In the traditional square of opposition, the four standard-form propositions about a subject-predicate pair are: A (universal affirmative, 'All S are P'), E (universal negative, 'No S are P'), I (particular affirmative, 'Some S are P'), and O (particular negative, 'Some S are not P').
Here S = 'animals with horns' and P = 'carnivores', and we are told the E proposition 'No animals with horns are carnivores' is false.
E and I are contradictories: they cannot both be true and cannot both be false - exactly one of them must be true. Since E is false, its contradictory I ('Some animals with horns are carnivores') must be true. This confirms code A.
A and E are contraries: they cannot both be true, but they can both be false. Knowing only that E is false does not by itself fix the truth value of A ('All animals with horns are carnivores') - it could be true or it could be false. So the truth value of the A proposition remains logically undetermined, which confirms code D.
I and O are subcontraries: they cannot both be false, but they can both be true. Since I has already been established as true, this relation places no constraint that would make O ('Some animals with horns are not carnivores') definitely true - O's truth value is likewise left undetermined, so code B (claiming O is definitely true) is incorrect.
Code C claims the A proposition is definitely false, but as shown above the contrary relation only tells us A and E cannot both be true; it does not force A to be false whenever E is false, so code C is also incorrect.
Therefore, only codes A and D validly follow from the falsity of the E proposition, making A and D only the correct answer.