# "Does everyone want coffee?" propositional logic problem

When three professors are seated in a restaurant, the hostess asks them: “Does everyone want coffee?” The first professor says: “I do not know.” The second professor then says: “I do not know.” Finally, the third professor says: “No, not everyone wants coffee.” The hostess comes back and gives coffee to the professors who want it. How did she figure out who wanted coffee?

The solution is that the first two professors who say they don't know if everyone wants coffee are the ones who want coffee.

However, I don't understand the logic. For example, the first professor may not want coffee, but does not know if the other professors want coffee, and thus can answer "I do not know." So how can the hostess take an answer of "yes" from that?

• The sad part of the story is that the hostess is a mathematics graduate. Aug 6, 2016 at 23:19

## 3 Answers

Suppose the first professor did not want coffee. Then, they would know that it is not the case that "everyone wants coffee" - so, when asked "Does everyone want coffee?," they would answer "no."

• But we don't know if any of the professors were speaking the truth in the first place. Still, I like your answer. Aug 7, 2016 at 5:40
• @DougSpoonwood That's kind of a non-sequitur. The problem has to assume that the professors are telling the truth. Otherwise, there's no way to conclude anything from what they say . . . Aug 7, 2016 at 6:01
• I feel like this would be a funny thing to do with some friends, I'm not sure if the waiter/waitress would enjoy it as much though. Aug 7, 2016 at 11:07
• The question was "how did she figure out who wanted coffee?" Maybe she didn't manage to figure out who wanted coffee, because she couldn't figure out how many of the professors told the truth. After all, maybe some of them worked in one of those controversial humanity departments, where many suspected sophists teach. Did we do an interview with her or the professors after this happened? Alright, maybe I'd better stop. You still have a good answer here. Aug 7, 2016 at 17:32
• If the first professor did not want coffee, he could still answer "I do not know" because, maybe, the second professor wants it. Also, when the 3rd professor says "no, not everyone wants coffee", he's not stating that he does not want coffee. He's only saying that not all of them want coffee but there is someone (or two) that want coffee. Dec 12, 2023 at 12:48

If he knew that he does not want coffee, then he could tell the waitress 'no', because he then knows that NOT everyone wants coffee. The only possibility that he doesn't know it when he is asked is that he does want coffee but does not know if the other two want some too. The same logic applies for the second professor. Only the third then has perfect knowledge and can say 'yes' if he also wants coffee, and 'no' if not.

The first professor wants coffee, because if he doesn’t want coffee then saying “no” would mean that the question is False — he is operating on the assumption that yes he himself does want coffee and isn’t sure if Professor 2 and Professor 3 also want coffee — which would be the requirement for the question to be True.

Professor 2 also wants coffee and for the same reason as Professor 1 — he knows now that Professor 1 wants coffee, and he himself wants coffee, but isn’t sure about Professor 3 — again, all three must want coffee for the original question to be True (note the use of "everyone" in the question).

Therefore Professor 3 is the only remaining variable that can negate the question. He then does negate the question with his statement — hence Professor 3 is the only one who does not want coffee.