Philosophical Beliefs

Ursole

You are given that:
Pure philosophers always tell the truth concerning their beliefs.
Applied philosophers always lie concerning their beliefs.
Sane philosophers beliefs are always correct.
Insane pilosophers beliefs are always incorrect.

Four philosophers {A,B,C,D}) have the following conversation:
A - I am insane
B - I am pure
C - I am applied
D - I am sane
A - C is pure
B - D is insane
C - B is applied
D - C is sane

Describe A,B,C, and D.

AKG

A says "I am insane." Now, no person can really think they're insane. A sane person will know they're sane, and an insane person will incorrectly believe themselves to be sane. Since no person can believe themselves to be insane, A must be lying, so A is applied. At this point, A could be sane or insane.

B says "I am pure." If B is pure, then he really believes he's pure, so he's sane. If he's applied, he believes he lied and believes he's applied, so again, he must be sane. B is sane, and could be pure or applied.

If C is pure, and really think's he's applied, he must be insane. Otherwise, if he's applied, and really thinks he's pure, he's wrong again and must be insane. C is insane.

D is pure, because everyone truly thinks their sane, so he must be telling the truth. Also, it's the only thing left after the first "round" of clues.

A - applied
B - sane
C - insane
D - pure

Now, working from the bottom, D truly thinks C is sane, which is wrong, so D is insane. B is sane, so he knows D is insane, and he says it, so he's pure. B is pure, and C is insane, so C thinks B is applied. Since C says B is applied, he's telling what he believes, so C is pure. A says C is pure, but since A is applied, he really thinks C is applied. But C is pure, so A is wrong, and thus insane. So:

A - applied insane
B - pure sane
C - pure insane
D - pure insane

The Bob

A = Applied, Sane
B = Pure, Sane
C = Pure, Sane
D = Applied, Sane

Likely not to be right but I did it quickly.

The Bob (2004 ©)

Ursole

Captain, this is not logical. [It is a personal opinion.]

.

AKG

Huh? No, it's perfectly logical. If A is sane, then A correctly knows himself to be sane. If A is insane, A incorrectly "knows" himself to be sane. Therefore, A necessarily "knows" himself to be sane. Q.E.D.

EDIT: I can see that my wording seemed rather colloquial, and may have thrown you off, but it was still a logically rigourous and sound argument. All philosophers believe themselves to be sane (based on the riddle's definitions of "philosopher" and "sane").

Ursole

Sorry, AKG. I forgot the wording of the puzzle.