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It is generally known that lecturers for some course either always tell the truth or always lie. Arthur and Ignaz are those lecturers. Arthur says: "Exactly one of us is lying". Ignaz says: "Arthur is telling the truth".

Which is the case? Who is telling the truth and who is lying?

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If

Arthur is telling the truth, then (from what Arthur says) Ignaz is lying but (from what Ignaz says) Ignaz is telling the truth. That's no good.

So

Arthur is lying. Then (from what Arthur says, which we now know to be false) both must be lying -- and indeed what Ignaz says is false.

Thus:

Arthur and Ignaz are both lying.

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Answer:

This is a classic one. The answer is that both of them are lying. Arthur is not telling the truth because he is lying when he says exactly one of them is lying. Ignaz is lying when he says Arthur is telling the truth. Therefore, the only possible answer is that both of them are lying.

Cases:

Both can't be telling the truth because the two statements are contradictory.
if Arthur is lying and Ignaz is telling the truth, then, obviously, it is again contradictory. So, that's not possible.
If Ignaz is lying, and Arthur is telling the truth, that's again not possible because Ignaz's statement gives information about Arthur. And if Ignaz is lying, then, Arthur has to be lying. Hence, the answer is that both of them are lying.(The Lying lecturers?)

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"Between Person A and me, exactly one person is lying" is equivalent to "The truth telling-status of Person A and me are different". "The truth telling-status of Person A and me are different" and "Person A is lying" are complementary; if someone claims one, we can conclude the other. Similarly, "The truth telling-status of Person A and me are the same" and "Person A is telling the truth" are complementary. So from Arthur's claim, we can conclude that Ignaz is lying, and from Ignaz's statement, we can conclude that Arthur's status is the same as Ignaz. Therefore, they are both lying.

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