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It is known that when logicians get drunk they may claim not to know the answer to a question when they should know it very well.

After a night of heavy drinking at the bar, three logicians decide that it's time leave. Before they walk out of the place, the barman asks "Is anybody able to drive back home?".

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

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

The third logician says "I don't know".

How many logicians must be drunk at a minimum? Provide reasoning.

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11  
Is it intended that a logician is unable to drive back home if and only if he is drunk? – Ninety-Three Jan 7 at 21:25
14  
People who supply answers like "the third logician does not know whether there is gas in his car" would probably be happier on a different site. – Malvolio Jan 7 at 23:35
39  
I don't know $~$ – humn Jan 7 at 23:50
5  
@CᴏɴᴏʀO'Bʀɪᴇɴ *lojician – Emrakul Jan 8 at 17:52
11  
Thank goodness none of them are drunk enough to think they can drive home. – Ian MacDonald Jan 8 at 18:02

13 Answers 13

up vote 73 down vote accepted

All three are drunk

If the first were sober s/he would answer yes. So the first is drunk.

If the second were sober s/he would answer yes. So the second is drunk.

If the third were sober s/he would answer yes. The third now has enough information to answer "no", but is too drunk to realise it.

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5  
Ahh, I was assuming it was possible to be sober and unable to drive. If they can drive iff they are sober, you're right. – frodoskywalker Jan 7 at 23:00
4  
Ah! Your last sentence makes sense of the first sentence of the question. +1 – Todd Wilcox Jan 8 at 14:08
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Although accepted, this answer is wrong. Suppose that all three logicians are stone-cold sober (I guess it's been a heavy night of diet Cokes) but none of them knows how to drive, or none of them has a car available. Although it is highly likely that somebody, somewhere in the world, is in a position to drive home, none of the logicians categorically knows that to be the case (it's possible, though massively, massively unlikely) that everybody in the world who knows how to drive is too drunk to drive, so they all answer "I don't know." – David Richerby Jan 10 at 21:51
2  
@CandiedOrange I don't think I'm redefining any words, though maybe I'm being a bit weaselly in trying to claim that "heavy drinking" doesn't necessarily mean "consuming large amounts of alcohol" or that that doesn't necessarily mean "too drunk to drive". However, if we interpret "heavy drinking" as necessarily implying "too drunk to drive", then the question is dumb: the answer has nothing to do with the drinkers being logicians or with the answers they give but it comes merely from the dull deduction that three people, after a night of heavy drinking, are too drunk to drive home. – David Richerby Jan 10 at 22:06
2  
Interesting fact: if we suppose every logician in the company known whether or not any other can drive, it can be proven that all three have to be drunk even if not-too-drunk logicians are allowed to be unable to drive. – Cthulhu Jan 11 at 14:32

At a minimum, one logician - the final one - must be drunk.

The first two can reasonably say they don't know if

  • they cannot drive but
  • they think one of the later logicians might

The last one knows none of the prior logicians can drive, their answer should be "yes" or "no" according to their own ability. The fact that they do not answer thus means they are drunk.

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1  
why can't the first 2 drive if they are not drunk? – njzk2 Jan 8 at 21:02
14  
@njzk2 I had assumed it is possible for a sober logician to be unable to drive. Many sober people can't drive ;-) – frodoskywalker Jan 8 at 21:07

At minimum? Zero.

The first logician was driven to the bar by his friend, who then biked home, leaving the car in a local garage. He himself never learned to drive, so he cannot drive home, and knows nothing about the other two. The second and third logicians both walked to the bar, but know how to drive. Each individually cannot drive because they have no car, so their answer to the bartender is conditional on whether the first logician came with a car, which was not revealed by his answer.

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2  
"If one more mathematician enters that building it will be empty!" – Michael Jan 10 at 21:17
    
This is now officially not funny. – OldBunny2800 Mar 9 at 21:06
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@OldBunny2800 I never even meant it to be funny - the question asked what the bare minimum was, and I answered accordingly. If I were trying to be funny, I'd hope I could do better than this! – Zerris Mar 9 at 23:51

The minimum is zero. All three logicians never learned to drive a car and there is one additional guest in the bar unknown to the logicians. The word "anybody" is not limited to the three logicians.

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2  
Nor is it limited to the people in the bar! – immibis Jan 9 at 12:34

Three. This is a conversation involving four persons: the barman and three logicians. If all three logicians claim that they do not know the answer to question "Is anybody able to drive back home?", they all must be drunk, because they very well know that the barman is sober and fully able to drive. The barman is "anybody". Being logicians they react only to the logic of the question and completely ignore the practical side of it.

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3  
How do they know the barman is able to drive? – JBentley Jan 9 at 5:39
1  
@JBentley drunks are able drive, just like they can keep a bar, such an act is however illegal. – Jasen Jan 9 at 11:57
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"because they very well know that the barman is sober and fully able to drive" such a strong statement without ANY indication for 1) that the barman can drive 2) that the barman is not drunk. – PascalvKooten Jan 9 at 13:21
    
@Jasen regardless of the barman's drunk/sober status and the illegality of driving while drunk, there's nothing in the question that shows us that the logicians know the barman can drive (i.e. has the ability to). The only way they'd know that is if they witnessed him driving on his way to the bar. – JBentley Jan 9 at 15:01

The answer is

zero

This is because the question is

poorly phrased. It is not known to the logicians what set of people 'anybody' is referring to. It could obviously refer to 'anybody' in the world, which would make the answer a trivial yes, It could also refer to just the 3 logicians, but being logicians they feel it's not their place to guess. So they answer with 'I don't know (could you elaborate on which people are part of 'anybody' in this case)'

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2  
I also think the answer is poorly phrased. As addition to your answer there are other factors that are uncertain. For example: do they all even have a driver's license? Do they know from each other who has a driver's license? Do they know from each other who is drunk? – Ivo Beckers Jan 8 at 11:46

Three. They have all been drinking heavily, and they all should know very well that none of them is in a state to drive. Any logician answering "I don't know" instead of "no" must be giving that answer because of being drunk. They all gave that answer, therefore they are all drunk.

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Zero. Imagine that each is at 99% of the legal limit, but how would they know for sure. Each would say "I don't know," because no one has measured their alcohol level, but perhaps none are legally drunk.

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I guess it must be as per the truth table in logic gates, if its 0 AND 0 , result/output is 0. Similar way , If first logician says "I don't know" AND so does the second one, the third logician as per the truth table says "I don't know" .

So the answer must be all the three logicians are drunk.

Just a funny thought..

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Three.

If a logician is sober, she will say "yes" because she herself is able to drive. So "I don't know" always means a logician is drunk, completely independent of everything else,

which is too simple and therefore makes me think I did not understand the question. Well, ok, there's one last wrinkle: is this situation even possible?

and the answer is yes, because although the third logician knew the answer was "no", she might decide to say "I don't know", and therefore there is no contradiction. Had she said "no", there would be a contradiction, but that's irrelevant.

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This is incorrect. Somebody who cannot drive home is not necessarily drunk. – David Richerby Jan 10 at 18:02

Zero

For all bodies, any is able to drive. Taking things very literally as a logician, they do not know the capabilities of all bodies in question, and thus cannot determine the answer.

The answer would be different if the question was "Are any of you able to drive back home?" At which point we could receive a more accurate number of drunken logicians. The first two would be able to speak for themselves with a yes or no answer, and the third would be able to speak conclusively based on the first two answers and their own knowledge with a yes or no answer.

However, as it is worded, they don't have enough information, and consequently don't know.

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Assuming they all have a driving license:

3 because if any could drive he would know.

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This is incorrect. Even if they all have driving licenses and they are all sober, they might be unable to drive home because they came by bus so don't have a car to drive. – David Richerby Jan 10 at 18:03
    
@DavidRicherby are you unable to drive if you don't have a car? – njzk2 Jan 10 at 18:27
    
The question asked is, "Is anybody able to drive back home?" I don't see how anyone who doesn't have access to a vehicle can reasonably claim that they're "able to drive back home". It's not a question about being able to drive in general but a specific question about being able to drive back home after leaving the bar. – David Richerby Jan 10 at 18:33
    
@DavidRicherby that's not how I understand the question. If you go this way, then they don't know anything about each other, and the question is trivially absurd – njzk2 Jan 10 at 23:38

None of them are necessarily drunk. If any logician was able to drive home, they would have said "yes." We conclude that each logician knows that they personally cannot drive home, but that could be because they don't know how to drive or they left their car at home, not necessarily because they're drunk. Each logician surely also feels that it is very likely that somebody, somewhere in the world, is capable of driving home. However, they do not know this for sure (some unlikely circumstance may have rendered everybody who knows how to drive unable to do so, or maybe everyone who is in a fit state to drive home is already at home) so they answer, "I don't know."

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