Suppose there are two identical twin brothers, one who always lies and the other who always tells the truth. Now, the truth teller is also totally accurate in all his beliefs; all true propositions he believes to be true and all false propositions he believes to be false. The lying brother is totally inaccurate in his beliefs; all true propositions he believes to be false, and all false propositions he believes to be true. The interesting thing is that each brother will give the same answer to the same question. For example, suppose you ask whether two plus two equals four. The accurate truth teller knows that it is and will truthfully answer yes. The inaccurate liar will believe that two plus two does not equal four (since he is inaccurate) and will then lie and say that it does; he will also answer yes.
Getting back to the twin brothers, two logicians were having an argument about the following question: Suppose one were to meet one of the two brothers alone. Would it be possible by asking him any number of yes-no questions to find out which one he is? One logician said, "No, it would not be possible because whatever answers you got to your questions, the other brother would have given the same answers." The second logician claimed that it was possible to find out. The second logician was right, and the puzzle has two parts: (1) How many questions are necessary?; and (2) more interesting yet, What was wrong with the first logician's argument? (Readers who enjoy doing logic puzzles might wish to try solving this one on their own before reading further.)
To determine which brother you are addressing, one question is enough; just ask him if he is the accurate truth teller. If he is, he will know that he is (since he is accurate) and truthfully will answer yes. If he is the inaccurate liar, he will believe that he is the accurate truth teller (since he is inaccurate in his beliefs), but then he will lie and say no. So the accurate truth teller will answer yes and the inaccurate liar no to this question.
Now what was wrong with the first logician's argument; don't the two brothers give the same answer to the same question? They do, but the whole point is that if I ask one person, "Are you the accurate truth teller?" and then ask another, "Are you the accurate truth teller?" I am really asking two different questions since the identical word you has a different reference in each case.
This is wrong. The writer is confusing the accurate perception of reality and lying. The writer is dealing with sanity and insanity, ie truth and untruth, not truth and a lie. The premise of a lie requires a level of communal objectivity in perception of reality, not individually subjective perception. Everyone sees and understands the way things are similarly, but when questioned on it they tell it as it is communally understood(eg. the sky is blue), or they tell it differently to how they know it is communally understood. The insane brother tells the opposite of what he knows as reality, but what he knows as reality is opposite to his brother. To an extent he is lying to himself, but for the purposes of the logic puzzle it does not work as an example, his example pushes one extent to prove the point they want, but does not push to the logical end point, that the insane brother sees the word truth as meaning lie, and yes meaning no, etc, etc. So the insane brother will think you are asking if they lie by asking if they tell the truth, will use the word they think means no, and will say yes. Unless they intend to always say the opposite 'knowingly' which means they know they are telling the opposite of what they know as true and will say their version of yes(our no), to spite them self. This is definitive of insanity(self-defeating behaviour). The logical puzzle therefore fails as logic measured against insanity you cannot ever derive a solution.
I received a letter from Brad Crane:
There is a similar puzzle to the one you put on your webpage. It goes like this:
Suppose there are twin brothers, one which always tells the truth and one which always lies. (So in this case they both know what is true and false, or as you put it, both are accurate in their knowledge.) What one yes-no question could you ask to either one of the brothers to figure out which one he is?
Just in case you want to try to solve it, I'm putting in space here.
Scroll to answer (question) below.
The question one could ask is, "If I were to ask your brother whether you always tell the truth, what would he say?" A reply of "no" means you are talking to the truth teller, a reply of "yes" means you are talking to the liar.
This case the logic is correct as it clearly states the liar knows the truth and consciously chooses to lie.
Another possible question is, "If I were to ask you whether you always tell the truth, what would you say?" In this case a reply of "yes" means you are talking to the truth teller and a reply of "no" means you are talking to the liar.
Both questions take advantage of the liar lying about what he or his brother would say, creating a double negative type situation.
This logic is wrong, it presumes the liar sometimes tells the truth, which destroys the logic of the problem and a possibility for a reasoned solution. The lying brother would also answer yes, they know they lie, but they say they do not, therefore they answer yes to the question. This presents the no win paradox of asking "do you tell the truth?" to either of them, both will answer yes.
Mike Schiraldi had this response
I have a much simpler solution to Brad Crane's problem. Just ask him a question which you already know the answer to! Ask him, "Do you exist?" for example.
That is one solution, but under the logic of the original quoted author, the liar is inaccurate in his belief of his own existence, so they will answer yes. Typically these problems involve a decision following the question and it is a life or death choice, so to ask either do they exist will leave the life/death issue unresolvable logically.
Stephan Hradek went farther
Of course Mike Schiraldi's answer is the best, regarding _that_ riddle, But I think, the original riddle is a bit more different. I know it like this:
Someone was sentenced to death, but since the king loves riddles, he threw this guy into a room with two doors. One leading to death, one leading to freedom. There are two soldiers, each one guarding one door. One of the guards is a perfect liar, the other one will always tell the truth. The man is allowed to ask one soldier one yes-no question and then has to decide, which door to take. Which one question can he ask to find the door to freedom?
The question you ask is: Which door will the other man tell me leads to freedom? If you asked the truth teller they will tell you the other man will tell you to take the door leading to death, if you asked the lie teller, they will tell you the truth teller would tell you to take the door leading to death, knowing the truth teller would give a true answer as to which door leads to freedom, so they will lie and say the door leading to death. Regardless of which one you ask, you should select the other door from the one indicated.
Deborah Smith has objected to Stephan's line of reasoning:
What Mr.Hradek is forgetting is that he will also be incorrect in his belief of his existance, he will have to say yes. And, yes he would have to be insane.
I agree, insanity is the only course.