Which of the following propositions are so related that they cannot both be false, though they can both be true?
A. Some poems are artworks.
B. All poems are artworks.
C. Some poems are not artworks.
D. Some artworks are poems.
Choose the correct answer from the options given below:

Correct option is B

Introduction: This question asks to identify a pair of propositions that are related in such a way that they cannot both be false, but they can both be true. This describes the subcontrary relationship within the traditional Square of Opposition in logic.

Information Booster: Let's first classify each given proposition using standard categorical forms, where the subject term (S) is "poems" and the predicate term (P) is "artworks."

• A. Some poems are artworks.

  • This is an I (Particular Affirmative) proposition. (Some S are P)

• B. All poems are artworks.

  • This is an A (Universal Affirmative) proposition. (All S are P)

• C. Some poems are not artworks.

  • This is an O (Particular Negative) proposition. (Some S are not P)

• D. Some artworks are poems.

  • This is also an I (Particular Affirmative) proposition. (Some P are S)

  • Note: Proposition D is the converse of Proposition A. I propositions convert validly, meaning if "Some S are P" is true, then "Some P are S" is also true, and vice versa. Therefore, A and D are logically equivalent.

Now, let's analyse the relationships between the propositions based on the condition: "cannot both be false, though they can both be true."

This condition precisely defines the subcontrary relationship. Subcontrary propositions are two particular propositions (I and O) with the same subject and predicate terms that cannot both be false, but can both be true.

Let's test the relevant pairs:

1. A. Some poems are artworks (I), and C. Some poems are not artworks (O):

   • Can they both be false? No. If "Some poems are artworks" (I) is false, it implies that "No poems are artworks" (E) is true. If "No poems are artworks" is true, then it must be true that "Some poems are not artworks" (O). Therefore, A and C cannot both be false.

   • Can they both be true? Yes. For example, consider the category of "writing." Some writing (poems) can be artworks (e.g., a beautifully crafted sonnet), and some writing (poems) might not be artworks (e.g., a simple limerick written for a casual purpose, or a rough draft). In this scenario, both "Some poems are artworks" and "Some poems are not artworks" could be true.

   • Conclusion: This pair (A and C) fits the subcontrary relationship perfectly.

Additional information:

• A and D Only: These are logically equivalent (I propositions that are converses of each other). If one is true, the other is true. If one is false, the other is false. They can both be false (e.g., if there are no poems that are artworks, and no artworks that are poems). So, they do not fit the "cannot both be false" criterion.

• B and C Only: B ("All poems are artworks" - A) and C ("Some poems are not artworks" - O) are contradictory propositions. Contradictories always have opposite truth values; they cannot both be true, and they cannot both be false. This does not fit the condition.

• A and B Only: A ("Some poems are artworks" - I) and B ("All poems are artworks" - A) are related by subalternation (A implies I). They can both be true (if all poems are artworks, then some are). However, they can both be false (e.g., if no poems are artworks, then both "All poems are artworks" and "Some poems are artworks" would be false). So, they do not fit the "cannot both be false" criterion.

Therefore, the correct options are A and C only.