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Three logicians walk into a bar. The barman says, 'Does everybody want a drink?'

The first logician says, 'I don't know.'

The second logician says, 'I don't know.'

What does the third logician say?

Please provide a clear explanation of why each of the logicians reply in the way they do.

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    $\begingroup$ I think this is a very old joke/puzzle. It appears various places in a variety of forms. $\endgroup$
    – Bob
    Commented Apr 19, 2015 at 12:31
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    $\begingroup$ As a former logician, I would like to point out that, in reality, the first logician would say "no." There is at least one person in the world who does not want a drink. $\endgroup$
    – David Richerby
    Commented Apr 19, 2015 at 20:26
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    $\begingroup$ spikedmath.com/445.html $\endgroup$
    – Abr001am
    Commented Apr 19, 2015 at 22:29
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    $\begingroup$ @DavidRicherby As another former logician, I would like to point out that, in reality, although the odds would definitely be in the favor of at least one person in the world not wanting a drink, it would still be an assumption unless he indeed knows at least one person in the world who does not want a drink... $\endgroup$
    – Warlord 099
    Commented Apr 20, 2015 at 14:16
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    $\begingroup$ @reirab I would love to know one person who has never ever in their life drank some water... There is nothing to imply that the drink has to be alcoholic (which I am assuming is your premise). I don't drink alcohol and I have been to bars with friends where I have had food and water. $\endgroup$
    – Warlord 099
    Commented Apr 20, 2015 at 20:47

2 Answers 2

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Question:

Does everybody want a drink?

The third logician answers:

"YES" if he wants a drink, "NO" if doesn't want it.

Reason:

The first says "I don't know" because he wants a drink, but doesn't know if everybody wants one. If the first didn't want a drink, he would have answered "No".
Same for the second, he wants a drink but doesn't know if the third wants one. So, the third answers "Yes" if he wants a drink, "No" if he doesn't.

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    $\begingroup$ He might also say "I don't know" if he interprets "everybody" as more than just the three logicians. $\endgroup$
    – Ian MacDonald
    Commented Apr 19, 2015 at 12:05
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    $\begingroup$ Actually, if "everybody" meant more than the logicians, the first one could safely answer "No", as the barman is not allowed to drink on the job. $\endgroup$
    – Nigralbus
    Commented Apr 21, 2015 at 12:19
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    $\begingroup$ @Nigralbus Not being allowed doesn't mean he doesn't want one! The logician would have to be aware of someone who never wants a drink, which seems trivial -- assuming a newborn doesn't want alcohol is a safe bet. $\endgroup$
    – Matthew Read
    Commented Apr 21, 2015 at 12:31
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    $\begingroup$ @MatthewRead But there is no reason to think that the drink has to be alcoholic... $\endgroup$
    – Warlord 099
    Commented Apr 21, 2015 at 13:14
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    $\begingroup$ If the first one really wants a drink, he will say "Yes", knowing that if the premise of all three being logicians is true, then their answers will not contradict. In that situation, either the premise is false or all three say "Yes" and the first one gets the drink he wanted.. $\endgroup$
    – Anon
    Commented May 20, 2015 at 22:23
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We'll name the logicians A, B, and C to simplify.

If A wants a drink, A doesn't yet know whether B and C want a drink. Thus A cannot answer "yes".

If A does not want a drink, A would say "no" because at least one of A, B, and C don't want a drink, making "does everybody want a drink" false.

By saying "I don't know" A indicates their personal desire for a drink, but his lack of knowledge about the other two.

B is essentially in the same situation. B can deduce that A wants a drink, but still doesn't know the preference of C, therefore must answer "no" if B doesn't want a drink, and must answer "I don't know" if B wants a drink.

C, however, can now deduce that A and B both want drinks.

So if the third logician wants a drink, they say:

Yes, everybody wants a drink.

If the third logician doesn't want a drink, they say:

No, not everyone wants a drink.

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    $\begingroup$ At which point the barman says, "Fooled you. I'm not thirsty." $\endgroup$
    – ruffin
    Commented Apr 20, 2015 at 19:14

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