Talk:Knights and Knaves

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Lying and truthtelling[edit]

As usually described, problems don't say what they mean. What is actually meant in a Knights and Knaves problem is that a knave negates the answer a knight would give, a la boolean logic. This is what makes asking about future questions work in those "one question" problems, because the knave is forced to double negate, meaning both the knight and the knave will give the same answer. It is most definitely possible to tell a lie that is not a negation of a true statement. I am the lying guard in the two paths/doors problem. if someone asks me "if i were to ask you if this path is correct, would you say yes?" and I give an answer of "I don't know", I have just told a lie. In fact it's an incredibly obvious lie, but it doesn't give the poor sap any information other than that i'm the liar. u mad, logician? u mad bro?74.211.60.216 (talk) 05:37, 13 June 2018 (UTC)

Tacos and Beer[edit]

Did anyone notice this rather strange edit? Haven't deleted it cause I'm wondering if this might be in some pop cultural joke reference that could be noted on this page. — Preceding unsigned comment added by 76.184.6.193 (talk) 20:35, 4 March 2013 (UTC)

It's just garbage. Herr Gruber (talk) 10:38, 9 March 2013 (UTC)

Smullyan's involvement and the 'Two Guards' puzzle[edit]

I'm not certain that Smullyan first did these, but he certainly does many of them. To my mind, the ancestor of these puzzles is the old story about the two guards on two doors, one with treasuer, one with a tiger, and one guard lies, one tells the truth, and you only have one question to ask. Does this old one have a specific name? Where does it originate? (it has a 1001 nights feel?) -- Tarquin 21:48 19 Jul 2003 (UTC)

Alternate solution to Question 2?[edit]

Regarding Question 2 - Without using any formal boolean algebra, a different solution came to me immediately - it appears to me that it could also be that both Bill and John are knaves (in this case, both Bill and John's statements are false, which is consistent with them both being knaves). Any problem with that solution? It seems a much simpler reasoning than the solution presented in section 1.5... Zoopee 09:09, 26 August 2006 (UTC)

I believe that's correct, they're both knaves, since they can't both be knights because of Bill's statement, they can't be a knave and a knight because then John's statement wouldn't be false, and they can't be a knight and a knave because then Bill's statement would be true. Which leaves us with two knaves. The solution to the puzzle isn't given in the article though, I think it was just meant as an example.--BigCow 21:18, 21 November 2006 (UTC)
yes, both of you are right, they are both knaves. I wrote the solution in the italian version but I can't write it in good english --Arirossa 20:55, 19 January 2007 (UTC)
On the other hand, if Bill is a knight and John is a knave, it works: Bill's statement that they are different is truthful, and, seeing as the requirements for John's statement to have meaning are unfulfilled, it could well be that he is lying and Bill's position has no bearing on John. The 2-knaves answer obviously works as well, so this has more than one answer.--68.100.78.205 20:06, 24 March 2007 (UTC)
I don't think that is correct. It doesn't work if Bill is a knight and John is a knave. The reason being, if John is a knave then the statement "If Bill is a knave, then John is a knight." is false. And when negated (assuming it is a conditional, not biconditional) it turns into "Bill is a knave and John is a knave." Thus if what John says is false, Bill can not be a knight. This is my math: (if not p, then q) = (p or q), so not(if not p, then q) = not(p or q) = not p and not q. --[I'm not a user] time: 11:52, September 23, 2007 —Preceding unsigned comment added by 24.12.147.8 (talk) 04:54, 24 September 2007 (UTC)
I have now edited the article to make it clearer why the situation doesnt work.Yarilo2 (talk) 01:28, 10 March 2008 (UTC)
I think Question 2 is unsolvable: Suppose Bill is a knave, then John is effectively saying that he is a knave. This is impossible, therefore Bill is a knight. This makes John's statement automatically true (P->Q is true if P is false), hence John is also a knight. But then John and Bill are not different, in contradiction to what Bill (who is a knight) says. Without Bill's statement, the question would be solvable (both are knights). 212.61.144.123 (talk) 09:09, 3 June 2008 (UTC)
I think you are confused by the boolean logic, Boolean logic states that if P->Q and !P (not P) then the STATEMENT "P->Q" is TRUE by if-part failure. HOWEVER, I think you are confused because you are thinking that this leads to John also being a knight, but this cannot be concluded because no light can be shed onto whether or not John can be a knight by his own statement. It is then that the solver must use Bill's statement as true because he can't (by logic) be a knave, and must be a knight. As his statement states taht he and John are different, it can only be that John is a knave since Bill is logically a knight. I hope this explains things some 97.101.32.93 (talk) 05:13, 17 June 2008 (UTC)
OK, let's go back to the Boolean algebra in the article then, if you don't believe my reasoning. There is a mistake in there, now that I have a good look at that. You will agree that !B->!J (John's statement, if we assume the definitions of B and J in the article) is equivalent in saying !(!B) | !J = !J | B. Or, alternatively, it is !(J & !B), if we keep the ANDs (DeMorgan translates the one into the other). This is definitely not the same as !J & !B, with which the statement was translated in the puzzle. If you calculate with these changes, you will arrive at "Bill and John are both knights", just as I stated earlier (which, by the way, proves that my earlier reasoning was valid). Then remains the fact that Bill, being a knight, states that he and John are different, which appears to be not true - they are both knights. So unless you want to apply Bill's statement to some other unnamed feature of John and Bill, Bill is a knight who lies, contrary to the definition of a knight and the problem is therefore unsolvable. 212.61.144.123 (talk) 14:32, 11 September 2008 (UTC)
John didn't say "Bill is a knave and I am not a knight" (contrary to what the boolean algebra states).
If John is a knight then Bill can be neither a knight nor a knave (either would make Bills statement self-contradictory), yet if John is a knave Bill's statement is consistent no matter what he is. So John would have to be a knave or we would have a paradox.
Given that John is a knave and his statement was a logical implication, this means that one or more of the following is true:
(i) Bill and John can't both be knaves
(ii) Bill and John can't both be knights
(iii) John can't be a knave while Bill is a knight
(iv) John can be a knight while Bill is a knave
The first two are consistent with Bill being a knight but the third and fourth cause contradiction. As a result the only consistent scenario is with Bill as a knight and John as a knave, where his statement is false for accepting the inconsistent as possible.
I recommend changing the question to the one relevant to the boolean algebra example, else make an algebra example elsewhere as this particular problem is fairly ugly looking in it's algebraic form.
82.4.230.85 (talk) 22:36, 11 September 2008 (UTC)


I think the correct answer to #2 is that it is impossible to determine who is a knight and who is a knave. Here is my proof:

The statement is "If Bill is Knight, then I am a Knight".

An "If...then" statement is false ONLY if the premise is true, but the conclusion is false. If both the premise and the conlcusion are false, the statement is true. If the premise is false and the conlcusion is true, the statement is true. If both the premise and conclusion are true, then the statement is true.

Applying this principle to the John's statement, the only way John's statement is a false is if Bill is a Knight, and John is a Knave. If both of them are Knaves, the statement is true. If both are Knights, the statement is true. If Bill is a Knave, and John is a Knight, the statement is true.

If they are both Knights, then John's statement is true, and it's therefore possible for John (as a Knight) to say it.

If Bill is a Knight and John is a Knave, then John's statement is false. It would be possible for John (as a Knave) to say it.

If Bill is a Knave and John is a Knight, then John's statement is true. It would be possible for for John (as a Knight) to say it.

If they are both Knaves, then John's statement is true. But it would not be possible for John (as a Knave) to say it.

So, the only combination we can exclude as a possibility is that both Bill and John are Knaves. Any other combination is possible.

Stoyen (talk) 22:46, 3 October 2008 (UTC)

Question 3[edit]

Answer to 3, in case someone can put it in boolean: John answers yes to the first question (if he answered no, he would have to be a knight and the logician would figure it out immediately) and yes to the second (if he answered no, the answer would remain unclear). He is a knave and Bill is a knight; allowing him to say no yes both times (since John and Bill are different, both yeses are untrue).--68.100.78.205 20:15, 24 March 2007 (UTC)

Well, Question 3 seems to be a trick question. I suspect it's just not worded such that it is very understandable. The Logician asks if both John and Bill are knights. John replies, either saying, "Yes," or saying, "No." In the case of the former, John is a knave as his assertion is impossible due to the nature of the puzzle. In the case of the latter, he is reaffirming one of the conditions of the puzzle. In this case, he would be telling the truth, and would thus be a knight.
From this we can see that in both cases, the Logician can identify the knight and the knave in the situation from his first question, contrary to what Question 3 states. I propose that Question 3 be reworded or removed.86.8.242.110 (talk) 15:37, 3 August 2008 (UTC)
Actually, it seems that the article doesn't state that John must either be a knight or a knave and Bill a knight if John is a knave or a knave if Bill is a knight. It seems I was mistaken. —Preceding unsigned comment added by 86.8.242.110 (talk) 16:46, 3 August 2008 (UTC)

It seems that the text by the answer to Question 3 is misplaced. Also, it is not possible for a knight or knave in this universum to change his mind - a statement is either true or false, so maybe the second line is false. From the first statement it can be inferred that John answered "yes"; a "No" would have proven that John is a knight and Bill is a knave. On the second question and assuming that John again answers either Yes or No, if John answers "Yes", then John is a knave and Bill is a knight, otherwise, the logician again has insufficient information. The third quistion is a dud; it is exactly the opposite of the second question, so you will get the same answers with Yes and No exchanged. 212.61.144.123 (talk) 14:56, 11 September 2008 (UTC)

Question 4's answer[edit]

For 4: The answer is "Are you the sort of person who would say that Someplaceorother is to the left?" Nobody would claim to be a knave, so if they say yes, it is to the left, and if no, it is to the right. The second problem posed has nothing to do with the question: "Does 2+2=4?" would do, or any obvious equivalent. --68.100.78.205 20:19, 24 March 2007 (UTC)

The question presumes, but does not state, that the knave is incapable of giving the obvious false answer of "I Don't know." Since the problem presupposes both know the answer, any answer of "I Don't know" is provably false, and thus a valid choice for a Knave. I am aware of no question that will gain information about which path to take that also precludes the answer of "I Don't know". If the knave is asked a question that has a truth value that he does not know the answer to, then he is trapped in a paradox, because he can't say he doesn't know, but doesn't know enough to give an answer that is true or false. The problem also does not state that either is under any obligation to obey any instruction to not answer "I don't know." 74.211.60.182 (talk) 16:47, 3 October 2016 (UTC)
I think the question to be to one of them could be: Would that other guy say that Someplace is to the left?
If you are asking the Knight, and Someplace is actually to the right, the Knight will answer YES. If you are asking the Knave and Someplace is actually to the right, the Knave will also say YES because we know that if the Knave were to speak the truth, he would say that the answer from another truth-teller would be NO.
If you are asking the Knight, and Someplace is actually to the left, the Knight will answer NO because he knows that the Knave would lie about the correct direction. If you are asking the Knave and Someplace is actually to the left, the Knave will also say NO because we know that if the Knave were to speak the truth, he would say that the answer from another truth-teller would be YES. But he will lie and say NO.
Having received YES, the traveler knows that Someplace is actually to the right. If the traveler received NO, the correct direction would be to the left.
I'm reasonably sure that one of the premises of this quiz has to be that you cannot ask a question to which you already know the answer. So asking if 2+2=4 would fall out as improper. I think that the second question would have to be: Would that other guy say you are a Knight?
If you are asking the Knight, he will answer NO since he knows the Knave will lie about him. If you are asking the Knave, he will respond YES because we know that if the Knave were to momentarily speak the truth, he would say that the answer from another truth-teller would be NO. But he will lie and say YES.
As a result if you hear NO, you are speaking to the Knight. If you hear YES, you have asked the Knave instead.
Implicit in this is the assumption that the Knights and Knaves all know who each of them are. I think this article should state that more clearly as a premise. Move over, I think that it should be made clear that you cannot ask a question to which you already know the answer. That would defeat the whole point of the puzzle. Dawgknot 21:42, 24 August 2007 (UTC)
In question 4, you can disregard one person completely to get the right path. "Does this path I'm pointing to lead to freedom being true, the same as you being a Knight?". In terms of Boolean equality, this holds true as follows: Let Q be the question, A and B being the two people, and L being the truth of the selected path leading to freedom. And hence we formulate the problem in terms of Boolean Equality as follows: Q ≡ A ≡ L ∧ A ≡ ¬B. From here, we already have a solution. Q ≡ (A ≡ L), which is the question stated above, where you take the path if the answer is Yes, or take the other path if the answer is No (That is, the truth value of the question Q). B can be completely disregarded, as it is unimportant in this scenario.
Let's do a case analysis too:
Case 1: Path is correct, and A is a Knight: A says YES (True ≡ True is True. Knight tells the truth and will agree).
Case 2: Path is wrong, and A is a Knight: A says NO. (False ≡ True is False. Knight tells the truth and will disagree).
Case 3: Path is correct, and A is a Knave: A says YES. (True ≡ False is False; Knave lies and will agree).
Case 4: Path is wrong, and A is a Knave: A says NO. (False ≡ False is True; Knave lies and will disagree).
And there you have it.
Regarding the next part of Question 4, I believe that stepping on the foot of the knight is cheating; you're just supposed to ask a question, and not do anything else. I personally would ask a question based on the premise that Q ≡ A ≡ (A ≡ True). That is, Question posed to A whether A is a Knight. Simplify this: Q ≡ (A ≡ A) ≡ True (Associativity of equivalence), then Q ≡ True ≡ True, and finally Q ≡ True. So just ask any question that the answer is already known, i.e. 1 + 1?, or "Am I asking you a question?, or "Is True 'True'?" (ha ha), or ask a logic puzzle :). 203.188.235.14 (talk) 06:28, 8 March 2008 (UTC)

Solution to Question 1[edit]

Here is the text in the article after establishing that John is knave.

Since knaves lie, and one statement is true, the other statement must be false

Wouldn't it be clearer, given the construction of the Question, to say:

Since knaves lie, then both statements cannot be true. Therefore, the other statement must be false

Also, is it an unspoken assumption that the knights and the knaves know among themselves which is which? That is to say that John and Bill know the truth about each other? If so, perhaps that should be premised. Dawgknot 21:00, 24 August 2007 (UTC)

More popular culture references[edit]

I think there's an episode of Samurai Jack that features a version of this puzzle, but I can't recall which episode. B7T 20:11, 2 December 2007 (UTC)

Also used in The Book of Lost Things by John Connolly. cryptoboy (talk) 19:54, 17 January 2015 (UTC)

Boolean Equality[edit]

Has anyone ever heard of this here? It's a very good read in regards to the problem of the Knights and Knaves. 203.188.235.14 (talk) 14:41, 18 January 2008 (UTC)

Intro[edit]

An early example of this type of puzzle involves three inhabitants referred to as A, B and C. The visitor asks A what type he is, but does not hear A's answer. B then says "A said that he is a knave" and C says "Don't believe B: he is lying!" To solve the puzzle, note that no inhabitant can say that he is a knave. Therefore B's statement must be untrue, so he is a knave, making C's statement true, so he is a knight and A is a knight.

I don't know where this example comes from, but I don't think it's true. We can't make any conclusions about A in this puzzle, the statements of the other two work whether A is a knight or a knave.

99.241.128.44 (talk) 06:44, 11 February 2008 (UTC)Alex

The example comes from "What is the name of this book?" by Raymond Smullyan, an excellent book if you like these kind of puzzles. In this book, the answer correctly states that A's type cannot be determined. The answer has now been fixed, I see, but it is still not entirely correct. The fact that A did not say that he is a knave (because nobody can), does not mean that he said that he is a knight. We simply don't know what A said, other than that it is not what B stated. Which makes A's type truly undecided. 212.61.144.123 (talk) 15:32, 11 September 2008 (UTC)

Also, there is no actual encyclopedic introduction to this puzzle. There is also no explanation of what "Knights and Knaves" literally refers to, and no explanation of the Knights DO and what the Knaves DO (knights "always tell the truth" while knaves "always lie" right?), which is obviously a crucial rule to include at the very beginning of the puzzle. —Preceding unsigned comment added by TurilCronburg (talkcontribs) 02:32, 22 October 2010 (UTC)

Question 1: fallacy?[edit]

Question one is really strange.

Look at this statement: "If John was a knight, he would not be able to say that he was a knave since he would be lying. Therefore the statement 'John is a knave' must be true."

Sentence one is right, but sentence one does not imply sentence two at all. If John was a knave, he would not be able to say that was a knave either, because he would be telling the truth. This contradicts with the fact that knaves always lie.

Also, although I'm not an expert on Boolean algebra, could an expert verify if the Boolean algebra performed is correct?

--Freiddie 12:22, 26 May 2008 (UTC)

It is very subtle: There are not two statements "John is a knave" and "Bill is a knave". In that case you would indeed get a contradiction. There is a single statement: "John is a knave AND Bill is a knave". John as a knight could not say that, because the statement would be untrue (since John would not be a knave), hence John must be a knave. Then the statement "John is a knave and Bill is a knave" must be untrue. Since John is a knave, then Bill cannot also be a knave (then the statement would be true), so Bill is a knight, and the statement is untrue (because Bill is not a knave, although John is). 212.61.144.123 (talk) 09:26, 3 June 2008 (UTC)
Oh I see now. The "AND" operator is also being said, but it's not assumed. Okay, I get it now. Thank you. --Freiddie 00:59, 5 June 2008 (UTC) —Preceding unsigned comment added by Freiddie (talkcontribs)
I'm going to reword the question to remove the ambiguity; as it stands it is very unclear. bokkibear (talk) 12:15, 15 December 2009 (UTC)

Solution for Question 2[edit]

The solution given for Question 2 is incorrect and the question itself is ugly and confusing. John didn't say "we are not of different kinds". John's asserting that the two person's states are dependent on each other, which is a higher level statement than simply "we are the same". Such a statement can be false simply because the dependency doesn't exist regardless of what states they actualy are.

This is my attempt at a 'solution': Purely based on Bill's statement, John can't be a knight because that would mean Bill is saying "I am a knave" which is paradoxical. Since John is a knave, Bill can be either a knight or a knave.

All we can know from John's statement being false is that one can be a knave (such as John) without the other also being a knave. The only way for John's statement to be provably false by itself would be if Bill were a knight, but as pleasing as this solution is it doesn't negate the possibility that Bill is a knave and that John's statement is false for other reasons such as admitting the possibilty of himself being a knave. He is of course a knave, but you can't be consistent while suggesting you might be lying.

Be careful with those implications; they make assertions about possibilities that in these puzzles don't exist.86.5.179.220 (talk) 03:07, 25 August 2011 (UTC)

At first I thought the second question was similar to the first and that Bill was a knight, but maybe this is not so. By the logic of the first question anyone who calls himself a knave is a knave (and always will be because they never trade places), so John is a knave by calling himself one, but Bill’s identity remains unknown because John does not declare Bill a knight or knave. This means Bill’s statement of being different offers no help. — Preceding unsigned comment added by 24.186.34.109 (talk) 21:45, 5 September 2011 (UTC)


Question 1: my thinking progress(realising my mistake) John and Bill are residents of the island of knights and knaves.

Question 1[edit]

John says: We are both knaves.

Who is what?

John's statement can't be true because nobody can admit to being a knave (see Liar paradox). Since John is a knave this means he must have been lying about them both being knaves, and so Bill is a knight...... However! after John admits being a knave we realise he can also tell the truth and since we are not told of anything that negates the possibility of them both being knaves or knights (that is because John could have told truth about Bill being a knave just like a told the truth about himself being a knave) it is possible for the other character (Bill) to be a knave as well.


The second writing was my thought but the simple fact that negates it is this: If john can only lie then it is not possible for him to admit be a person who only lies. Therefore the lie must be in the other part of his statement (which makes the entire statement a lie and then it doesn't matter that john told the truth about himself being a liar in the first part of the statement.) which says that bill is a liar. After writing this I thought: "then perhaps he is a knight (which can only tell the truth" But then I remembered that it is not possible for a knight to say he is a knave for that would be a lie.


All this mistake came after I read the riddle and it's answer and started thinking this: (this is only true if john uses two separate statements for this, the statements being: "I am a knave" and "the second character is a knave". That is because John already lies about himself being.

one could say it is a "smart ass" riddle since it's solution is only possible if you refer to the statements/the two parts of the same statement as one.

Yuval. — Preceding unsigned comment added by 46.116.154.228 (talk) 01:13, 18 January 2012 (UTC)

Labyrinth[edit]

The version in Labyrinth does have an issue as a riddle since one of the two guards actually explains the conditions to Sarah; that means he has to be the knight if the terms of the puzzle are accepted as true, so it's rather self-defeating. Accepting he's the knave would mean he was lying about the two being a knight and a knave. (It can also be assumed that the two lower heads are not knights or knaves since one says they don't know the answer and the other agrees with him). Herr Gruber (talk) 10:45, 9 March 2013 (UTC)

Problem with the second solution to 3[edit]

The "if I asked you would you say yes" example has the problem that it requires the knave to tell the truth, even though he can't do that. The most likely get-out would be he would negate the resulting answer again on the basis it was the true answer and he cannot tell the truth (or that in complex questions he need not lie about every aspect of the answer as long as his answer is a generally a lie), and say "no" if his door leads to freedom and "yes" if it does not; he would then be telling a lie about a lie about a lie.

The "would he tell me this was the correct door?" solution does not require this (it is not a complex question and the knave's answer in both cases is the actual opposite of the true answer), and therefore is a stronger solution to the problem. Herr Gruber (talk) 03:38, 5 December 2015 (UTC)

Firstly, I believe the "if I asked you would you say yes" solution to be stronger as it doesn't require that each person has knowledge of what the other would say, only what they themselves would say.
Secondly, I see no such issue with the knave saying he would say "yes" (when his path leads to freedom). You seem to be confusing the answer to the question "what would you answer?" with the answer to the question itself: He wouldn't be telling the truth if he says that he would say "yes" to a question to which he must actually say "no". The knave has no such freedom to, as you put it "negate the resulting answer again" and say that he wouldn't say "yes" as this would be the truth (the one thing the knave must not say) and this is evidenced by the fact that asking him if his path leads to freedom when it in fact does requires him to not respond "yes". Nbrader (talk) 21:25, 17 January 2016 (UTC)
The knave cannot tell the truth, but once you start asking complex questions you complicate how he is allowed to deal with that. It is up to him, not you, how he parses this and what he considers to be a lie. He can either only lie about one part of the complex question (since you were, after all, told you could ask one question, not one nested in another, so he can legitimately choose to only reply to part of your question since you're cheating) and say no when the answer is yes, directly lie about his response and truthfully identify the lie he would initially tell (he says yes because he would actually say no because the answer is yes, even though an even number of negations also means he's technically telling the truth), or lie about the lie about the lie and say no when the answer is yes (he says no because he would say yes, because he would say no because the answer is yes).
Perhaps the most powerful problem, though, is the assumption that the knave must truthfully identify his lie about his answer to a question he isn't asked to directly reply to. He could instead use the false premise that his original answer would have been the truth to address the complex question:
"If I were asked that question, I would lie about it and say no, because the answer is yes" -> "Therefore, I will pretext my reply on the falsehood that I would tell the truth about it, which I would not do, and use yes as my initial answer" -> "Therefore, I will give an answer that is the opposite of my false initial answer" -> "No."
The "what would he answer" example only requires that the knight and knave understand which one of the two they are and that the other is not that one, and only involves a legitimate simple one-part yes / no question. Since there is no way for the knight and knave to not know who they are, they can probably guess which the other is, even if we accept that they've somehow never done this to anyone else to observe the other's behaviour. Herr Gruber (talk) 10:53, 22 January 2016 (UTC)
You are arbitrarily introducing behavior that the knave has never before been stated to have (and which are incompatible with the statement "the knave only tells lies"): You assert "he can legitimately choose to only reply to part of your question since you're cheating" when nowhere in the original problem is this given. Furthermore, you've introduced a notion of 'complex question' that was also never mentioned and which I consider ill-defined. I have difficulty seeing how a question about what they themselves would answer is any more complex or "one nested in another" than a question about what the other person would answer. As such your argument is self-defeating.
If instead you follow the simple rule originally given- that a knave only tells lies and that a knight only tells truths -and understand that a truth is a truth regardless of it's subject matter and similarly a lie is a lie no matter what the lie is about, then the problem has a clear and well-defined resolution. Nbrader (talk) 22:53, 2 February 2016 (UTC)
Addendum: I'll put it this way (using bold to highlight similar parts): Do you agree that if X happens then "X happens" is the truth? It follows that if asking a person Q results in them responding 'Yes' then "asking a person Q results in them responding 'Yes'" is the truth. For example, if asking a person "Does your path lead to freedom?" results in them responding 'Yes' then "asking a person "Does your path lead to freedom?" results in them responding 'Yes'" is the truth.
Hence, if you ask a person "Would asking you "Does your path lead to freedom?" result in you responding 'Yes'?" results in them responding 'Yes' and asking them "Does your path lead to freedom?" results in them responding 'Yes' then the first respond was the truth.
On the other hand, if you ask a person "Would asking you "Does your path lead to freedom?" result in you responding 'Yes'?" results in them responding 'Yes' and asking them "Does your path lead to freedom?" results in them responding 'No' then the first respond was not the truth.
If a person is required to not tell the truth then they cannot respond in the former way (in the case where the path does in fact lead to freedom) as the first response was the truth, evidenced by the second response. If that same person must also answer either 'Yes' or 'No' to every question then they must answer in the latter way as it is the only permitted answer to give. Both of these conditions are imposed on knaves and therefore he answers in the latter way. The other response (to the question "Does your path lead to freedom?") in each case is determined by whether or not the path actually leads to freedom and each combination of knave and path results in a comparable situation to the one outlined above. Nbrader (talk) 23:27, 2 February 2016 (UTC)
Addendum 2: NB: Throughout this we assume that the knight and knave both have perfect knowledge about the things they are asked to avoid some legitimate philosophical problems. As soon as you break this assumption the term 'lie' becomes ambiguous as it may refer to contradicting reality or contradicting belief. Nbrader (talk) 23:45, 2 February 2016 (UTC)
The problem is you're trying to ask them a two part question: "If I asked you this, what would you say?" The knave doesn't have to tell the truth about what the answer to the first question would be, so he can answer with a lie about the truth rather than a lie about a lie. Since you're asking a conditional question, he can tell a conditional lie. It's a bad example, and it's also not the one most commonly used (Labyrinth, one of the most well-known uses of the knight and knave puzzle, uses "what would he say?" Herr Gruber (talk) 23:49, 2 February 2016 (UTC)
When asked the question "Would asking you "Does your path lead to freedom?" result in you responding 'Yes'?" you don't respond with two answers: You respond with one answer. It is a question like any other yes/no question, the responses of which having a simple method of verification: To ask the question "Does your path lead to freedom?" and compare the response with the one they said they'd give. That doesn't mean you must verify it or that you are even permitted to do so. It simply means that it has a well-defined truth-value. If he answers "Yes" when he would indeed answer "Yes" to the inner question then he is telling truth and so cannot be a knave. He is not, as you are suggesting, permitted to give such an answer if he is a knave. This is a stated assumption of the problem and I don't know how I can make it any clearer to you.
I'm familiar with the film "Labyrinth" and the solution it gives. Yes, this might give that solution some cultural importance but this in no way prevents it from being a needlessly complicated solution. Aside from requiring perfect knowledge of another persons mind, which I see as a bigger assumption than knowing one's own mind, but if indeed the question is "what would he say?" then that's even worse as it relies on the possible responses of "Yes" or "No" being interpreted not as agreement/rejection but as referring to the responses themselves. Compare and contrast: "Give a word with three letters!" (with possible responses including "Yes" and "Dog" as words) and "Are you a dog?" (with only the possible responses "Yes" and "No" as states of agreement).Nbrader (talk) 00:11, 3 February 2016 (UTC)
It isn't needlessly complex, it's a very simple solution based on a non-conditional question which is based on how a truth table works, and merely requires the knave to know he is not the knight and the knight to know he is not the knave. Since you yourself say they must have perfect knowledge or the puzzle falls apart, you can't cite this as a problem. Yours is instead a conditional question that requires the knave to behave in a very specific way which gives a non-conditional answer. The thing is, he can tell a series of lies (he is the knight, he would tell the truth, so he lies about the truth) which gives the opposite answer, which is a conditional untruth. You can't demand an answer less complex than your question is and expect it to actually work. Since "Would he say this door leads to freedom?" is not a conditional, it's reasonable to expect a simple answer to it. Herr Gruber (talk) 00:19, 3 February 2016 (UTC)
They must have perfect knowledge about whatever they are asked. This means that if you want a simpler situation, where the people involved don't require unlikely knowledge then it is your responsibility to ask a question that doesn't require it. That doesn't mean you aren't able to ask questions that entail that they have unlikely knowledge. There are even questions that can never exist in such a situation such as "Is your answer to this question a lie?".
I'm not sure what you mean by conditional truth. In what sense is one truth conditional and another not in your examples? You seem to be introducing at least one more value into your truth table other than 'True' and 'False' which you may label 'Conditionally True'. This is not the two-valued logic of which I'm making use.
Also, in your explanation of how your preferred solution is simpler, you made a tacit assumption that there is exactly one knight and one knave. This assumption is not required by my preferred solution. In this way and others, mine is simpler.Nbrader (talk) 00:46, 3 February 2016 (UTC)
How "unlikely" is it that the knight and knave, who stand next to each other and have presumably done this before, don't know what the other answers? You're creating a problem out of thin air here, there's no logical reason to think they don't understand how their puzzle works.
Your version is overcomplicated by the fact that the question is explicitly presented as a conditional: "assuming X, Y?" Since the knave can assume a false X, he can start from the basis that he is actually the knight. His statement is therefore false for an assumed value of X, and is therefore a conditional falsehood.
Also, the door puzzle is built on the assumption that one speaker is a knight and the other is a knave, that's not my "tacit assumption" at all. And how is your solution "simpler" when it requires over twice as much explanation? Herr Gruber (talk) 00:53, 3 February 2016 (UTC)
How unlikely? More unlikely than them knowing themselves. My problem with the Labyrinth solution is not that it doesn't work, but that it relies on more than it has to and this makes it harder to grasp and utilize.
The Labyrinth solution assumes the exact same thing implicitly. It shouldn't be counted against my solution that it has been stated more explicitly. I could easily state it as implicitly as you have done in your solution: "What would you say?".
The assumption was tacit in this conversation till I mentioned it (which was the only reason I said it was tacit).
Don't make the mistake of thinking my solution more complicated simply because I try harder to explain things in detail than you do.Nbrader (talk) 01:07, 3 February 2016 (UTC)
It's not unlikely at all, though. There's nothing improbable about two guys who stand near each other as part of their job knowing who the other person actually is and what they do, I'm not sure why you regard that as some vast leap of faith rather than something that would basically have to be true unless they'd been parachuted in there five minutes beforehand with a note saying which door was which.
The solution in Labyrinth is phrased as a simple question, which means it can be expected to receive a simple answer. You're using a conditional and expecting to still receive a simple answer, which is a weaker solution.
The actual biggest problem with 3 in fiction is usually the doorkeepers explain the puzzle, often alternating in a way that means the knave has to be telling the truth (Labyrinth, notably, actually gets this right since the one who explains the puzzle is the knight if it's assumed Sarah went through the door that was not certain death and the lower heads aren't knights or knaves). This means that you could easily beat most instances of this puzzle just by directing your question at the one who explained it since either he is the knight or there is no reason to believe any of the rules you were just told apply. Herr Gruber (talk) 01:12, 3 February 2016 (UTC)
All I meant by 'more unlikely' is relying on more assumptions because every time you make a new assumption, it presents an additional opportunity for at least one assumption to fail. Though all assumptions may hold for this puzzle, the answer given is less likely to work on similar puzzles.
In the film, she says "Answer yes or no. Would he tell me that this door leads to the castle?". How is this any less 'conditional' (as you put it) than my question: "Would asking you "Does your path lead to freedom?" result in you responding 'Yes'?"? If anything, her question allows misleading answers such as answering "No" because he assumes you never actually asked the question in your hypothetical scenario (she only asked if he would answer, not that she asked). I stated my question as I did to try and avoid that kind of ambiguity. If you like I can restate it like hers: "Would you say that your path leads to freedom?"
You're wrong about the film being consistent with regards to the knight/knave explaining the puzzle too: The knight agrees with the knave that you can only ask one of them. The knight can never agree with the knave in a knights and knaves puzzle.
I'd like to have acknowledgement that my solution is indeed a solution. I personally prefer my solution as it still works in a similar puzzle without there necessarily being 1 knight and 1 knave and without them having to know anything about each other. Nbrader (talk) 13:14, 3 February 2016 (UTC)
But the solution is to an example where there is necessarily one knight and one knave (that's in the description), so it just adds a layer of pointless complexity and more ways to get out of it since you overstate the question to the point it's an explicit conditional question which could reasonably be given a conditional answer. Sarah does not ask an if-then question, she just asks what the other guard would say. This provides a lot less space for bullshitting because it's a much simpler way to state the question and so leaves fewer ways to work around it by creative interpretation. Sure, you still have "no, he wouldn't tell you" rather than "no, it's the wrong door," (though you could assume they'll both give you charitable answers if you're not obviously trying to cheat around the "one question only" rule) but your example is like an RPG genie wish where it has seven hundred different conditions attached to it to try to stop the GM giving you a cloak of giant strength that currently has a giant in it or that you lose immediately after getting or that belongs to someone else, and it still doesn't stop him doing it.
And if you asked the knight and knave "would you say your path leads to freedom" you're not going to get anywhere, because to that question the knave would give the opposite answer to the knight. You have to have the truth about a lie / lie about the truth to make sure the answer you get is always false, since you can't get a true answer out of the knave, so it needs to be what the other one would say.
The knave does make a statement which is partially false ("you can't ask us, you can only ask one of us" is a contradiction, and so cannot be completely true), though I do admit I forgot that line. Herr Gruber (talk) 14:37, 3 February 2016 (UTC)
Being explicit might make the sentence harder to parse but you shouldn't confuse that with logical complexity: A poorly defined question introduces huge complexity.
In the non-Labyrinth question you get a lie about a lie or a truth about a truth, resulting in yes if the door of the person you ask leads to freedom (regardless of whether knight/knave). So it does still work, contrary to what you thought. The knave does not give the opposite answer to the knight in either of the two suggested solutions. They don't have to answer the question differently because the same question asked to a different person is then about a different subject, resulting in a different proposition.
I did notice that about the knave explaining the rules but interpreted that as "You can't ask us both". I'm happy to agree that this makes it unclear whether they were being consistent or not. You could give the movie the benefit of the doubt here. Nbrader (talk) 08:42, 4 February 2016 (UTC)
I move that we edit the article to introduce interesting points from this discussion that we agree on. I move that this includes both solutions suggested here. Nbrader (talk) 09:02, 4 February 2016 (UTC)

Third Opinion[edit]

A third opinion has been requested. The discussion has been so lengthy that it isn't obvious what the question is. Please state the question concisely. Robert McClenon (talk) 18:04, 3 February 2016 (UTC)

Largely whether the classic solution to the third puzzle on the page or this apparently "simpler" overcomplicated gibberish should be used, the latter being much harder to explain, requiring the knave lie twice in the same answer rather than just saying something that isn't true, and introducing excessive complexity into the conditions of a puzzle that only functions if it's kept extremely simple. Nbrader's at about the point the knight can start lying to him because he didn't ask whose freedom the door led to. Herr Gruber (talk) 01:29, 4 February 2016 (UTC)
I'm not even sure how to read that last sentence. I certainly am not suggesting the knight can lie. You on the other hand have suggested multiple times that the knave can choose to tell the truth, which is something I've always denied.
The main point of dispute is on how to edit the article. I would suggest we should at least present both solutions to problem 3. Herr Gruber doesn't seem to consider one of these solutions to be valid whereas I do. Furthermore, I see this solution as simpler whereas he does not. The only point in favour of including the solution appearing in 'Labyrinth' is cultural, whereas the other solution is logically simpler.Nbrader (talk) 08:09, 4 February 2016 (UTC)
The two suggested solutions are as follows (in their simple+ambiguous form):
"Would he tell me that this door leads to the castle?" - Labyrinth Solution
"Would you tell me that this door leads to the castle?" - Simpler Solution
Additionally to suggesting the simpler solution (and as a separate, secondary point) I also proposed making the question more explicit by rephrasing it thusly:
"Would asking you "Does your path lead to freedom?" result in you responding 'Yes'?"
I would agree that this is slightly harder to read but it avoids some pedantic issues where they might validly assume you didn't ask a question in the given hypothetical scenario. This doesn't change the intended meaning of the question, but simply makes it clearer what is meant (which is one point Herr Gruber doesn't seem to be understanding and is confusing with the first point). Nbrader (talk) 09:10, 4 February 2016 (UTC)
I'd say the best option is completeness - presenting both sounds like the best go. We could even include both and just let the article contradict itself a bit - something like this is bound to have differing views after all. TheLogician112 (talk) 05:43, 5 February 2016 (UTC)