Friday, September 15, 2017
Satan, Cantor, and Infinity p. 16
I am reading Raymond Smullyan's "Satan, Cantor, and Infinity", and on page 16 I find myself in some disagreement with the solution to the puzzle, although I agree with the final result. Here is the puzzle as stated (bear in mind that this story takes place on an island where there are only Knights, who always tell the truth, and Knaves who always lie):
The case concerns a stolen horse. There are four suspect -- Andrew, Bruce, Clayton, and Edward. The authorities know for sure that one and only one of these four is the thief. The first three have already been found and put in custody, but Edward cannot be found anywhere. The trial will have to proceed without him.
First, the judge pounded his gavel and asked a highly relevant question: "Who stole the horse?" He got the following replies:
Andrew: Bruce stole the horse.
Bruce: Clayton stole the horse.
Clayton: It was Edward who stole the horse.
Then, quite unexpectedly, one of the three defendants said, "The other two are lying."
The judge thought for a bit, then pointed to one of the three and said, "Obviously, you didn't steal the horse, so you may leave the court."
The acquitted man was happy to comply, and so only two defendants were left on trial.
The judge then asked one of the remaining two whether the other was a knight, and, after receiving an answer (yes or no), knew who stole the horse. What did the judge decide?
Now, here is the answer from the book:
First we must determine whom the judge immediately acquitted. Suppose it was Andrew. If Andrew is a knight, then Bruce must be guilty and Andrew is innocent. If Andrew is a knave, then it is false that Bruce and Clayton both lied; at least one of the told the truth. This means that either Clayton is guilty (if Bruce told the truth), or Edward is guilty (if Clayton told the truth); in either case, Andrew would be innocent. And so if it was Andrew who made the second statement, he is innocent, regardless of whether he is a knight or a knave. The judge, of course, would have realized this and acquitted him.
However, if either Bruce or Clayton made the second statement, the judge could not have found grounds to acquit anyone. If Bruce spoke, the judge could only tell that either Bruce is a knight and Clayton is guilty, or Bruce is a knave and either Bruce or Edward is guilty. If it was Clayton who spoke, then the judge could only tell that either Clayton is a knight and Edward is guilty, or Clayton is a knave and either Bruce or Clayton is guilty. Since the judge did make an acquittal, it must have been Andrew who spoke and was acquitted.
Thus the remaining defendants were Bruce and Clayton. One of these two, by answering the judge's last question, either claimed that the other defendant was a knight, or that the other was a knave. If the former, then the two defendants are the same type (both knights or both knaves); if the latter then the two are different types. Suppose the latter. Then Clayton may be a knight and Bruce a knave, in which case Edward is guilty (because Clayton said he was), or Bruce may be a knight and Clayton a knave, in which case Clayton is guilty. However, the judge couldn't have known which, and hence couldn't make a conviction. Therefore, one of the two must have claimed that the other was a knight (by answering yes to the judge's question). The judge then knew that they are of the same type. They can't both be knights (since their accusations conflict), so they are both knaves and their accusations are both false. Neither Clayton nor Edward stole the horse. Andrew, as we know, has already been acquitted. So it was Bruce who stole the horse.
The main issue I have with this analysis is that I think it is irrelevant who said "The other two are lying", and that the judge could conclude that Andrew is innocent regardless of who said it (it doesn't specify that he released the one who said it, just that he released "one of the three"). I also think that you don't have to use the process of elimination to conclude that Bruce is guilty.
Once the three have testified, we can quickly conclude that at most only one is telling the truth, since all three statements contradict one another. That leaves either one truth-teller, or all three are lying. Now, when someone states "The other two are lying", that eliminates the possibility that all three are lying. The case that the statement is true should be obvious. If the person saying it is lying, then the statement means that is is not true that both the others are lying, so at least one is telling the truth. Since there can't be two truth-tellers because of the conflicting statements, then the statement means that exactly one person is telling the truth. Given that one of three original statements is true, we know that it must be either Bruce, Clayton, or Edward who is guilty, so Andrew cannot be guilty. This conclusion does not require that Andrew said "The other two are lying".
In looking at Smullyan's analysis of the statement, he says that "If Bruce spoke, the judge could only tell that either Bruce is a knight and Clayton is guilty, or Bruce is a knave and either Bruce or Edward is guilty. If it was Clayton who spoke, then the judge could only tell that either Clayton is a knight and Edward is guilty, or Clayton is a knave and either Bruce or Clayton is guilty. Since the judge did make an acquittal, it must have been Andrew who spoke and was acquitted." I think he missed the fact that if either Bruce or Clayton spoke, Andrew is still exonerated. If Bruce spoke, either Clayton, Bruce, or Edward are guilty. If Clayton spoke, either Clayton, Bruce, or Edward are guilty. In neither case is Andrew guilty, so it does not follow that Andrew must have been the one to speak when the judge rightly concluded that Andrew was innocent.
My other point about Smullyan's conclusion is just that you don't have to use the process of elimination to settle on Bruce. Since we know that only one person is telling the truth, we have three possibilities left:
Bruce is a knight, Andrew and Clayton are knaves, Clayton is guilty
Clayton is a knight, Andrew and Bruce are knaves, Edward is guilty
Andrew is a knight, Bruce and Clayton are knaves, Bruce is guilty
As Smullyan explains above, the judge must have received a "yes" in reply to the second question indicating that both Bruce and Clayton are knaves. You can now directly conclude that Bruce is guilty because if Bruce and Clayton are both knaves, then the knight -- the lone truth-teller -- must be Andrew, who said that Bruce is guilty.
The case concerns a stolen horse. There are four suspect -- Andrew, Bruce, Clayton, and Edward. The authorities know for sure that one and only one of these four is the thief. The first three have already been found and put in custody, but Edward cannot be found anywhere. The trial will have to proceed without him.
First, the judge pounded his gavel and asked a highly relevant question: "Who stole the horse?" He got the following replies:
Andrew: Bruce stole the horse.
Bruce: Clayton stole the horse.
Clayton: It was Edward who stole the horse.
Then, quite unexpectedly, one of the three defendants said, "The other two are lying."
The judge thought for a bit, then pointed to one of the three and said, "Obviously, you didn't steal the horse, so you may leave the court."
The acquitted man was happy to comply, and so only two defendants were left on trial.
The judge then asked one of the remaining two whether the other was a knight, and, after receiving an answer (yes or no), knew who stole the horse. What did the judge decide?
Now, here is the answer from the book:
First we must determine whom the judge immediately acquitted. Suppose it was Andrew. If Andrew is a knight, then Bruce must be guilty and Andrew is innocent. If Andrew is a knave, then it is false that Bruce and Clayton both lied; at least one of the told the truth. This means that either Clayton is guilty (if Bruce told the truth), or Edward is guilty (if Clayton told the truth); in either case, Andrew would be innocent. And so if it was Andrew who made the second statement, he is innocent, regardless of whether he is a knight or a knave. The judge, of course, would have realized this and acquitted him.
However, if either Bruce or Clayton made the second statement, the judge could not have found grounds to acquit anyone. If Bruce spoke, the judge could only tell that either Bruce is a knight and Clayton is guilty, or Bruce is a knave and either Bruce or Edward is guilty. If it was Clayton who spoke, then the judge could only tell that either Clayton is a knight and Edward is guilty, or Clayton is a knave and either Bruce or Clayton is guilty. Since the judge did make an acquittal, it must have been Andrew who spoke and was acquitted.
Thus the remaining defendants were Bruce and Clayton. One of these two, by answering the judge's last question, either claimed that the other defendant was a knight, or that the other was a knave. If the former, then the two defendants are the same type (both knights or both knaves); if the latter then the two are different types. Suppose the latter. Then Clayton may be a knight and Bruce a knave, in which case Edward is guilty (because Clayton said he was), or Bruce may be a knight and Clayton a knave, in which case Clayton is guilty. However, the judge couldn't have known which, and hence couldn't make a conviction. Therefore, one of the two must have claimed that the other was a knight (by answering yes to the judge's question). The judge then knew that they are of the same type. They can't both be knights (since their accusations conflict), so they are both knaves and their accusations are both false. Neither Clayton nor Edward stole the horse. Andrew, as we know, has already been acquitted. So it was Bruce who stole the horse.
The main issue I have with this analysis is that I think it is irrelevant who said "The other two are lying", and that the judge could conclude that Andrew is innocent regardless of who said it (it doesn't specify that he released the one who said it, just that he released "one of the three"). I also think that you don't have to use the process of elimination to conclude that Bruce is guilty.
Once the three have testified, we can quickly conclude that at most only one is telling the truth, since all three statements contradict one another. That leaves either one truth-teller, or all three are lying. Now, when someone states "The other two are lying", that eliminates the possibility that all three are lying. The case that the statement is true should be obvious. If the person saying it is lying, then the statement means that is is not true that both the others are lying, so at least one is telling the truth. Since there can't be two truth-tellers because of the conflicting statements, then the statement means that exactly one person is telling the truth. Given that one of three original statements is true, we know that it must be either Bruce, Clayton, or Edward who is guilty, so Andrew cannot be guilty. This conclusion does not require that Andrew said "The other two are lying".
In looking at Smullyan's analysis of the statement, he says that "If Bruce spoke, the judge could only tell that either Bruce is a knight and Clayton is guilty, or Bruce is a knave and either Bruce or Edward is guilty. If it was Clayton who spoke, then the judge could only tell that either Clayton is a knight and Edward is guilty, or Clayton is a knave and either Bruce or Clayton is guilty. Since the judge did make an acquittal, it must have been Andrew who spoke and was acquitted." I think he missed the fact that if either Bruce or Clayton spoke, Andrew is still exonerated. If Bruce spoke, either Clayton, Bruce, or Edward are guilty. If Clayton spoke, either Clayton, Bruce, or Edward are guilty. In neither case is Andrew guilty, so it does not follow that Andrew must have been the one to speak when the judge rightly concluded that Andrew was innocent.
My other point about Smullyan's conclusion is just that you don't have to use the process of elimination to settle on Bruce. Since we know that only one person is telling the truth, we have three possibilities left:
As Smullyan explains above, the judge must have received a "yes" in reply to the second question indicating that both Bruce and Clayton are knaves. You can now directly conclude that Bruce is guilty because if Bruce and Clayton are both knaves, then the knight -- the lone truth-teller -- must be Andrew, who said that Bruce is guilty.
# posted by Mark Wutka @ 6:21 AM 
