Statements : All cups are glasses. Some glasses are bowls. No bowl is a plate. Conclusions : I. No cup is a plate. II. No glass is a plate. III. Some plates are bowls. IV. Some cups are not glasses.

The correct answer is: Only II and III follow.

The first statement, “All cups are glasses,” can be expressed in propositional logic as $A \subseteq B$. The second statement, “Some glasses are bowls,” can be expressed as $B \cap C \neq \emptyset$. The third statement, “No bowl is a plate,” can be expressed as $C \cap P = \emptyset$.

The first conclusion, “No cup is a plate,” can be derived from the first and third statements by modus tollens. The second conclusion, “No glass is a plate,” can be derived from the second and third statements by modus tollens. The third conclusion, “Some plates are bowls,” cannot be derived from the given statements. The fourth conclusion, “Some cups are not glasses,” cannot be derived from the given statements.

Here is a more detailed explanation of each option:

• Option A: None follows. This is incorrect because the second conclusion, “No glass is a plate,” can be derived from the given statements.
• Option B: Only either I or III follows. This is incorrect because both the first and second conclusions can be derived from the given statements.
• Option C: Only II and III follow. This is correct because the second and third conclusions can be derived from the given statements.
• Option D: Only III and IV follow. This is incorrect because the fourth conclusion cannot be derived from the given statements.