Which of the following statements are logically equivalent? A. All saints are…
2023
Which of the following statements are logically equivalent?
A. All saints are materialists.
B. No saints are non-materialists.
C. All non-materialists are non-saints.
D. No materialists are saints.
Choose the most appropriate answer from the options given below
Answer: D. A, B and C only — ConceptTwo categorical statements are logically equivalent when they have the same truth value for every possible interpretation of their classes. Obversion…
- A.
B, C and D only
- B.
A and C only
- C.
A and B only
- D.
A, B and C only
Show answer & explanation
Correct answer: D
Concept
Two categorical statements are logically equivalent when they have the same truth value for every possible interpretation of their classes.
Obversion replaces the predicate with its complement and changes affirmation to negation; contraposition swaps subject and predicate while complementing both. These operations preserve equivalence when applied validly.
Application
Let S denote the class of saints and M denote the class of materialists.
Statement A becomes ∀x(S → M): every saint belongs to M.
Statement B says that no S is outside M, so it becomes ∀x(S → ¬(¬M)), which simplifies to ∀x(S → M).
Statement C becomes ∀x(¬M → ¬S), the contrapositive of ∀x(S → M).
Statement D becomes ∀x(M → ¬S). This is a different disjointness condition: an individual who is both S and M satisfies A but violates D.
Cross-check
Statement | Set or implication form | Relationship |
|---|---|---|
A | S ⊆ M | Original inclusion |
B | S ∩ (not M) = ∅, so S ⊆ M | Same inclusion as A |
C | not M ⊆ not S | Contrapositive of A |
D | M ∩ S = ∅ | Disjointness, a different condition |
Result
Therefore, the logically equivalent statements are A, B and C only.