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A competent logician is walking down an unfamiliar road to reach his home, until he reaches a fork. Stood by each of the routes is a twin, who both look identical to each other. He has been warned of these twins by a passerby: one of them always tells the truth, and the other always lies.

Being a competent logician, he asks a single question of one of the brothers to find the correct route home. Out of curiosity he continues to ask questions until he discovers which of the twins lies and which one tells the truth.

In asking his questions, the logician discovers that these twins, whilst mystically cursed to only ever address the truth from a single perspective, have been given the ability to speak the truth or lie about everything, even if they have no knowledge of the subject.

The logician gets excited by this, and considering that he is not in a rush as he has no further plans for that day, and he knows his cat at home can fend for itself for a while, he spends the next several hours interrogating the twin that tells the truth, occasionally clarifying with his brother, by constructing a series of carefully phrased yes/no questions.

He learns the answers to some of life's greatest questions in subjects he previously had no idea about, such as quantum physics, religion and philosophy. He also discovers a great many interesting secrets about his friends and family, and also learns facts that would help him to impress the person that he has a crush on.

This is until the logician reaches a stumbling block. He asks the truth teller a question that seems straightforward, but upon asking the twin the same question, he observes that he gives an answer that is the same as his brother's answer. Confused, he attempts ask the same question again, and finds that the twins again seem to agree on a single result.

The logician reasons to himself:

Logically, this should be impossible. As both twins separately know exactly what the truth is, if one of them always lies and the other always tells the truth, then they should never appear to be in agreement on the answer to a particular question.

He fears that his skills for reasoning and logic have become a poisoned chalice, as he can no longer trust any of the things that he has been told by either of the twins.

Disheartened, the logician believes he has been swindled by the passerby that gave him the warning about these twins, and feels that he has wasted an entire day. Despite everything earlier appearing to make sense to him, he decides that the information he has learned is useless, and promptly forgets everything he was told that day, before walking the rest of the journey home.

What question could the logician have asked?

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    $\begingroup$ There are several questions that would produce this result - anything of the form "What would your brother say if I asked him..." for instance, which is the solution you referenced at the beginning. I'm not quite sure what you're asking for, to be honest. $\endgroup$ – Deusovi Jul 19 '16 at 12:08
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    $\begingroup$ There are a couple of valid answers now. Is this a "guess what I am thinking puzzle?" $\endgroup$ – Trenin Jul 19 '16 at 14:27
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    $\begingroup$ Am I right to assume he asked a question that has an objective answer, meaning he did not ask an opinion or ethical question? $\endgroup$ – Zircon Jul 19 '16 at 17:27
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    $\begingroup$ What happens in the event that the answer to the question is indeterminant. What would each of the twins answer if you asked "Given Euclids first 4 postulates (excluding the parallel postulate), if two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, do the two line intersect each other?" Since we don't have the parallel postulate, the answer is neither yes nor no, but it depends on the particular space. $\endgroup$ – Shufflepants Jul 19 '16 at 17:59
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    $\begingroup$ In some sense, the truthful twin could say 'yes' because that is true in Euclidean space. And the liar could say 'yes' as well as it is not true in a hyperbolic space. $\endgroup$ – Shufflepants Jul 19 '16 at 18:02

12 Answers 12

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It could be something where the answer could go either way. Given that the question deliberately mentions a cat, the traveler asks something like

Is my cat alive?

or

Is light a particle?

These statements can be both true and false. If the logician is unfamiliar with the concepts, then he may think it a relatively simple question with a definitive answer.

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  • $\begingroup$ Comments are not for extended discussion; for discussion on the validity of the question and the Copenhagen interpretation, please see chat. $\endgroup$ – Aza Jul 21 '16 at 13:58
  • $\begingroup$ If the first one is the intended answer, the question should have been: "What's the name of the logician?" to make it one layer deeper, and also gives a clue on what kind of answer is expected. $\endgroup$ – justhalf Oct 21 '16 at 2:40
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Might be

Do you always tell the truth? (Both will say yes)

or

Do you ever lie? (Both will say no)

not exact:

What time is it, to the second? (T will give the time, F will say the same, but it won't be that time anymore)

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    $\begingroup$ The issue with this is that it's essentially asking the two brothers different questions, as they have different subjects. You're asking A "does A always tell the truth?" and asking B "does B always tell the truth?" to which the answer will always both be yes. However if you asked both A and B if A always told the truth, the answers would be different. I'll try to clarify the question a little, as it isn't very clear. $\endgroup$ – Mike.C.Ford Jul 19 '16 at 15:29
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Although I am not a competent logician, it seems to me that any question with a binary answer, that asks how the other twin will answer it, will have the same answer when asked of both. For example, if x=true, then the question "How will your twin evaluate x?" will get a response of false from each twin.

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  • $\begingroup$ Actually that's a paradox. Think carefully. If the liar says false than he would telling the truth... $\endgroup$ – The Great Duck Jul 20 '16 at 5:40
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    $\begingroup$ @TheGreatDuck I don't get it. The liar says that the truther will say false, when he would say true. Therefore if the liar says false he will not be telling the truth. ? $\endgroup$ – BobRodes Jul 20 '16 at 5:49
  • $\begingroup$ But the rather will say false so then when the false says false he will be telling the truth. It's an example of a question that is neither true nor false. A is truther. B is liar. A says B is false. Then B says A is true, but now A is lying. A now says true. So B says false.......... $\endgroup$ – The Great Duck Jul 20 '16 at 5:56
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    $\begingroup$ @TheGreatDuck You're being confused right now. Look at the question, it isn't self-referential at all and perfectly fine. The question isn't "What would your twin answer to this question", but "What would your twin answer to 'Water is wet'?" $\endgroup$ – Anon Jul 20 '16 at 6:25
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Seems like you need a question that

has a different answer the second time it's asked.

So, what about

"Have I asked this question before?"
Ask the truthteller first: he says 'No.'
Then ask the liar: The truth is 'Yes', so he'll say 'No'.

If you want to avoid any pronouns, as they complicate the 'same question' stipulation, you could ask
'Have I ever said ____?', where the blank is something you've never said. Same reasoning as above.

INCORRECT
This solution doesn't work because

the puzzle states that the asker asks the pair of questions again, and on this second pass, both answerers (instead of just one) have already heard the question. Thus, if the question was "Have I asked this question before?" the true answers will be "No", "Yes", "Yes", "Yes", and the stated answers (truthteller first) will be "No", "No", "Yes", "No".

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    $\begingroup$ "He attempts to ask the same question again...the twins seem to agree" - But this time they would say "yes" and "no". $\endgroup$ – Zircon Jul 19 '16 at 17:20
  • $\begingroup$ @Zircon The truths of my question will be 'No' and 'Yes'. The stated answers will be 'No' and 'No'. $\endgroup$ – kayzeroshort Jul 19 '16 at 17:54
  • $\begingroup$ This issue makes me realize there is some ambiguity in the wordage of the question. When the OP wrote "Confused, he attempts ask the same question again, and finds that the twins again seem to agree on a single result," I had assumed that meant he asked each of them again, in the same manner he had been. However, if he only asked the lying twin again, then you would be right. $\endgroup$ – Zircon Jul 19 '16 at 18:10
  • $\begingroup$ Ah, now I see that my solution won't work on the second time around... $\endgroup$ – kayzeroshort Jul 19 '16 at 18:47
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    $\begingroup$ @kayzeroshort Nonsense! Simply put "an odd number of times" into that question and you're good to go. $\endgroup$ – mr23ceec Oct 20 '16 at 12:39
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Asking, or answering the question changes the state of the object of the question.

Do I want to ask another question?

First asking the Truthful Twin

The twin answers. NO. Because the Logician has indeed run out of questions he wants the answers to.

Asking the Lying Twin

The Liar responses. NO. Because the answer causes the Logician to become confused, and naturally he wants to figure out what is going on.

Asking the Truthful Twin again.

Answers. Yes. Because he is confused, bemused and wants to get to the bottom of this.

Asking the Liar again.

Answers. YES. Because the truth is, he does not want to ask another question, because he "fears that his skills for reasoning and logic have become a poisoned chalice".

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I am not sure that there can be a definitive answer (or question, as the case may be). What happens if the question-asker is not omniscient?

For instance, the question "Will my sister ever marry?," where the question-asker knows of his spinster sister, but not his lovely illegitimate half-sister. In this case, the truth-telling twin could be referring to the known sister (saying "No"), and the liar could be referring to the other (saying "No" as well).

Other questions that could fall into this category

Is the sky blue? (which planet are we referring to?)

Is whole milk good for you? (depends on age)

Is the answer to "what is the answer to life, the universe and everything" really 42? (maybe?)

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I can't think of any question involving a fact that will work as a solution other than a fact about which the truth changes by simply asking the question. For instance:

"Have I asked this question an odd number of times?" Each time the question is asked the truth changes (yes,no,yes,no,...)

However, most of these types of questions just come across as two different questions anyway because everyone involved in the conversation understands that additional information has been added, and hence the logician would not be surprised by the answer.

Subjective questions can definitely have both twins having different opinions (thus answering the same), but since we know nothing of the twins personalities there is no logical way to come up with a subjective question that we could be sure they would have the same answer to. The only exception to this would be the man asking the twins the following:

"Do you think I am a truthful person". On the topic of telling the truth we have adequate information to make a reasonable assumption that the truthful twin will think he is not a truthful person and the lying twin will think he is a truthful person. Therefore the truthful twin will always answer no, and the lying twin will think yes but also answer no (as he lies).

However, even the response to this question shouldn't really cause the logician to be very surprised as he is also well informed of this exception. And perhaps it also slightly breaks the rule of the questions being different because first you are asking person A about person A's opinion and then asking person B about person B's opinion.

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  • $\begingroup$ Subjective questions aren't an issue, since these twins know the absolute truth about everything and so they prove the existence of an absolute truth. Thus subjective questions are only subjective to us mortals who don't know the absolute truth. As in the question, these twins give the truth about Physics, Religion, and Philosophy. $\endgroup$ – GreySage Jul 19 '16 at 19:07
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    $\begingroup$ Sorry subjective might have been the wrong word, relative is the word I meant to use. Is something big or small, hot or cold, etc. Since most things are not at an absolute end of the scale, there is not a 'correct' answer. I suppose if they knew everything, they could know exactly what the answer means to the man asking the question, and answer in terms of his understanding of the relativity involved. I guess you could also argue its technically an incomplete question which a logician would never ask. A question of relativity should be completed by giving another thing for which to compare. $\endgroup$ – Kachimaru Jul 19 '16 at 19:12
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The answer is quite simple really and makes perfect sense:

How will I die?

Reasoning

The answer from each brother will be "I cannot answer that". Simply put, the brothers cannot see the future and therefore cannot answer true or false.

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  • $\begingroup$ ....except when the answer is "painfully, when you ask me again", but I guess you would not be interested in the statement of the liar (2nd round) in that case... (And the story wouldn't have continued) $\endgroup$ – BmyGuest Jul 20 '16 at 6:29
  • $\begingroup$ @BmyGuest exactly. We assume the brothers are not evil or malicious, and therefore won't randomly kill. Plus, considering they answered all those questions... They must be either very nice people or cursed to stand there all day and only speak truth to questioners. $\endgroup$ – The Great Duck Jul 20 '16 at 18:35
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I'm not sure if I got the question right but

If I'm not mistaken the question would actually be the first question he asked: What would your brother say, if I asked him if the left (right doesn't matter) way is the correct way to get home? -> Both will answer the same (and you figure the right way by going in the opposite direction)

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Say, the girl he has a crush on is called Amanda. Thanks to the twins, he's asked a question, e.g. does she like flowers, the answer was yes. Now he asks the true twin:

Is it possible to make Amanda fall in love with me, using the conclusion from the previous question? (= c0 : Amanda likes flowers). Answer is No; (=> c1 = Amanda will not be won over by flowers.)

To False twin:

Is it possible to make Amanda fall in love with me, using the conclusion from the previous question? (c1 = Amanda cannot be won over by flowers.). False twin knows that if he does not approach with flowers, he "will" approach via the correct way i.e. the truth is yes. He lies and says "No" (=> c2 = Amanda can be won, but not by flowers). However, the logician is perplexed, because he is not aware of this variable, as it is not made known to him, therefore based only on the information he already has, c1 cannot possibly be used to make things better as far as he's concerned.

To True Twin:

Is it possible to make Amanda fall in love with me, using the conclusion from the previous question? (c2 = Amanda can be won, but not by flowers). Knowledge of this fact alone does not help in itself. So twin says no. (=> c3 = Knowing that amanda can be won, doesn't help you to know how) Logician agrees, but for a different reason: he thinks the previous conclusion was useless.

To False Twin:

Is it possible to make Amanda fall in love with me, using the conclusion from the previous question? (c3 = Knowing that amanda can be won, doesn't help you to know how). The knowledge that the previous knowledge is not useful, can be used to make the logician ask about different approaches instead. So the true answer is yes. False twin lies and says "no". Logician is confused, because he expected a yes, since knowing that a useless conclusion doesn't help shouldn't help in itself.

We've now had four "noes" in a row which made sense for the twins. However, they don't make sense for the logician, because from the 2nd question he reaches the opposite conclusions each time, due to lack of future knowledge, and judging only from present variables. Therefore the logician loses confidence, in the absence of knowledge about knowledge.

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  • $\begingroup$ They both answer from the same objective view. The liar would be using a different interpretation in that context. $\endgroup$ – The Great Duck Jul 21 '16 at 1:58
  • $\begingroup$ If the logician had the ability to ask them both at the same time, then yes, their answers would be a Yes/No pair; the point is that each time the question is asked, it refers to new information added in the previous answer by the other twin, so the difference in interpretation by the two twins for the 'same' question is irrelevant. $\endgroup$ – Tasos Papastylianou Jul 21 '16 at 11:13
  • $\begingroup$ The crux is that the logician has a different interpretation from the answering twin at each point in time, because their interpretation relies on knowledge they have, whereas the logician's interpretation relies on knowledge that he does not have, and does not know he doesn't have, and therefore has no reason to assume exists, so he interprets the context based only on known information and known unknowns. $\endgroup$ – Tasos Papastylianou Jul 21 '16 at 11:16
  • $\begingroup$ But the twins are omniscient. They know and see all relevant to answering. Therefore, they cannot change their interpretation as all information in the entire continuum of the universe has been taken into account. $\endgroup$ – The Great Duck Jul 21 '16 at 19:18
  • $\begingroup$ They could only give different answers if the question were in fact true and false. It would have to be an answer whose negation is itself. $\endgroup$ – The Great Duck Jul 21 '16 at 19:19
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The question could be:

Are you a man?

if both twins are of the opposite sex, and assuming that the male twin is telling the truth, he will reply yes and the other twin (female) will lie and say yes too.

The same goes the other way.

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    $\begingroup$ This would work from the logic, but I don't think a perfect logician would be very surprised/perplexed. Even if he can't distinguish the sex of each twin, he would clearly realize that he can not easily distinguish the sex of each twin and hence would have no issue with getting the answers. $\endgroup$ – BmyGuest Jul 20 '16 at 18:47
  • $\begingroup$ Technicallly they are not the same question. That is like saying f(x) is true, and f(y) is false, so there is an inconsistency. We want -f(x) = f(x). $\endgroup$ – The Great Duck Jul 21 '16 at 1:57
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Wouldn't...

Does your twin lie?

...do?

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  • $\begingroup$ No because it isn't the same question each time. There is a variable in the question that changes. $\endgroup$ – The Great Duck Jul 20 '16 at 18:36

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