Which of the following propositions are so related that they cannot both be true, although they can both be false?
A. No professors are materialists
B. Some professors are materialists
C. All professors are materialists
D. Some professors are not materialists
Choose the correct answer from the options given below:

Correct option is A

A and C only are the propositions that cannot both be true but can both be false. This is because proposition A ("No professors are materialists") and proposition C ("All professors are materialists") are direct negations of each other. If no professors are materialists (A), it is impossible for all professors to be materialists (C). However, they can both be false if some but not all professors are materialists, which would neither fully exclude nor include all professors as materialists, aligning with the logical middle ground.
Information Booster:
(b) B and D only. Proposition B ("Some professors are materialists") and D ("Some professors are not materialists") can both be true simultaneously if there is a division within the group of professors where some are materialists and some are not.
(c) B and C only. These cannot both be true since if all professors are materialists (C), there cannot be just some who are materialists (B); however, both could be false if no professors are materialists.
(d) A and D only. These can both be false, but they can also both be true, as A (no professors are materialists) does not conflict with D (some professors are not materialists) — in fact, A implies D.