## 16. The Fish-Footman and the Frog-Footman

- All this is very interesting, - said Alice, only the two cases you told me are quite different.
- Of course, my dear! And the moral of the whole thing is: to be or not to be is not the same as to be and not to be.

Alice tried to understand what the Duchess meant, but the Duchess interrupted her thoughts.

- I have two footmen, - said the Duchess, - the Fish-Footman and the Frog-Footman. You must have seen them?
- Oh yes! - answered Alice, vividly recalling the unimaginable rudeness of the Frog-Footman.
- The Fish-Footman believes, - continued the Duchess, - that he and the Frog-Footman are alike in everything, that is, they are either both in their right minds or both not in their right minds. So, my dear, tell me which of my footmen is not in his right mind.

Alice couldn't understand why she should answer such tricky questions, but the task itself seemed interesting to her.

Alice tried to solve it, but it wasn't that easy!

- I'm afraid I can't solve your problem, - said Alice. - I can determine whether one of your footmen is in his right mind, but I can say nothing about the other.
- Oh, you're simply a genius and did a wonderful job! - exclaimed the Duchess, hugging Alice. From what I told you, nothing can be said about the other footman. It's a pity - I don't know which one is the other.

About which of the Duchess's footmen do you know that he is in his right mind - the Fish-Footman or the Frog-Footman?
16. The Butler-Perch and the Tadpole. The information provided in the problem does not allow us to determine whether the Butler-Perch is in his right mind or not, but we will prove that the Tadpole must be in his right mind. We will reason as follows.

There are two possibilities: either the Butler-Perch is in his right mind, or he is not. We will show that in either case, the Tadpole must be in his right mind.

Suppose the Butler-Perch is in his right mind. Then he judges everything correctly. This means that the Tadpole is indeed similar to the Butler-Perch in everything. Therefore, the Tadpole is in his right mind.

On the other hand, suppose the Butler-Perch is not in his right mind. Then he judges everything incorrectly, so the Tadpole is completely dissimilar to the Butler-Perch. Since the Butler-Perch is not in his right mind,

the Tadpole, in contrast to him, must be in his right mind.

Thus, in any case (whether the Butler-Perch is in his right mind or not), the Tadpole must be in his right mind.

What if the Butler-Perch considered the Tadpole not similar in everything, but completely dissimilar to himself? What would the Butler-Perch be in this case - in his right mind or not?

Answer: In this case, the Tadpole should be out of his mind. The proof of this statement I leave to the reader as an independent exercise.