Which of the following two statements cannot both be true but can both be…

Concept: the square of opposition

Traditional categorical logic sorts every statement of the "S–P" kind by two things at once — its quantity (universal, speaking about the whole class, or particular, speaking about part of it) and its quality (affirmative or negative). That gives exactly four standard forms: 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".

The square of opposition records how each pairing of these forms is allowed to share truth values:

• Contrary pair — the two universals, A with E. They cannot both be true, but they can both be false.

• Contradictory pairs — A with O, and E with I. Exactly one member of the pair is true, so such a pair can neither be true together nor false together.

• Subcontrary pair — the two particulars, I with O. They can both be true, but they cannot both be false.

The condition set by the stem — "cannot both be true but can both be false" — is word for word the definition of a contrary pair. So the whole task reduces to finding the two universal statements among the four given.

Application to this question

Classify each given statement by its standard form:

The two universals are therefore "All monkeys are apes" (A-form) and "No monkeys are apes" (E-form) — statements A and D. Check both halves of the required condition directly on them:

1. They cannot both be true. If every monkey is an ape, then it is simply false that no monkey is an ape; the two claims affirm and deny the same predicate of the whole class, so at most one of them can hold.

2. They can both be false. Take the mixed situation in which some monkeys are apes and some monkeys are not. In that situation "All monkeys are apes" is false, and "No monkeys are apes" is false as well — the two are false together, which is exactly what the stem allows.

Cross-check: why the other pairings do not qualify

• "All monkeys are apes" with "Some monkeys are not apes" (A-form with O-form) is a contradictory pair: one of the two must be true, so they can never be false together.

• "Some monkeys are apes" with "No monkeys are apes" (I-form with E-form) is the other contradictory pair: again exactly one is true, so being false together is impossible.

• "Some monkeys are apes" with "Some monkeys are not apes" (I-form with O-form) is a subcontrary pair: in the mixed situation used above both of them are true, so this pairing fails the "cannot both be true" half as well.

Both halves hold, so the required pair is statements A and D.