Since my other Raymond Smullyan puzzle was so popular, I’m adding another one. As always, you can find his riddles in his books in much greater depth.
A perfect logician goes to visit the island of knights and knaves. Knights always tell the truth and knaves always lie. There is no one else on the island (besides the logician). The logician is “perfect” in two ways:
- He never misses anything. If there is a way to logically deduce something, he will.
- He never makes a mistake. Anything he proves to be true really is true; he never proves anything false.
While he is visiting the island, one of its inhabitants makes a startling statement from which we (the reader) can deduce that the inhabitant must be a knight but the logician can never prove that he is.
Give an example of such a statement.
For extra credit, if the reader is able to deduce that the inhabitant making the statement must be a knight, why can’t the logician figure it out? He is, after all, way smarter than we are 
He’s infallible!
Please use rot13 to encode your answers so as not to ruin the puzzle for anybody else.
Tags: knights and knaves, logic, Raymond Smullyan
Oh, as you know Raymond Smullyan never uses “tricks” in his puzzles. The answer indeed relies solely on logic. As a hint, “gur nafjre unf gb qb jvgu Tbqry’f vapbzcyrgrarff gurberz.”
I think I got something…
“Lbh unir ab jnl gb xabj jrgure V nz n xavtug be n xanir onfrq ba guvf fgngrzrag”
Since it’s wrong, I’m going to un-obfuscate it: “You have no way to know whether I am a knight or a knave based on this statement.”
The interesting thing is that the statement alone is undecidable. Someone has to speak it before we can tell what the rules are on the island. If a knight speaks it, it’s true, and the logician can’t figure it out. However, if a knave speaks it, it’s false, in which case the logician will know that it’s a knave. Since both cases are possible, it doesn’t solve the riddle.
Come to think of it, it might not even be possible for that phrase to be spoken, since both cases are possible. It’s circular logic, yes? Nobody can say that statement until somebody has said that statement in order to establish who is allowed to say that statement. Either that, or the first person who says it decides. I’m not sure.
Well, I thought about it again last night and I realised this statement was not correct, but not for the same reasons as you. This would lead to a contradiction if the guy was a knight, and this is why I came up with this question : not being able to determine wether the guy is a knight or a knave would make him tell the truth (first contradiction), therefore the logician would know that he is a knight, and this deduction would make his statement a lie (second contradiction), which would be ok (no more contradiction… I loose)
The problem with this statement is that it is valid if a knave tells it, and the logician can deduce that.
What is wrong about your reasonning, is that you assume different deductions from the logician based only on what kind of guy talks to him. He does not know that a priori, so his deduction must only be based on the statement itself. There is no way that he can decide in one case but not in the other, because it is the same statement, and he will experience the exact same thing, and therefore get exactly the same set of informations in both cases.
My initial goal was that the statement would lead to a contradiction for the logician in both cases, and therefore that he would really not be able to know what kind of guy he is dealing with, letting us know that he is a knight. I failed.
But I will think about it later. Great puzzle, by the way. on the contrary to the vampire thing, which is totally ruined by the meta question thing, this one seems to be meta-question-proof
…maybe.
There are no contradictions. The statement is either true, false, or neither (paradox). If a knight says it, it’s true and the logician doesn’t know who he’s talking to, no problem there. If a knave says it, it’s false and the logician figures out it’s a knave, no problem there.
There is not enough information for us to know which type of person is able to say that statement (who knows, there may be a sign on the island that says knaves make that statement, or a knight may have previously told us). However, as soon as the statement is made and the logician gives his answer, we will know. If he says, “Ah, you are a knave!” then we know it’s a knave, obviously. If instead he says, “I have no idea who you are,” then we (the reader) know that it must be a knight but the logician has no idea.
In other words, you’ve half-solved the riddle, the only problem is that it only works half the time.
“What is wrong with your reasoning, is that you assume different deductions from the logician based only on what kind of guy talks to him.” I’m right for the same reason you think I’m wrong. Ha ha. The irony. The statement itself is not true or false. The statement is only true or false in the context of the person saying it. It’s like the statement, “I am a man.” Is that true or false? It happens to be true because I’m saying it, but it could easily be false if somebody else is saying it.
Here’s a good example. “All inhabitants of this island drink lemonade.” Is that true or false? There is no way to tell from the statement itself. However, if somebody makes that statement and the logician says, “Ah, quite right you are, my good friend,” then now we know.
Remember that we live in a strange world, here. If a knight says, “It’s going to rain tomorrow,” then it’s going to rain tomorrow. How could he know that? I have no idea, but he does, and he always tells the truth. In fact, what people say can directly affect their environment. Very strange, but true.
It’s kind of like the philosophical question, “Is never being wrong equivalent to being all-powerful?” Because whatever you say must come true…
I think I have a better way of explaining it.
“You have no way of knowing whether I am a knight or a knave based on this statement.”
1. Assume the statement is true.
2. It must be a knight saying it, because it’s true.
3. The logician won’t be able to figure out he’s a knight, because the statement is true.
4. Once the logician realizes he can’t figure it out, he will know the statement is true.
5. The logician now figures out it’s a knight because the statement is true.
6. PARADOX!
7. Therefore, the statement cannot be true and it must be a knave saying it.
8. Because the statement is false, the logician is able to prove that it’s a knave.
The problem with your reasoning is that you are assuming the logician will be able to figure out that he can’t figure it out. He might not. Let’s say he spends five days thinking about it and doesn’t come up with an answer. He may still come up with the answer on the sixth day. He can never be sure that he will never be able to answer it. Step 4 above never happens and there is no contradiction.
It’s like saying, “Nobody has been able to build a quantum computer, therefore it’s not possible.” Who knows, it might be possible tomorrow.
Well, I don’t really know how to present this without locking both of us in a prove-the-other-wrong loop, but I get your point, and I really think you are missing the main part of mine. First of all, I want to remind you that I know my answer is wrong.
The statement refers only to the fact that the guy is either a knight or a knave. It has not much to do with the lemonade, raining, quantum computer, or you being a man problems, which refer to informations that are external to the problem data set. Coming from one of the island inhabitants, those statements you give reduce to “this statement is true”, or simply “I am a knight”.
The problem states that Knight always tell the truth and knaves always lie. This implies that both only make statements they are sure about. Neither of them can state that it’s gonna rain tomorrow if he can’t know for sure wether it is gonna happen or not, because this would neither be truth or lie, it would be guess, which is not the same thing (by the way, a knave could say “I know that it’s raining tomorrow”, which is not the same thing as “it’s raining tomorrow”). And by reading your message, it gives me the impression that you think knight and knave can actually say whatever they want, making their statements become true or false afterwards. I certainly disagree with that.
The problem also states that the logician is perfect. That means that, given a set of informations, he will be able to take a perfect deduction. So if the statement actually does not allow him to deduce the nature of the guy, he will know it.
The previous argument confirms that if a knight or a knave says that the logician has no way to know if he is a knight or a knave, that means he is also a perfect logician, else it would neither be truth or lie, just guess.
Knowing that, an inhabitant who would make this statement would either know that the logician will never be able to make this mind, or that he will. So, if I follow you, in both case, the logician won’t be able to determine wether he is speaking with a knight or with a knave, and the reader will be able to know the statement was true, thus my original solution to the problem works. So I actually are trying to prove myself wrong, and you are trying to prove me right… weird.
So, let me get this straight. A statement is either true, false, or unknowable with the data we are given. Amongst true and false statements, there are some which are unknowable with the data the logician is given.
According to you, which of these statements are true, which are false, which are unknowable to us, which are true but unknowable to the logician, and which are false but unknowable by the logician ?
1) the guy is a knight
2) the guy is a knave
3) the logician can figure out wether the guy is a knight or a knave.
In my point of view, 1 is false, 2 is true, and 3 is true. A point of view that would make my solution win would be to say that 1 is true, 2 is false and 3 is true but unknowable to the logician.
If you tell exactly what is your point ov view, that would help me understand it better, I think.
“The problem also states that the logician is perfect. That means that, given a set of informations, he will be able to take a perfect deduction. So if the statement actually does not allow him to deduce the nature of the guy, he will know it.”
The fact is that’s not true. Actually, it’s provably untrue. Undecidability or independence http://en.wikipedia.org/wiki/I.....cal_logic) is really quite fascinating. Sometimes you can prove things, sometimes you can’t, sometimes you can prove that you can’t prove things, and sometimes you can’t. People have been trying for years to tell if P = NP is decidable or not without success http://en.wikipedia.org/wiki/P_%3D_NP_problem
Nowhere has it been stated that, “The logician can prove whether or not he can prove things.” That’s why Raymond Smullyan chose this puzzle in order to demonstrate what Godel’s incompleteness theorems mean. See my post here: http://www.philipbrocoum.com/?p=133 The main consequences for this riddle are as follows:
1. There exist true things that the perfect logician can never prove.
2. The perfect logician does not know that he is perfect.
The perfect logician is unable to use the fact that he is perfect in his logic in order to solve this riddle. As outsiders, beyond the scope of the world he lives in, we are able to deduce the nature of the inhabitant because we know things that the logician doesn’t (can’t) know. I just gave away the answer to the “extra credit” part of the riddle, but no matter.
That’s why I’ve been trying to “prove you right”. Your answer is almost correct. As I said, it’s half-correct. Half the time, the reader can deduce that it’s a knight making the statement but the logician can’t. The other half of the time, both the reader and the logician know that it’s a knave, which is why it doesn’t quite solve the riddle.
The point I was trying to make with the lemonade and the raining and all of that is that we have to be very careful not to tell an inhabitant of this island what he can or can’t say. Just because we lowly human beings haven’t figured out how to accurately predict the weather doesn’t mean the knights haven’t. If a knight says, “It will rain tomorrow,” then you can be 100% sure that it will rain tomorrow. It doesn’t matter how he knows, that’s not the point. In other words, the statement, “it will rain tomorrow,” does not necessarily preclude the possibility of that person being a knight.
1) the guy is a knight
2) the guy is a knave
3) the logician can figure out wether the guy is a knight or a knave.
All three can be either true or false, that’s the problem. If a knight says the statement, only number one is true. If a knave says the statement, numbers two and three are true. We can’t know which type of person could say that statement.
That’s why I at first suspected your statement was a paradox. It’s not clear that it’s even knowable whether knights or knaves are the ones who can make that statement (it can only be one or the other, not both). However, once the statement is made, the foundation is made for the universe.
“Half the time, the reader can deduce that it’s a knight making the statement but the logician can’t. The other half of the time, both the reader and the logician know that it’s a knave, which is why it doesn’t quite solve the riddle.”
Wait… what ?
The flaw in this sentense is so huge I don’t really know how to point it out without sounding ridiculous. Let’s try…
The logician meets a guy. He only knows the guy is either a knight or a knave. The guy makes a statement. The logician makes his deduction, based on the information he has.
What you are saying is that, whereas the logician is experiencing the exact same thing, hearing the exact same statement, and having the exact same set of information, the nature of the guy will influence the deduction the logician can make ? Well… I would say this is obviously wrong, but as it is obviously not so obvious, it just is wrong. The logician, with a given set of informations, will get to only one conclusion or non conclusion. On his point of view, there is only one situation.
“The point I was trying to make with the lemonade and the raining and all of that is that we have to be very careful not to tell an inhabitant of this island what he can or can’t say. Just because we lowly human beings haven’t figured out how to accurately predict the weather doesn’t mean the knights haven’t.”
That is very precisely my point. The only thing we know for sure is that whatever a knight says, he has enough elements to know it is true, and whatever a knave says, he has enough elements to know it is false. So, if a knight or a knave says that the logician can’t figure out what kind of guy he is, then it is either true or false. In one case the logician will actually figure out the thing with the set of information he has, in the other case he won’t. The key is that no matter if the guy is a knight or a knave, the logician has the exact same set of informations, and is either able or unable to know the answer in a finite amount of time.
Two different conclusions can be drawn from this, depending on the fact that the perfect logician knows he is perfect or not.
If he knows he is perfect, considering that the guy is a knight will lead to a contradiction, making this case impossible, therefore only a knave could have said that. If he does not know it, then he will never be able to figure out wether he is able to draw a conclusion or not, therefore the guy was right, he is a knight.
So, as both my objections come down to rejecting just one of your main arguments, I guess everything reduces to that : One set of information given to a perfect logician leads to only one conclusion or lack of conclusion. If you disagree with that, we’re done talking. If you think that there is something more subtle that I did not get, maybe you could try to point out the error in my logic directly.
“One set of information given to a perfect logician leads to only one conclusion or lack of conclusion.”
True. The problem is, we don’t know which conclusion it will be until the logician answers. Hence the “half the time” thing. Both are possible depending on which “universe” we are in. It’s kind of like the axiom of choice http://en.wikipedia.org/wiki/Axiom_of_choice which can never be proven true or false and practically makes no difference. Half the time it’s true and half the time it’s false depending on what “universe” the mathematician chooses to work in.
I’m sorry you don’t like it. You’ll have to take it up with Kurt Godel, Raymond Smullyan, or the professor of logic at your nearest university. It kind of reminds me of quantum mechanics and how everything is random. In the words of Richard Feynman, “I don’t think in terms of what I like or don’t like. I think in terms of what is and what isn’t. If you don’t like it, you can go to a different universe!”
It really does not matter wether we know what is the conclusion or not. The point is the conclusion does not depend on the nature of the guy.
It does not matter either what is provable or not in the logician reasonning. Even your changing of the logician’s perfection definition afterwards does not change anything about that. The real nature of the guy does not affect anything in the logician’s reasonning or ours. Only the statement. Therefore, same statement, same reasonning.
Even if there was a quantum-type unpredictability in the logician’s reasonning, the nature of the guy is not what determines it. It only allows him to speak the sentence or not.
Believe me, this Feynman citation could be mine. You may want to avoid being so condescending if you want to be taken seriously. You don’t imagine the number of guys which would not accept some quantum properties even after seing Aspect’s work… that’s something which makes me sad for our species, not as hard as religion, but still.
And please stop trying to drown me in things which have nothing to do with the problem, and which you should know I understand at least as good as you, based on what you say about it.
So, as Raymond Smullyan is the only person you cite whose work I am not very familiar with (thank you about the logic teacher, this was a really good one), I think it would be the time to disclose the actual solution to the problem, which I am sure makes sense, It is really no use anymore to try to stand by your initial error, I am done trying to understand your point of view, I have no doubt about “the loop” anymore.
But this was fun, at least at the beginning. Seriously, interresting problem. I hope my use of your language was not too much painful to read.
‘Even your changing of the logician’s perfection definition afterward does not change anything about that.’ = you don’t get it.
The fact that he’s perfect implies these two things:
1. There exist true things that the perfect logician can never prove.
2. The perfect logician does not know that he is perfect.
Well… I insist… I am done speaking about this wrong solution of mine. At least with you. I told you it did not change anything, you did not even bother giving me something new.
By the way, Here is something new. As I was able to take a few minutes to think about an actual solution to the problem without being bothered by the fact that you don’t get what I say, I came up with a solution which works. As my first proposition led to a paradox only if the guy was a knight (I know… you don’t agree with this… believe me, I don’t care), I realised it was not so hard to solve this symetry problem. I don’t bother putting my explanations in rot13, because I am pretty sure this won’t be the solution you are waiting for, because it follows the exact same logic I have been following from the beginning.
“You can’t come to the conclusion that I am a knight”.
And the problem is solved : The logician still can’t come to the conclusion that the guy is a knight, because it would be the same contradiction as before. But this time, he can’t come to the conclusion that he’s a knave either, which would be a contradiction too. Therefore, the logician can’t make up his mind, but we can : the guy was right, so he is a knight.
Well, please don’t bother to explain me why I am wrong, if it is the case and that I don’t figure it out by myself within a few hours, only a comparison with the actual solution could work
“I never lie.”
here is an idea:
If the logician is perfect, and we (the readers) can make a correct deduction that he cannot make, that must mean that we have access to some piece of information that the logician does not possess. There is no information in the question which tells us how much of the situation the logician is aware of.
For instance, does the logician know that he is perfect? How could he? If he admits that his logic may be flawed or incomplete, then there is no way for him to deduce through logic that it is not, as this would be circular reasoning. There is no way for any logician to prove that his logic is perfect, as to do this would itself require logic. Therefor, because the reader is told explicitly that the logician is perfect, all that is required is for somone to approach the logician and say, “You are a perfect logician.” The reader knows this statement to be true, and therefor can identify the stranger as a knight, but the logician has no way to prove or disprove that statement.
I think you have to assume that the logician doesn’t know he’s perfect, as that is the only peice of information that we have that the logician doesn’t. If we are supposed to make a more perfect, purely logical, deduction then we must have more information than is possible for the logician to have.
Philip, both the statements, “You have no way to know whether I am a knight or a knave based on this statement.” as well as \You are a perfect logician.\ meet the parameters of an example of the requested statement that only the reader would know is a true statement.
Whereas \I am not lying.\ or \I always tell the truth.\ or \I am telling the truth.\ or \I never lie.\ would stump both the reader and the logician, and would be a completely impossible statement to figure out, as both a liar and a truthful person would say the exact same thing.
The point of the riddle is not to stump the logician, as the riddle expressly STATES THAT HE IS STUMPED. The point of the riddle is to come up with a phrase that only the reader would know is true, and the logician would be stumped.
The only phrase that works is one that somehow states that the logician will be stumped, as it is the only information the reader has that the logician doesn’t. In this world, the Knight (as we know him to be) would tell the logician that “You have no way to know whether I am a knight or a knave based on this statement.\ (or something else stating he cannot figure something out) and the Knight (again, as we know him to be) would continue walking, living happily ever after.
The logician will try to figure out if he was a knight or a knave until the day he dies, because the RIDDLE TELLS US HE CANNOT FIGURE IT OUT AND NEVER EVER DOES. But we do.
You have tried to make this much more complicated and meta-physical than it was intended to be, and should get your pompous head out of your ass.
I apologize, the only statements that would fit the parameters are ones that would say that the logician will be stumped by the statement, or that say the logician is a perfect logician, as those are the only two peices of information we are given that the logician does not have.
Sumakit ulo ko sa inyong dalawa
a very special island is inhabited only by knights and knaves. Knights always tell the truth while knaves always lie. you meet two inhabitants: Kenneth and rachel. Kenneth says “rachel and I are both knights or both knaves.” and Rachel claims “kenneth and I are the same”
who is a knight and who is a knave?