# B A logical problem from Smullyan’s Logical Labyrinths book

#### pickmenot

Summary
Please help me understand whether it’s me who is not seeing something obvious or it’s something wrong with the problem itself.
Hello

I’m going through this book and it contains many puzzles and solutions. I’ve encountered a problem that makes no sense to me. I’d spent a considerable amount of time on it before I finally gave up and looked at the solution, where I discovered essentially the deductions identical to mine.

In the solution the author starts with an assumption that one of the two opposite possibilities is true, reasons until he encounters a contradiction, and declares that, since we encountered it, the opposite must be true.

But if you explore the opposite assumption it also leads to a contradiction. I suspect that there's is something wrong with the problem itself, but would still like a confirmation.

The problem and the solution are in attachment. And these are preliminary conditions:

There are two types of people on an island: knights and knaves. Knights always tell the truth and knaves always lie. Everyone is mute and uses cards to signal YES or NO. The cards are RED and BLACK, but we don’t know which is which.

Thanks

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#### PeroK

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I did this without using the contradiction argument.

First I wrote down all 8 possibilities for A, B and C to be Knights or Knaves.

A and B answered differently to the question of guilt, so that means A and B must be different. That rules out 4 of the possibilities. Using T and L for truthteller and liar, instead of knight and knave, we have four possibilities:

ABC
TLT
TLL
LTT
LTL

In the first case B is a liar and he responds to the question about whether A & C are the same with a Red card. They are the same, so Red must mean No in this case. That means that A, a truthteller, said the defendent was guilty and B, a liar, said he was innocent, hence he must be guilty.

You can reason this way for all four cases to determine what the cards mean and the verdict in each case:

ABC
TLT (Red - no; Guilty)
TLL (Red - yes; Innocent)
LTT (Red - no; Innocent)
LTL (Red - Yes; Guilty)

Finally, we have the strange question. If C is a truthteller and shows a Red card, then Red means Yes. That rules out the first and third options above. If C is a liar, then Red means No. That rules out the second and fourth options.

So, yes, the problem is wrong somehow.

I originally got confused about who answered the last question. I thought it was B again and the problem does have a solution.

But, when I changed it to C, there is no solution.

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#### sysprog

The defendant is innocent.

The point of the 3rd question is that it eliminates the difference in answers between the knight and the knave -- both will answer the same -- red if the defendant is innocent, black if he is guilty.

If red means yes, then A is a knight, B is a knave, and C will answer yes, even though he's a knave, because he will lie about how he would answer the innocence question.

If red means no, then A is a knave, B is a knight, and C will answer no, because he's a knight, so he will truthfully say no to the question asking whether he would say no to the question of innocence.

If the defendant were guilty, C would have shown black, no matter whether he is knight or knave, no matter whether black means yes or black means no.

#### PeroK

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If the defendant were guilty, C would have shown black, no matter whether he is knight or knave, no matter whether black means yes or black means no.
Ah! That's what's meant. But, the judge doesn't say which question he is asking. He just says "this question". I took that to mean simply a question about the red and black cards. But, I guess, "this question" refers to the innocence or guilt.

Moreover, given that the previous question (he asked B) was "are the other two of the same type" it would be equally valid to consider that as "this question".

The last part of the problem is not just poorly worded but completely ambiguous. The judge in no way specifies which question he is asking C.

PS Moreover, as far as I can see, the final question makes all the other questions redundant. That question alone is sufficient to determine guilt or innocence.

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#### sysprog

PeroK said:
Ah! That's what's meant. But, the judge doesn't say which question he is asking. He just says "this question". I took that to mean simply a question about the red and black cards. But, I guess, "this question" refers to the innocence or guilt.
Yes.
Moreover, given that the previous question (he asked B) was "are the other two of the same type" it would be equally valid to consider that as "this question".
There aren't 2 other questions before the Court, and there are 2 other witnesses; it seems to me that the puzzle makes sense only if 'the other two' refers to the other 2 witnesses, which are the only elements in the puzzles that have a type, (you could view the defendant as being of 2 possible types, innocent or guilty, but there is only 1 defendant).
The last part of the problem is not just poorly worded but completely ambiguous. The judge in no way specifies which question he is asking C.
The question could be reworded as "If I were to ask you 'is the defendant innocent', would you then show red?".
PS Moreover, as far as I can see, the final question makes all the other questions redundant. That question alone is sufficient to determine guilt or innocence.
Without the prior questions, the judge would not know that red meant innocent and black meant guilty -- he would know only that both knight and knave would answer the same, and that which way either would answer, would hinge on what the witness knew about the meanings of the colors, and what he knew about the defendant.

If you imagine that the first question the judge asked was to witness A, and he asked "if I were to ask you 'is the defendant innocent', would you show red?", the answer to that question, red or black, would not by itself establish innocence or guilt.

From the red answer, the judge would know that if red meant yes, the answer to the innocence question would be yes, no matter whether the witness was knight or knave. But if red meant no, the answer to the innocence question would be no.

If the judge were to ask A a second question, "are you a knight?", if A were a knight, he would answer with whichever color meant yes, and so would the knave. The judge would then know which color meant yes, and would then from the first question know whether the defendant was innocent.

#### pickmenot

Ah! That's what's meant. But, the judge doesn't say which question he is asking. He just says "this question". I took that to mean simply a question about the red and black cards. But, I guess, "this question" refers to the innocence or guilt.
No, this is wrong. I think the judge is very clearly specifies which question he asks — by asking it :-) The question obviously contains a self-reference. It’s a logic book after all and we’re meant to take formulations at face value, not to read into it something that’s isn’t there.

If it really were the case then the question would’ve been formulated something like this: “are you the type who could flash the red card if the suspect is innocent?”, and without specifically telling us that the question was “curious”. The type of question sysprog is talking about is no longer curious at this point in the book.

Plus it makes the rest of the problem redundant. Why would the book from a logician of the highest calibre contain such stupid tricks for the single purpose of obfuscating the answer? And why then the reasoning in the solution is completely different from sysprog’s interpretation?

My interpretation is that the question is curious only because it’s paradoxical: it can be answered only with the red card. The judge knew this, and the red card didn’t tell him anything new, rather it was the fact of the answer that told him which of the two states the situation is in.

If C answers, it means that one of these alternatives is true: 1) C is a knight and Red=Yes; 2) C is a knave and Red=No.

If C doesn’t answer, it means that one of the opposite alternatives holds: 3) C is a knave and Red=Yes; 4) C is a knight and Red=No.

That is the purpose of the final question, and it was meant to be a piece of the puzzle, not the only thing required for solution.

#### PeroK

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The point of the 3rd question is that it eliminates the difference in answers between the knight and the knave -- both will answer the same -- red if the defendant is innocent, black if he is guilty.
The question could be reworded as "If I were to ask you 'is the defendant innocent', would you then show red?".
I agree with this, hence:

1) Let's assume that the defendant is guilty; that the respondent is a knight (truthteller); and that Red means no:

The knight knows the defendant is guilty so he would answer "no", which is red in this case. The answer to the full question is "Yes", which in this case is black.

A black card would be shown.

2) Now, let's assume that the defendant is guilty; that the respondent is a knight (truthteller); and that Red means yes:

The knight would answer "no", which is black in this case. The answer to the full question is, therefore, "no", which is black.

A black card would be shown.

3) Now, let's assume that the defendant is guilty; that the respondent is a knave; and that Red means no:

The knave would answer "yes" to the question of innocence, which is black. So, asked if he would show a red card, his answer should be no, but he would say "yes", by showing a black card.

A black card would be shown.

4) Now, let's assume that the defendant is guilty; that the respondent is a knave; and that Red means yes:

The knave would answer "yes" to the question of innocence, which is red. So, asked if he would show a red card, his answer should be yes, but he would say "no", by showing a black card.

A black card would be shown.

In all four cases, guilt implies a black card.

Likewise, in all four cases of innocence a red card is shown.

Therefore, this question alone is sufficient.

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#### PeroK

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No, this is wrong. I think the judge is very clearly specifies which question he asks — by asking it :-) The question obviously contains a self-reference.
You'll have to ask whoever wrote the book what he meant. In any case, it isn't clear. You and I have interpreted it one way and @sysprog has interpreted it another way.

Certainly, C would have been entitled to ask the judge "what are you asking me exactly"?

I'm honestly not sure.

In any case, there is a much better problem with the one witness as I've described above.

#### pickmenot

@PeroK Well, there's been only the first edition, and it's known to contain many typos and mistakes :-)

... ... ...

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#### sysprog

sysprog said:
The question could be reworded as "If I were to ask you 'is the defendant innocent', would you then show red?".
I agree with this, hence:

1) Let's assume that the defendant is guilty; that the respondent is a knight (truthteller); and that Red means no:

The knight knows the defendant is guilty so he would answer "no", which is red in this case. The answer to the full question is "Yes", which in this case is black.

A black card would be shown.

2) Now, let's assume that the defendant is guilty; that the respondent is a knight (truthteller); and that Red means yes:

The knight would answer "no", which is black in this case. The answer to the full question is, therefore, "no", which is black.

A black card would be shown.

3) Now, let's assume that the defendant is guilty; that the respondent is a knave; and that Red means no:

The knave would answer "yes" to the question of innocence, which is black. So, asked if he would show a red card, his answer should be no, but he would say "yes", by showing a black card.

A black card would be shown.

4) Now, let's assume that the defendant is guilty; that the respondent is a knave; and that Red means yes:

The knave would answer "yes" to the question of innocence, which is red. So, asked if he would show a red card, his answer should be yes, but he would say "no", by showing a black card.

A black card would be shown.

In all four cases, guilt implies a black card.

Likewise, in all four cases of innocence a red card is shown.
I have to agree with you here.

Going through the four cases of innocence:

In the case in which the defendant is innocent, the witness is a knave, and red means no, the knave knows he would show red (say no) in response to the innocence question, so he lies, and says in response to the question 'would he show red (say no) in response to the innocence question' that no, he would not show red (say no), and therefore displays a red (no) card.​
In the case in which the defendant is innocent, the witness is a knave, and red means yes, the knave knows he would show black (say no) in response to the innocence question, so he lies, and says in response to the question 'would he show red (say yes) in response to the innocence question' that yes, he would show red (say yes), and therefore displays a red (yes) card.​
In the case in which the defendant is innocent, the witness is a knight, and red means no, the knight knows he would show black (say yes) in response to the innocence question, so in response to the question 'would he show red (say no) in response to the innocence question' that no, he would show black (say yes), and therefore displays a red (no) card.​
In the case in which the defendant is innocent, the witness is a knight, and red means yes, the knight knows he would show red (say yes) in response to the innocence question, so in response to the question 'would he show red (say yes) in response to the innocence question' that yes, he would show red (say yes), and therefore displays a red (yes) card.​

If the judge were to switch the color, he could still determine innocence or guilt from the single question: if he were to ask "If I were to ask you 'is the defendant innocent', would you then show black?", a black response would mean innocent, and red would mean guilty.

Looking at just one of the eight cases, and substituting black for red in the question:

In the case in which the defendant is innocent, the witness is a knight, and black means yes, the knight knows he would show black (say yes) in response to the innocence question, so in response to the question 'would he show black (say yes) in response to the innocence question' that yes, he would show black (say yes), and therefore displays a black (yes) card.​

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#### sysprog

From my earlier post:
sysprog said:
If you imagine that the first question the judge asked was to witness A, and he asked "if I were to ask you 'is the defendant innocent', would you show red?", the answer to that question, red or black, would not by itself establish innocence or guilt.

From the red answer, the judge would know that if red meant yes, the answer to the innocence question would be yes, no matter whether the witness was knight or knave. But if red meant no, the answer to the innocence question would be no.
Re-examining this in light of @PeroK's analysis, with which I just agreed:

If the defendant is innocent, the witness is a knight, and red means yes, and the judge asks "if I were to ask you 'is the defendant innocent', would you show red?", the knight knows he would show red regarding the embedded innocence question, and truthfully answers the question in which the innocence question is embedded by showing red.

If the defendant is innocent, the witness is a knight, and red means no, and the judge asks "if I were to ask you 'is the defendant innocent', would you show red?", the knight knows he would not show red, and truthfully answers the question in the negative by showing red.​

So my earlier answer was partially incorrect. The innocence or guilt of the defendant would not depend on whether red meant yes or no. The source of my error was in not recognizing that the judge's question not only embedded the question of innocence, which eliminated the difference between the knight and and the knave, but also used the witness' own knowledge of the yes or no meaning of the colors, which made whichever color the judge referred to mean innocent.

If the defendant were innocent, and the question had been "if I were to ask you 'is the defendant innocent' would you show the card that means yes?", and the witness were a knight, and red meant yes, the answer would be red, but if red meant no, the answer would be black.

If the defendant were innocent, and the question had been "if I were to ask you 'is the defendant innocent' would you show the card that means yes?", and the witness were a knave, and red meant yes, the knave's false answer would be red, but if red meant no, the knave's false answer would be black.​

The embedding of the question 'is the defendant innocent' suffices to eliminate the difference between knight and knave, but does not by itself suffice to allow discernment of innocence or guilt.

When the question in which that embedded question is embedded asks 'would you show red?', rather than asking 'would you show the card that means yes?', the judge is correctly recognizing that he can thereby eliminate not only the difference between knight and knave, but also his need to know which of red or black means yes or no, and so can discern by the single question whether the defendant is innocent or guilty.

"A logical problem from Smullyan’s Logical Labyrinths book"

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