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Premise: These are like the knight-knave puzzles; only now we are visiting insane asylums, where there are only patients and doctors. Moreover, each of them are either insane or sane. Sane inhabitants of an asylum are entirely accurate in their beliefs (they believe all and only true statements and disbelief all and only false ones), whereas insane inhabitants are totally inaccurate in their beliefs. Nonetheless they are all honest--whatever they say, they believe. You may reason as you like, but you must write down your reasoning.

Questions: You visit an asylum and ask an inhabitant “Are you a patient?” The inhabitant replies “I believe so.” Is anything necessarily wrong with this asylum?

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    $\begingroup$ Is the implication that doctors are sane and patients are insane? $\endgroup$
    – Julian Rosen
    Commented Oct 2, 2015 at 17:41
  • $\begingroup$ Yes. It exists. $\endgroup$
    – Joshua
    Commented Oct 2, 2015 at 22:53

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Not necessarily. That is a reply that could be made by an insane patient.

Anything an insane inhabitant says is false. When an insane inhabitant says "I believe [X]", then the truth is that he does not believe [X]. And since an insane inhabitant disbelieves only true statements, this means that [X] is true.

In a more symbolic representation,

insane_believes(X) -> !X
insane_person_says(X) -> insane_believes(X)
Therefore
insane_person_says(insane_believes(X)) -> insane_person_says(!X) -> !!X -> X

Therefore, the response you hear to your question could come from

  • a sane patient
  • an insane patient

The former is an injustice, since sane people should not be patients. But the latter is perfectly normal for an asylum. So there is not anything necessarily wrong with the asylum.

If the inhabitant had instead replied "I am a patient", then he could be either a sane patient or an insane doctor. Both of these are injustices, so it is a good thing that is not what the inhabitant said.

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    $\begingroup$ There's a problem with your second paragraph. Everyone is honest - "whatever they say, they believe." So when an insane inhabitant says "I believe [X]" the truth is that he or she believes [X]. $\endgroup$
    – Rob Watts
    Commented Oct 2, 2015 at 16:31
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    $\begingroup$ They believe that they believe [x], but they don't actually believe [x]. $\endgroup$
    – Kevin
    Commented Oct 2, 2015 at 16:32
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    $\begingroup$ Wow. Being insane must be incredibly confusing. $\endgroup$
    – user1618143
    Commented Oct 2, 2015 at 16:38
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    $\begingroup$ @user1618143, I believe so, yes. $\endgroup$
    – Kevin
    Commented Oct 2, 2015 at 16:38
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    $\begingroup$ This is incorrect. Insane patients don't lie; they tell the truth. It's just that they have an inverted concept of reality. When the patient says, "I believe [X]", they are truthfully saying that they believe [X] to be true. If they are insane, then [X] is actually false. Your last line should be insane_person_says(insane_believes(X)) -> insane_person_says(!X) -> !X because what insane people say is true; it's just what they believe that is false. $\endgroup$
    – GentlePurpleRain
    Commented Oct 2, 2015 at 17:26
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Given:

  • Sane person must believe true statements, insane must believe false statements
  • Person will accurately say what they believe
  • Person says they believe they are a patient

Sane Doctor

  • Does not believe "I am a patient" because it's false
  • Does not believe "I believe I am a patient" because it's false

Insane Doctor

  • Does believe "I am a patient" because it's false
  • Does not believe "I believe I am a patient" because it's true

Sane Patient

  • Does believe "I am a patient" because it's true
  • Does believe "I believe I am a patient" because it's true

Insane Patient

  • Does not believe "I am a patient" because it's true
  • Does believe "I believe I am a patient" because it's false

The person we are talking to must be a patient, but they could either be sane or insane and we have no way of knowing which from the information provided. Also, I don't know what criteria is used to determine if "anything is wrong with the asylum," but this situation is possible given the information provided.

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This is further to @Kevin's answer; I was having trouble understanding it, and thought I would reason it out more. If this helps you understand it, please upvote Kevin's answer, as he actually figured it out.

No, there is nothing necessarily wrong the asylum. The inhabitant could be an insane patient.

The inhabitant believes the statement, I believe I am a patient. Note that this is different than saying she believes the statement, I am a patient.

If she is an insane patient, then she is saying that she believes the statement, I believe I am a patient to be true. Since anything an insane person believes is necessarily false, then the statement I believe I am a patient is false.

Thus the inverse is actually true: I believe I am not a patient. This means that she believes the statement I am not a patient to be true.

Since anything an insane person believes is necessarily false, then the statement I am not a patient is false, and she is in fact a patient.

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