## Liars and Truthtellers - Possible trick question/paradox

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greenpixel
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Joined: Mon Mar 30, 2009 10:08 pm UTC

### Liars and Truthtellers - Possible trick question/paradox

This is a question from an example for the Cambridge admissions test for philosophy.
I can't get an answer I'm happy with, A, C, and D all cancel each other out, but B cancels itself out, which is the only thing separating it from the others.

Albert says "Everything Caroline says is true"
Betty says "Everything I say is false"
Caroline says "Everything David says is true"
David says "Everything Caroline says is false"

Who is the only person who can be telling the truth?

I think that you can answer Betty if you don't include her statement in the things she says, but as it's in quotes I can't think this would be acceptable.

Do we think this is a trick question?

EDIT: Sorry, I posted this in the wrong board! Could a mod please put it in logic puzzles?

Lukeje
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Location: United Kingdom.

### Re: Liars and Truthtellers - Possible trick question/paradox

Spoiler:
Albert is the only one who could be telling the truth.

Edit: whoops, yeah BeerBottle's right.
Last edited by Lukeje on Fri Sep 18, 2009 9:53 am UTC, edited 2 times in total.

BeerBottle
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### Re: Liars and Truthtellers - Possible trick question/paradox

Spoiler:
David is the only one who can be telling the truth.

Given that only one person is telling the truth - if Albert is telling the truth then so is Caroline, but we know only one person is telling the truth. So it can't be Albert. It can't be Betty, as she cancels herself out. If Caroline is telling the truth then so is David, but we know only one person is telling the truth. So it can't be Caroline. So David is telling the truth - If David is telling the truth, then this implies that Caroline is lying when she says everything David says is true. But that doesn't mean that nothing David says is true.

jaap
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### Re: Liars and Truthtellers - Possible trick question/paradox

Spoiler:
Lukeje wrote:Albert is the only one who could be telling the truth.

No, David is.

Lukeje
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Location: United Kingdom.

### Re: Liars and Truthtellers - Possible trick question/paradox

Spoiler:
jaap wrote:
Lukeje wrote:Albert is the only one who could be telling the truth.

No, David is.
I know; see my edit.

Nitrodon
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### Re: Liars and Truthtellers - Possible trick question/paradox

The fact that only one person is telling the truth is in the conclusion, not the assumptions. Hence, it's sort of (but not really) cheating to utilize that information. Even without that information,
Spoiler:
if Caroline is telling the truth, then David is telling the truth and thus Caroline never tells the truth, which is still a contradiction, so all is well.

d0nk3y_k0n9
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### Re: Liars and Truthtellers - Possible trick question/paradox

greenpixel wrote:Albert says "Everything Caroline says is true"
Betty says "Everything I say is false"
Caroline says "Everything David says is true"
David says "Everything Caroline says is false"

Spoiler:
Case A: If Albert were telling the truth, then so would Caroline, and thus, so would David. However, David says that Caroline is lying, and so we have a contradiction since Caroline can't be both lying and telling the truth.

Case B: If Betty were telling the truth, then she would be lying, and we have a contradiction.

Case C: If Caroline were telling the truth, then so would David. We would have the same contradiction as in Case A.

Case D: If David were telling the truth, then Caroline would be lying. If everything Caroline says is false, then it is easy to mistakenly assume that means that "Everything David says is true" negated means that "Everything David says is false". However, properly negating Caroline's statement results in "Not everything David says is true," which means that he can still be telling the truth.

Therefore, David is the only one who can possibly be telling the truth, because every other possibility results in a contradiction.

TheChewanater
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### Re: Liars and Truthtellers - Possible trick question/paradox

greenpixel wrote:This is a question from an example for the Cambridge admissions test for philosophy.
I can't get an answer I'm happy with, A, C, and D all cancel each other out, but B cancels itself out, which is the only thing separating it from the others.

Albert says "Everything Caroline says is true"
Betty says "Everything I say is false"
Caroline says "Everything David says is true"
David says "Everything Caroline says is false"

Who is the only person who can be telling the truth?

Albert said that before Caroline said anything. She could have always said the truth beforehand, so Al wasn't lying. I think this makes
Spoiler:
Albert's case possible, but correct me if I'm wrong.

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Tass
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### Re: Liars and Truthtellers - Possible trick question/paradox

TheChewanater wrote:
greenpixel wrote:This is a question from an example for the Cambridge admissions test for philosophy.
I can't get an answer I'm happy with, A, C, and D all cancel each other out, but B cancels itself out, which is the only thing separating it from the others.

Albert says "Everything Caroline says is true"
Betty says "Everything I say is false"
Caroline says "Everything David says is true"
David says "Everything Caroline says is false"

Who is the only person who can be telling the truth?

Albert said that before Caroline said anything. She could have always said the truth beforehand, so Al wasn't lying. I think this makes
Spoiler:
Albert's case possible, but correct me if I'm wrong.

Spoiler:
Albert does not only consider past cases in his statement. He may believe that his statement is true at the time he says it, but it still turns out to be false.
It didn't ask who is not lying, but who is telling the truth (relevant if we consider saying a false statement that you believe true, not to be lying).

Puck
Posts: 615
Joined: Tue Nov 27, 2007 7:29 pm UTC

### Re: Liars and Truthtellers - Possible trick question/paradox

It's important to note that the assumption that only one person is telling the truth is not a part of this problem, and not necessary to reach the solution:
Spoiler:
B cannot be telling the truth, as ipso facto it generates a contradiction. B must be false.

If C were true, then D would be true, but D says that C is false. Contradiction; C must be false.

If C is false, then A is false.

Therefore D is the only statement that could be the truth; note that it isn't necessarily true, as C could lie sometimes and tell the truth sometimes. But A, B and C are provably false.
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