Craig then told the company what each had said.
“That explains it perfectly!” said Thor. “Moreover, it follows from their having said what they did that Cyrus was the one you met in the morning. And interestingly enough, if you hadn’t met Alexander in the afternoon, you could never have known whether the one you met in the morning was Cyrus or Alexander.”
Craig thought the matter over and realized that Thor was right.
What statements could these two outsiders have made which fulfill all of the above conditions?
• Epilogue—a Philosophical Puzzle •
The next morning when Craig was wide awake and recalling his dream, he wondered whether he had been logically inconsistent in his sleep. “The trouble is this,” thought Craig: “In my dream I believed that Thor was a god and that gods always tell the truth. Yet Thor told me that I wasn’t dreaming. Now how could Thor, who tells the truth, say that I wasn’t dreaming when in fact I was? Wasn’t this an inconsistency on my part?”
Would you say that Craig’s dream was logically inconsistent?
SOLUTIONS
1 · One statement that works is: “I am not a knight.” If the speaker were a knave or a demon, then it would be true that he was not a knight, but knaves and demons don’t make true statements. Therefore the speaker was neither a knave nor a demon, hence he was a knight or a god and his statement was true. Since it was true, then he really is not a knight; hence he must be a god.
2 · A statement that works is: “I am a demon.” Obviously no demon can claim he is a demon, so the speaker is not a demon. Therefore his statement was false and since he is not a demon, he must be a knave.
Incidentally, this and the last puzzle are essentially the same as Problems 4 and 5 of Chapter 1, the puzzles about the prizes.
3 · This is a bit more tricky: A statement that works is: “I am either a god or a knave.” That could be said by a god, since a god is either a god or a knave; it could also be falsely said by a demon. It couldn’t be said by a knight, because a knight would never lie and claim that he is either a god or a knave, and it couldn’t be said by a knave, because a knave would never admit to the true fact that he is either a god or a knave. And so the speaker must be either a god or a demon, but there is no way to tell which.
4 · The speaker’s first statement was obviously false, for if it were true, a god would have once claimed that the speaker was a demon, which would mean that the speaker really was a demon, but no one who tells the truth can be a demon. Since the first statement was false, so was the second statement, since it was made by the same speaker. Therefore a knight did once claim that the speaker was a knave, hence the speaker really is a knave.
5 · The speaker’s second statement was obviously a lie, because no truth-teller would ever say that he sometimes claims to be a demon. Therefore the first statement was also a lie, hence the speaker does sometimes claim to be a knave, hence he must be a demon.
6 · Many solutions are possible; here is one. Let us call the two beings A and B. Now, suppose A and B make the following two statements:
A: B is a knight.
B: A is not a knight.
A is either telling the truth or lying.
Case 1—A is telling the truth: Then B really is a knight, hence his statement is true, hence A is not a knight, therefore A must be a god, since he is telling the truth.
Case 2—A is lying: Then B is not a knight, since A says he is. Also, since A is lying, then A is certainly not a knight, hence B’s statement is true. Therefore B is telling the truth, but is not a knight, hence B is a god.
So if Case 1 is true, A is a god; if Case 2 is true, then B is a god. There is no way to tell whether A is telling the truth or lying.
7 · Again let us call the two beings A and B. The following statements would work:
A: Both of us are knaves.
B: Both of us are demons.
It is obvious that both are lying. Since A is lying, they are not both knaves. Since B is lying, they are not both demons. Therefore one is a knave and one is a demon, but there is no way to tell which one is which.
8 · A statement that works is: “I am either a knave or a demon or the god Thor.”
If the speaker were either a knave or a demon, then it would be true that he is either a knave or a demon or the god Thor. This would mean that a knave or a demon made a true statement, which is not possible. Therefore the speaker is neither a knave nor a demon, hence his statement is true. Hence he must be the god Thor.
9 · Here is one possible solution.
MORNING SPEAKER: “I am neither a knight nor a god.”
AFTERNOON SPEAKER: “I am either a knave or a demon.”
No inhabitant of the region could make either of those statements. No knight or god could claim that he is neither a knight nor a god; no knave or demon could make the true statement that he is neither a knight nor a god. As for the second statement, obviously no knight or god would claim to be either a knave or a demon and no knave or demon would admit to being a knave or a demon. Therefore both were outsiders; namely, Cyrus and Alexander. The statement of the morning speaker was true and the statement of the afternoon speaker was false. Since Cyrus never makes false statements, he couldn’t have been the afternoon speaker. Thus he was the morning speaker.
• Discussion of the Epilogue •
As I see it, Craig’s dream was not necessarily inconsistent. If Craig had actually believed in the dream that he was dreaming, then the set of his beliefs during his dream would have been inconsistent, since the following propositions are indeed logically contradictory: (1) Thor is a god; (2) Gods make only true statements; (3) Thor stated that Craig was not dreaming; (4) Craig was dreaming.
The contradiction is obvious. However, there is no evidence that Craig at any time of his dream believed that he was dreaming, although at one point he wondered whether he might be dreaming. Craig presumably believed that he was awake, and this belief, though false, was perfectly consistent with the other beliefs of his dream.
Curiously enough, if Craig had formulated the belief that he was dreaming, then this belief, though correct, would have created a logical inconsistency!
8
In Search of the Fountain of Youth
INTRODUCTION
Arthur Reynolds, Esq., was in search of a recipe for immortality. He read a great deal of occult and alchemical literature but could find nothing of practical value. Then he heard of a great Sage of the East who was reputed to be a specialist in this area. At great expense, he made a lengthy journey and finally found the Sage.
“Is it really possible to live forever?” he asked the Sage.
“Oh, quite easily,” replied the Sage, “provided you do just two things.”
“What are the two things?” asked Reynolds eagerly.
“First of all you must never make false statements; in fact, you must resolve to make only true statements from now on. That’s a small price to pay for immortality, isn’t it?”
“Oh, definitely!” replied Reynolds. “But what about the second thing?”
“The second thing is that you say now: ‘I will repeat this statement tomorrow.’ If you do just these two things,” concluded the Sage, “then I guarantee that you will live forever!”
A Question for the Reader: Is it really true that if you do these two things you will live forever? The answer is given in the text that follows, but the reader might like to think about this before reading further.
Reynolds thought about this for a bit. “Oh, of course!” he said suddenly. “If I do those two things, then I will certainly live forever, because if I truthfully say now, ‘I will repeat this sentence tomorrow,’ then tomorrow I will again say, ‘I will repeat this sentence tomorrow,’ and if I am truthful tomorrow, then I will say the same thing again the next day, and so on throughout all eternity.”
“Exactly!” said the Sage with a triumphant smile.
“But the solution is not a practical one!” prote
sted Reynolds. “How can I truthfully say that I will do something tomorrow if I don’t know for sure whether I’ll even be alive tomorrow?”
“Oh, you want a practical solution,” said the Sage. “I didn’t realize that. No, I’m not very good at practical solutions; I deal mainly with theory. But a practical solution? The only thing I can think of is the Fountain of Youth. Have you considered searching for it?”
“The Fountain of Youth?” cried Reynolds incredulously. “Why, I’ve read about it in history books, and I know that many have sought it, but does it exist in reality or only in the imagination?”
“That I do not know,” replied the Sage, “but if it does exist, I know as likely a place as any where it can be found.”
“What place is that?” asked Reynolds.
“The Island of Knights and Knaves,” replied the Sage. “I cannot guarantee that the fountain is there, but if it is anywhere, that island is as likely a place as any.”
Reynolds thanked the Sage and forthwith departed for the Island of Knights and Knaves.
IN SEARCH OF THE FOUNTAIN OF YOUTH
And so we are back to the Island of Knights and Knaves. Reynolds arrived without mishap, and his adventures began.
1 • A Preliminary Incident
On the first day, Reynolds met a native who made a statement. Reynolds then realized that if the native was a knight, the Fountain of Youth must be on the island, but if the native was a knave, there was no way of telling whether or not the Fountain of Youth was on the island.
What statement could the native have made?
2 • A Grand Metapuzzle
Reynolds’s next adventure was far more interesting—it involves as profound a logic puzzle as any I have ever come across.
On the following day, Reynolds came across two natives A and B and said to them, “Please tell me whatever you know about the Fountain of Youth. Is it on this island?” The two natives then made the following statements:
A: If B is a knave, then the Fountain of Youth is on this island.
B: I never claimed that the Fountain of Youth is not on this island!
Reynolds thought about this for a bit and said, “Please, now, I want a definite answer! Is the Fountain of Youth on this island?” One of the two answered—he either said yes or he said no—and Reynolds then knew whether or not the Fountain of Youth was on the island.
Some months later, Reynolds told the above facts to Inspector Craig. (He was a good friend of Craig’s and knew of his fondness for logic puzzles.) Craig said, “It is obviously impossible for me to deduce from the facts you have given me whether or not the Fountain of Youth is on this island. You haven’t told me whether A or B was the one who answered your second question, or what answer he gave. Whichever one it was who answered, suppose the other one had answered instead. Do you know if you could then have decided whether or not the fountain is on the island?”
Reynolds thought about this and finally told Craig whether or not he knew whether he could have decided had the other one answered the second question instead.
“Thank you,” said Craig. “I now know whether or not the Fountain of Youth is on the island.”
Is the Fountain of Youth on the island?
SOLUTIONS
1 · One possibility is that the native said: “I am a knight and the Fountain of Youth is on this island.” If he was a knight, then the fountain is obviously on the island. If he was not a knight, then what he said is false, regardless of whether or not the fountain is on the island, and there would be no way of telling whether or not the fountain was there.
2 · B said that he had never claimed that the Fountain of Youth was not on the island. If B is a knave, then B has previously claimed that the Fountain of Youth is not on the island, and since he is a knave, the fountain is on the island! So we now know that if B is a knave, the Fountain of Youth is on the island. Well, this is just what A said; therefore A must be a knight. And so we now know the following two facts:
Fact 1: A is a knight.
Fact 2: If B is a knave, the Fountain of Youth is on the island.
Of course Reynolds, who can reason as well as you and I, realized these two facts also.
Now, we are not told who answered Reynolds’s second question, or whether the answer was yes or no, and so there are four possible cases that we must analyze.
Case A1: A claimed that the fountain was on the island (by answering yes).
In this case, Reynolds, knowing that A is a knight, would know that the fountain was on the island.
Case A2: A claimed that the fountain was not on the island (by answering no).
In this case, Reynolds would have known that the fountain was not on the island.
Case B1: B claimed that the fountain was on the island (by answering yes).
In this case, Reynolds would know that the fountain was on the island by reasoning as follows: “Suppose the fountain is not on the island. Then B is a knave for having just affirmed that it is. But by Fact 2, if B is a knave, the fountain is on the island. This is a contradiction, hence the fountain must be on the island after all (and also B must be a knight).”
Case B2: B claimed that the fountain was not on the island (by answering no).
In this case, Reynolds could not possibly know whether or not the fountain was on the island; B could be a knave who falsely claimed that the fountain was not on the island and who also falsely claimed that he never claimed that the fountain was not on the island, or he could be a knight who truthfully claimed that the fountain was on the island and who also claimed that the fountain was not on the island. And so in this case, Reynolds couldn’t decide.
However, we are given that Reynolds did decide, and therefore Case B2 is ruled out. So we now know that one of three cases—A1, A2, B1—is the one that holds, and we can henceforth forget about the fourth case B2.
At this point, we must take into account the second part of the story—the conversation Reynolds had with Inspector Craig. It is important to realize that Craig did not ask Reynolds whether if the other one had answered the question, Reynolds could have decided; Craig asked Reynolds whether he knew whether or not he could have decided. Let us see how Reynolds would reason in response to Craig’s question. Of course Reynolds knows which of the three cases—A1, A2, B1—is the actual one, although we don’t (at least not yet), and so we must see how Reynolds would reason in each of the three cases.
Case A1: Reynolds would reason thus: “A is a knight; A said that the fountain is on the island; the fountain is on the island. Now suppose B had answered my second question. I don’t know whether he would have answered yes or no. Suppose he had answered yes. Then I would have known that the fountain was on the island, because I would have reasoned that if the fountain were not on the island then B is a knave for claiming it was. But I also know (Fact 2) that if B is a knave, the fountain is on the island and so I would have gotten a contradiction from the assumption that the fountain is not on the island. Therefore, if B had answered yes, I would have known that the fountain was on the island. But suppose he had answered no? Then I would have had no way of knowing whether or not the fountain was on the island; I would have reasoned that he could be a knight and the fountain was not on the island, or he could be a knave and the fountain was on the island. And so if he had answered no, then I couldn’t have decided whether or not the fountain was on the island.
“In summary, had B answered yes, I could have decided; had he answered no, I couldn’t have decided. Since I have no way of knowing what answer B would have given, I have no way of knowing whether I could have decided or not.”
Case A2: In this case, here is how Reynolds would reason: “The fountain is not on this island; A told me this and A is a knight. Now, suppose B had answered instead. Well, B is a knight, because I have already proved that if B were a knave, then the fountain would be on this island, which it isn’t. Since B is a knight, then had he been the one to answer, he would also have answered no. But then I couldn’t
have known that he was a knight, and so I would have had no way of knowing whether the fountain is on the island or not. In brief, had B been the one to answer, then I definitely could not have decided whether or not the fountain was on the island.”
Case B1: This is the simplest case of all! In this case, B is the one who really answered, hence Reynolds would already know that if A had been the one to answer, he could have decided about the fountain, because he already knew that A was a knight.
We now see that if Case A1 were the actual one, then Reynolds could not know whether he could have decided, hence his answer to Craig would be: “No, I don’t know whether I could have decided had the other one been the one who answered my second question.”
If Case A2 is the actual one, then Reynolds would have told Craig: “Yes; I do know whether or not I could have decided.” Reynolds, in fact, even knows that he couldn’t have decided.
If Case B1 is the actual one, then Reynolds would again have told Craig: “Yes, I know whether or not I could have decided.” He in fact even knows that he could have decided.
Therefore, if Reynolds answered yes to Craig’s question, then either Case A2 or Case B1 holds, but there is no way we or Craig could tell which, hence Craig couldn’t have known whether or not the Fountain of Youth was on the island. But we are given that Craig did know, hence Reynolds must have answered no and Craig then knew that Case A1 was the only possibility and that the fountain was on the island.
To Mock a Mocking Bird Page 6
