Forever Undecided
Page 11
Of course the above argument generalizes as follows:
Theorem 2. Given any propositions k and C, if a reasoner of type 4 believes BC⊃C and believes k≡(Bk⊃BC), then he will believe C.
3
Again a reasoner of type 4 has the prior belief that if he believes that the cure will work, then the cure will work. This time the native says to him: “Sooner or later you will believe that if I am a knight, then the cure will work.” We will see that again the reasoner will believe that the cure will work.
More generally, we will prove the following theorem.
Theorem 3. For any propositions k and C, if a reasoner of type 4 believes BC⊃C and believes k≡B(k⊃C), then he will believe C.
The proof of Theorem 3 is facilitated by the following two lemmas, which are of interest in their own right.
Lemma 2. Suppose that for some proposition q, a native says to a reasoner of type 4: “You will believe q.” Then the reasoner will believe: “If he is a knight, then I will believe that if he is a knight, I will believe that he is a knight.” (Stated more abstractly, for any propositions k and q, if a reasoner of type 4 believes k≡Bq, then he will believe k⊃Bk.)
Lemma 3. Given any proposition p, suppose a native says to a reasoner of type 4: “You will believe that if I am a knight, then p is true.” Then the reasoner will believe: “If he is a knight, then I will believe p.” (Stated more abstractly, if a reasoner of type 4 believes k≡B(k⊃p), then he will believe k⊃Bp.)
How are Lemmas 2 and 3 and Theorem 3 proved?
Proof of Lemma 2. This is Problem 9 of Chapter 11, which we have already solved, but I wish to give a knight-knave version of the proof, which is particularly intuitive. The native has said: “You will believe q.” The reasoner then reasons: “Suppose he is a knight. Then I will believe q. Then I’ll believe that I believe q, hence I’ll believe what he said, hence I’ll believe he’s a knight. Therefore, if he is a knight, then I’ll believe he’s a knight.”
Proof of Lemma 3. We shall use Lemma 2 to facilitate this proof.
The native has said: “You will believe that if I’m a knight, then p.” Let q be the proposition “If I’m a knight, then p.” The native has told the reasoner that he will believe q, and so by Lemma 2, the reasoner will believe that if the native is a knight, then he (the reasoner) will believe that the native is a knight. And so the reasoner reasons: “Suppose he is a knight. Then I’ll believe that he is a knight. Then I’ll believe what he says—I’ll believe Bk⊃p. I’ll also believe Bk (I’ll believe that I believe he is a knight). Therefore, if he is a knight, then I’ll believe Bk⊃p and I’ll believe Bk, hence I will also believe p. And so if he is a knight, then I will believe p.”
Proof of Theorem 3. I will give a knight-knave version of the proof. The native has said: “You will believe that if I am a knight, then the cure will work.” By Lemma 3, the reasoner will believe: “If he is a knight, then I will believe that the cure will work.” The reasoner then continues: “Also, if I believe that the cure will work, then the cure will work. Therefore, if he is a knight, the cure will work. I now believe that if he is a knight, then the cure will work. He said I would believe that, hence he is a knight. And so he is a knight, and in addition (as I have proved), if he is a knight, then the cure will work. Therefore the cure will work.”
At this point the reasoner will believe that the cure will work.
Discussion. One can also obtain Theorem 3 as an easy corollary of Theorem 1 by the following argument. Suppose we are given propositions k and C such that a reasoner of type 4 believes k≡B(k⊃C) and believes BC⊃C. We are to show that he will believe C. Since he believes k≡B(k⊃C), he must also believe (k⊃C)≡(B(k⊃C)⊃C)), which is a logical consequence of k≡B(k⊃C)). Then he believes the proposition k′≡(Bk′⊃C), where k′ is the proposition k⊃C. Since he also believes BC⊃C, then he will believe C by Theorem 1.
The following curious exercise illustrates self-reference carried to its extreme.
Exercise 4. Suppose a native of the island says to a reasoner of type 4: “Sooner or later you will believe that if I am a knight, then you will believe that I am one.”
(a) Prove that the reasoner will believe that the native is a knight.
(b) Prove that if the rules of the island really hold, then the native is a knight.
Solution. This is an easy consequence of Lemma 2.
(a) The native has asserted B(k⊃Bk). Thus there is a proposition q—namely, k⊃Bk—such that the native has claimed that the reasoner will believe q. Then by Lemma 2, the reasoner will believe k⊃Bk. Then, since he is normal, he will believe B(k⊃Bk)—he will believe what the native said. Hence he will believe that the native is a knight.
(b) Since the reasoner believes Bk, he certainly believes k⊃Bk, hence the native’s statement was true. And so the native is a knight (if the rules of the island really hold).
The essential mathematical content of the above exercise is that for any proposition k, if a reasoner of type 4 believes k≡B(k⊃Bk), then he will also believe k. If also k≡B(k⊃Bk) is true, so is k.
DUAL FORMS
Problems 1, 2, and 3 (more generally, Theorems 1, 2, and 3) have their “dual” forms, which are rather curious.
1°. (Dual of Problem 1)
Again, a reasoner of type 4 comes to the island already believing that if he believes that the cure will work, then the cure will work. He meets a native who says to him: “The cure will not work and you will believe that I am a knave.”
Prove that the reasoner will believe that the cure will work.
2°. (Dual of Problem 2)
Like Problem 1°, except that the native says: “You will believe that I’m a knave, but you will never believe that the cure will work.” Show that the same conclusion follows (the reasoner will believe that the cure will work).
3°. (Dual of Problem 3)
This time the native says: “You will never believe that if I am a knave, then the cure will work.” (Alternatively, he could say: “You will never believe that either I am a knight or that the cure will work.”) Show that the same conclusion follows.
Solution to Problem 1°. We could prove this from scratch, but it is easier to take advantage of Theorem 1, which we have already proved.
The native has asserted (~C&B~k), and so the reasoner believes k≡(~C&B~k). But we know k≡(~C&B~k) is logically equivalent to ~k≡~(~C&B~k), which in turn is logically equivalent to ~k≡(B~k⊃C). Therefore the reasoner believes ~k≡(B~k⊃C), and so he believes a proposition of the form p≡(Bp⊃C)—p is the proposition ~k—and so, by Theorem 1, if he believes BC⊃C, he will believe C.
The solutions of Problems 2° and 3° can likewise be obtained as corollaries of Theorems 2 and 3, respectively. We leave the verification to the reader.
Exercise 5. Suppose a native says to a reasoner of type 4: “If you ever believe I’m a knight, then you will be inconsistent.” Is it possible for the reasoner to believe in his own consistency without becoming inconsistent? (Hint: Use Theorem 2.)
Exercise 6. Suppose a reasoner of type 4 believes that if he believes the cure will work, then it will work. Suppose we now have a native who says to him: “If you ever believe that you will believe I’m a knight, then the cure will work.”
Will the reasoner necessarily believe that the cure will work?
Exercise 7. Suppose the native instead says: “You will believe that if you ever believe I’m a knight, then the cure will work.”
Will the reasoner necessarily believe that the cure will work?
Exercise 8. The following dialogue ensues between a student and his theology professor:
STUDENT: If I believe that God exists, then will I also believe that I will be saved?
PROFESSOR: If that is true, then God exists.
STUDENT: If I believe that God exists, then will I be saved?
PROFESSOR: If God exists, then that is true.
Prove that if the professor is accurate and if the student believes
the professor, then God must exist and the student will be saved.
Exercise 9. The following strengthening of Theorem 3 can be proved. A reasoner of type 4 comes to the island for the cure and has the prior belief that if he should ever believe that the cure will work, then it will. He asks a native: “Will I ever believe that if you are a knight, then the cure will work?” The native replies: “If that is not so, then the cure will work.” (Alternatively, he could have replied: “Either that is so or the cure will work.”) The problem is to prove that the reasoner will believe that the cure will work. (Stated more abstractly, if a reasoner of type 4 believes k≡(CvB(k⊃C)) and believes BC⊃C, then he will believe C.) The proof of this is facilitated by first proving the following two facts as lemmas:
(1) For any propositions p and q, suppose a native says to a reasoner of type 4: “Either p is true or you will believe q.” Then the reasoner will believe: “If the native is a knight, then I will believe that either p is true, or that the native is a knight.”
(2) For any propositions p and q, suppose a native says to a reasoner of type 4: “Either p is true or else you will believe that if I am a knight, then q is true.” The reasoner will then believe: “If the native is a knight, then either p is true or I will believe that q is true.”
Exercise 10. The last exercise has the following dual. Again, a reasoner of type 4 believes that if he should believe that the cure will work, then it will. He now meets a native who says: “The cure doesn’t work and you will never believe that either I’m a knight or that the cure works.” Prove that the reasoner will believe that the cure will work.
• 16 •
The Rajah’s Diamond
THOSE OF you who have read the magnificent story “The Rajah’s Diamond,” by Robert Louis Stevenson, will recall that at the end, the diamond was thrown into the Thames. Recent research, however, has revealed that the diamond was subsequently found by an inhabitant of the Island of Knights and Knaves who was vacationing in England at the time. One rumor has it that he then took the diamond to Paris and died shortly after. According to another, he took the diamond back home. If the second version is correct, then the diamond is somewhere on the knight-knave island.
A reasoner of type 4 decided to follow up on the second rumor in hopes of finding the diamond. He reached the island and believed the rules of the island. Also, the rules of the island really held. There are five different versions of what actually happened; each is of interest, and so I will relate them all.
1. The First Version
According to the first version, when the reasoner reached the island, he met a native who made the following two statements:
(1) “If you ever believe that I am a knight, then you will believe that the diamond is on this island.”
(2) “If you ever believe that I am a knight, then the diamond is on this island.”
If this version is the correct one, is the diamond on the island?
2. The Second Version
According to a second, slightly different version, the native, instead of making the two statements reported above, made the following two statements:
(1) “If I am a knave and if you ever believe that I’m a knight, then you will believe that the diamond is on this island.”
(2) “I am actually a knight, and if you ever believe this, then the diamond is on this island.”
If this second version is correct, is there sufficient evidence to conclude that the diamond must be on the island?
3. The Third Version
The third version is particularly curious. According to it, the native made the following two statements:
(1) “If you ever believe that I’m a knight, then the diamond is not on this island.”
(2) “If you ever believe that the diamond is on this island, then you will become inconsistent.”
If this third version is correct, what conclusion should be drawn?
4. The Fourth Version
According to the fourth version, the native made only one statement:
(1) “If you ever believe I’m a knight, then you will believe that the diamond is on this island.”
The reasoner, of course, could get nowhere. He then discussed the whole affair with the Island Sage, who was known to be a knight of the highest integrity. The Sage made the following statement:
“If the native you spoke to is a knight and if you ever believe that the diamond is on this island, then the diamond is on this island.”
If this fourth version is correct, what conclusion should be drawn?
5. The Fifth Version
This is like the last version, except that the native said:
(1) “You will believe that if I am a knight, then the diamond is on this island.”
The Sage made the same statement.
Assuming that these five versions are equally probable, what is the probability that the Rajah’s diamond is on the Island of Knights and Knaves?
Remarks. The mathematical content of the last two problems constitutes strengthenings of Theorems 2 and 3 of the last chapter—see the discussion following the solutions.
SOLUTIONS
We let k be the proposition that the native is a knight. We let D be the proposition that the diamond is on the island.
1. Since the native made the two statements which he made, then the reasoner will believe the following two propositions:
(1) k≡(Bk⊃BD)
(2) k≡(Bk⊃D)
And the reasoner will therefore certainly believe the following two weaker propositions:
(1)′ (Bk⊃BD)⊃k
(2)′ k⊃(Bk⊃D)
As we will see, the fact that the reasoner believes even (1)′ and (2)′ is enough to solve the problem.
Since he believes (2)′, then by Lemma 1 of the last chapter, he will believe Bk⊃BD. Believing this and believing (1)′, he will then believe k. Believing k and believing (2)′, he will then believe Bk⊃D. Also, since he believes k, he will believe Bk, and hence he will believe D. Therefore BD is true. Hence Bk⊃BD is true, and since (1) is true (the rules of the island really hold), then k is true (the native is really a knight). Then since (2) is true, the proposition Bk⊃D is true. Also Bk is true (we have seen that the reasoner will believe k), and thus D is true. Therefore the diamond is on the island.
2. According to this version, it does not follow from the native’s two statements that the reasoner will believe the propositions (1) and (2) of the solution to the last problem, but it does follow that he will believe the weaker propositions (1)′ and (2)′ (see the note below). But, as we have seen in the solution of the last problem, this is enough to guarantee that the diamond is on the island.
Note: If a native of a knight-knave island says: “If I am a knave, then X,” it logically follows that if X is true, the native must be a knight (because if X is true, then any proposition implies X, hence it is true that if the native is a knave, then X, but a knave couldn’t make such a true statement). This is why the reasoner (who believes the rules of the island) will believe (1)′. As for (2)′, it is obvious that if a native says, “I am a knight and X,” it follows that if the native is a knight, then X must be true.
3. Step 1: The reasoner believes k≡(Bk⊃~D)—by virtue of the native’s first statement. Then by Lemma 1 of the last chapter, the reasoner will believe Bk⊃B~D. And so the reasoner reasons: “If I ever believe that he is a knight, then I will believe that the diamond is not on the island. If I should also believe that the diamond is on the island, then I will be inconsistent. Therefore, if I should ever believe he is a knight, then his second statement is true. And, of course, if his second statement is true, then he is a knight. This proves that if I should ever believe that he is a knight, then he really is a knight.”
Step 2: The reasoner continues: “So suppose I believe he’s a knight. Then he really is a knight, as I have just shown, hence his first statement Bk⊃~D is true. Also, if I believe he’s a knight, then Bk is true, and thus ~D is true. Therefore, if I
believe he’s a knight, then the diamond is not on the island. He said just this in his first statement, and so he is a knight.”
Step 3: The reasoner continues: “Now I believe he is a knight and I have already shown that Bk⊃~D, hence ~D is true—the diamond is not on this island.”
Step 4: The reasoner now believes that the diamond is not on the island. Therefore, if he should ever believe that the diamond is on the island, he will be inconsistent. This proves that the native’s second statement was true and therefore the native is in fact a knight. And so the native’s first statement was also true—Bk⊃~D is true. But Bk is true (as we have proved), and so ~D is true. Therefore the diamond is not on the island.
4. Step 1: The native asserted Bk⊃BD, and so the reasoner believes k≡(Bk⊃BD). Then by Lemma 1 of the last chapter (taking BD for p), the reasoner will believe Bk⊃BBD, and so the reasoner reasons: “Suppose I ever believe that the native is a knight. Then I will believe BD. Then I will believe k and I will believe BD, hence I will believe k&BD. I also believe the Sage’s statement (k&BD)⊃D, so if I ever believe k&BD, then I will believe D. Therefore, if I ever believe k, I will also believe D—Bk⊃BD is true. This is what the native said, hence he is a knight.”
Step 2: The reasoner continues: “I now believe k—Bk is true. Also Bk⊃BD is true (as I have proved), hence BD is true (I will believe the diamond is on the island). Thus k is true and BD is true, so k&BD is true. Then, by the Sage’s statement, D must be true—the diamond is on this island.”
Step 3: The reasoner now believes D, so BD is true. Then Bk⊃BD is certainly true, hence the native is really a knight. Thus k and BD are both true, so k&BD is true. Hence, by the Sage’s statement, D must be true—the diamond is on the island.