Who's who and whom to believe? logic?

Question

Alice: I am insane.

Bob: I am pure.

Charlie: I am applied.

Dorothy: I am sane.

Alice: Charlie is pure.

Bob: Dorothy is insane.

Charlie: Bob is applied.

Dorothy: Charlie is sane.

You are also given that:

* Pure mathematicians tell the truth about their beliefs.

* Applied mathematicians lie about their beliefs.

* Sane mathematicians beliefs are correct.

* Insane mathematicians beliefs are incorrect.

Describe the four mathematicians.

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Answer 1

Suppose Alice is Pure; then if she is Sane, her belief in her own insanity must be incorrect, so she is Insane. On the other hand, if she is Insane, then her belief in her own insanity is correct, so she is Sane. Therefore, Alice must be Applied.

Suppose Dorothy is Applied; then she believes in her own insanity. If she is Sane, then she holds a false belief, so she cannot be Sane. On the other hand, if she is Insane, she holds a true belief, so she cannot be Insane. Therefore, Dorothy must be Pure.

Suppose Bob is Insane. Then, if he is Applied, he believes himself to be Applied, but this is a false belief, so he must be Pure; on the other hand, if he is Pure, then he believes himself to be Pure, but this is a false belief, so he is Applied. Therefore, Bob is Sane.

Suppose Charlie is Sane. Then, if he is Applied, he believes himself to be Pure and this is a true belief, so he must be Pure; on the other hand, if he is Pure, then he believes himself to be Applied, and this is a true belief, so he's Applied. Therefore, Charlie is Insane.

Dorothy is Pure, so she believes that Charlie is sane. Therefore Dorothy is Insane.

Bob claims that Dorothy is Insane, but he cannot believe otherwise, because then he would hold a false belief, which is impossible because he is Sane. Therefore, Bob is Pure.

If Charlie is Applied, then he believes Bob is Pure, but this is a true belief, which is impossible, because Charlie is Insane. Therefore, Charlie is Pure.

Now, Alice believes that Charlie is Applied, which is a false belief, so she must be Insane.

The summary:

Alice: Applied / Insane

Bob: Pure / Sane

Charlie: Pure / Insane

Dorothy: Pure / Insane