Which of the following statements are so related that they can neither both be true nor can they be false together?

A) No saints are materialists.

B) All saints are materialists.

C) Some saints are not materialists.

D) Some saints are materialists.

Choose the correct answer from the options given below:

The correct answer is A & D only.

To understand why this is the correct choice, we need to analyze the statements using the classical Square of Opposition in traditional logic. The Square of Opposition describes the logical relationships between four types of categorical propositions:

1. A-proposition (Universal Affirmative): "All saints are materialists." (Statement B)
2. E-proposition (Universal Negative): "No saints are materialists." (Statement A)
3. I-proposition (Particular Affirmative): "Some saints are materialists." (Statement D)
4. O-proposition (Particular Negative): "Some saints are not materialists." (Statement C)

The question asks for statements that "can neither both be true nor can they be false together." In logic, this relationship is known as a contradictory relationship. Contradictory statements always have opposite truth values: if one is true, the other must be false, and vice versa.

According to the Square of Opposition, contradictory relationships exist diagonally across the square:
- Between A ("All saints are materialists") and O ("Some saints are not materialists") – i.e., statements B and C.
- Between E ("No saints are materialists") and I ("Some saints are materialists") – i.e., statements A and D.

Therefore, statements A and D are contradictories. If it is true that "No saints are materialists" (A), then it must be false that "Some saints are materialists" (D). If it is false that "No saints are materialists" (A), then it must be true that "Some saints are materialists" (D). They can neither be true together nor false together, confirming that A & D only is the correct option.