Which of the following propositions are so related that they cannot both be…

The question asks for propositions that cannot both be true but can both be false. This describes the logical relationship known as 'Contraries' in the Square of Opposition.\nIn traditional logic, contraries are universal propositions where one affirms and the other denies a predicate of the same subject. Specifically, Proposition (A) 'No professors are materialists' is a Universal Negative (E-type), and Proposition (C) 'All professors are materialists' is a Universal Affirmative (A-type).\nIf it is true that no professors are materialists, then it must be false that all professors are materialists. Conversely, if all professors are materialists, then it is false that no professors are materialists. Thus, they cannot both be true. However, if some professors are and some are not materialists, then both (A) and (C) would be false. This matches the condition given in the question.\nOption B pairs 'Some professors are materialists' (Particular Affirmative) and 'Some professors are not materialists' (Particular Negative), which can both be true. Option C pairs a Particular Affirmative and Universal Affirmative, which can both be true. Option D pairs a Universal Negative and Particular Negative, which are subalterns and can both be true. Therefore, only (A) and (C) satisfy the condition of being contraries.