Which of the following statements is logically equivalent to the statement “No…

Concept

A universal negative proposition, “No S are M,” says that the two classes have no common member: S ∩ M = ∅.

Since set intersection is symmetric, S ∩ M and M ∩ S describe the same set. Therefore, a universal negative proposition permits simple conversion.

Application

1. Let S denote saints and M denote materialists. The given statement becomes S ∩ M = ∅.

2. Exchange the class names. Symmetry gives M ∩ S = ∅.

3. In words, M ∩ S = ∅ means “No materialists are saints.”

Contrast

• “Some materialists are saints” asserts M ∩ S ≠ ∅, so it introduces a common member.

• “All non-materialists are non-saints” states Mc ⊆ Sc; by contraposition, this is S ⊆ M, an inclusion claim.

• “Some non-materialists are non-saints” asserts Mc ∩ Sc ≠ ∅, an existence claim about objects outside both classes.

Cross-check

Reversing the converted statement again returns “No saints are materialists,” confirming that the two universal negative forms have identical truth conditions.

Therefore, the logically equivalent statement is “No materialists are saints.”