If the statement “all lions are carnivorous animals” is given as true, which…

Concept. In classical logic four categorical forms are tied together by fixed truth relations on the square of opposition:

• Universal affirmative — All S are P

• Universal negative — No S are P

• Particular affirmative — Some S are P

• Particular negative — Some S are not P

Contradictories can never share a truth value, so the truth of one forces the other to be false. Contraries can never both be true, so the truth of one again forces the other to be false. A universal is the superaltern of the matching particular, so the truth of the universal guarantees the truth of that particular. A particular affirmative also converts simply: interchanging its subject and predicate preserves its truth.

Applying it. The given statement “All lions are carnivorous animals” is a universal affirmative and is taken as true. Each listed claim is then settled by its relation to it:

1. “No lions are carnivorous animals” is the universal negative built from the same two terms, and a universal affirmative and its universal negative are contraries. They cannot both be true, so this claim is forced to be false.

2. “Some lions are carnivorous animals” is the particular affirmative standing under the given universal — its subaltern. The truth of the universal carries down to the particular, so this claim comes out true.

3. “Some lions are not carnivorous animals” is the particular negative, the direct contradictory of the given universal. Contradictories always take opposite truth values, so this claim is forced to be false.

4. “Some carnivorous animals are lions” is obtained by converting the particular affirmative established in the previous step, and a particular affirmative converts simply, so this claim also comes out true.

Cross-check. Read it through the members rather than the forms. If the universal is true, not one lion falls outside the carnivorous animals, so there is no lion left to support “some lions are not carnivorous animals”, and the sweeping denial “no lions are carnivorous animals” is ruled out just as firmly. The same reading positively supplies at least one carnivorous lion, which is what keeps the two particular affirmative claims standing — classical logic assumes the subject class is not empty, and that assumption is exactly what licenses the step from the universal down to the particular.

Result. Exactly two of the listed claims are immediately inferred to be false — “No lions are carnivorous animals” and “Some lions are not carnivorous animals” — that is, A and C.