, E yielding O and A yielding I. Independently of the square, an E proposition is simply convertible: “No S are P” and “No P are S” always assert the same thing.

Application — Put S = squares and P = circles. The proposition given as true, “No squares are circles”, is the E form on those two terms, so each listed statement is settled by reading off its relation to that E.

1. “All squares are circles” is the A form on the same two terms. A is the contrary of E, and contraries can never both be true, so a true E drives this statement to false.

2. “Some squares are circles” is the I form on the same two terms. I is the contradictory of E, and contradictories always carry opposite truth values, so a true E drives this statement to false as well.

3. “Some squares are not circles” is the O form on the same two terms. O is the subaltern of E, and a true universal carries its own particular down with it, so this statement comes out true and therefore cannot be inferred false.

4. “Some circles are not squares” is an O form on the converted terms. Convert the given E into “No circles are squares”, which says exactly the same thing and is equally true, then take its subaltern; this statement also comes out true and therefore cannot be inferred false.

Cross-check — Picture the two classes as regions with no overlap whatsoever, which is precisely what “No squares are circles” asserts. An empty intersection refutes every claim of overlap, while it leaves “Some squares are not circles” and “Some circles are not squares” both satisfied. Existential import is not a worry here either: squares exist and circles exist, so the two non-empty classes legitimise subalternation and the contrariety of A and E, and the modern Boolean reading returns the same verdicts. The four relations line up like this:

Result — Exactly two of the listed statements are immediately inferable as false: “All squares are circles” and “Some squares are circles”. The selection naming that pair reads “(B) and (C) Only”.