I Believe I Am Mad
Edgar Abercombie reached an island where some of the inhabitants are totally mad, and all their beliefs are wrong! The other inhabitants, who are sane, are accurate in their judgments; all their beliefs are correct.
Everyone is completely truthful. Since a mad person believes (wrongly) that he is sane, he will tell you "I am sane".
Which inhabitants can say "I believe I am mad"?
This problem is taken from Raymond M. Smullyan's book Logical Labyrinths.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
2 solutions
I believe I understand the problem, but perhaps I am mad!
Joking aside, I think the problem with this problem is in how to interpret "they always tell you honestly what they believe". A lot of people seem to interpret it [I would say, justifiably] as "they accurately report their beliefs", but after reading the solution, it's obvious it was intended as "they only make statements which they believe to be true".
It's that semantic tomfoolery that leads people down completely different logical derivations.
Log in to reply
I broke up that sentence. I wanted to make it clear that this problem isn't about truthteller/liars, but is a very different concept.
How does it look? It shouldn't feel like semantic tomfoolery.
If a mad man says, 'I believe I am mad', since all his beliefs are wrong, meaning he is actually sane?
Log in to reply
No, read your statement carefully. Do not jump steps in the logical reasoning of the mad person.
Since his beliefs are wrong, then he does not believe he is mad, so he believes he is sane. It doesn't not (immediately) imply that "he is actually sane".
Log in to reply
For more head banging - https://eniuss.files.wordpress.com/2015/09/logical_labirints.pdf, page 27, 4.1.
However its all in the propositions which is what makes this confusing.
The solution in the book says this which is confusing too, - "Indeed, if you ask a mad inhabitant: “Are you mad?” he will say no, but if you ask him: “Do you believe you are mad?” he will answer yes (since he doesn’t really believe he is mad)."
Now in the proposition question do you believe you are mad the author says he will respond yes, why does he have to? Its not a faulty switch problem in mad man is it? Since he is mad and has false beliefs, he believes he is not mad, but does he have to answer in the negative to what he believes too? To clarify he will always say what he believes, and he believes he is not mad, so he can say, "I believe I am not mad", or "I am sane". To put it differently a mad man(or for that matter any person on the island) can only say one of two statements, "I believe I am not mad" or "I believe I am mad". But they will only say what they believe. And since a mad man thinks he is sane, he can only say "I believe I am not mad" , so will a sane man say the same implying that he is sane or he is not mad.
Log in to reply
@Siva Bathula – @Calvin Lin Also I think maybe I am wrong or right, it doesn't matter, too much head spinning on every response, and counter response to these sort of logic questions. However my argument is around this fact - any person will stick to and say what they believe in. The proposition and negative switch to false beliefs is somewhat confusing to me.
Log in to reply
any person will stick to and say what they believe in
That is not true. A mad person will say "I am sane" because he makes the true statement that he believes he is sane.
The true version is instead
More generally, if anyone on this island says "I believe X", then X is a true statement.
Log in to reply
@Calvin Lin – That's what even the author concluded at the end. I think the confusion will be erased if you can convince me why a mad man can't say 'I believe I am sane'.
Log in to reply
@Siva Bathula – Read the solution. I laid out all of those cases of a mad/sane man saying "I believe I am mad/sane".
Log in to reply
@Calvin Lin – Yeah that's what is frightening, the head spinning that might arise out of trying to understand the logic. I went through the author's argument on the pdf link I posted, and it was indeed head spinning and I am tired to go through yours now, hence the flooding comments. I will respond again later. Let the confusion from community pore in.
@Calvin Lin – BTW will the statement change if I were to add another layer of 'I believe'?.
Log in to reply
@Siva Bathula – Yes. The mad person will say "I believe I believe I am sane".
The guiding principle is " If anyone says "I believe X", then X is a true statement. In this case, it is true that the mad person believes he is sane (even though he cannot say that statement).
@Siva Bathula – Or is it that he can actually say both, he being mad, LOL.
So when a mad person says "I believe I am mad " , that's a wrong belief so he is not mad which is not possible. Its really very confusing !!!!
Log in to reply
No, read carefully. Do not jump steps.
Suppose a mad man says "I believe I am mad", then this means that the mad man believes "I believe I am sane". Does a mad man believe he is sane, or he is mad?
Log in to reply
Okay I got your approach to the problem but what if somebody approaches the problem in the other way . So where is my logic going wrong? I can't find the mistake in my logic , that's the problem.
Log in to reply
@Nashita Rahman – The error is in
So when a mad person says "I believe I am mad " , that's a wrong belief so he is not mad
As laid out above, the correct conclusion is
So when a mad person says "I believe I am mad " , that's a wrong belief so he believes he is not mad.
Log in to reply
@Calvin Lin – Oh yes , I got it. Thanks alot.
The beginning of the problem states "everyone is truthful" which contradicts your statement that "A mad person can only say statement which he believes are false.". As the problem is written, a mad person would believe he was sane and therefore say what he believed was true, and therefore claim to be sane.
Log in to reply
Ok, I see the logic of how "I believe" changes it. However, the first sentence of your explanation is still contradictory.
Log in to reply
Ah yes. That's an error. I've fixed that paragraph. It should be "A mad person can only say statement which he that are false."
How is the quick solution now?
Essentially, you're saying if a person A says statement X, then the correct interpretation is "A believes X is true". While it's a reasonable interpretation, I can also argue that the interpretation "X is true" is also plausible.
"You say what you believe" - V i s h a l
You say 'I'm sane' because you believe in the statement. You say 'I believe I'm mad' because you believe in the statement but you belief in the statement 'I believe I'm mad' is wrong/ false. Correct/true belief is the counter statement 'I believe I'm n o t mad'..
The key point here is to think in terms of true and false.
As long as a statement is true, either of the two can say it, but if it's false, they can't.
A mad person can say he's sane, although he's not, because for him it's the truth.
But he is also allowed to say that he believes he is mad, because mentioned at the very beginning is that all mad people have wrong beliefs! "I believe I am mad" implies that a person has the luxury of believing that he (believes he is mad), which he doesn't have , because he's mad in the first place.
So given sentence is true, only for him, but false in reality.
So a madman can say it.
A sane person cannot say that, because he has perfect judgement.
Quick version: Given a true statement, a mad man believes the false version of it, and that's what he truthfully says. So, a mad person can only say statements that are false.
A mad believes that he is sane (which is false), and will say "I am sane". He believes that he is mad (which is false), and will say "I believe that I am mad".
Complete details:
A sane person has the right belief. Since a sane person is sane, he believes he is sane.
Let's consider what it means for him to make the statement "I believe I am sane". He is expressing the right belief, which means that he actually believes he is sane. This is true, so he can say it.
Similarly, if he says "I believe I am mad", that is the right belief, which means he actually believes he is mad, which is not true. Thus he can't say it.
A mad person has the wrong belief. Since a mad person is mad, he believes he is sane.
However, let's consider what it means for him to make a statement. If he says "I believe I am sane", it is the wrong belief, which means that he actually believes he is mad. This is not true, so he can't say it.
Similarly, if he says "I believe I am mad", that is the wrong belief, which means he actually believes he is sane, which is true, so he can say it.
Put another way, a mad person believes that he believes he is mad. Though he doesn't believe he is mad, yet he believes that he does believe he is mad. (Yes, this is maddening).
More generally, if anyone on this island says "I believe X", then X is a true statement. For the mad person, since X is true, he would believe "not X".