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Topic: Where Smullyan went wrong (Read 8122 times) |
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Jonathan the Red
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I first encountered the work of Raymond Smullyan when I was in sixth grade (which was more years ago than I like to remember) when I picked up, on a whim, a copy of What is the Name of This Book? in my local library. I was instantly hooked, and I've been a big fan of logic puzzles ever since.
In WitNoTB?, Smullyan introduces his Knights, Knaves, and Normals. Knights always tell the truth, Knaves always lie, and Normals can do either. He poses many puzzles that are set on islands populated by Knights, Knaves, and Normals, from the easy to the difficult to the sublime.
In the subchapter titled "How to Marry a King's Daughter", Smullyan tells the story of a king who wants his daughter to marry a nice normal Normal, not one of those goody-goody Knights or devious scoundrel Knaves. Smullyan asks:
- How can you make a true statement that will convince the King you are a Normal?
- How can you make a false statement that will convince the King you are a Normal?
- How can you make a statement that will convince the King you are a Normal, but he won't know whether it's true or false?
These problems are not terribly difficult, and the solution is left as an exercise.
Then Smullyan tells the story of a different King, one who is not so kindly disposed towards Normals. In fact, he sees them as wishy-washy girly-men, unreliable and untrustworthy. A Knight will always be true, and a Knave is wholly reliable as long as you remember to believe the exact opposite of what he says, while a Normal will deceive you when you least expect it. He wants his daughter to marry anyone but a Normal. Smullyan asks: how many statements must you make to convince the King?
Smullyan's answer was: no statements you make can ever convince the King you are no Normal. Since a Normal can say anything, any statement you make could be made by a Normal. There is no way to prove your abnormality to the King.
Smullyan was wrong! There is a subtle flaw in his argument, and in fact it is quite possible for a non-Normal to prove it. So the puzzle for you is:
- How can you, in one statement, convince the King you are a Knight?
- How can you, in one statement, convince the King you are a Knave?
- How can you, in one statement, convince the King you are not a Normal, but leave him unable to deduce whether you are a Knight or a Knave?
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| « Last Edit: Apr 25th, 2005, 3:31pm by Icarus » |
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Eric Yeh
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Re: NEW PROBLEM: Where Smullyan went wrong
« Reply #1 on: Aug 3rd, 2002, 9:59pm » |
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Jon,
Smullyan is a great guy, is he not?? My favorite of his was "This Book Needs No Title", which I have since been sad to discover is no longer in print. Ah well, at least I have my original copy somewhere.
Hmm, I also seem to recall finding some errors in his books at some points, but I no longer remember where -- perhaps it was the same as the one you found. In any case, I haven't had a chance to read your example very carefully yet, but will definitely do so tomorrow.
Goodnight for now!
Eric
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tim
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Re: NEW PROBLEM: Where Smullyan went wrong
« Reply #2 on: Aug 4th, 2002, 12:06am » |
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My first guess would be a variant on Epimenides' Paradox. After constructing suitable assertions, I find that it works. I won't spoil the fun too much by actually posting examples 
The flaw in Smullyan's reasoning is that by his conditions, Normals are not free to say anything they like. Their statements must be either true or false. That makes them unlike real people, and hence not really normal at all.
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Eric Yeh
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Re: NEW PROBLEM: Where Smullyan went wrong
« Reply #3 on: Aug 5th, 2002, 8:07pm » |
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Yep, it took me a little while to dig up my copy of WitNoTB, but finding it (and the problem -- god, it's such a hidden chapter!!!) confirmed what I suspected before: Smullyan really set himself up to be proven wrong on this one!! If he just presented the "pf" in a section, we might have quickly read it and idly accepted it as the truth. But instead he presents it as a problem, for which you have to flip several pages for the supposed solution. Furthermore, it's the first in the section, which really makes one try hard to answer it! So this is one reason why multiple people find the same error.
As far as the soln, I won't be as generous as Tim, partially bc Ollie is breathing down my neck to post more solns. But I guess I will at least make it black "to prevent spoilage", as the author would say.
I don't know who Epimedides is and am too lazy to look it up, but I'll guess it's the same thing I want to use: any contradiction such as C = "This statement is false." If N = "I am a normal" then for a normal N=>C is a contradiction, so it cannot be said. (I'm slightly abusing my notation and letting C control the entire statement.) Since the statement is true otherwise (since the premise is false), only a Knight can make the statement. That answers part 1.
Of course, an inversion should give you the opposite effect; the only caveat is that you actually then need to flip the inside to maintain the contradiction. ~(N=>~C) (equivalently N*C) works for part 2.
Finally, you can throw in an extra clause to allow the randomness, e.g. (N=>C)*~Kv where Kv = "I am a knave". This last term has the profile TTF for N/Kt/Kv, which is precisely what we need to multiply by.
Happy now Ollie?
Best,
Eric
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zarathustra
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Re: NEW PROBLEM: Where Smullyan went wrong
« Reply #4 on: Sep 9th, 2002, 8:58am » |
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Woh, where does it say that normals can't speak contradictions? "Knights always tell the truth, Knaves always lie, and Normals can do either." Word choice is important here as with any riddle and it says that Normals can tell the truth or lie, but not that they can only tell the truth or lie. Also, If you think of a normal as a (somewhat) normal person like you or me, we would certainly be able to speak contradictions. I agree with Smullyan on this one, clearly it is strongly implied that normals can make any statment, and therefore there is no possible answer.
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Eric Yeh
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Re: NEW PROBLEM: Where Smullyan went wrong
« Reply #5 on: Sep 9th, 2002, 1:03pm » |
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Zari,
To me, the statement "Knights always tell the truth, Knaves always lie, and Normals can do either" does contain the strong implication that "either" is all the Normals can do, i.e. that we have been given a complete specification. If not, it is an unclear specification of how a Normal behaves, and that would be a problem. For example, if he is allowed to do anything he wants (as a true "normal" person, per your writing), he could just remain silent when I ask him a question, or other strange things. Supposing you are correct about his intention, I would claim it was poorly expressed in this case.
Eric
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Brett Danaher
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Re: NEW PROBLEM: Where Smullyan went wrong
« Reply #6 on: Oct 4th, 2002, 8:45am » |
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I'm sorry, I don't understand your notation c=>n, so I don't know what statement you mean. Is it "I am a normal is a false statement"?
Because I do not see why a normal could not say that. He would be lying, but he's allowed. What does he say that causes a contradiction?
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James Fingas
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Re: NEW PROBLEM: Where Smullyan went wrong
« Reply #7 on: Oct 4th, 2002, 10:03am » |
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The notation A => B is short form for "A implies B", or "if A, then B". So Eric is saying:
A Normal would be making a contradiction if he were to say "If I am a Normal, then this statement is false".
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Doc, I'm addicted to advice! What should I do?
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Eric Yeh
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Re: NEW PROBLEM: Where Smullyan went wrong
« Reply #8 on: Oct 11th, 2002, 3:27pm » |
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Oops, just saw this msg -- sorry Brett. Yes, James' comment if correct. P=>Q is F iff P is T and Q is F. Hence can also be written as "~P or Q".
Best,
Eric
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Jonathan_the_Red
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Re: NEW PROBLEM: Where Smullyan went wrong
« Reply #9 on: Oct 11th, 2002, 4:43pm » |
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For what it's worth, my solutions to these were:
Knight: If I am normal, this sentence is false.
Knave: I am normal and this sentence is false.
Either: I am normal if and only if this sentence is false.
...which have a symmetry I quite like.
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| « Last Edit: Oct 11th, 2002, 4:43pm by Jonathan_the_Red » |
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My arcade cabinet
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Eric Yeh
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Re: NEW PROBLEM: Where Smullyan went wrong
« Reply #10 on: Oct 11th, 2002, 6:23pm » |
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Indeed, that's a nice symmetry 
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Jim Harvey
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Re: NEW PROBLEM: Where Smullyan went wrong
« Reply #11 on: Apr 25th, 2005, 6:11am » |
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Just because you put two statements in one sentence does not make it one statement. A normal, since he can lie, and he can tell the truth could say. I am normal. I am not a normal. Two statements, a contradiction, this does not prove that the one telling me is not a normal, only that he tells the truth, and then doesn't.
While it's true a knights COULD say that, it doesn't restrict the normal from making the statement. If you are normal, then the if statement is made true, so it would mean that "I am normal and this statement is false", these are just two separate statements and I don't see how a normal could not say this. I don't think this is a proper answer and I don't think that this puzzle should be posted on the site.
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rmsgrey
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Re: NEW PROBLEM: Where Smullyan went wrong
« Reply #12 on: Apr 25th, 2005, 9:12am » |
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on Apr 25th, 2005, 6:11am, Jim Harvey wrote:Just because you put two statements in one sentence does not make it one statement. A normal, since he can lie, and he can tell the truth could say. I am normal. I am not a normal. Two statements, a contradiction, this does not prove that the one telling me is not a normal, only that he tells the truth, and then doesn't.
While it's true a knights COULD say that, it doesn't restrict the normal from making the statement. If you are normal, then the if statement is made true, so it would mean that "I am normal and this statement is false", these are just two separate statements and I don't see how a normal could not say this. I don't think this is a proper answer and I don't think that this puzzle should be posted on the site. |
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In Logic, it's generally accepted that you can form compound propositions out of simple propositions by using various logical connectives (And, Or, Iff, Implies, Not, Xor, etc...). If you don't allow the construction of compound propositions, instead insisting that all propositions must be simple, then things generally become rather confusing.
For example, "this statement is false" is usually treated as a compound proposition, made of "this statement", "is", and "false". If you require it to be treated as a simple proposition rather than a compound proposition, then it could be true or it could be false, depending on how you choose to assign truth to it. For it to be paradoxical, you need its truth value to be based on the meaning of its component parts.
The statements: "I am normal" and "I am not normal" stated by the same person in either order establish him as normal. The statement "I am normal and I am not normal" only establishes that the speaker is not a knight - the presence of an "and" in the statement merges the two statements into one compound statement whose overall truth is dependent on both the truth of the component statements, and the rules for joining statements with an "and".
The statement "If I am normal then this statement is false" is equivalent (accoridng to the normal rules for "if ... then ...") to "I am not normal or this statement is false". When spoken by a normal, this is equivalent to "{false} or this statement is false" which is equivalent to "this statement is false".
The question then is whether a "normal" can ever make the statement "this statement is false" - a statement which is neither true nor false. If every statement a "normal" makes has to be either true or false, then the puzzle in his thread is solvable. If a "normal" can say anything he likes regardless of whether it's true, false or paradoxical, then the puzzle is insoluble.
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Grimbal
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Re: Where Smullyan went wrong
« Reply #13 on: Apr 27th, 2005, 8:05am » |
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To prove he is a knight:
"If I'm not lying, I am a Knight".
If he were lying, the premisce would not be met and the sentence would be true. So, he is not lying, and therefore he is a Knight.
A nicer formulation is: "If I'm not mistaken, I am a Knight."
It only works if you understand "being mistaken" as applying to the "if ... then ... " sentence and not to "I am a knight".
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| « Last Edit: Apr 27th, 2005, 1:15pm by Grimbal » |
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rmsgrey
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Re: Where Smullyan went wrong
« Reply #14 on: Apr 28th, 2005, 10:05am » |
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If I'm not lying, I'm a knave.
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Jim_Harvey
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Re: Where Smullyan went wrong
« Reply #15 on: Apr 28th, 2005, 2:26pm » |
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Actually, in logic, if an if statement is not satisfied, then it is true, so a knave could never state an if statement that wasn't true.
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rmsgrey
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Re: Where Smullyan went wrong
« Reply #16 on: Apr 29th, 2005, 4:57am » |
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on Apr 28th, 2005, 2:26pm, Jim_Harvey wrote:| Actually, in logic, if an if statement is not satisfied, then it is true, so a knave could never state an if statement that wasn't true. |
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I was trying to provide a counter-argument to Grimbal's post without actually (re-)posting the argument.
If there is some entity that can say "If I'm not lying, I'm a knave", then Grimbal's post doesn't provide a proof.
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| « Last Edit: Apr 29th, 2005, 4:59am by rmsgrey » |
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Three Hands
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Re: Where Smullyan went wrong
« Reply #17 on: Apr 29th, 2005, 5:23am » |
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There is some question as to whether the normal analysis for "if...then" statement (that it is only false when the antecedent is true and the consequent false) actually translates to "if...then" statements in a linguistic sense. Granted, it's the best version of a potential truth-table analysis we can have, but given that the defence of such an analysis is based in linguistic conventions, I would tend towards not really accepting that kind of analysis.
From the normal analysis of "if...then" statements, since the "If I'm not lying" part of the statement requires that the antecedent be true (as if it were false then the statement would be true, making the antecedent true), this forces the consequent to be true, and so a statement "If I'm not lying, then I'm a knave" could never logically be stated, since it creates a paradox. From an analysis which makes the truth value of "if...then" statements undetermined when the antecedent is false, then rmsgrey's statement could be used, and Grimbal's would not prove that the person talking is a Knight. I suspect, however, that the normal analysis is what is intended for this puzzle, and so Grimbal's statement would prove that the individual is a Knight.
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Grimbal
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Re: Where Smullyan went wrong
« Reply #18 on: Apr 29th, 2005, 6:15am » |
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If you prefer, "If I'm not lying, I am a Knight" can be restated as "I am lying or I am a Knight". The analysis is simpler. For anybody not a Knight, it is like saying "I am lying".
But of course, rmsgrey could also say "I am lying". So it is no proof unless you accept there is some magic in that kingdom that makes it so that every sentence is either true or false, and that every Knight tells the truth however deep the consequences. That magic doesn't apply to the forum, obviously.
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Deedlit
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Re: Where Smullyan went wrong
« Reply #19 on: Apr 29th, 2005, 6:32am » |
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Somehow I'm not following the latest objections. Of course if someone is allowed to make contradictory statements, then all bets are off, but for someone who has to say something true or false, Grimbal's statement could only be made by a knight. 
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Jim_Harvey
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Re: Where Smullyan went wrong
« Reply #20 on: Apr 29th, 2005, 12:17pm » |
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The only phrase in question is the phrase "I am lying", who could possibly say they are lying? It is an incomplete sentence, what are you lying about? If you said, I am hoping, is that a paradox? Just because you don't know what you're hoping for?
Arg1: Lish Mish Dish
Arg2: Arg1 is a lie
Arg3: Arg3 is a lie
Arg4: Arg5 is a lie
Arg5: _________
See, the only problem with Arg3 is that there is no argument whatsoever. To make the statement, "this statement is a lie," well, there is no statement made to make it a lie, it is the lack of an argument that makes it not a lie, and not a truth. Lish Mish Dish, is also not a lie, and it is not the truth, wouldn't you agree? Could a normal say Lish Mish Dish? It is neither true, nor false. So could you make a statement, "if I've lied before, lish mish dish" to prove that I'm a knight?
There is nothing saying that a Normal can say only truths and lies, it just states that he can say either, not only, so I think it is within the power of the Normal to say a senseless statement, even ones without an actual argument such as "this statement is a lie." This statement is really no different than Arg4, it is no paradox, only a senseless statement, and there is nothing saying a normal can't say senseless statements.
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Grimbal
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The original post asked to prove you are a Knight, a Liar (a Knave) or a Normal. Here is the answer. I put it in gif form because of the diagrams.
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FredFnord
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Re: Where Smullyan went wrong
« Reply #22 on: May 29th, 2005, 11:23pm » |
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There is another answer to the riddle as stated, even if you assume that normals can utter any sentence (sentence fragment, word, contradiction, truth, falsehood, meaningless sound, whatever) in existence.
At least, it answers the first and second questions. I must admit, I'm at a loss for the third.
Anybody?
-fred
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Deedlit
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Re: Where Smullyan went wrong
« Reply #23 on: May 30th, 2005, 1:13am » |
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The same methods that convice the king you are not a knight and/or not a knave under the stricter definition of normal, will also convince him under the more inclusive definition. The difference is the latter will allow a few more possibilies that a normal could say.
However, under the more inclusive definition of normal, it is impossible to ever convice the king that you are not a normal; whatever set of responses you give, a normal could give the same set of responses.
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Ajax
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Re: Where Smullyan went wrong
« Reply #24 on: May 30th, 2005, 2:15am » |
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I don't think there is an answer for the second problem. A normal guy can say whatever he wants without wondering if he's lying, telling the truth or being paradoxical. It's not a case of black or white; it can be grey. A normal guy cannot be restricted from saying whatever a knight or a knave may say. After all, we wants to marry the princess and get the dowry; Why would he let anyone else get it?
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