In detail

Sincere, sane, crazy and liars

Sincere, sane, crazy and liars

There is a town inhabited by sincere people and other liars, where half of the population is crazy and the other half is sane. A sane person believes that all statements are true
true and believes that all false statements are false. A madman, on the other hand, believes that all true statements are false and believes that all false statements are true.

Thus, before the question: Two plus two is four? I would answer:

The sincere sane would answer: Yes
The sincere insane would answer: No
The sane liar would answer: No
The crazy liar would answer: Yes

In the example, the crazy liar would believe that 2 + 2 is not 4 and falsely answer that it is.

Once, we visited that town and as soon as we arrived we met a couple. The guide told us that one was sincere and the other liar, but did not know if they were crazy or sane.

Then each of the members of the couple told us the following:

Him: We're both crazy.
She: That is not true.

Which of the two is the liar?

Solution

We have 4 valid possibilities:

a) He is a sane liar and she sincere rope
b) The crazy liar and she sincerely crazy
c) He sincere sane and she liar rope
d) The mad truthful and she crazy liar

Let's analyze each case separately.

to) He is a sane liar and she is sincere rope. If we assume that he is a sane liar then he would be lying to say that they are both crazy. She being honest would have to be
necessarily a rope so that your phrase is true "That is not true" because otherwise we would have a contradiction.

b) He is a crazy liar and she is sincerely crazy. If we assume that he is a crazy liar then he would be telling the truth by saying that they are both crazy. She is therefore sincere and
necessarily crazy, which makes sense to say "That is not true."

c) He is sincere sane and she is a liar rope. If we assume that he is sane, his phrase "We are both crazy" would be true, but it turns out that he is not crazy, with which we reach a
contradiction.

d) He is sincere crazy and she is a crazy liar. If we assume that he is crazy, his phrase "We are both crazy" would be false, so the woman must necessarily be sane. As the woman is a liar rope, her phrase "That is not true" is true, but since she is lying, she really meant that it was true, with which there is contradiction.

Therefore the only possible solutions of the enigma are the to and the b and in both cases he is the liar and she is the sincere.