A puzzle of identical twins

Following is an excerpt from
5000 B.C. and Other Philosophical Fantasies
by R.Smullyan

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.

The situation is reminiscent of an incident I read about in a textbook on abnormal psychology: The doctors in a mental institution were thinking of releasing a certain schizophrenic patient. They decided to give him a test under a lie detector. One of the questions they asked him was, "Are you Napoleon?" He replied, "No." The machine showed that he was lying!

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.

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.

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.

Enjoying your page as always,

Brad Crane

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.

Stephan Hradek went farther

Hi Alexander.

To my understanding, this is wrong:

> 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. 

Imagine you are the liar and you get asked "Are you the truth-teller". Since you have a misconception regarding the world, you believe you are the truth teller. So you will _lie_ and say "no". _BUT_! Don't you reflect about your answers? So thinking "Yes" but saying "No" will have to lead you to the conclusion, that you are a liar! So you will either go mad ;-) Or will get at least on right concept regarding the world, or better: regarding yourself. So the liar will also have to say "yes".

But there is still one question, you could ask - provided, each brother knows about the other: "Is your brother the liar?" The liar thinks "yes" -> Says "no". The truth-teller thinks "yes" -> Says "yes".

What do you think?

Regarding the other riddle:

> 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? 

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?

So the target is _not_ to find the liar/truth-teller, but to decide upon the answer, one of them gives. I think you want to solve it by yourself - or you already know it? Nevertheless: Ask me if you want me to solve the riddle.

There are also some more, similar riddles, I remember. They are from a book, I once had. I can't remember it's title, nor whom I gave it - and never got it back from. Nevertheless. Maybe it would be interesting for you, to put one of the riddles - the one I can remember - onto your page?

There is a great underground empire where there live knights. There are two kind of knights; Day-knights and Night-knights. Day-knights tell the truth by day and lie at night. Night-knights, of course, do it the other way round. Since it is an underground empire, you can't tell the time of day by any natural means. But the knights always know whether it's day or night. So, as a visitor to this strange world, you once meet a knight who tells you "I'm a day-knight and it's night". Since you forgot to wind up your watch a few days ago, you don't know the time. But can you tell from this sentence, whether it's day or night? And can you tell, whether the knight is a Day-knight or a Night-knight?

Hope you like the two riddles. I'm currently trying to figure out, what the title of that book was. When I found it, I'll tell you.

regards
--
Stephan Hradek

(Lyman Hurd has gracefully informed us that Derek's misplaced book is none other than To Mock a Mockingbird by R. Smullyan.) Deborah Smith has objected to Stephan's line of reasoning:

I had to write to tell you that Stephan Hradek's line of reasoning has a flaw. He said, "If he is an inaccurate liar he will believe that he is an accurate truth teller (since he is inaccurate in his beliefs), but then he will lie and say no."

What Mr.Hradek is forgetting is that he will also be incorrect in his belief of his existence, he will have to say yes. And, yes he would have to be insane.

Kueh Shuiqiu continues the discussion:

This is regarding Stephan Hradek's rebuttal and his two riddles.

Hradek says that the liar will have to say yes when asked if he were the truth teller because if he reflected on his answers he would realize that he was unable to tell the truth and thus go insane. There are two reasons why this argument does not hold water. The first is that, though the inaccurate liar may eventually go insane, he would still abide by the rules of the riddle at the time it was asked. For all intents and purposes, he is already insane, and going further insane might not affect the way he answers questions at all. The second reason this argument doesn't change the solution to the puzzle is that if he did reflect on his answers and think himself to be insane, his inaccurate nature would lead him to believe that he was, in fact, sane.

Regarding his follow up riddles,

"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"

One might respond to this in a manner similar to Brad Cane's solution to his riddle, and ask one guard which the other would say lead to freedom. The lying guard would realize that the truth teller would tell which door lead to freedom, and then point to the door that lead to death. The truth-telling guard would recognize that the lying guard would lie about which door was which, and point to the door that leads to death. After asking, one would only need to choose the opposite of whichever door he was directed at.

And his second riddle,

"There is a great underground empire where there live knights. There are two kind of knights; Day-knights and Night-knights. Day-knights tell the truth by day and lie at night. Night-knights, of course, do it the other way round. Since it is an underground empire, you can't tell the time of day by any natural means. But the knights always know whether it's day or night. So, as a visitor to this strange world, you once meet a knight who tells you "I'm a day-knight and it's night". Since you forgot to wind up your watch a few days ago, you don't know the time. But can you tell from this sentence, whether it's day or night? And can you tell, whether the knight is a Day-knight or a Night-knight?"

If it were night, the Day-knight would be lying, so already know that this person is not being honest, meaning you can't be talking to a Day-knight during the day, or a Night-knight during the night. (In other words, it's either a Night-Knight during the day, or a Day-knight during the night.) Furthermore, the person, who we know is being dishonest, says that he is a Day-knight, meaning that the opposite is true and that the person you are talking to is a Night-knight, and it is day-time.


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