# Three people sitting together of which two are truthful and one is a liar. Can the truthful detect the liar?

How can this question be approached philosophically?

EDIT: A liar in this case can produce multiple versions of a statement.

• I don't understand the edit. Do you mean that the liar's behavior is random? – Hunan Rostomyan Apr 24 '14 at 7:23
• Well, the liar can produce multiple versions, now whether all of them are false, or only some of them, I am not sure. Its how its given in the test that I am trying to solve. – user6214 Apr 24 '14 at 7:26
• So we've moved from a liar to a skilled dissembler? In that case, no we cannot detect the liar without further information about the parameters under which the liar lies. – virmaior Apr 24 '14 at 7:27
• @virmaior Okay, I guess I will go with a No, rather then "Unsolvable". Hope its right, I will share if I get the right answer. – user6214 Apr 24 '14 at 7:29
• Pick a person, ask her the same question n times. The liar's answer will vary. But that's not an actual solution, because it's possible that the random-liar behave like a truthful person. So the answer I believe is a no. – Hunan Rostomyan Apr 24 '14 at 7:29

This is just the well-known two brothers puzzle. It is easy to solve if you can assume that the truthful is incapable of lying, that the liar is incapable of telling the truth, and that both are aware of the other's character (and that each knows the other knows).

Then ask either of the two "would the other person describe you as a liar?". If the answer is "no", you are talking to the liar. If the answer is "yes", you are addressing the truthteller. The solution works by effectively routing the answer through one truth and one lie to yield a known truth value.

If the starting assumption that all three of you know there are two truthtellers and one liar, and that each person knows what he is, but that you don't know which the others are, the problem is harder but still soluble with the same basic pattern. Your question would now be "would the other person say that I would describe you as a liar?" which routes you through two truths and a lie.

The problem is harder again, and probably insoluble, in the case that the liar can tell the truth if he wishes, since he could then theoretically duplicate any answer the truthgiver could give. There's a knowledge differential in this case, since the liar knows his two colleagues are both truth-tellers, but it's all to the advantage of the liar, so it's hard to see how to exploit that.

Edit: I didn't closely read the earlier answers, so I see that I've partially duplicated the ones ahead of mine (although I hope I've added a little explanatory value).

It also occurs to me that the question "is there a liar among us?" would do the trick, since the truth-teller would say yes, and the liar would say no.

• You certainly have added to the discussion. The routing idea, for instance, is key I think and I'm glad it's been pointed out. Thank you. – Hunan Rostomyan Apr 24 '14 at 16:55
• I think this does add valuably. – virmaior Apr 25 '14 at 9:55

Ask both people before you the following question: "Am I truthful?". The liar will respond "no", because he should say the opposite of what is the case (he knows that he is a liar and there is only one liar, therefore the other two people are truthful, including you). The truthful will respond "I don't know", because he really doesn't.

• What if the liar does not always lie, he can change his statement or produces different versions of a statement. – user6214 Apr 24 '14 at 7:17
• This answer is very good. @user6214 the liar must be a liar for this exercise to work. It didn't say he is sometimes truthful and sometimes a liar. If that were the case and only one question was asked, if he answered truthfully he would not be a liar. – Cynapse Apr 24 '14 at 13:52

I like user132181's answer very much, but here is how I would approach it. Suppose you're one of the truthful ones, so that of the remaining two, one is a liar (L), the other tells the truth (T). How can you detect the liar? Well, let's be clear on what exactly the two types do:

T will answer "yes" to the question: "Is φ true?" iff φ is true.

L will answer "yes" to the question: "Is φ true?" iff φ is false.

Consider the following question:

(Q) Would T answer 'no' to 'is 1 = 0?'?

Pick one of the two people P and ask them Q. Let's analyze what will happen.

• T will answer 'yes' to (Q) iff T would answer 'no' to 'is 1= 0?'. Since T would answer 'no' to 'is 1 = 0?', T would answer 'yes' to Q.

• L will answer 'yes' to (Q) iff T would not answer 'no' to 'is 1 = 0?'. T would answer 'no' to 'is 1 = 0?', so L would answer 'no' to Q.

So you can ask person P question Q and if the answer is 'yes', P is a truth-teller, if 'no' then P is a liar.

Yes we can. The answer is simple. The one trying to find out if the other one is a liar is using memory - long term memory.

He is comparing past vocalizations with present actions and or vocalizations.

If there is an inconsistency found then either one is a liar or the other has a bad memory. Or both have a bad memory.

A lie is always in reference to something. To actual events or an actual statement. You cannot just have one sentence and based on that sentence determine if it is a lie or not. One sentence in itself is just a statement.

Truth --> a = b lie --> a not= b

So you need at least two items to compare. But then you still have the problem of bad memory.

I guess thats when we started inventing writing. Then at least you can compare reality then to reality now. But of course the past reality can be faked too. So you need to figure out if the reality then is a real reality then or a fake reality then.

• What's up with all the down votes? – DisplayName Apr 24 '14 at 22:01
• It's not clear what you're talking about. "He is using memory": one of the truth-tellers is using memory?; what does that have to do with anything? "That's why we have it in the first place": what is this "it" referring to? "If it does not match up": what is this "it" referring to? "There is an inconsistency...": which is? And then there is that murdering business...I don't even know what to ask about that. – Hunan Rostomyan Apr 24 '14 at 23:32
• ok thanks for the feedback, it was all making sense in my mind, I totally forgot we did not all share the same consciousness (yet) :P – DisplayName Apr 25 '14 at 8:57