An Analysis of the “Sudden Drill Paradox”

29,827 characters2006.06.24

An Analysis of the “Sudden Drills Paradox”

Abstract:
The “sudden drills paradox,” also called the “surprise exam paradox,” the “unexpected hanging paradox,” and so on, is briefly surveyed and analyzed in this article in non-technical language. I reexamine the reasoning premises and steps that may give rise to the paradox, and attempt to offer a way of understanding it.

Keywords: sudden drills paradox exam paradox epistemic paradox contradiction

Origin

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

Within the coming week, there will be an air-raid drill. In order to verify whether preparations for wartime readiness are sufficient, no one has been told in advance on which specific day the drill will take place. Therefore, this is a sudden drill.

The Swedish mathematician Åkebom realized 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 drill would be known in advance to occur on Sunday, and so it would not be sudden; therefore, Sunday is ruled out. By the same token, Saturday can also be ruled out. Since it has already been determined that the drill cannot be held on Sunday, then among the remaining six days, if it is held on Saturday, it still would not be sudden. Proceeding in this way, the same line of reasoning can be used to rule out Friday, Thursday, and so on, all the way to Monday. Åkebom thus concluded that a sudden drill satisfying the conditions of the announcement is impossible.

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

This paradox has many different variants, such as the “hanging paradox” and the “exam paradox.” Quine, Shaw, Montague, Kaplan, and many other logicians have discussed and studied it in depth. Montague and Kaplan pointed out, on the basis of a highly plausible understanding of the assumptions from which this paradox and the related “surprise test” paradox are derived—or what we may call the “background elements” of knowledge—that “they can achieve a status comparable to that of the liar paradox and Richard’s paradox, and can, like them, lead to significant technical progress.”[②]

Still, although those logicians’ analyses of this problem are certainly profound, they often come with an excessive degree of technicality. This is necessary for argumentative rigor, of course, but it makes them difficult for non-specialists to grasp. Their analyses reveal the paradox’s importance for logic, but they do not show what it actually means for the way we think in everyday life. Therefore, I will attempt here to sort through and analyze the paradox again in a more intuitive, and not overly technical, language—hoping that these efforts are still worthwhile.

1. Where exactly is the problem?

Since a paradox has arisen, the first thing to do is, of course, to reflect on the premises and process of the whole reasoning. If it turns out that the entire reasoning process is flawed, and we can find where the error lies, then the paradox is dissolved.

So, is there a contradiction in Åkebom’s reasoning?

The drill cannot be held next Sunday, because then the drill would be known in advance to occur on Sunday, and so it would not be sudden; therefore, Sunday is ruled out; by the same token, Saturday can also be ruled out, …

Given that after six uses of “by the same token” any drill on any day next week has been ruled out, thereby producing the paradox, many people naturally think that the “problem” with this reasoning lies in the “by the same token.” But I think that if there is a problem with this reasoning, it occurs already at the first step!

Let us look again at the Swedish Broadcasting Corporation’s announcement: “Within the coming week, there will be an air-raid drill, … no one has been told in advance on which specific day the drill will take place, …”

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

If, on Saturday, one can infer that “the drill cannot be held next Sunday,” then one should not ignore the first premise either—“Within the coming week, there will be an air-raid drill.” According to this condition, and since no drill has occurred in the first six days, one can infer that “the air-raid drill will take place on Sunday.” Thus, a contradiction has already been generated. In other words, following the earlier line of thought, once the first six days have passed without an air-raid drill, we will infer both that the drill will occur on Sunday and that it is impossible for the drill to occur on Sunday—a pair of contradictions.

Put differently, on Saturday, because following Åkebom’s line of thought we can “infer” that a drill cannot occur on Sunday, we may “assume” that there will be no drill on Sunday—so even if the drill does occur on Sunday, it will still be “sudden” for Åkebom![③]

A similar situation arises on any day before Sunday. For example, on Thursday, the reasoning Åkebom carries out can be as follows:

Φ:

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

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

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

(3) (Assumption) The announcement will be fulfilled.

(4) From (2), (3), it is impossible for the drill to occur on Sunday; otherwise (2) would be violated, that is, on Saturday I could infer that the drill would take place the next day.

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

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

(7)
From (4), (5), (1), and (0), that is, since the drill cannot occur on Saturday or Sunday; no drill has occurred from Monday to Thursday; but there must be one drill this week; therefore, the drill will occur on Friday. … (6) and (7) are contradictory!

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

(9) But… for instance, when Friday arrives, the drill is suddenly held, and before that I certainly had not inferred that it would be held, so (3) was not wrong. … (8) and (9) are contradictory!

(10) …Which other premise is wrong?

So, where exactly does the contradiction come from: the announcement itself, or the process of reasoning? If it lies in the reasoning process, then is it in the first step, (4)? Or in the recursive steps (5) and (6)? Or in the reasoning in (7), namely the failure to derive the contradiction between (6) and (7), and thus the failure to derive (8)? Or is (9) wrong, so that (3) is indeed not satisfied? Or is it the case that there is no error in the reasoning up to (9), and that somewhere earlier there remains hidden a premise that ought to be reduced to absurdity?

Before further discussion, it is necessary to clarify the concepts involved in the original problem.

2. Restatement and refinement

First of all, a “relatively minor weakness” of the sudden drills paradox is that the drill might never be held at all. “To remedy this weakness, Michael Scriven recast the paradox in the form of an egg experiment. His analysis was published in 1951 in the British journal Mind: in front of you there is a row of boxes, ten in all, labeled 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; that much is beyond question. Your friend says: ‘Open the boxes one by one, and I guarantee that you will unexpectedly find an egg in one of them.’”[⑤]

I think that the “unexpected egg paradox” is no substantive improvement over the “sudden drills paradox.” 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 puts the egg in the last box, then in every sense he will be doomed to fail. Similarly, in the sudden drills paradox, if a whole week passes without a drill, then one can conclude that no air-raid drill has occurred during that week, and thus the first statement of the announcement has not been carried out. Therefore, if the announcer does not hold the drill, then in every sense he too will be doomed to fail. But the question is: once we know that the announcer knows that if he does not arrange the drill he will be doomed to fail, is that enough to conclude that he will not arrange the drill on the last day? And in the egg paradox, the question is: once I know that my friend knows that if he puts the egg in the last box he will be doomed to fail, is that enough to conclude that he will not put the egg in the second-to-last box? In other words, the “improvement” in the egg paradox merely shifts the problem a bit later, by matching the case where “no drill is held on any day” to the case where “the egg is in the last box,” and the case where “the drill is held on the last day” to the case where “the egg is in the second-to-last box”… offset by one position in this way, the two paradoxes are still completely equivalent!

If we examine reasoning Φ, Michael Scriven’s improvement lies in being able to eliminate the extra premise “the announcement says” from (1), so that “there will be an air-raid drill this week” can be objectively true independently of the announcement. But the parenthetical clause before (2) can never be removed, so as a whole, premises (1), (2), and (3) remain unchanged.

Second are the words “sudden” and “unexpected,” which are obviously ambiguous. For example, if we mean psychological suddenness or unexpectedness—say, rolling a 6 on a die—does that count as “unexpected”? For many people, this would not count as “unexpected,” because any face on a die is possible. Likewise, if people recognize that tomorrow there may or may not be a drill, then no matter on which day the drill is held, it is not unexpected; the announcement will always fail.

However, it is enough to replace words like “sudden” and “unexpected” with more precise expressions to ensure the paradox is bona fide.

In fact, when I stated “reasoning Φ” above, I had already clarified the formulation of the paradox to some extent. The so-called “sudden” means either “P or not-P cannot be logically inferred,” or “P should be logically inferred, but in fact not-P holds.” The latter statement 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 sense is more appropriate in the sudden drills paradox will be discussed later.

Let us first look at a rigorous formulation that draws on the research of Montague and Kaplan:[⑥]

Announcement: “Unless you know that this announcement is false, the drill will take place on one day next week, and on the day before the drill takes place you do not know that ‘on the basis of this announcement’ ‘the drill will take place tomorrow’ is true (or, that is to say, you cannot infer from this announcement on the day before the drill that the drill will take place the next day)”

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

Announcement: “Unless you know that this announcement is false, the drill will take place tomorrow, and you cannot now 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 named Zhang San; oh, you still don’t know what my name is.” I will return later to analyze this.

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

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

This is, in fact, a variant of the so-called “knower paradox.” The above simplification reveals the profound connection between the “sudden drills paradox” and the “knower paradox.” Zhang Jianjun pointed out: “Any solution that can handle the knower paradox can also handle the surprise exam paradox, but not vice versa.”

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

As Zhang Jianjun said, if the knower paradox can be solved, it should provide a solution to a series of related paradoxes such as the “sudden drills paradox.” But conversely, perhaps one can bypass the “knower” paradox and find a way to resolve the “sudden drills paradox” directly. In what follows, I will attempt to explore such a direct approach to the “sudden drills paradox.”

3. Reexamination of the reasoning

After briefly sorting out the problem and the concepts, let us return to “reasoning Φ” and inspect it line by line:

(0), (1), (2) — all are objective facts, so there should be no doubt.

(3) — is an assumption, generally not something to doubt. However, not every random “assumption” is reasonable. For example:

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

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

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

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

The above reasoning is obviously absurd! The problem is that the referent of “the above reasoning” keeps changing; the “above reasoning” in the second line is already a different thing from that in the first line. A reasonable “assumption” should evidently refer to something that does not keep changing as the reasoning proceeds. Fortunately, it seems that the “announcement” involved in (3) is not mutable, because we have already restated the announcement in a more rigorous form above. Now the target of suspicion is the reasoning process, so for the moment we may take the announcement to be clear and set aside doubts about (3).

(4) — this first step is the most questionable. Suppose that after the first six days have passed without a drill, one really can infer that Sunday must be a drill day (leaving aside how that inference is made). Then if the announcer arranges the drill for Sunday, he will indeed be doomed to fail. However, at the same time, if the announcer does not arrange the drill for Sunday, he will also be doomed to fail because he would violate (1)! In other words, with only one day left, the announcer will violate the announcement whether he chooses to hold the drill or not!

But at this point, what reason do I have to “be able to infer that Sunday must be a drill day”? Because at this moment, for the announcer, either way means failure; both holding and not holding the drill mean failure. So even if we assume that the announcer will certainly “do his utmost” to ensure the prophecy succeeds, when both choices lead to failure, what reason do I have to “infer” that he must choose this kind of failure rather than that kind? It seems that I still cannot determine the announcer’s choice, which contradicts our assumption (that after the first six days have passed without a drill, I can indeed infer that Sunday must be a drill day)!

Thus, the most immediate conclusion is: the assumption is not valid. That is to say, even after six quiet days have passed, I still do not have a method to infer that Sunday must be a drill day!

Notice that the above reductio ad absurdum proceeds as follows: from the assumption “I can infer his arrangement” we derive “I cannot infer his arrangement,” therefore the assumption is wrong — this is direct and clear; it can be contrasted with the following reductio: from “I can infer that he made such an arrangement” we derive “I can infer that he did not make such an arrangement,” therefore the assumption is wrong, namely I cannot infer. The latter reductio is intuitively obviously valid too, but “if one can infer P then one can infer not-P, therefore one cannot infer” is still somewhat indirect. By contrast, the reductio used above—“if P then not-P, therefore not-P”—is beyond doubt.

However, in reaching the above reductio, one still uses a less direct inference: “He faces the same dilemma either way, so I cannot determine his choice, that is, I cannot infer which choice he will make.” Still, this inference is very intuitive—compared with the multi-layered reasoning used in this paradox, such as “I know that he knows that I can infer that he will certainly…, so…”[⑧] it is worlds apart.

Given this, it is reasonable to conclude that “even after six peaceful days, there is still no method for inferring in advance that Sunday must be a drill”! So we can now focus the point of criticism on (4), and see just what it is that makes (4) seem “apparently reasonable”?

The key to this step of reasoning lies in inferring a conclusion from a premise of the form “I cannot infer…”. Perhaps this will shed some light on understanding this kind of reasoning; or perhaps it is merely a slight detour. In what follows, let us turn instead to another variant of the “sudden drill paradox.”

IV. Variants of the paradox and other issues

Compare the following two problems:

1. A mathematical puzzle:

Students A, B, and C stand in a line, with A at the front. Each of them wears a hat, and those standing farther back can only see the hats of the people in front of them, not their own nor those behind them. It is known that the three hats worn by them are drawn from among 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 answers: “I don’t know.” The teacher then asks whether B knows the color of his own hat, and B also says, “I don’t know.” Finally the teacher asks A, and A says, “I know!” How did A know the color of his own hat?

This is not a paradox, 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 among the hats numbered 1, 2, 3, and 4; A chooses 2, B chooses 3. C says to A and B: “The statement I’m about to make is correct, right—neither of you can infer whose number is larger!” The two answer in unison: “Yes.” Then they answer in unison again: “No!” Here A’s and B’s reasoning is quite clear: A will think, since B cannot infer whose number is larger, then B must not have chosen 1 or 4, and since I took 2, B must have chosen 3. … The question is, is C’s statement correct or not? Before C tells A and B the statement, it is correct, but once he tells A and B, it is no longer correct!

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

Two people on a train, A and B, each pick a number, and then whisper it to C. C stands up and announces: “I’ve reached my station; the two positive integers you told me are different, and neither of you can infer which number is larger.” Then C gets off the train.

A and B continue their journey in silence: A’s chosen number is 157. He thinks: “Obviously B did not choose 1; if he had chosen 1, then he would know that the number I chose is larger than his, because C has just said that the numbers we chose are different. Equally obvious is that B also knows I did not choose 1. Yes, 1 can be ruled out. Neither of us would choose it; the smallest number that is possible is 2. But if B had chosen 2, he ought to know that I did not choose 2, so 2 must also be ruled out…”

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

Taking the inspiration from the mathematical puzzle above—perhaps C’s statement really was correct originally, but once 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 immediately made the situation clear. Of course, one can also design more complicated mathematical puzzles, for example ones that require repeatedly proclaiming: “None of you can infer it” — “You still can’t infer it” … “You still cannot infer it” — “Aha! Someone has inferred it!”

Each time one says “none of you can infer…,” a new piece of information is added, and it may be only after N such “repeated” pieces of information have been added that the situation becomes clear—there are quite a few such mathematical problems, so I will not belabor the point.

In other words, the force of these newly added pieces of information is “one-time only”: they can be used only for one round of reasoning, that is, for analyzing the situation before this information was announced. Yet precisely while people are carrying out that analysis, the situation has already changed.

In the Hollis paradox, is the problem also like this? Based on the information C provides, A is indeed entitled to infer that B did not choose 1; but in the subsequent chain of reasoning, is it still legitimate to continue using this assertion made by C?

However, the situation in the Hollis paradox is not that simple. If C’s announcement were “you can ‘never’ infer whose number is larger,” then what? In other words, 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 even this way of stating it is still not quite clear. The problem is, what does “this information” refer to?—If it refers only to the first half of the sentence X, namely “you cannot infer whose number is larger now,” then, since we take C’s entire information A to have a “one-time only” effect, because the situation changes at the same time that A is announced to A and B. But here A contains a nesting, so it is equivalent to being usable “twice.” That is, something like saying “none of you can infer it” — “you still cannot infer it.” Under this condition, A is indeed entitled to reason that B cannot have chosen 2! But if the information C provides contains only one layer of “nesting,” then the reasoning A draws from it can only go as far as 2! If one wants A to be entitled, on the basis of the conditions given by C, to reason without end, then C’s information must contain infinite “nesting,” that 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, if we combine the Hollis paradox with the earlier mathematical puzzle, we can transform it into a formulation that does not require infinite “nesting”:

A and B each draw one number from the eight numbers 1 through 8; each can see only the number in his own hand. A’s number is 3, 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, according to correct reasoning, one can fully anticipate A and B’s answers, the pattern in which C asks and A and B answer in unison can be changed into C’s directly making assertions. Then the situation is—C says to the two: “Neither of you knows whose number is larger!” … “After hearing what I just said, you still do not know whose number is larger!”

At this point, A already knows that B’s number is larger. If at this point D were to say to E: “A and B still do not know whose number is larger!”, then E would think D was mistaken, since A has already arrived at the correct judgment. However, if it is still C who says to A and B: “After hearing my previous two sentences, you still do not know whose number is larger!”, then A is probably going to get confused—“Clearly I have already inferred from the first two sentences that my number is smaller than B’s; how can you then say that I don’t know? If C’s third sentence is correct, then the first two sentences must be wrong; if the first two sentences are not wrong, then the third sentence must be wrong. But I cannot determine which of them is wrong, so… am I forced to admit that all three of C’s sentences are correct?! Because I have no way of determining which sentence is wrong, it really was never true that I knew whose number was larger…”

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

Then C says a fourth sentence: “After hearing my previous three sentences, none of you still knows whose number is larger!” For A, who has by now become utterly disoriented, and for B, who has reached an incorrect conclusion, C’s fourth sentence is indeed correct, and B begins to get confused as well……

V. Returning to the “sudden drill paradox”:

Why, then, do the reasoning processes of A and B above fall into confusion? The reason is that C has supplied extra information, while A and B cannot determine whether, and at what point, C’s information became extra.

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

Returning to the earlier question, what exactly does “sudden” mean? Does it mean
“cannot reason out P or not-P,” or “reason out that it should be P when in fact it is not-P”? As noted above, the latter statement is relatively stronger; for example, rolling a 6 on a die is “unexpected” in the first sense, but not yet in the latter.

But here their relative strength seems 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 shall set aside (4) for the moment—if we skip over (6), then (7) has already reached the conclusion “the drill will take place on Friday,” while (6) reaches the opposite conclusion. Therefore, this is not a case of “cannot reason out P or not-P,” but rather of “reasoning out both P and not-P.” That is to say, for every day of this week, we can “reason” that “the drill will take place tomorrow,” only for the subsequent reasoning to arrive at the opposite conclusion.

The question here is: is it “from contradiction, everything follows,” or “from contradiction, nothing follows”? When we infer P and then negate that inference, does that still count as having “inferred P”? If it still counts, then the announcement in fact cannot be satisfied! Because on the basis of that announcement, on each day we can infer that there will be a drill tomorrow, so no matter on which day the drill is held, it is something that has already been inferred. But if, when one infers P and then negates that inference, the result is that one cannot infer P, then in “reasoning Φ,” although (4) infers the conclusion that there cannot be a drill on Sunday, once (5) and (6) are further combined, from the conjunction of (0), (1), (5), and (6)—that is, “the drill will be held on one of Friday, Saturday, or Sunday; the drill is not on Saturday; the drill is not on Friday”—one can obtain (4’): “the drill will be held on Sunday,” thereby negating the inference of (4). According to what was just said, we should then hold that (4) cannot be inferred.

If “sudden” is to be understood as “reasoning out that it should be P when in fact it is not-P,” one must likewise look at how “contradiction” is understood to yield anything. If, when we infer P and then negate that inference, it still counts as “inferring P,” then on each day we can infer on the basis of this announcement that there will be no drill tomorrow, and thus any day on which the drill is held would be “sudden” (similarly, in fact, on each day we can also infer that there will be a drill tomorrow, so any peaceful day is also “sudden”). But if one holds that when one infers P and then negates that inference the resulting conclusion is 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 what number a die will show, but would think that no matter what number is rolled, it does not count as “sudden”; likewise, no matter on which day the drill is arranged, it also does not count as surprising.

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

	├P∧﹁P则仍然├ P	├P∧﹁P then leads to “not ├ P”
突然=不能推理出P或者非P	Announcement not satisfied	Announcement satisfied
突然=推理出应是P而事实却是非P	Announcement satisfied	Announcement not satisfied

And in everyday reasoning, these two levels of distinction are often muddled together, and that is precisely what causes reasoning to fall into confusion.

Bibliography

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

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

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

[①] The above follows Zhang Jianjun, 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 highly vague; in the next section the author will give a more precise account of it

[④] This deduction has been rewritten by the author. According to Aickbohm’s deduction, item (7) should read: “From (4), (5), (6), and (0), it follows that no day is possible for the drill, thereby contradicting (1).” But I have swapped 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 Reasoning, trans. Li Daqiang, Beijing Institute of Technology Press, 2005, p. 131

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

[⑦] Zhang Jianjun, 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 the drill; and because B knows that “A knows that B will not fail to arrange the drill,” B knows that if it is arranged for Sunday, A will know in advance; so because A knows that “B knows that if it is arranged for Sunday, A will know in advance,” A knows that B will not arrange the drill for Sunday……The foregoing reductio ad absurdum is clearly much more explicit than these reasonings.

[⑨]The reasoning is very simple: because if the hats of A and B are one black and one white, then C can know that the hat on his own head can only be red; therefore, at least one of A and B is wearing a red hat—after hearing C’s answer, B can know this, 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, B would then know that his own hat must be red. Therefore, after hearing the answers of B and C, A knows that his own hat must be red!

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

Latest Comments

apostar 2007-12-19 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.
Gu 2007-12-19 22:10:56 Anonymous 125.34.50.182
Did you finish reading my article? Although I’m quite ashamed of this piece, I believe I still did quite a bit of thinking on the basis of everyday language, and this paradox is not as simple as it seems at first glance.
That friend’s line of thought is indeed pretty good, but it still doesn’t seem to solve the paradox, and neither did I. Solving the paradox is by no means easy.
Gu 2007-12-20 09:54:58 Anonymous 123.112.82.142
I forgot to say that later I really did find a way to completely solve paradoxes (including this paradox and Russell’s paradox, and so on), namely “intuitionism.” Because intuitionist mathematics thoroughly expels the “actually infinite,” it truly resolves all kinds of logical paradoxes once and for all. Although the “price” intuitionism pays is rather high, these costs are not paid specifically to overcome paradoxes, but are entirely based on philosophical considerations. Apart from taking refuge in intuitionism, if one wants to solve paradoxes once and for all without paying much of a price, I think that is probably not possible.
Allen

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

Not bad, much better than the one on the first floor
igeli2008-11-21 17:09:40
The emergence of some paradoxes is masked by exploiting loopholes in people’s everyday language that are extremely hard for people to notice. For example, with the strengthened version of the egg problem, under reasoning that accords with what ordinary people can understand, B’s proposition: “You can open the boxes one by one in numerical order, and I dare say that before seeing the egg, you cannot possibly infer which box the egg is in!” is false.
This is because if I have opened 9 boxes and still not found the egg, then before seeing this egg, I can be certain that it is in the 10th box.
What B says is a kind of everyday language, whose logical loophole in our lives is something we are long accustomed to and widely use.
If we revise B’s statement: “You can open the boxes one by one, and I dare say that at most you may have to open 9 before you can infer which box the egg is in!”
This time, B’s proposition is rigorous.
The essence of that drill is also similar to this; I call this kind of question a bounded problem.
A bounded problem needs boundary constraints in order to be complete.
For example: I think of a natural number between 1 and 100, and you cannot guess it before I tell you what it is—that proposition is false.
But: I think of a natural number, and you cannot guess it before I tell you what it is—that proposition is correct; this is an unbounded problem.
This place is too small, 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, you might end up calculating the wrong result.
Gu Dui2008-11-21 18:06:39
The emergence of paradoxes is often due to loopholes in language; that is not wrong. But the comment above doesn’t seem to have grasped the interesting point of why this paradox is a paradox in the first place. Of course, I’m not very interested in other discussions about similar paradoxes, unless you are making comments directly aimed at my article~
That said, isn’t the greatest feature of my article precisely that it almost never uses those operators and formulas! The only symbols that appear are in the final table, right? I added that only to save space (otherwise one line wouldn’t fit on A4 paper); besides, the content of that table has already been expressed in words earlier in the text.