An Analysis of the “Sudden Drill Paradox”

29,908 characters2006.06.24

An Analysis of the “Sudden Drill Paradox”

Abstract:
The “sudden drill paradox,” also known as the “unexpected exam paradox,” the “unexpected hanging paradox,” and so on, is briefly sorted through and analyzed here in nontechnical language. The inferential premises and steps that may lead to the paradox are reexamined, and an attempt is made to offer a way of understanding the paradox.

Keywords: sudden drill paradox exam paradox epistemic paradox contradiction

Origins

What was called the “sudden drill problem” was a difficult question that had been circulating for several years at the time. During the Second World War, the Swedish Broadcasting Corporation aired an announcement:

Within the coming week, an air-raid drill will be held. In order to verify whether preparations for war are adequate, no one has been told in advance the exact day of the drill, and therefore this is a sudden drill.

The Swedish mathematician Aikebo noticed that this announcement had a strange property: given the conditions stated in the announcement, the drill could not be held on next Sunday, because then the fact that the drill would occur on Sunday would already be known in advance, and thus it would not be sudden; therefore Sunday is ruled out. By the same token, Saturday can also be ruled out: since the drill has already been determined not to take place on Sunday, then among the remaining six days, if it were held on Saturday it would still not be sudden. Continuing in this way, the same reasoning can be used to eliminate Friday, Thursday, all the way back to Monday. On this basis, Aikebo inferred that a sudden drill satisfying the conditions of the announcement was impossible.

Yet in the early hours of Wednesday of the second week, the air-raid siren sounded and the drill was “suddenly” held……[①]

This paradox has many different variants, such as the “hanging paradox” and the “exam paradox.” Quine, Shao, Montague, Kaplan, and other logicians have all discussed and studied it in depth. Montague and Kaplan pointed out that, on the basis of an understanding of the assumptions already derived for this paradox and the related “surprise examination paradox,” or what we call “background knowledge elements,” “they can attain a status comparable to that of the liar paradox and Richard’s paradox, and can, like them, lead to important technical advances.”[②]

However, although the logicians’ analyses of this problem are certainly profound, they are often excessively technical. This is necessary for rigor in argument, of course, but it also makes them difficult for non-specialists to grasp. Their analyses reveal the paradox’s importance for logic, but do not show what it actually means for our everyday ways of thinking. Therefore, I will here offer some further sorting-through and analysis of the paradox in a more intuitive, and not overly technical, language—hopefully these efforts still have some value.

I. Where Did Things Go Wrong?

Since a paradox has appeared, the first thing to do, of course, is to reflect on the premises and process of the entire argument. If it turns out that the whole chain of reasoning is defective and we can locate where the problem lies, then the paradox will be dissolved as well.

So, does Aikebo’s reasoning contain a contradiction?

The drill cannot be held next Sunday, because then the fact that the drill would occur on Sunday would already be known in advance, and thus it would not be sudden; therefore Sunday is ruled out. Likewise, Saturday can also be ruled out, …

Since after six applications of “likewise” every day of next week is ruled out as a possible day for the drill, thereby giving rise to the paradox, naturally many people would think that the “problem” with this reasoning lies in the “likewise” step. But I think that if there is a flaw in this reasoning, it should already appear at the very first step!

Let us look again at the Swedish Broadcasting Corporation’s announcement: “Within the coming week, an air-raid drill will be held, … no one has been told in advance the exact day of this drill, …”

Notice that our reasoning has two usable premises: first, “Within the coming week, an air-raid drill will be held”; second, “not known in advance.”

If, on Saturday, one can infer that “the drill cannot be held next Sunday,” then please do not ignore the first premise either—“Within the coming week, an air-raid drill will be held.” Given this condition, and because no drill has occurred in the first six days, one can infer that “the air-raid drill will be held on Sunday.” Thus a contradiction has already arisen. That is to say, following the previous line of thought, when no air-raid drill has occurred in the first six days, we will infer both that the drill will be held on Sunday and that the drill cannot possibly be held on Sunday—this pair of contradictions.

Put another way, on Saturday, because according to Aikebo’s line of reasoning we can “infer” that the drill cannot be held on Sunday, we can “believe” that there will be no drill on Sunday—so even if the drill does take place on Sunday, it will still be “sudden” for Aikebo![③]

A similar situation exists on any day before Sunday. For example, on Thursday, the reasoning Aikebo carries out may be:

Φ:

(0) Today is Thursday, and no drill has been held in the previous four days.

(1) The announcement says: there will be one air-raid drill this week.

(2) The announcement says: on the day before the drill, I cannot infer that the drill will be held the next day.

(3) (Assumption) The announcement will be fulfilled

(4) From (2) and (3), the drill cannot take place on Sunday; otherwise (2) would be violated, i.e. on Saturday I could infer that the drill would take place the next day.

(5) From (2), (3), and (4), the drill cannot take place on Saturday.

(6) From (2) to (5), the drill cannot take place on Friday.

(7)
From (4), (5), (1), and (0), that is, since a drill cannot take place on Saturday or Sunday; there has been no drill from Monday to Thursday; yet there must be one drill this week; therefore, the drill will take place on Friday. …… (6) and (7) contradict each other!

(8) The assumption in (3) is false.[④]

(9) But … for example, when Friday comes, the drill is suddenly held, and before that I truly did not infer that it would be held, so (3) was not wrong. …… (8) and (9) contradict each other!

(10) ……Which other premise is wrong?

So, is the cause of the contradiction the announcement itself, or the reasoning process? If it is the reasoning process, then is it in the first step (4)? Or in the iterative steps (5) and (6)? Or in the reasoning at (7), meaning that the contradiction between (6) and (7) does not follow, and thus (8) does not follow either? Or is it that (9) is wrong, so that (3) is indeed not satisfied? Or is it that the reasoning up to (9) contains no error at all, and the premise that should be reduced to absurdity lies hidden earlier on?

Before proceeding further, it is necessary to make the concepts in the original problem explicit.

II. Rephrasing and Improvement

First of all, the “somewhat minor weakness” of the sudden drill paradox is that the drill might not take place at all. “To remedy this weakness, Michael Scriven reformulated the paradox in the form of an egg experiment; his analysis was published in 1951 in the British journal Mind: There are ten boxes in a row in front of you, numbered 1 through 10. You turn around, and your friend hides an egg in one of the boxes. The egg is certainly in one of the boxes; this much is beyond doubt. Your friend says: ‘Open the boxes one by one, and I guarantee that you will unexpectedly discover the egg in one of them.’”[⑤]

I think that, compared with the “unexpected egg paradox,” the “sudden drill paradox” is not substantially improved. In the egg paradox, what can be determined is this: when the first nine boxes have been opened, I can conclude that the egg must be in the last box, so if my friend has put the egg in the last box, he will inevitably lose in every sense. But similarly, in the sudden drill paradox, if a whole week passes without any drill being held, then one can conclude that no air-raid drill was held that week, and thus the first item of the announcement was not carried out; therefore, if the announcer does not hold a drill, he too will inevitably lose in every sense. But the issue is this: if we know that the announcer knows that if he does not arrange a drill he will inevitably lose, is that enough to conclude that the announcer will not arrange the drill for the last day? And in the egg paradox, the issue is this: if I know that my friend knows that if he puts the egg in the last box he will inevitably lose, is that enough to conclude that he will not put the egg in the penultimate box? In other words, the “improvement” of the egg paradox merely eases the problem a little, namely by matching the case where “the drill does not take place on any day” to the case where “the egg is in the last box,” and the case where “the drill takes place on the last day” to the case where “the egg is in the penultimate box”… With this one-position offset, the two paradoxes are still completely equivalent!

Examining reasoning Φ, Michael Scriven’s improvement lies in the fact that one can remove the added premise “the announcement says” from (1), so that “there will be one air-raid drill this week” can objectively be true independently of the announcement; but the parenthetical note before (2) can never be removed, so taken as a whole, premises (1), (2), and (3) do not change at all.

Second, there are the words “sudden” and “unexpected,” which are obviously ambiguous. For example, if we are talking about psychological suddenness or unexpectedness, suppose someone rolls a 6 on a die—does that count as “unexpected”? For many people, that would not count as “unexpected,” because any face can come up in a die roll. Likewise, if people realize that tomorrow the drill may or may not happen, then no matter on which day the drill is held, it will not be unexpected, and the announcement will always fail.

However, it is enough to replace words like “sudden” and “unexpected” with more precise formulations to ensure the paradox’s “genuine article” status.

In fact, when I stated “reasoning Φ” above, I had already clarified the formulation of the paradox. Here, “sudden” means either “logically unable to infer P or not-P,” or “logically inferring that P should be the case when in fact not-P is the case.” The latter formulation is relatively stronger: for example, rolling a 6 on a die counts as “unexpected” in the former sense, but not yet in the latter; still, both formulations are fairly strict. Which meaning is more appropriate in the sudden drill paradox will be discussed later.

Let us first look at a strict formulation that refers to the studies of Montague and Kaplan:[⑥]

Announcement: “Unless you know this announcement to be false, the drill will take place on some day next week, and on the day before the drill you do not know, ‘based on this announcement,’ that ‘the drill will take place tomorrow’ is true (or, to put it another way, you cannot, on the day before the drill, infer from this announcement that the drill will take place the next day).”

This announcement can be simplified to a case with only one day remaining:

Announcement: “Unless you know this announcement to be false, the drill will take place tomorrow, and right now you cannot infer from this announcement that the drill will take place tomorrow.”

This announcement is similar to Zhang San saying: “I’m telling you—I’m called Zhang San; oh, you still don’t know what my name is.” I will return later to analyze this.

The earlier simplification can even be taken further, so that the set of possible drill dates becomes the empty set, namely:

Announcement: “Unless you know this announcement to be false, the drill will take place on an impossible day, and…” This amounts to “Either you know this announcement to be false, or a contradiction holds,” that is, “You know that this sentence is false.”

This is, in fact, a variant of the so-called “knower paradox.” The simplifications above reveal the profound connection between the “sudden drill paradox” and the “knower paradox.” Zhang Jianjun points out: “Any solution capable of handling the knower paradox can also handle the unexpected exam paradox, but not vice versa.”

There has been much research on the knower paradox. In 1962, Montague had already clarified “the strict connection between the liar paradox and the knower paradox.”[⑦] Subsequently, people constructed many other paradoxes involving propositional attitudes such as “belief”; I will not go into that here.

As Zhang Jianjun says, if the knower paradox is solved, it should provide solutions for a whole series of related paradoxes such as the “sudden drill paradox.” But conversely, perhaps one can also bypass the “knower” paradox and find a way to dissolve the “sudden drill paradox” directly. In what follows, I will try to search for a way of directly handling the “sudden drill paradox.”

III. Reexamining the Reasoning

After briefly sorting through the problem and the concepts, let us turn back to reasoning Φ and inspect it line by line:

(0), (1), (2) —— all are objective facts, and should raise no doubt.

(3) —— is an assumption, and normally there is no need to doubt it. However, not every random “assumption” is reasonable; for example:

1) The above reasoning is more than two sentences long. —— assumption

2) The above reasoning is only one sentence long. —— objective fact

3) 1 and 2 are contradictory, therefore the assumption does not hold.

4) The above reasoning is no more than two sentences long. —— reductio ad absurdum

The above reasoning is obviously absurd! The problem is that the referent of “the above reasoning” is constantly changing; the “above reasoning” that appears in the second line is already something different from that in the first line. A reasonable “assumption” should obviously refer to something that does not keep changing as the reasoning proceeds. Happily, it seems that the “announcement” involved in (3) is not unstable, because earlier we already rephrased the announcement in relatively strict terms. Now the target of suspicion is the reasoning process, so for the moment we may take the announcement as clear, and thus set aside doubts about (3).

(4) —— this first step is the one most worth doubting. Suppose that after getting through the first six days without any drill, one really can infer that Sunday must be the drill day (leaving aside how this is inferred). Then the announcer would be doomed if he arranged the drill for Sunday. However, at the same time, the announcer would also be doomed if he did not arrange the drill for Sunday, because he would violate (1)! In other words, when only one day remains, no matter whether the announcer chooses to hold the drill or not to hold it, he will violate the announcement!

But at this point, what reason do I still have for “being able to infer that the drill must be on Sunday”? For to the announcer, either way is a failure: holding the drill and not holding the drill are both failures. Even if we assume that the announcer will certainly do his “utmost” to ensure the prophecy succeeds, when both options are failures, what grounds do I have for “inferring” that he will necessarily choose this kind of failure rather than that kind of failure? It seems that I still cannot be certain of the announcer’s choice, and this contradicts our assumption: that after getting through the first six days without any drill, one really can infer that Sunday must be the drill day!

Thus the most direct conclusion is: the assumption is false. That is to say, even after six quiet days have passed, I still do not have a way to infer that the drill must take place on Sunday!

Notice that the above reductio proceeds from the assumption “I can infer his arrangement,” leading to “I cannot infer his arrangement,” so the assumption is wrong—that is direct and concise. It can be compared with the following reductio: from “I can infer that he made such an arrangement” to “I can infer that he did not make such an arrangement,” therefore the assumption is wrong, that is, I cannot infer it. Although the latter reductio is also intuitively obvious, “if one can infer P then one can infer not-P, therefore one cannot infer” is still somewhat less direct; the reductio used above—“if P then not-P, therefore not-P”—is beyond doubt.

However, in arriving at the above reductio, one still used a somewhat indirect inference: “he is facing the same kind of lose-lose dilemma, so I cannot determine his choice, which is to say I cannot infer that he will make one choice rather than another.” But this inference is quite intuitive—compared with the layered and nested reasoning used in this paradox, such as “I know that he knows that I can infer that he will surely……, so……,”[⑧] the difference is literally worlds apart.

In light of this, it is reasonable to conclude that “even after six calm days have passed, there is still no way to infer that a drill must take place on Sunday”! So we can focus our doubts on (4) and see what exactly makes (4) seem “apparently reasonable”?

The key point of this step of reasoning is that a conclusion is inferred from a premise of the form “I cannot infer…”. Perhaps this will be illuminating for understanding this kind of reasoning, or perhaps it is merely a detour; in what follows, let us turn instead to another variant of the “sudden drill paradox.”

IV. Variants of the paradox and other problems

Compare the following two problems:

1. A mathematical puzzle:

Students A, B, and C stand in a line, with A in front. Each of them wears a hat, and the person farther back can only see the hats of the people in front, not their own hat nor the hats of those behind them. It is known that the three hats they wear are taken from two red hats, one black hat, and one white hat. The teacher first asks C, who is at the back: “What color is your own hat?” C replies: “I don’t know.” The teacher then asks B whether he knows the color of his hat, and B also says “I don’t know.” Finally the teacher asks A, who says, “I know!” How did A know the color of his own hat?

This is not a paradox at all, but a mathematical problem with a correct answer. A’s hat must be red.[⑨]

This problem can be rewritten in the following form, which may offer some inspiration for the paradox to be discussed below:

A and B each take one number from the hats numbered 1, 2, 3, 4; A picks 2, and B picks 3. C says to A and B: “Is the following statement true—that neither of you can infer whose number is larger?” The two answer in unison: “Yes.” Then they immediately answer in unison again: “No!” Here A and B’s line of reasoning is quite clear: A thinks, since B cannot infer whose number is larger, then B’s number cannot be 1 or 4, and since I took 2, B must be 3. … The question is, then: is C’s statement true or not? Before C tells A and B the statement, it is true; but once it is told to them, it becomes false!

2. Another variant of the “sudden drill paradox” (that is, the Hollis paradox)[⑩]

Two people on a train, A and B, each choose a number and then whisper it to C. C stands up and announces: “I’m getting off at the next station. The two of you have told me two different positive integers, and neither of you can infer whose number is larger.” Then C gets off the train.

A and B continue their journey in silence: A’s number is 157. He thinks: “Obviously B did not choose 1. If he had chosen 1, then he would know that my number is larger than his, because C has just said that our two numbers are different. Equally obviously, B also knows that I did not choose 1. Right, 1 can be ruled out. The smallest possible number either of us could choose is 2, but if B had chosen 2, then he should know that I did not choose 2, so 2 is also ruled out…”

If his journey is long enough, he can rule out every number.

Taking the hint from the mathematical puzzle above—perhaps C’s statement really was true at first, but when he told A and B that statement, because the statement itself became new information for A and B’s reasoning, the situation changed!

In the earlier mathematical puzzle, the addition of new information made the situation immediately clear. Of course, one can also design more complicated mathematical puzzles, for example ones that require repeated declarations: “None of you can infer it”—“None of you still can infer it” … “None of you still can infer it”—“Ah! Someone has inferred it!”

Each declaration of “none of you can infer” is the addition of a new piece of information, and it may be that only after adding N such “repeated” pieces of information does the situation become clear—there are quite a few such mathematical problems, and I won’t go into them here.

That is to say, the effect of these newly added pieces of information is “one-off”: they can only be used for one round of reasoning, namely for analyzing the situation before that information was announced; but at the very moment people begin to analyze on that basis, the situation has already changed.

In the Hollis paradox, is the situation the same? A, based on the information C has given, is indeed entitled to infer that B did not choose 1; but is it still legitimate to continue using C’s assertion in the subsequent reasoning?

However, the situation in the Hollis paradox is not so simple. If C’s announcement were “You will ‘never’ be able to infer whose number is larger,” then what? That is to say, this requires C to state his claim more strictly—namely A = “Not only can you not infer whose number is larger now; even if you know ‘this information,’ you still cannot infer whose number is larger!”

But this way of stating it is still not very clear. The question is, what does “this information” refer to? — If it refers only to the first half, X, namely “you cannot infer whose number is larger now,” then, because we think the whole information A provided by C always has a “one-off” effect, the situation changes the moment A is told to A and B. But here A contains a “nesting,” so it is equivalent to being able to be “used twice.” That is something like saying, “None of you can infer it”—“None of you still can infer it.” Under this condition, A really is entitled to deduce that B cannot choose 2! But if the information given by C contains only one layer of “nesting,” then the reasoning A can carry out on that basis can only go as far as 2! If one wants A to be entitled to reason endlessly on the basis of the conditions given by C, then C’s information must contain infinite “nesting,” which is to say, the “this information” in A must refer to A itself!

3. A combination of the mathematical puzzle and the Hollis paradox:

However, combining the Hollis paradox with the earlier mathematical puzzle allows us to recast it in a form that does not require infinite “nesting”:

A and B each draw one number from the eight numbers 1 through 8, and each can only see the number in his own hand. A’s number is 3, and B’s is 5. C asks them: “Whose number is larger?” The two answer in unison: “I don’t know!” After a while, C asks again: “Whose number is larger?” and the two again answer in unison: “I don’t know!” But at this point, A already knows the answer.

Because correct reasoning can fully anticipate the answers A and B give, the pattern of C asking questions and A and B answering in unison can be turned into C directly making a declaration. Then the situation is this—C says to the two of them: “Neither of you knows whose number is larger!” … “After hearing what I just said, neither of you still knows whose number is larger!”

At this point, A already knows that B’s number is larger. If at this moment D were to say to E: “A and B still do not know whose number is larger!” then E would think D was wrong; clearly, A has already arrived at the correct judgment. However, if at this moment it is still C who says to A and B: “After hearing my previous two statements, neither of you still knows whose number is larger!” then A is probably going to get confused—“Clearly, based on the first two statements I already inferred that my number is smaller than B’s; how can you now say that I don’t know? If C’s third statement is correct, then the first two statements must be wrong; if the first two statements are correct, then the third statement must be wrong. But I cannot tell which of them is wrong, so… am I forced to admit that all three of C’s statements are correct?! Because I have no way of determining which statement is wrong, it is indeed true that I have never really known whose is larger…”

Just as A is getting confused, B’s reasoning continues normally. Under C’s three assertions, B has already ruled out, in sequence, A being 1, 8, 2, 7, 3, 6; since the number in B’s hand is 4, B infers that the number in A’s hand is 5. Unfortunately, his reasoning is wrong.

Then C says a fourth statement: “After hearing my previous three statements, neither of you still knows whose number is larger!” For A, who is already completely dazed, and B, who has already reached a mistaken conclusion, C’s fourth statement is indeed correct, and B begins to get confused as well…

V. Back to the “sudden drill paradox”:

Why do A and B’s reasoning in the above case fall into confusion? The reason is that C has provided extra information, while A and B cannot determine whether, or at what point, C began providing information that was superfluous.

In the sudden drill paradox, what the announcement provides is something of the form “You cannot infer in advance …” Using an announcement that contains this information, we can infer “the drill cannot be on Sunday,” “the drill cannot be on Saturday,” and so on. Like A in the previous example, our reasoning is correct for the first several steps, but we do not know at exactly which step the information “You cannot infer in advance …” becomes superfluous, so that all subsequent reasoning falls into confusion.

Returning to the question raised earlier: what exactly does “sudden” mean? Is it “unable to infer P or not-P,” or is it “infer that it should be P when in fact it is not-P”? As mentioned above, the latter statement is relatively stronger; for example, rolling a 6 with a die is “unexpected” in the first sense, but not yet in the second.

But here the strength relation between them seems to be reversed. In fact, after using C’s first two pieces of information, A has already reached the correct conclusion that B’s number is larger, only to negate it afterward. In the “reasoning Φ” of the sudden drill paradox—we’ll set aside (4) for the moment—if we skip over (6), then (7) has already reached the conclusion “the drill is on Friday,” while (6) reached the opposite conclusion. So here the issue is not “unable to infer P or not-P,” but rather “both infer P and infer not-P.” In other words, on every day of this week, we can “infer” that “tomorrow there will be a drill,” only for the subsequent reasoning to reach the opposite conclusion.

The question here is: does “contradiction implies everything” hold, or does “contradiction implies nothing” hold? When we infer P and at the same time negate that inference, does it still count as “inferring P”? If it still counts, then the announcement in fact cannot be satisfied! Because based on this announcement, on every day we can infer that there will be a drill tomorrow, so no matter on which day the drill takes place, it is something that has already been inferred. But if inferring P and then negating that inference leads to the conclusion that one cannot infer P, then in the “reasoning Φ,” although (4) inferred the conclusion that the drill cannot be on Sunday, when combined with the further inference of (5) and (6), the conjunction of (0), (1), (5), and (6)—namely, “the drill will take place on one of Friday, Saturday, or Sunday; the drill will not be on Saturday; the drill will not be on Friday”—can yield (4’): “the drill will take place on Sunday,” thereby negating the inference of (4). According to what was just said, we should then conclude that (4) cannot be inferred.

If “sudden” is to be understood as “infer that it should be P when in fact it is not-P,” then one must likewise look at how “contradiction implies what.” If, when we infer P and then negate that inference, it still counts as “inferring P,” then on every day we could, on the basis of this announcement, infer that there will be no drill tomorrow, and thus a drill on any day would be “sudden” (similarly, in fact, on every day we can also infer that there will be a drill tomorrow, so any calm day is also “sudden”). And if one thinks that inferring P and then negating that inference leads to the conclusion that one cannot infer P, then the announcement is in fact again unsatisfiable, because on any day we can infer neither “there will be a drill tomorrow” nor “there will not be a drill tomorrow,” just as we cannot infer how many will come up on a die but would think that whatever number does come up does not count as “sudden,” and that no matter which day the drill is scheduled for, it does not count as unexpected.

To sum up, the relation between different understandings of “sudden” and of “what contradiction implies” and whether the announcement can be satisfied is as follows:

	├P∧﹁P则仍然├ P	├P∧﹁P则导致“并非├ P”
突然=cannot infer P or not-P	The announcement is not satisfied	The announcement is satisfied
突然=it can be inferred that it should be P, but in fact it is not-P	The announcement is satisfied	The announcement is not satisfied

And in everyday reasoning, these two levels of distinction are often mixed together, which is what causes reasoning to fall into confusion.

Reference Works

Zhang Jianjun: *An Introduction to the Study of Logical Paradoxes*, Nanjing University Press, 2002

[U.S.] William Poundstone: *The Labyrinths of Reason—Paradoxes, Puzzles, and the Fragility of Knowledge*, trans. Li Daqiang, Beijing Institute of Technology Press, 2005

[U.S.] Martin Gardner: *The Unexpected Hanging—and Other Mathematical Diversions*, trans. Hu Leshi, proofread by Qi Minyou, Shanghai Education Press, 2003

[①]The above draws on Zhang Jianjun, *An Introduction to the Study of Logical Paradoxes*, Nanjing University Press, 2002, pp. 193–194

[②]Ibid., p. 207

[③]Note that the concept of “sudden” here is quite vague; in the next section I will give a more precise account of it

[④]This line of reasoning has been rewritten by me. According to Eklbom’s derivation, item (7) should read: “From (4), (5), (6), and (0), infer that there can be no possible day for an exercise, thereby contradicting (1).” But I have switched the positions of (6) and (1)! The significance of this adjustment will become clear later in the text.

[⑤] [U.S.] William Poundstone: *The Labyrinths of Reason*, trans. Li Daqiang, Beijing Institute of Technology Press, 2005, p. 131

[⑥]See Zhang Jianjun, *An Introduction to the Study of Logical Paradoxes*, Nanjing University Press, 2002, p. 206

[⑦]Zhang Jianjun, *An Introduction to the Study of Logical Paradoxes*, Nanjing University Press, 2002, p. 208

[⑧]For example: A knows that B will certainly do his utmost to ensure the prophecy succeeds, so A knows that B will not fail to arrange an exercise; and because B knows that “A knows B will not fail to arrange an exercise,” B knows that if it is arranged for Sunday, A will learn of it in advance; so because A knows that “B knows that if it is arranged for Sunday, A will learn of it in advance,” A knows that B will not arrange the exercise for Sunday … The foregoing reductio ad absurdum is obviously much more explicit than these reasonings.

[⑨]The reasoning is quite simple: if A and B’s hats are one black and one white, then C can know that the only possibility for the hat he is wearing is red; therefore, at least one of A and B is wearing a red hat—after hearing C’s answer, B can know this point, but B still does not know the color of his own hat; that is to say, A’s hat cannot be black or white, because if it were, then B would know that his own hat must be red. Therefore, after hearing B and C’s answers, A can know that his own hat must be red!

[⑩]See [U.S.] William Poundstone: *The Labyrinths of Reason*, trans. Li Daqiang, Beijing Institute of Technology Press, 2005, p. 132

Latest Comments

apostar2007-12-19 20:39:15 Anonymous 222.205.106.237
http://blog.sina.com.cn/s/blog_50154f7901007m32.html
There’s an answer here that may be better.
古2007-12-19 22:10:56 Anonymous 125.34.50.182
Did you finish reading my article? Although I am very ashamed of this piece, I do believe that, on the basis of using everyday language, I still did quite a bit of thinking. This paradox is not as simple as it seems at first glance.
That friend’s thinking is indeed quite good, but it seems not to have resolved the paradox, and neither have I. Resolving a paradox is by no means that easy.
古2007-12-20 09:54:58 Anonymous 123.112.82.142
I forgot to mention that later I really did find a method for completely resolving paradoxes (including this paradox and Russell’s paradox, and so on), namely “intuitionism.” Because intuitionist mathematics thoroughly banishes “actual infinity,” it truly and once and for all resolves all kinds of logical paradoxes. Although the “price” paid by intuitionism is too high, that price is not something paid specifically to overcome paradoxes; it is entirely based on philosophical considerations. Apart from taking refuge in intuitionism, if one wants to resolve paradoxes once and for all without paying much of a price, I think that is not very likely.
Allen

2008-05-21 18:47:17 Anonymous 218.19.175.248

Not bad, much better than the first-floor comment
igeli2008-11-21 17:09:40
Some paradoxes arise because they use loopholes in everyday language that are extremely hard for people to notice, and these loopholes are concealed. For example, with the enhanced egg problem, under reasoning that ordinary people can all understand, B’s claim—“You can open the boxes one by one in numerical order; I dare say that before you see the egg, it is impossible for you to infer which box the egg is in!”—is false.
This is because if I open 9 boxes and still do not find the egg, then before I see the egg, I can be certain that it is in the 10th box.
What B says is a kind of everyday language; its logical loophole is something habitual in our lives and widely used.
Let me revise B’s statement: “You can open the boxes one by one; I dare say that at most you may have to open 9 boxes before you can infer which box the egg is in!”
This time, B’s proposition is rigorous.
The essence of that exercise is similar to this as well; I call this kind of problem a bounded problem.
A bounded problem needs boundary restrictions in order to be complete.
For example: I think of a natural number from 1 to 100; before you say that number, you cannot guess it—the proposition is false.
But: I think of a natural number; before you say that number, you cannot guess it—the proposition is correct; this is an unbounded problem.
There isn’t much room here, so I can only write this much.
One additional remark: for this kind of logical judgment, try not to use those operators and formulas—ordinary people can’t really understand them, and besides, they may end up producing the wrong result.
古雴2008-11-21 18:06:39
Paradoxes often arise because of loopholes in language; that is not wrong. However, the person above does not seem to have grasped the interesting point in what makes this paradox a paradox. Of course, I’m not very interested in other discussions about similar paradoxes anymore, unless you make comments directly aimed at my article~
By the way, isn’t the greatest feature of this article of mine precisely that it hardly uses those operators and formulas? The only place where symbols appear is probably that final table, right? That was added only to save space (otherwise one line on an A4 page would not be enough), and the content of that table had already been stated in words earlier in the text.