Which of the following propositions are so related that they cannot both be true, although they may both be false?

A. All frogs are amphibians.

B. Some frogs are amphibians.

C. Some frogs are not amphibians.

D. No frogs are amphibians.

Choose the most appropriate answer from the options given below:

The propositions that cannot both be true but can both be false are A and D.

Key PointsExplanation:

• Proposition A, "All frogs are amphibians" is an A-type proposition in the form of "All A are B", meaning that the subject "frogs" is entirely included in the predicate "amphibians".
• Proposition D, "No frogs are amphibians" is an E-type proposition in the form of "No A are B", meaning that the subject "frogs" is completely excluded from the predicate "amphibians".
• Since A and D are contradictory propositions, they cannot both be true, but both can be false, i.e., there could be some frogs that are amphibians and some frogs that are not amphibians.
• Propositions B and C are not contradictory and can be true or false independently of each other.

Additional Information

In terms of the square of opposition

• Propositions A and D are contradictories as they cannot both be true, although they may both be false.
• Proposition A "All frogs are amphibians" belongs to the "A" category, which is the universal affirmative proposition, meaning that it affirms a universal relationship between two terms.
• In this case, it affirms that all frogs belong to the class of amphibians.
• Proposition D "No frogs are amphibians" belongs to the "E" category, which is the universal negative proposition, meaning that it denies a universal relationship between two terms.
• In this case, it denies that there is any frog that belongs to the class of amphibians.

Since A and D have opposite truth values, they are contradictories and cannot both be true.