Which of the following statements are logically equivalent? A. Some beverages…

CONCEPT

An existential categorical statement such as ‘Some S are not P’ means that at least one object belongs to S and lies outside P: there exists an x for which S(x) and not P(x). A complementary class name can be expanded as a negation, and a double negation returns the original class.

APPLICATION

1. Statement A has the form: there exists an x such that Beverage(x) and not Liquid(x).

2. Statement D uses ‘non-liquid’ for ‘not liquid’, so it has the same form as A.

3. Statement C has the form: there exists an x such that not Liquid(x) and not(not Beverage(x)). Removing the double negation and commuting the conjunction gives Beverage(x) and not Liquid(x), the same form as A.

4. Statement B has the form: there exists an x such that Beverage(x) and Liquid(x), so it describes membership inside the liquid class rather than outside it.

CROSS-CHECK / CONTRAST

• A, C, and D each assert the existence of a beverage outside the class of liquids.

• B asserts the existence of a beverage inside the class of liquids.

Therefore, the logically equivalent statements are A, C, and D only.