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A common constraint in logic problems is the one restricting questions you may ask to those that can only be helpfully answered with a "yes" or "no".

But in some cases, you might ask someone a question where "yes" or "no" are the only answers if you know the answer, but where some respondents may lack the necessary information to know.

Example: You meet one "truth-teller" and one "random-respondent", who is "50-50" to reply yes or no to any given question. You don't know who's who, but they do. If you're limited to "yes/no" questions, can you ask one of them what the other person would say? (The problem is that the truth-teller doesn't know, and knows he doesn't know; his only possible honest answer is "I don't know."

What, if anything are the conventions in cases where "yes/no" is a constraint, but the issue of questions the respondent can't answer is not explicitly addressed? Presumably, they must fall into one of the following:

  1. You're not allowed to ask a question where the respondent might not be able to reply. This is simple, but... seems somehow more odd to me than the other arbitrary ridiculous rules, although I have no idea why that is.
  2. You are allowed to ask questions like that, in which case, you must also assume they are allowed to convey "I don't know" (assume they have a veracity type that would make them honest).

I was deliberately using a simple example, but the puzzle that this comment is on is what inspired the question. (I haven't solved it yet, so please don't share whether it's relevant to that particular solution; I'm more interested in what assumptions, if any are generally applied to other problems like this.

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  • $\begingroup$ More rigorously speaking, what I think you're looking at is whether a set of inputs fully determines the outputs, without knowing what the inputs are. If it does, you know, and if it doesn't, you don't know. For instance, if I asked "I tell you the first, second, and third byte of a three-byte number. Do you know the number?" you'd say 'yes.' If I asked you to tell me what it is, you'd say 'no clue.' $\endgroup$ – Aza Apr 20 '15 at 17:58
  • $\begingroup$ I would suggest that the simplest resolution is probably to have puzzles simply prohibit head-exploding questions. Alternatively, they could specify that anyone answering a question is entitled to regard the answer as being 'yes' if they have no reason to believe it less truthful than a 'no' response, and vice versa; if to the best of the answerer's knowledge either answer would be equally truthful, the answerer may answer in whatever fashion would be most vexatious. $\endgroup$ – supercat Apr 30 '15 at 22:48
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For the situation mentioned and most other situations I would say that there is a contradiction in the question's formation. That is, there are three contradictory givens: the truth teller will always answer your question, the truth teller's answer will always be true, and the truth teller will always answer either yes or no. Thus it should be up to the author to fix the mistake.

As for how I think one should deal with an ill-formed problem of this type: I would ask for clarification. Until the author replies to resolve the contradiction, I would first try to solve the problem by resolution 1, and only afterwards by resolution 2, possibly with a goal of improving the limitations of the problem (eg, asking only 2 questions instead of 3).

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  • $\begingroup$ I think relaxing the first statement to "The truth teller will always answer your question, if it is possible" - will make the smallest change to the rules and could therefore be chosen, if the question is not amended. - so he will simply not answer at all, if asked something he doesn't know. $\endgroup$ – Falco Apr 22 '15 at 10:52
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    $\begingroup$ @Falco, I think that allowing the truth teller simply not to answer a question to which he does not know the answer is equivalent to letting him answer "I don't know", because you can tell that the truth teller is not answering your question. Adding the third option makes a lot of questions uninteresting or easier, as you can get more information from one of 3 possible responses than one of 2. Usually the author's intention is that you shouldn't ask any unanswerable questions. $\endgroup$ – Ben Frankel Apr 22 '15 at 11:42
  • $\begingroup$ An alternative method which ensures only 2 outcomes is to say "he tells the truth unless he doesn't know in which case he says "no". $\endgroup$ – IanF1 Apr 23 '15 at 7:01
  • $\begingroup$ @IanF1: Saying that undecidable questions will be answered "no" doesn't necessarily solve the problem, since determining whether a particular question is decidable may itself constitute an undecidable problem. Allowing the answerer to say either "yes" or "no", whichever would be most vexatious, in cases where neither "yes" nor "no" is the clear answer would avoid the issue. $\endgroup$ – supercat Jul 12 '15 at 19:26
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If I don't know is a possible response, the asker will include it, otherwise you must act as if they can't answer in this way.

It is up to the askers to include all the necessary details required to answer the puzzle, if they do not, it's very low quality. Answerers should never have to make assumptions on a whole genre of a specific puzzle.

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