Which of the following statements are logically equivalent? A. No P is S B.…

Concept

Categorical propositions can be compared by translating them into set or predicate notation. Two statements are logically equivalent only when they impose exactly the same condition, including the same existential commitment.

Application

1. A. ‘No P is S’ means ∀x(P(x) → ¬S(x)), equivalently P ∩ S = ∅. It is universal and does not assert that any S exists.

2. B. ‘Some S are non-P’ means ∃x(S(x) ∧ ¬P(x)).

3. C. ‘Some non-P are not non-S’: ‘not non-S’ simplifies to S, so this means ∃x(¬P(x) ∧ S(x)).

4. D. ‘Some S are not P’ means ∃x(S(x) ∧ ¬P(x)).

Contrast

• B, C, and D all reduce to the same existential condition: at least one object is S and not P.

• A states a universal exclusion between P and S and has no existential commitment, so it is not equivalent to the other three.

Cross-check

If S is empty, A is true because P and S do not overlap, but B, C, and D are false because no S-object exists. Conversely, any witness that is S and not P makes B, C, and D true together. This confirms the distinction.

Therefore, B, C, and D only are logically equivalent.