1
The above argument is fallacious! Can the reader spot the fallacy? (Hint: Despite the fact that the reasoner knows propositional logic as well as you and I, there is something we know that the reasoner doesn’t know. What is it?)
Solution. I told you that the reasoner is always accurate; I never said that he knew he was accurate! If he knew he was accurate (which in fact he can’t know), we would get a paradox. You see, part of our proof that the native is a knight used the assumption that the reasoner is always accurate; if the reasoner made the same assumption, then he could likewise prove that the native is a knight, thus making the native a knave.
Let us now retract the assumption that the reasoner is always accurate in his judgments, but let us suppose that the reasoner believes that he is always accurate. Thus for any proposition p, the reasoner believes that if he should ever believe p, then p must be true. Such a reasoner we will call a conceited reasoner. Thus a conceited reasoner is one who believes that he is incapable of believing any false proposition.
And so we retract the assumption that the reasoner is always accurate and replace it with the assumption that the reasoner believes that he is always accurate. Do we then get a paradox? No, we don’t; we instead have the following more interesting result.
Theorem 1. Suppose a native of a knight-knave island says to a reasoner of type 1: “You will never believe I’m a knight.” Then if the reasoner believes himself always accurate, he will lapse into an inaccuracy—i.e., he will sooner or later believe something false.
2
Prove Theorem 1.
Solution. The reasoner reasons: “Suppose he is a knave. Then his statement is false, which means that I will believe he is a knight. But if I ever believe he is a knight, he must really be one, because I am not capable of making mistakes [sic!]. So if he is a knave, he is a knight, which is not possible. Therefore he is not a knave; he is a knight.”
At this point, the reasoner believes the native is a knight. Since the native said that the reasoner would never believe that, then the native is in fact a knave. So the reasoner now has the false belief that the native is a knight.
The interesting thing is that if the reasoner had been more modest and had not assumed his own infallibility, he would never have been driven into the inaccuracy of believing the native a knight. The reasoner has been justly punished for his conceit!
PECULIAR REASONERS
Let us say that a reasoner is accurate with respect to a given proposition p if the reasoner’s believing p implies that p is true; in other words, if it is either not the case that he believes p, or it is the case that he believes p and p is true. We will say that the reasoner is inaccurate with respect to p if he believes p and p is false.
It is a noteworthy fact about the last problem that the reasoner was inaccurate with respect to the very proposition about which he believed himself to be accurate—namely, the proposition that the native is a knight. By believing that he was accurate with respect to that proposition, he finally came to believe that the native was a knight, thus making the native a knave. Of course our proof that the native is a knave rested on our assumption that the rules of the island held. Suppose we retract this assumption but continue to assume that the reasoner believes that the rules of the island hold; does it still follow that the reasoner will believe some false proposition? Of course the reasoner will still believe that the native is a knight, although the native said he never would; but if the rules of the island don’t hold, then the native is not necessarily a knave. However, even though our proof of Theorem 1 fails if we retract the assumption that the rules of the island hold, we have the following more startling proposition:
Theorem 2. Suppose an inhabitant of the island says to a reasoner of type 1: “You will never believe that I’m a knight.” Suppose the reasoner believes that the rules of the island hold. Then regardless of whether the rules really hold or not, if the reasoner is conceited, he will come to believe some false proposition.
3
Under the assumption of Theorem 2, what false proposition will the reasoner believe?
Solution. By the same method of proof we followed for Theorem 1, the reasoner will come to believe that the native is a knight. This belief is not necessarily false (since the rules of the island don’t necessarily hold), but then the reasoner continues: “Since he is a knight and he said I would never believe he is a knight, then what he said must be true—namely, that I don’t believe (i.e., it’s not the case that I believe) he’s a knight. So I don’t believe he is a knight.”
At this point, the reasoner believes that the native is a knight and also believes that he doesn’t believe that the native is a knight. So he has the false belief that he doesn’t believe the native is a knight (it’s false, since he does believe that the native is a knight!).
The conclusion of the above problem is really quite weird. The reasoner believes that the native is a knight and also believes that he doesn’t believe that the native is a knight. Now, this does not involve a logical inconsistency on the part of the reasoner, though it certainly does involve a psychological peculiarity. We shall call a reasoner peculiar if there is some proposition p such that he believes p and also believes that he doesn’t believe p. This condition of course implies that the reasoner is inaccurate (because he believes the false proposition that he doesn’t believe p). And so a peculiar reasoner is automatically inaccurate, but not necessarily inconsistent.
Let us say that a reasoner is peculiar with respect to a given proposition p if he believes p and also believes that he doesn’t believe p. A reasoner is then peculiar if and only if there is at least one proposition p with respect to which he is peculiar. If a reasoner is peculiar with respect to p, then he is inaccurate, not necessarily with respect to p, but with respect to Bp.
We now see that even if we remove the assumption that the rules of the island actually hold, if the reasoner believes that they hold and he is of type 1 and a native tells him that he will never believe the native is a knight, then if the reasoner believes that he is accurate with respect to the proposition that the native is a knight, his belief forces him to be inaccurate with respect to the proposition that he believes the native is a knight. If the rules of the island do actually hold, then the reasoner will also be inaccurate with respect to the proposition that the native is a knight—i.e., he will believe that the native is a knight, whereas the native is really a knave.
Of course this same problem can be formulated in the context of the student and his theology professor. Suppose the student is a reasoner of type 1 and his professor says to him: “God exists if and only if you will never believe that He does.” Suppose also that the student believes that if he ever believes that God exists, then God does exist. This will mean: (1) If the student believes the professor, then he will wind up believing that God exists and also believing that he doesn’t believe that God exists. (2) If the professor’s statement is also true, then God doesn’t exist, but the student will believe that God does exist.
Both versions of this problem are special cases of the following theorem.
Theorem A. Suppose a reasoner is of type 1 and that there is a proposition p such that he believes the proposition p≡~Bp and also believes that Bp⊃p. Then it follows that:
(a) He will believe p and also believe that he doesn’t believe p (he will be peculiar with respect to p).
(b) If also p≡~Bp is true, then p is false, but he will believe p.
Proof. (a) He believes p≡~Bp, hence he believes Bp⊃~p (which is a logical consequence of p≡~Bp). He also believes Bp⊃p (by hypothesis), hence he must believe ~Bp (which is a logical consequence of Bp⊃~p and Bp⊃p). But he also believes p≡~Bp, hence he must believe p (which is a logical consequence of ~Bp and p≡~Bp). And so he believes p and he also believes ~Bp (he believes that he doesn’t believe p!).
(b) Suppose also that p≡~Bp is true. Since ~Bp is false (he does believe p!), then p, being equivalent to the false pro
position ~Bp, is also false. Therefore he believes p, but p is false.
Exercise. Suppose a native says to a conceited reasoner of type 1: “You will believe that I am a knave.” Prove: (a) If the reasoner believes that the rules of the island hold, then he will believe that the native is a knave and also that he doesn’t believe that the native is a knave. (b) If also the rules of the island really hold, then the native is in fact a knight.
REASONERS OF TYPE 1*
By a reasoner of type 1*, we shall mean a reasoner of type 1 with the added property that for any propositions p and q, if he ever believes the proposition p⊃q, then he will believe that if he ever believes p then he will also believe q. In symbols, if he ever believes p⊃q, then he will also believe Bp⊃Bq.
Let us note that if a reasoner of type 1 does believe p⊃q, then it is true that if he ever believes p, then he will believe q—i.e., Bp⊃Bq is a true proposition (if he believes p⊃q). What a reasoner of type 1* has that a reasoner of just type 1 doesn’t have is that if he believes p⊃q, then not only is the proposition Bp⊃Bq a true one, but he correctly believes Bp⊃Bq. Thus a reasoner of type 1* has a shade more “self-awareness” than a reasoner who is only type 1.
We continue to assume that the reasoner, who is now of type 1*, believes the rules of the island and hears all statements made to him, and so whenever the native asserts a proposition p, the reasoner believes the proposition k≡p, where k is the proposition that the native is a knight.
The following fact will be quite crucial:
Lemma 1.† Suppose the native asserts a statement to a reasoner of type 1*. Then the reasoner will believe that if he ever believes that the native is a knight, he will also believe what the native said.
Problem 4. How is this lemma proved?
Solution. (The proof is really quite simple!) Suppose the native asserts the proposition p to the reasoner. Then the reasoner believes the proposition k≡p. Then he also believes k⊃p, because k⊃p is a logical consequence of k≡p. Then, since he is of type 1*, he will believe Bk⊃Bp.
I have called a reasoner “conceited” if he believes in his own infallibility. I would hardly regard a person’s belief that he is not peculiar as an act of conceit; indeed, to assume that one is not peculiar is a perfectly reasonable assumption. I’m not even sure whether it is psychologically possible for a person to be peculiar. Could a person really believe something and also believe that he doesn’t believe it? I doubt it. Yet it is not logically impossible for a person to be peculiar.
At any rate, to have confidence in one’s own nonpeculiarity is far more reasonable than to have confidence in one’s complete accuracy. Therefore the following theorem is a bit sad.
Theorem 3. Suppose a native says to a reasoner of type 1*: “You will never believe that I am a knight.” Then if the reasoner believes that he is not (and never will be) peculiar, he will become peculiar!
Problem 5. Prove Theorem 3.
Solution. The proof of this is a bit more elaborate than any other proof so far.
Assume that the native makes this statement and that the reasoner believes that he is incapable of being peculiar. Since the native made the statement, then by Lemma 1, the reasoner will believe that if he ever believes that the native is a knight, he will also believe what the native said. And so the reasoner reasons: “Suppose I should ever believe that he is a knight. Then I’ll believe what he says—i.e., I’ll believe that I don’t believe he is a knight. And so I will then believe he’s a knight and I will also believe that I don’t believe he is a knight. This means I will be peculiar. Therefore, if I ever believe he’s a knight, I will become peculiar. Since I will never be peculiar [sic], then I will never believe that he’s a knight. Since he said I wouldn’t, his statement is true, and so he is a knight.”
At this point the reasoner has come to the conclusion that the native is a knight, and a bit earlier he came to the conclusion that he doesn’t believe that the native is a knight. He has thus lapsed into peculiarity.
Of course the above proposition can be stated and proved in the following more general form:
Theorem B. For any reasoner of type 1*, if he believes any proposition of the form p≡~Bp (“p is true if and only if I will never believe p”), then he cannot believe that he is not peculiar, unless he lapses into peculiarity.
Moral. If you are a reasoner of type 1*, and you wish to believe that you are not peculiar, you can avoid becoming peculiar by simply refusing to believe any proposition of the form “p if and only if I will never believe p.”
In particular, if you ever visit the knight-knave island (or what you have been told is a knight-knave island) and a native tells you that you will never believe that he is a knight, then your wisest course is to refuse to believe that the rules of the island hold.
Later in this book, however, when we come to the study of mathematical systems and talk about provability in the system rather than beliefs of a reasoner, we will see that the analogue of the option of not believing p≡~Bp will not be open.
* * *
† A lemma is a proposition proved not so much for its own sake as for help in proving subsequent theorems. You might say that a lemma is a proposition that is not “dignified enough” to be called a theorem.
• Part V •
THE CONSISTENCY PREDICAMENT
• 11 •
Logicians Who Reason About Themselves
WE ARE getting close to Gödel’s consistency predicament. But first, we need to consider reasoners of higher degrees of self-awareness than those of just type 1.
ADVANCING STAGES OF SELF-AWARENESS
We will now define reasoners of types 2, 3, and 4, which represent advancing degrees of self-awareness. Reasoners of type 4 play a major role in the dramas that will unfold.
Reasoners of Type 2. Suppose a reasoner of type 1 believes p and believes p⊃q. Then he will believe q. This means that the proposition (Bp&B(p⊃q))⊃Bq is true for a reasoner of type 1. However, the reasoner doesn’t necessarily know that this proposition is true. Well, we define a reasoner to be of type 2 if he is of type 1 and believes all propositions of the form (Bp&B(p⊃q))⊃Bq. (He “knows” that his set of beliefs—past, present, and future—is closed under modus ponens. For any propositions p and q, he believes: “If I should ever believe p and believe p⊃q, then I will also believe q.”)
To emphasize a point, reasoners of type 2 have a certain “self-awareness” not necessarily present in reasoners of type 1. A reasoner of type 1 who believes p and believes p⊃q will sooner or later believe q; reasoners of type 2 also know that if they ever believe p and believe p⊃q, they will believe q.
Reasoners of Type 3. We will say that a reasoner is normal if for any proposition p, if he believes p, then he believes that he believes p. (If he believes p, then he also believes Bp.) By a reasoner of type 3, we shall mean a normal reasoner of type 2.
Reasoners of type 3 have one more stage of self-awareness than those of type 2.
Reasoners of Type 4. A normal reasoner doesn’t necessarily know that he is normal. If a reasoner is normal, then for any proposition p, the proposition Bp⊃BBp is true (if he believes p, then he believes Bp), but the reasoner is not necessarily aware of the truth of Bp⊃BBp. Well, we will say that a reasoner believes that he is normal if for every proposition p, he believes the proposition Bp⊃BBp. (For every proposition p, the reasoner believes: “If I should ever believe p, then I will believe that I believe p.”)
A reasoner of type 3 is in fact normal. By a reasoner of type 4, we mean a reasoner of type 3 who knows that he is normal. Thus for any proposition p, a reasoner of type 4 believes the proposition Bp⊃BBp.
As we have remarked, reasoners of type 4 play a major role in this book. Let us review the conditions defining a reasoner of type 4.
(1a) He believes all tautologies.
(1b) If he believes p and believes p⊃q, then he believes q.
(2) He believes (Bp&B(p⊃q))⊃Bq.
(3) If he believes
p, then he believes Bp.
(4) He believes Bp⊃BBp.
SOME BASIC PROPERTIES OF SELF-AWARE REASONERS
We will now establish a few basic properties of reasoners of types 2, 3, and 4 that will be used throughout the remaining chapters.
First, a simple observation about reasoners of type 2: For any proposition p and q, the proposition (Bp&B(p⊃q))⊃Bq is logically equivalent to the proposition B(p⊃q)⊃(Bp⊃Bq)—because for any propositions X, Y, and Z, the proposition (X&Y)⊃Z is logically equivalent to Y⊃(X⊃Z), as the reader can easily verify—and so any reasoner of type 2 believes all propositions of the form B(p⊃q)⊃(Bp⊃Bq). Conversely, any reasoner of type 1 who believes all propositions of the form B(p⊃q)⊃(Bp⊃Bq) must be of type 2. Let us record this as Fact 1.
Fact 1. A reasoner of type 1 is of type 2 if and only if he believes all propositions of the form B(p⊃q)⊃(Bp⊃Bq).
Suppose now a reasoner of type 2 believes B(p⊃q). He also believes B(p⊃q)⊃(Bp⊃Bq), according to Fact 1, and being of type 1 (since he is of type 2) he will then believe Bp⊃Bq—which is a logical consequence of B(p⊃q) and B(p⊃q)⊃(Bp⊃Bq). And so as an obvious consequence of Fact 1 we have the following corollary: If a reasoner of type 2 believes B(p⊃q), then he will believe Bp⊃Bq.
Now we come to some less obvious facts about reasoners of type 2.
1
Show that for any reasoner of type 2 and any propositions p, q, and r:
Forever Undecided Page 7