When Abercrombie reached his destination, he found himself on a very strange island, indeed! All the inhabitants of this island are completely truthful—they always tell you honestly what they believe, but the trouble is that half the inhabitants are totally mad, and all their beliefs are wrong!
The other half are totally sane and accurate in their judgments; all their beliefs are correct.
Problem 4.1 (Let’s Be Careful!). We shall start with a very tricky puz-zle, but also one that illustrates a basic principle. Is it possible for an inhabitant of this island to say: “I believe I am mad”?
Problem 4.2 (A Simple Form of the Nelson Goodman Principle).The Nelson Goodman principles for the islands of Chapters 1 and 2 involve questions that are rather convoluted and unnatural. Well, on the present island, there is a much more natural-sounding yes/no question you could ask to ascertain any information you want. For example, if you want to find out whether a given native is married or not, there is a relatively natural-sounding question you can ask—one, in fact, having only six words. What question would work?
Problem 4.3. When Abercrombie got settled on this island, he first inter-viewed three siblings named Henry, Dianne, and Maxwell. Henry and Dianne made the following statements:
Henry: Maxwell believes that at least one of us is mad.
Dianne: Maxwell is sane.
What type is each?
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26 I. Be Wise, Generalize!
Problem 4.4. Next, Abercrombie interviewed Mary and Gerald, a mar-ried couple, together with their only child, Lenore. Here is the dialogue that took place.
Abercrombie (to Gerald): I heard that your wife once said that all three of you are mad. Is that true?
Gerald: No, my wife never said that.
Abercrombie (to Lenore): Did your father once say that exactly one of you three is sane?
Lenore: Yes, he once said that.
Abercrombie (to Mary): Is your husband sane?
Mary: Yes.
What type is each?
Problem 4.5. Abercrombie’s next interview was a bit more puzzling. He met a married couple, Arthur and Lillian Smith. Arthur was the only one who said anything, and what he said was: “My wife once said that I believe that she believes I am mad.”
What can be deduced about either one?
Problem 4.6. Abercrombie next interviewed eight brothers named Arthur, Bernard, Charles, David, Ernest, Frank, Harold, and Peter. They made the following statements:
Charles: Arthur is mad.
David: Bernard and Charles are not both mad.
Ernest: Arthur and David are alike, as far as sanity goes.
Frank: Arthur and Bernard are both sane.
Harold: Ernest is mad.
Peter: Frank and Harold are not both mad.
From this confusing tangle, it is possible to determine the madness or sanity of one of the eight. Which one, and what is he?
Problem 4.7 (A Metapuzzle). Before Abercrombie left the island, one of the sane inhabitants, whose name was David, told him of a trial he had attended some time ago. The defendant was suspected of having stolen a watch. First, the judge (who was sane) said to the defendant: “I have heard that you once claimed that you stole the watch. Is that true?” The defendant answered (either yes or no). Then the judge asked: “Did you steal the watch?” The defendant then answered (either yes or no) and the judge then knew whether he was innocent or guilty.
“What answers did the defendant give?” asked Abercrombie.
4. Mad or Sane? 27
“I don’t quite remember,” replied David. “It was quite some time ago.
I do, however, remember that he didn’t answer no both times.”
Was the defendant innocent or guilty?
Solutions
4.1. Many of you will say that it is not possible. You will reason that a sane person knows he is sane, hence does not believe he is mad, and a mad person erroneously believes that he is sane, hence does not believe the true fact that he is mad. Thus, no inhabitant can believe he is mad.
Well, so far, so good. It is indeed true that no inhabitant can believe he is mad. And since the inhabitants honestly state what they be-lieve, then no inhabitant can say that he is mad. But I didn’t ask you whether an inhabitant can say that he is mad, nor did I ask whether an inhabitant can believe that he is mad; what I asked you was whether an inhabitant can say that he believes that he is mad, and that’s a different story!
Look, a mad person doesn’t believe that he is mad, and so it is false that he believes he is mad, but since he believes false propositions, then he also believes that one—he believes that he believes he is mad!! Thus, he doesn’t believe he is mad, yet he believes that he does believe he is mad. And, being honest, he could indeed say that he believes he is mad. Indeed, if you ask a mad inhabitant: “Are you mad?” he will say no, but if you ask him: “Do you believe you are mad?” he will answer yes (since he doesn’t really believe he is mad).
The fact of the matter is that given any true proposition, he will dis-believe the proposition, but also dis-believe that he dis-believes the propo-sition! Conversely, whatever a mad person believes that he believes must be true. (Also, of course, whatever a sane person believes that he believes must be true.) Thus, whatever any inhabitant, mad or sane, believes that he believes must be true. Also, if an inhabitant believes that he doesn’t believe a certain proposition, then the propo-sition must be false. (This is obvious for a sane inhabitant, but if the inhabitant is mad, then it is false that he doesn’t believe the proposi-tion (since he erroneously believes that he doesn’t believe it), which means that he does believe it, and hence it is false.)
Let us record two of the things we have just learned:
Fact 1. When an inhabitant believes that he believes something (whatever that something is), that something must be true.
28 I. Be Wise, Generalize!
Fact 2. When an inhabitant believes that he doesn’t believe some-thing, that something must be false.
4.2. For the Island of Knights and Knaves of Chapter 2, to find out if an inhabitant is married, you can ask: “Are you a knight if and only if you are married?” For the island of the present chapter, you can ask instead: “Are you sane if and only if you are married?” Also, the question “Are you the type who can claim that you are married?”
would work for this island as well as the islands of Chapters 2 and 3. For the present island, however, a much more economical and natural-sounding question is possible. All you need ask is “Do you believe you are married?” If he answers yes, then he believes that he believes he is married, hence he really is married (by Fact 1, stated and proved in the solution of the last problem). If he answers no, then he believes that he doesn’t believe that he is married, hence he is not married (by Fact 2).
Thus, in general, to find out if something is the case, you ask the inhabitant if he believes that the something is the case. For example, if you want to know if there is gold on the island, all you need to ask is “Do you believe there is gold on the island?”
Neat, eh?
4.3. Suppose Henry is sane. Then his statement is true; hence Maxwell really does believe that at least one of the three is mad. If Maxwell were mad, then it would be true that at least one is mad, and so mad Maxwell would have a true belief, which is not possible. Hence, Maxwell is sane (still under the assumption that Henry is sane).
Then, also, Dianne is sane (since she correctly believes that Maxwell is sane); hence all three are sane, contrary to Maxwell’s sane belief that at least one is mad! Thus, it is contradictory to assume that Henry is sane. Thus Henry is mad.
Since Henry is mad, what he said is false, so Maxwell doesn’t re-ally believe that at least one of the three is mad; he believes that all three are sane. But his belief is wrong (since Henry is mad), and so, Maxwell is mad. Hence Dianne is also mad (since she believes that Maxwell is sane). Thus, all three are mad.
4.4. Step 1. Gerald and Mary are alike, as far as their sanity goes.
Reason. Mary believes that Gerald is sane. If Mary is sane, her belief is correct, hence Gerald is also sane. If Mary is mad, her belief is wrong, which means that Gerald is not sane, but mad.
Step 2. Lenore must be mad.
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Reason. Suppose Lenore were sane. Then her statement would be true, hence Gerald did once say that exactly one of the three is sane, but this leads to a contradiction because:
(1) If Gerald is sane, so is Mary (by Step 1); hence all three are sane, so it is false that exactly one is sane, but sane people here don’t make false statements.
(2) On the other hand, if Gerald is mad, then so is Mary (Step 1) and Lenore is then the only sane one, so it is true that exactly one is sane, but mad inhabitants don’t make true statements.
Thus Lenore can’t be sane: she is mad.
Step 3. Suppose Gerald is mad. Then, so is Mary (by Step 1); hence all three are mad. Then Mary, who is mad, never did make the true statement that all three are mad; hence Gerald was right when he denied that Mary did, but mad people here don’t make true state-ments! Thus, it is contradictory to assume that Gerald is mad. Hence Gerald is sane and so is his wife (by Step 1). Thus, the mother and father are both sane, but their daughter Lenore is mad.
4.5. Suppose Arthur is sane. Then Lillian did once claim that Arthur believes that Lillian believes that Arthur is mad. Suppose Lillian is sane. Then Arthur believes that Lillian believes that Arthur is mad. Since Arthur is sane, then Lillian does believe that Arthur is mad, and since Lillian is sane, then Arthur is mad, contrary to the assumption that Arthur is sane.
Suppose Lillian is mad. Then Arthur doesn’t really believe that Lil-lian believes that Arthur is mad. Since Arthur is sane (by assump-tion), then it is false that Lillian believes that Arthur is mad. But Lillian is mad, and since she doesn’t believe that Arthur is mad, then Arthur is mad, again contrary to the assumption that Arthur is sane.
Thus, Arthur must be mad. Hence Lillian never did say what Arthur said she said, and so nothing can be deduced about Lillian.
4.6. I will prove that Peter must be sane.
Step 1. Arthur and David cannot both be mad.
Reason. Suppose David is mad. Then, contrary to what he said, Bernard and Charles are both mad. Since Charles is mad, then, con-trary to what he said, Arthur must be sane. This proves that if David is mad, Arthur is sane; hence Arthur and David cannot both be mad.
Step 2. Frank and Harold cannot both be mad.
30 I. Be Wise, Generalize!
Reason. Suppose Harold is mad. Then Ernest must be sane; hence Arthur and David are really alike, as far as their sanity goes. But Arthur and David are not both mad (as we proved in Step 1), so they are both sane. Hence, Frank’s statement was true, so Frank is sane.
This proves that if Harold is mad, then Frank is sane, so Harold and Frank are not both mad.
Step 3. Therefore, Peter’s statement was true, so Peter is sane.
4.7. Since we know that the defendant didn’t answer no both times, there are only three cases to consider.
Case 1: He Answered Yes Both Times. It could be that he is sane and stole the watch and did once claim that he stole it. But it is also possible that he is mad and never stole it and also never claimed that he did. The judge would then have no way of knowing whether the defendant was innocent or guilty.
Case 2: His First Answer Was No and His Second Was Yes. Then it could be that he is sane and guilty but never claimed that he had stolen the watch, but it is also possible that he is mad and never stole the watch, but once claimed that he had.
Again, the judge would have no way of knowing which of these possibilities held.
Case 3: His First Answer Was Yes and His Second Was No. In this case, could he be sane? No, because then he would be innocent of the theft (by virtue of his second answer) but also would have once claimed that he had stolen the watch, which would be a false claim, hence not possible for a sane inhabitant. Therefore, he must be mad, hence he did steal the watch (since he indicated that he hadn’t) but also never claimed that he had stolen it (since he indicated that he did make such a claim). This is the only possibility, and the judge would then know that the defendant was guilty.
Since the judge did know, Cases 1 and 2 are out; hence it must be Case 3 that actually held. Thus the defendant was guilty (and also mad).