As per square of opposition, if the statement 'All apes are mammals' is given…

Concept

In the traditional square of opposition, categorical propositions are classified as A, E, I, and O according to quantity and quality.

An A-proposition is contrary to the corresponding E-proposition and contradictory to the corresponding O-proposition. With the traditional existential assumption, its truth also entails the corresponding I-proposition.

Application

1. The given statement, ‘All apes are mammals’, is an A-proposition: universal affirmative.

2. Statement A, ‘No apes are mammals’, is the corresponding E-proposition. Because A and E are contraries, truth of the given A-proposition makes statement A false.

3. Statement B, ‘Some apes are mammals’, is the corresponding I-proposition. Under the traditional square, truth descends from A to I, so statement B is true.

4. Statement C, ‘Some mammals are apes’, is the simple conversion of statement B. A particular affirmative converts validly, so statement C is true.

5. Statement D, ‘Some apes are not mammals’, is the corresponding O-proposition. A and O are contradictories, so truth of the given A-proposition makes statement D false.

Contrast

• ‘A only’ omits statement D, although D is the contradictory O-proposition.

• ‘D only’ omits statement A, although A is the contrary E-proposition.

• ‘A and D only’ contains exactly the E- and O-propositions forced false by the given truth assignment.

• ‘A, C and D only’ includes statement C, even though C follows from the true particular affirmative.

Cross-check

As a set check, all apes lie inside the set of mammals and at least one ape exists. Therefore no ape lies outside mammals, while some ape lies inside mammals; reversing that particular relation also gives some mammal that is an ape.

Result

Therefore, the statements immediately inferred to be false are A and D only.