Understanding Logical Equivalence in Statements

The question asks us to identify which statements among the given options are logically equivalent. Logical equivalence means that two statements have the same truth value under all possible circumstances. We need to analyze each statement regarding 'poems' and 'artworks'.

Analyzing Statement A

Statement A: "All poems are artworks."

This is a universal affirmative statement. In propositional logic, we can represent it as:

$\forall x (P(x) \rightarrow A(x))$

Where $P(x)$ means 'x is a poem' and $A(x)$ means 'x is an artwork'.

Analyzing Statement B

Statement B: "No non-poems are non-artworks."

This statement is about the complements of the sets 'poems' and 'artworks'. Let $\neg P(x)$ represent 'x is not a poem' and $\neg A(x)$ represent 'x is not an artwork'.

The statement "No S are P" is logically equivalent to "All S are not P" (Obversion).

Applying obversion to Statement B (where S = non-poems, P = non-artworks):

"All non-poems are not non-artworks."

This simplifies to: "All non-poems are artworks."

In propositional logic: $\forall x (\neg P(x) \rightarrow A(x))$

This statement asserts that if something is not a poem, it must be an artwork. This is different from statement A.

Analyzing Statement C

Statement C: "Some poems are artworks."

This is a particular affirmative statement. In propositional logic:

$\exists x (P(x) \wedge A(x))$

This statement asserts the existence of at least one poem that is also an artwork. It does not make a claim about *all* poems.

Analyzing Statement D

Statement D: "All non-artworks are non-poems."

This is also a universal affirmative statement.

In propositional logic: $\forall x (\neg A(x) \rightarrow \neg P(x))$

Determining Logical Equivalence

We need to check which statements are equivalent. Let's focus on Statement A and Statement D.

• Statement A: $\forall x (P(x) \rightarrow A(x))$
• Statement D: $\forall x (\neg A(x) \rightarrow \neg P(x))$

Statement D is the contrapositive of Statement A. The rule of contraposition states that a conditional statement ($p \rightarrow q$) is logically equivalent to its contrapositive ($\neg q \rightarrow \neg p$).

Therefore, Statement A and Statement D are logically equivalent.

Comparing Other Statements

Now let's compare Statement B and Statement C with A and D.

• Statement A ($\forall x (P(x) \rightarrow A(x))$) claims all members of set P are in set A.
• Statement B ($\forall x (\neg P(x) \rightarrow A(x))$) claims all members not in P are in A. This is different. For example, if P={dogs} and A={mammals}, "All dogs are mammals" (A) is true, but "All non-dogs are mammals" is false (e.g., cats are non-dogs but are mammals, but consider rocks - they are non-dogs and non-mammals). A more precise example: Let P={1}, A={1,2}, U={1,2,3}. A is true. B says "All {2,3} are {1,2}" which is false.
• Statement C ($\exists x (P(x) \wedge A(x))$) is a particular statement, while A and D are universal. They cannot be equivalent unless the domain is restricted in specific ways not mentioned.

Since A and D are logically equivalent (as they are contrapositives of each other), and B and C are not equivalent to A and D, the correct option includes only A and D.

Conclusion

Based on the analysis of logical forms and rules of equivalence (specifically contraposition), statements A and D are logically equivalent.

• Statement A: All poems are artworks.
• Statement D: All non-artworks are non-poems.

These two statements hold the same logical meaning.