Which of the following statements are logically equivalent to the statement…

Concept

An existential affirmative claim has the form ∃x(P(x) ∧ S(x)): at least one object belongs to both classes. Conjunction is commutative, and double negation returns the original predicate; changing a predicate to its complement or using universal exclusion changes the claim.

Application

1. Statement A has the form ∃x(S(x) ∧ P(x)). By commutativity of conjunction, this is the same as ∃x(P(x) ∧ S(x)).

2. Statement B has the form ∃x(¬S(x) ∧ ¬¬P(x)), which simplifies to ∃x(P(x) ∧ ¬S(x)). It requires a professor outside the spiritualist class, so it is not equivalent to the original existential intersection.

3. Statement C has the form ¬∃x(S(x) ∧ P(x)). It says the intersection is empty and therefore contradicts the original claim that the intersection has at least one member.

4. Statement D has the form ∃x(P(x) ∧ ¬¬S(x)), which simplifies to ∃x(P(x) ∧ S(x)). It is therefore equivalent to the original statement.

Cross-check and contrast

• A, B & D only includes B, whose simplified form is P(x) ∧ ¬S(x).

• A & D only retains exactly the converse and double-negation forms of the original claim.

• B & C only combines a complement-based existential claim with universal exclusion.

• C & D only combines a contradiction of the original claim with an equivalent double-negation form.

Result

Hence, the logically equivalent statements are A and D, so the answer is A & D only.