Identical twins, sort of

[bonanova]

Identical twins Cal and Hal look alike, act alike and seem to talk alike. If asked whether 2+2=5, both would say No. "Is the sky blue?" would get a Yes from them both. Yet one always tells the truth, while the other always lies. That is because the truthteller has an accurate view of the world, and truthfully presents it, while the liar has a totally inaccurate world view (he believes the sky is not blue) and wrongfully presents that view. Thus they will give the same response to the same question.

Joe and Moe, themselves twins and also logicians of some repute, pondered this strange behavior one day. They wondered: if they 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? Joe said, "No, it would not be possible because whatever answers you got to your questions, the other brother would have given the same answers." Moe claimed that it was possible to find out. Moe was right, and the puzzle has two parts: (1) How many questions are necessary?; and (2) more interesting yet, What was wrong with Joe's argument?

- Puzzle due to R. Smullyan.

[Rainman]

The problem with Joe's argument is that the twins are two different people. Any one honest and accurate person would answer exactly the same as if he were dishonest and inaccurate. But ask the twins a question regarding their own identity, and they will have a different base answer which allows you to distinguish one from the other.

[the_count]

Might be able to do it with one question like...

[spoiler='Identical twins, sort of']

Q: Are you a liar?

....The accurate truth-teller rightly believes himself to be honest and thus answers "no"...the inaccurate liar wrongly thinks himself an honest man, but nonetheless is compelled to lie and thus responds with "yes". These responses distinguish the two.

However, if the above is properly reasoned, I still do not know if the given explanation counts as a concise answer for the second portion of the question

Edited December 27, 2013 by the_count

[bonanova]

Might be able to do it with one question like...

[spoiler='Identical twins, sort of']

Q: Are you a liar?

....The accurate truth-teller rightly believes himself to be honest and thus answers "no"...the inaccurate liar wrongly thinks himself an honest man, but nonetheless is compelled to lie and thus responds with "yes". These responses distinguish the two.

However, if the above is properly reasoned, I still do not know if the given explanation counts as a concise answer for the second portion of the question

Hi Count, and welcome to the Den.

Your question solves the first part of the puzzle.

To answer the second part, think of some other questions and determine what they have in common.

[the_count]

Thanks for the warm welcome bonanova...I shall ponder your hints further

[Unraveler]

The same question but in reverse; are you a truthteller. I think that would produce the same result.

[Nins_Leprechaun]

Hopefully this should answer the second question

Joe's argument relies on the fact that when the confused lieing twin answers a question he ends up negating his answer twice (once due to confusion or world view, once more to make it a lie) and thus ends up answering honestly. However Joe forgot that these two different people will have different honest answers to the some questions, such as whether or not they are trying to lie, or if their name is Hal.

Edited January 15, 2014 by Nins_Leprechaun

[bonanova]

The problem with Joe's argument is that the twins are two different people. Any one honest and accurate person would answer exactly the same as if he were dishonest and inaccurate. But ask the twins a question regarding their own identity, and they will have a different base answer which allows you to distinguish one from the other.

Rainman identifies the crucial point. It's a subtle one. If overlooked, we have a paradox. It's basically this: "you" has different meanings when Joe and Moe are addressed. Because the pronoun stands for a different noun in the two cases, the questions are not identical. Even though the wording itself is identical.