A Preliminary Analysis of the “Sudden Drill Paradox”
Xingding posted on 2006-05-21 20:45:48
I don’t know whether there is any future in writing this line of thought out… In any case, this topic could either serve as an assignment for the summer course on “paradox studies,” or as the assignment for the current “philosophy of logic” course~ But to turn this piece of writing into a proper paper will still take some effort…
A Preliminary Analysis of the “Sudden Drill Paradox,” (Continued)
Xingding posted on 2006-05-25 17:24:18
But in any case, I still have not really dissolved this paradox. In fact, the extent to which I have analyzed it should already have been analyzed by Quine. If we approach this issue from the subject’s perspective and add one more condition, for example: “I know he never lies,” or “I am certain the announcement will not be violated.” Then the paradox arises again.
A Preliminary Analysis of the “Sudden Drill Paradox,” (Continued 2)
Xingding posted on 2006-05-27 01:01:32
Recently I’ve been constantly refuting myself and thinking things over again; even my dreams are not peaceful. Studying paradoxes is really rather painful…
First let me make a few additions, and then I’ll turn my spear toward the Hollis paradox, offering an account of the so-called new variant of the sudden drill paradox. At the very least, it connects the Hollis paradox with the problems of the liar paradox and set-theoretic paradoxes (that is, it includes “self-reference”), and may be opening up a new line of thought compared with my earlier articles.
A Preliminary Analysis of the “Sudden Drill Paradox,” (Continued 3)
Xingding posted on 2006-05-29 16:52:26
I feel that thinking about paradoxes is a good way to batter one’s self-confidence, because one keeps coming up with answers that feel pretty good, only to have me refute them myself very quickly.
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According to the doc, repost the entire version of this preliminary analysis
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A Preliminary Analysis of the “Sudden Drill Paradox”
Cognitive paradox research originated in a paper published in 1948 by the British scholar O’Connor on the “sudden drill problem.”
The so-called “sudden drill problem” was a puzzling issue that had circulated 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 war preparations are adequate, no one knows in advance the specific day of this drill. Therefore, this will be a sudden drill.
The Swedish mathematician Åkebom realized that this announcement had a peculiar property: given the conditions stated in the announcement, the drill cannot be held on Sunday next week, because then the drill would be known in advance to occur on Sunday, and thus would not be sudden; Sunday is therefore ruled out. By the same token, Saturday can also be ruled out, since once it is determined that the drill cannot be held on Sunday, then among the remaining six days, holding it on Saturday would still not count as sudden. Proceeding in this way, the same line of reasoning can 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 was impossible.
Yet in the early hours of Wednesday of the second week, the air-raid siren sounded and the drill was “suddenly” carried out… (The above refers to Zhang Jianjun, An Introduction to the Study of Logical Paradoxes, pp. 193–194)
This paradox has many different variants, such as the “hanging paradox,” the “exam paradox,” and so on. Quine, Shaw, Montague, Kaplan, and other logicians have conducted in-depth research on it; Zhang Jianjun, in his An Introduction to the Study of Logical Paradoxes, also provides detailed related discussion. Their views are very incisive, and in the course of studying this paradox many profound advances and insights have also been achieved,
So then, what meaning is there in my analyzing this paradox again? I think there still is some—those logicians’ analyses of this problem are certainly profound, but they often carry an excessively high degree of technicality. Although that is necessary in order to make the argument rigorous, it is still hard for ordinary people to grasp. Their analyses reveal this paradox’s importance for logic, but they have not shown what it means for our everyday ways of thinking.
Therefore, from a nonprofessional perspective, I will carry out some further preliminary analysis of this paradox from my own point of view—I hope these efforts will still be meaningful.
Of course, my analysis will still need to rely on the tools of formal logic, especially “epistemic logic” on the basis of modal logic. Before introducing epistemic logic, I will first reexamine the reasoning involved in this paradox through intuitive thinking:
Since a paradox has appeared, the first thing to do, of course, is to reflect on the entire process of reasoning. If one finds that the whole process of reasoning is flawed, and where exactly the problem lies, then the paradox is dissolved.
So, is there a contradiction in Åkebom’s reasoning?
The drill cannot be held on Sunday next week, because then the drill would be known in advance to occur on Sunday, and thus would not be sudden; therefore Sunday is ruled out; similarly, Saturday can also be ruled out, …
Given that after six uses of “similarly,” any day next week is ruled out as a day for the drill, thereby producing a paradox, many people naturally assume that the “problem” with this reasoning lies with “similarly.” But I do not see it that way.
I think that if there is a flaw in this reasoning, it appears at the very first step! Or rather, this paradox already arises at the first step of the reasoning!
Let us look again at the Swedish Broadcasting Company’s announcement: “下周内将举行一次防空演习，……事先并没有任何人知道这次演习的具体日子，……”
Notice that our reasoning has two usable premises: first, “there will be an air-raid drill sometime next week”; second, “it is not known in advance.”
If, on Saturday, one can infer that “the drill cannot be held on Sunday next week,” then please do not overlook the first premise—“there will be an air-raid drill sometime next week.” Given this condition, and since no drill has occurred in the previous six days, one can infer that “the air-raid drill will be held on Sunday.” A contradiction has thus already arisen. That is to say, according to the earlier line of thought, when there has been no air-raid drill in the previous six days, we will simultaneously infer both that the drill will be held on Sunday and that it is impossible for the drill to be held on Sunday—this pair of contradictions.
Put another way, on Saturday, because following Ackerman’s line of thought we can “infer” that the drill cannot possibly be held on Sunday, we may “assume” that it will not be held on Sunday—so even if the drill were in fact held on Sunday, it would still count as “sudden” for Ackerman!
Let us turn back and see again how the conclusion “the drill will not be held on Sunday” is inferred:
In fact, the previous discussion has already mentioned that “Ackerman thus concludes that a sudden drill satisfying the announcement’s conditions cannot occur.”—If this inference is correct, then it has still not demonstrated that “a sudden drill not satisfying the announcement’s conditions cannot occur”！ That is to say, on Saturday, even if one can say that “it is impossible for a drill satisfying the announcement’s conditions to be held on Sunday,” one still cannot conclude that “a drill not satisfying the announcement’s conditions” will not occur. This means that, on Saturday, whether there will actually be a drill on Sunday remains uncertain to us!
At this point the analysis seems a bit muddled. But that is all right; since we already know that some kind of “fault” may exist at the very first step of the reasoning, let us focus on this step next.
Let us simplify the original paradox—unlike the usual practice in logic, where “one week” is simplified to “three days,” I will extract only “one day” from “one week”:
Suppose it is now Saturday, and no drill has been held in the previous six days; then one of the broadcasting company’s conditions, “there will be an air-raid drill sometime next week,” becomes A = “there will be an air-raid drill tomorrow”; while the second condition, “it is not known in advance,” is B = “(we) do not know that there will be an air-raid drill tomorrow”! The question is: are A and B contradictory?
If A and B do lead to a contradiction, then, as Ackerman says, the broadcasting company’s announcement is unreasonable; yet on the other hand, the drill can indeed be held without violating the announcement, and that is what creates the paradox. But are A and B really contradictory?
Here, we introduce modal logic to help us understand this issue.
We take the most basic normal system K of modal logic as our basis, that is, we add the operator “□” to first-order propositional logic, together with axiom K and the necessitation rule N: ├α then ├□α. We then introduce operators such as “∧,” “∨,” and “◇” by definition; I will not go into them here.
In short, interpreting “□” as “knows” rather than “necessarily” is the application of generalized modal logic. (See He Xiangdong et al., Generalized Modal Logic and Its Applications, p. 87 ff.; as a rule, “epistemic logic” uses the symbol K in place of □, but here I still use □, also because in the discussion that follows I do not draw a very clear distinction between “knowing” and “believing.”)
The conditions A and B can easily be translated into this logical language—using the proposition p to represent “there will be an air-raid drill tomorrow,” then “(we) do not know that there will be an air-raid drill tomorrow” is simply ┐□p.
The issue is obvious at a glance: can p∧┐□p lead to a contradiction? Clearly not, because p→□p does not hold!
What p∧┐□p can at most lead to is ◇p; no matter what ◇ is understood to mean in “epistemic logic,” it remains an uncertain state. Even under certain extension systems of K—even in the “largest” system that has not yet collapsed, namely “S5,” we may further infer “□◇p,” but that still does not help—in fact, from Monday to Saturday we each day “know that tomorrow may have a drill,” but if the drill really does happen on that very day, then for us it is still “sudden,” and it is no different on the final day.
That is to say, after such a simple “translation,” the paradox seems to disappear!
The only question left is: is this “translation” of the sudden-drill problem into the language of generalized modal logic fully appropriate? So next I will translate the inferential rules in that logical language back into intuitive language to see what they actually mean.
First let us look at rule N: “├α then ├□α.” This can be understood as follows: when we can, without premises, infer a conclusion solely on the basis of those axioms we ourselves accept, then we “know” it—this is relatively easy to understand. But the statement that “β├α implies β├□α” is false in modal logic; what does that mean in ordinary language? That is to say, what do “premises” mean in logical language?
Referring to the “sudden drill paradox,” it seems we can understand “premises” as declarations and announcements made by others. “Knowing” is inseparable from the “subject”; only when a conclusion is inferred entirely on the basis of knowledge and beliefs (axioms) that belong to the subject and have been accepted by the subject itself do we count as “knowing” it. And any assertion (proposition or formula), whether it is an assertion or declaration (premise) made by someone else, if it cannot be absorbed by the subject into its own cognition or beliefs, cannot be said to be “known.”
However, in general, the “premises” of an inference are often also reliable objective facts. For example, in the “sudden drill paradox,” we may also regard the broadcasting company’s announcement as a reliable fact, or to give a clearer example: “All men are mortal, Socrates is a man ├ Socrates will die”—what is it that we “know” in this inference?
According to the “epistemic logic” we have set up, from the inference “All men are mortal, Socrates is a man ├ Socrates will die,” we still cannot say that we “know that Socrates will die”; so what is it that we “know”?
Actually, this is not hard to understand—if we do not know that all men are mortal, or do not know that Socrates is a man, then we certainly do not know that “Socrates will die”; what we actually “know” is only that “if all men are mortal and Socrates is a man, then Socrates will die.” If we are to say that we “know that Socrates will die,” then we are required already to “know that all men are mortal and Socrates is a man.” Then do we “know” that “all men are mortal”? Hume long ago pointed out that there is no logically fully rigorous proof for such assertions!
Hume’s problem has long since revealed the limits of strictly logical demonstration. Of course, we may still “know” that “all men are mortal,” but that requires us to absorb certain things that cannot be strictly demonstrated into our own “beliefs,” that is, to use them as “axioms” that need no further proof.
If the broadcasting company’s announcement were “you know that there will be an air-raid drill sometime next week, but you do not know in advance the exact day of this drill,” then that would undoubtedly be contradictory. But in that case, the problem would lie in the announcement itself (fortunately, the broadcasting company did not say that). Because “knowing” is inseparable from the subject, no one is qualified to assert that another person “knows” something. Only when Ackerman actually takes the announcement “there will be an air-raid drill sometime next week” as an article of faith, as an axiom of his own logical system, does the “paradox” arise; yet no one (even if someone wanted to, he would have no right) has stipulated that Ackerman must believe the broadcasting company’s announcement beyond doubt. If Ackerman were to say afterward, “I know that there was a drill last week,” or if he said “I know that the announcement said there will be a drill next week,” that would seem more reasonable; but to hear the announcement and then declare “I know that there will be a drill next week” is far too rash. Moreover, if we assume that “I know that there will be a drill next week” is true, then according to the second clause of the announcement, logical deduction yields a contradiction. The reasonable choice, then, should be to reexamine the premises—either “I know that there will be a drill next week” or the announcement; one of the premises must be wrong. If one insists that even after judging the announcement to be false, one still maintains that “I know that there will be a drill next week” is true, then this is no longer a matter of paradox, but a matter of stubbornness… In short, the making of the “paradox” is not the announcement’s fault.
Finally, let me add an explanation of one issue: since the broadcasting company’s announcement has no right to claim that the listener “knows p,” why can it claim that the listener “does not know p”? In fact, under ordinary circumstances one also has no authority to assert that another person “does not know” something; the saying “You are not me—how do you know I do not know the joy of fish?” is not entirely without merit. However, in some situations one can indeed make the judgment that “you do not know something”—for example, when control over that matter is in my hands. Here one must presuppose the existence of “free will.” If we acknowledge “free will,” then when the power to decide on which day the drill will be held lies entirely with us, we are qualified to say that others “do not know on which day the drill will be held”!
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But in any case I still have not really dissolved this paradox. In fact, the extent of the analysis I have reached ought to have been reached by Quine long ago. If one starts from the subject’s standpoint and adds another condition, such as: “I know he never lies,” or “I am sure the announcement will not be violated.” Then the paradox arises anew.
In order to display the sharpness of this paradox more fully, below I will use another variant of the paradox to analyze it—the “unexpected egg” (from Skriven, see “The Unexpected Hanging,” pp. 4~5)
There are 10 empty boxes, numbered from 1 to 10. While A is turned away, B puts an egg into one of the boxes, and says to A: “You may open the boxes in numerical order. I dare say that before you see the egg, you cannot possibly infer which box it is in!”
So A can reason as follows: If B’s prediction is correct, then if the egg is in box 10, when I open box 9, since all the previous boxes have been empty, I can conclude that the egg is in box 10, and therefore the egg cannot be in box 10. Likewise, if the egg is in box 9, then when I have opened 8 empty boxes, and since the egg cannot be in box 10, I can conclude that the egg is in box 9; therefore the egg is not in box 9… and so on, until in the end the egg will be unable to appear in any box at all!
The above is nothing more than a rephrasing of the sudden-exam paradox, but the “unexpected egg” problem has a certain special feature: when I analyzed the sudden-exam paradox before, I pointed out that even after six days of calm and uneventful passage, one still cannot conclude that the exam will necessarily be held on the seventh day. Because from p one cannot infer □p, this does not conflict with the announcement’s ┐□p.
However, here, once B has hidden the egg, whether the egg is indeed placed in some box, and exactly which box it is in, have already been determined! For example, B may allow A to perform a verification—namely, weigh the total weight of these 10 boxes (individual weighing not allowed). And when A finds that the total weight of the 10 boxes is exactly equal to the weight of 10 empty boxes plus one egg, that is enough to conclude that one of these 10 boxes must contain an egg! In other words, here A really can know in advance that “there is an egg in one of the boxes.” That is to say, no stubborn gullibility is needed; once he has opened 9 empty boxes, even if A no longer believes what B said, as long as he trusts the balance, he can at this point already derive “□p,” and this truly conflicts with “┐□p”! So it seems that at this point, if B unfortunately happens to have put the egg in box 10, his prediction is doomed to fail!
It seems that for this strengthened version of the paradox, it is hard to sort out the problem by focusing only on the last box. So what about the second step of the reasoning, that is, the case of box 9?
Redefine the logical symbols at this point: let p mean “the egg is in box 9,” q mean “the egg is in box 10,” □a mean that A “knows” before opening box 9; and □b mean that A “knows” after opening box 9 but before opening box 10.
Thus, because of the conclusive evidence provided by the balance, after opening 8 empty boxes, A clearly knows that the egg is not in box 9 but in box 10, that is, “□a(┐p→q)” (and also □a(p→┐q) is true). Of course, □b(┐p→q) is naturally also true.
B’s prediction can be expressed as “┐□ap∧┐□bq”.
Since □b(┐p→q)→(□b┐p→□bq) is precisely obtained from axiom K, and since □b(┐p→q) holds, we get □b┐p→□bq. And if we assume q is true, then p is false, and there is no reason to deny that □b┐p is true at this point. Hence we obtain □bq, which contradicts B’s prediction “┐□ap∧┐□bq”.
If B’s prediction is true, then the assumption “q is true” does not hold, so q is false, that is, p is true, and then…
The problem is that can we derive “□ap” from “if B’s prediction is true then p is true,” that is, from “┐□ap∧┐□bq→p,” or, in other words, from “□a(┐□ap∧┐□bq→p)”? It seems that this is where the problem lies!
As mentioned earlier, we understand “□a” as something the subject is convinced of (an axiom), or a conclusion reached by the subject through inference that requires no presuppositions (a theorem). But here, to derive “□ap,” one would also need “□a(┐□ap∧┐□bq)” to do it. In other words, the problem is still that A always lacks sufficient reason to be sure that B’s prediction is true! Even if, in this improved paradox, A can be certain of the first announcement in the “sudden exam” problem, namely “there will be an exam next week,” that is, in this problem, can clearly know “there is an egg in some box,” it still is not enough to be sure that B’s prediction will definitely “succeed.”
In fact, when only two boxes remain, if we assume that A really can determine that the egg is in box 9, then B’s prediction fails; however, likewise, if B puts the egg in box 10, the result at worst is also the failure of the prediction. Since both are failures, why can’t B possibly put the egg in box 10? In that case, A’s judgment that “the egg is in box 9” would be wrong, a contradiction! —This is only a contradiction, not a paradox, because when a contradiction appears in the course of reasoning it means that we need to deny one of the premises; that is to say, the premise “if we assume that A really can determine that the egg is in box 9” is false, and the conclusion is that A cannot determine that the egg is in box 9.
Although here one must consider the situation of the last two boxes, A’s reasoning still goes wrong from the very first step; that is, although when A opens 9 empty boxes in sequence he can determine that the egg is in box 10, before the earlier boxes have been opened, it is still not enough to determine that the egg is not in box 10.
Then what if we consider it from B’s standpoint? If A and B are both “sufficiently smart people,” is it actually possible for B to put the egg in box 10?
Mentioning “sufficiently smart” inevitably brings to mind mathematical questions about “optimal strategy” and “game theory.” Perhaps wandering around in mathematics will bring some inspiration to thinking about this paradox.
In mathematical problems, we often encounter conditions such as “if both people are sufficiently smart” and “both people know that the other is sufficiently smart.” For example, the most famous “prisoner’s dilemma” problem is exactly like this; these problems are precisely typical cases based on reasoning and decision-making under the expectation of the reasoning the other side will adopt. Another more complicated problem is, for example, the “pirate booty division problem”—the reasoning process for solving these problems all requires the condition that “the other side will definitely adopt the optimal strategy”; they all have definite answers, and people usually believe that the solutions to these mathematical problems are rigorous and reasonable. Here one can see that reasoning “under the condition of expectations about the other side’s behavior” is not necessarily unreasonable.
But in many cases, one side in a game cannot have a definite “optimal strategy”; the choices of both sides will be indeterminate. In mathematical terms, this is a game problem “without a saddle point.” The most ordinary example of this kind of game is precisely “rock-paper-scissors.” Both A and B are convinced that the other will try his best to adopt the “optimal strategy,” yet the fact is that this “optimal strategy” is not any specific choice. If, for A, playing “rock” is the optimal strategy, then B should just keep playing “paper.” So for A, the so-called “optimal strategy” is precisely to choose rock, paper, or scissors with equal probabilities of one-third each.
Next let us look at a slightly more complicated game problem—A and B agree to choose simultaneously one character from “heads” or “tails”; if both choose “heads,” then A loses 3 yuan; if both choose “tails,” then A loses 1 yuan; if the two sides choose differently, then B loses 2 yuan. (See The Mathematical Principles of Chance, pp. 126~127)
At first sight, one might think: “This game is unfair, because when B chooses tails the result is favorable to B, so he will choose tails every time, and then A can only win 1 yuan or lose 2 yuan each time.” But thinking again, since A knows that B is a smart person, A can expect B to choose tails, and then A can also choose tails every time, and then every time it is A who wins…
The above reasoning is obviously wrong, but B does indeed have the advantage, and there really is an “optimal strategy,” namely “choose heads with probability 3/8”! In this way, B can win on average 1/8 yuan per round. Even if the clever A knows the strategy B adopts, A will still inevitably lose.
“The unexpected egg” can also be rewritten as a game problem:
First consider the case of two boxes. After B hides an egg in one of the boxes, he says to A: “You may choose to open the first box to take a look, or you may choose to ‘guess’ that the egg is in the next box; you have only one chance to guess. If you have not guessed before opening the box containing the egg, or if you guess wrong, then I will win 1 yuan; if you guess right, then the money is yours.”
So, who has the greater chance of winning, A or B? What are their respective optimal strategies?
The situation of this game for A is as shown in the table below:
|
|
Yi puts the egg in the first box |
Yi puts the egg in the second box |
|
Jia chooses to open the first box and take a look |
-1 |
1 |
|
Jia guesses that the egg is in the first box |
1 |
-1 |
At this point the problem is as simple as rock-paper-scissors: Yi’s best strategy is to decide with a 50% probability whether to put the egg in one of the boxes, and Jia’s best strategy is likewise to decide with a 50% probability whether to open the first box and look, while the odds of winning are evenly split between the two sides.
Now let us consider the case of three boxes. Yi first has two choices—put the egg in the first box or not put it there at all—and Jia can also choose either to open the first box first or to directly guess that the egg is in the first box. If Yi chooses to put the egg in the first box, or if Jia chooses to guess directly, then the game will be decided in the first round. If Yi does not put the egg in the first box and Jia opens the first box, then the situation returns to the state with two boxes. According to the previous analysis, in this state the two sides have equal chances of winning. Thus the situation is as shown in the table below:
|
|
B puts the egg in the first box |
B does not put the egg in the first box |
|
A chooses to open the first box and take a look |
-1 |
0 |
|
A guesses that the egg is in the first box |
1 |
-1 |
At first glance, B’s choosing not to put the egg in the first box would clearly be at a disadvantage, and A’s choosing to open the first box and look would also be clearly unfavorable. So would the clever B then necessarily choose to put the egg in the first box and thereby be guessed correctly by the clever A? Obviously not. Similar to the earlier “positive-negative” game, B’s best strategy at this point is to put the egg in the first box with a probability of 1/3, and to turn back and adopt the best strategy for the two-box case with the remaining 2/3, namely 50%. In other words, B’s best strategy is exactly like the game of rock-paper-scissors: place the egg in one of the boxes with equal probability. Then, whether A chooses to open the first box （1/3×-1+2/3×0）, or to guess directly （1/3×1+2/3×-1）, he will on average lose 0.333 yuan!
By induction, it is easy to generalize to the case of N boxes,
|
|
B places the egg in the first box |
B does not place the egg in the first box |
|
A chooses to open the first box and take a look |
-1 |
（2/N －1） |
|
A guesses that the egg is in the first box |
1 |
-1 |
The conclusion is that B’s best strategy is simply to place the egg randomly in one of the N boxes, and the amount of money A loses will never be less than 1－2/N! Allowing A to open the boxes one by one is no different from letting him make a blind guess straightaway!
What insight does this game above offer for analyzing our “paradox”?
We find that in the “unexpected egg” paradox, although the situations are very different, A’s reasoning likewise contains an unreasonable assumption—that is, if we assume that B is “smart enough” and “will do his utmost to maintain the prediction,” then he must either definitely make or definitely not make some particular choice.
Consider the following obviously fallacious reasoning: “We play a heads-or-tails game. Because you are smart enough and are trying your best to beat me, you will certainly choose the optimal strategy, namely choose tails. Then I will also choose tails to beat you. Since you are smart enough to infer that I will choose tails, then you will choose heads to beat me. So I should also choose heads. Then you will choose tails again……”
In the “unexpected egg,” B is likewise “smart enough,” but he is not trying his hardest to win money; rather, he is trying his hardest to make his prophecy come true. But the mistake in A’s reasoning is also this: “If B is trying his hardest to win and is smart enough, then he certainly will not put the egg in box X……” But we already know that the “best strategy” need not be any fixed choice; in order to preserve his prophecy, B only needs to casually put the egg into some box.
But the problem still has not dissolved, because for a smart enough B, he would certainly know that putting the egg in the last box is a sure path to defeat, and this is different from the choice in a game-theoretic problem that is “more likely to lose”; once he chooses to put the egg in the last box, there is no reason that can stop A from using the evidence of the balance scale after opening 9 boxes to “know” that the egg must be in the last box. Then in B’s “best strategy,” even if the choice is an uncertain random one, the optimal probability for putting it in box 10 should be 0%! Because B is not merely trying to increase his chances of winning; rather, he ought to hope for, and indeed be able to guarantee, certain victory—for example, by putting the egg in box 2…… But does A have the right, by saying “the probability that my clever B chooses to put the egg in box 10 is 0%,” to know in advance that the egg will not be in box 10? Once we begin to waver over the last box, the whole situation will again sink into confusion; this step must not be lightly conceded.
How to hold the line on the position that “A’s reasoning went wrong from the very first step” is something I have not yet settled on. Perhaps we can think this way: even if one “knows that the probability of B choosing to put the egg in the last box is 0%,” one still does not “know that B did not put the egg in the last box.” In fact, in probability theory, “zero probability” does not mean “impossible,” and an event with probability 1 is not the same as a “necessary event”! For example, if one randomly picks a natural number, then because the natural numbers are infinite, the “probability” of picking any particular natural number is zero; yet we cannot say “1 cannot be picked.” If we could say that, then we could likewise conclude in order that “2 cannot be picked,” “3 cannot be picked,” and so on to infinity. Therefore, it should be held that from “the probability of 1 being picked is zero” one cannot infer that “what is picked is not 1.” Thus B can choose such a strategy: “Randomly pick a natural number; if it is 142857, then put the egg in box 10; otherwise put the egg in box 1.” In that case, in mathematical terms, A cannot prove that he “knows the egg is not in box 10,” even though he is very certain of this. Of course, the issue here is whether “randomly picking a natural number” is actually possible, but even if it really were possible, this mathematical trick seems merely to be cleverly dodging the issue.
The focus of this paradox has always been what the word “know” (be certain, affirm) actually means. The following statement likewise sharply reflects this paradox:
“You do not believe this statement,”
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Perhaps some modification of the rules of the “logic of knowledge” could dissolve these paradoxes; for example, if from “□α├⊥” one could not infer “├┐□α,” and likewise from “┐□α├⊥” one could not infer “□α,” then these paradoxes could also be avoided. This also fits reality better, because in reality people, even when a certain belief or fact has already become incompatible with other facts or beliefs they themselves accept, still “firmly believe” or “know” the matter. But logic requires rigor, and how to construct a reasonable logical system in which “┐□α├⊥” does not entail “□α” is a problem. Before that, the sophistry that “zero probability” ≠ “impossible” may perhaps also explain the issue to some extent.
2006年5月25日
2006年5月25日
But a further rewriting of the paradox will again render the above discussion ineffective. For example, B declares: “You cannot be very ‘certain’ in advance that the egg will appear in the next box.” — By “very certain,” one means, for instance, that we are “very certain” that the genes of two different people will not be exactly the same. Then is B’s assertion itself credible? Intuition tells us that this statement is indeed worthy of being “very certain” — as long as, say, B casually tosses the egg into the first, second, or third box. At the same time, we are also “very certain” that B “is a rational person who will do his utmost to ensure the success of his prophecy.” Then are we “very certain” that B “cannot put the egg in box 10”? This also seems worthy of certainty. Then what about box 9……
Changing our angle of approach may perhaps be helpful.
Compare the following two questions:
1. A primary-school math olympiad problem:
Three students, A, B, and C, stand in a line, with A at the front. Each of them is wearing a hat, and the people farther back can see only the hats of those in front of them; they cannot see their own hat or the hats of the people behind them. It is known that the three hats they are wearing are chosen 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 answers: “I don’t know.” The teacher then asks B whether he knows the color of his own hat, and B also says, “I don’t know.” Finally the teacher asks A, who says, “I know!” So how did A know the color of his own hat?
The reasoning in this problem is very simple: if A’s and B’s hats were one black and one white, then C would be able to know that the hat he was wearing could only be red; therefore, at least one of A and B must be wearing a red hat — after hearing C’s answer, B would then know this fact, 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 be able to know that his own hat must be red. Therefore, after hearing B’s and C’s answers, A knows that his own hat must be red!
This problem can be rewritten in the following form:
A and B each take one hat from among three hats, of which hat1 and hat2 are red, and hat3 is black. Both of them can only see the hat they have taken; the teacher asks: “Do you know what color the other person’s hat is?” The two answer in unison: “No!” After a little while, the two answer in unison again: “We know!” — There is no paradox in the above description; it is easy to see that both of them have taken red hats.
Finally, let us rewrite it once more; perhaps this will provide some inspiration for the paradox to be discussed below:
A and B each take one hat from among four hats numbered1,2,3, and4; A chooses2, and B chooses3. C says to A and B: “The following statement of mine is correct, right? — You both cannot infer whose number is larger!” The two answer in unison: “Yes.” Immediately afterward, they answer in unison again: “No!” Here A and B’s line of reasoning is quite clear: A will think, since B cannot infer whose number is larger, then what B chose cannot be number1 or number4; and I took number2, so B must be number3. … The question is, is C’s statement actually correct or not? Before C tells A and B his statement, the statement is correct; but once he tells A and B, it is no longer correct! Is this a new kind of paradox?
II. Another Variant of the “Sudden Drill Paradox” (That Is, the Hollis Paradox)
On a train, two people, A and B, each choose a number and then whisper it to C. C stands up and declares, “My stop has arrived. The two different positive integers you two told me are different, and neither of you can deduce whose number is larger.” Then C gets off the train.
A and B continue their journey in silence: A’s chosen number is 157, and he thinks: “Clearly 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 two numbers we chose are different. Equally clearly, B also knows that I did not choose 1. Yes, 1 can be ruled out. Neither of us would choose it; the smallest possible number is 2. But if B chose 2, he ought to know that I did not choose 2, and so 2 is ruled out too……”
If his journey is long enough, he can rule out every number. (The Labyrinth of Reasoning, p. 132)
The lesson drawn from the above math problem is this—perhaps C’s statement was indeed correct to begin with, but once he told A and B that statement, the statement itself became new information for A and B’s reasoning, and the situation changed!
In the previous math problem, the addition of new information immediately made the situation clear. Of course, one can also devise more complicated mathematical puzzles, for example requiring repeated declarations: “None of you can figure it out” — “Still none of you can figure it out” … “Still none of you can figure it out” — “Ah! Someone has figured it out!”
Each declaration of “none of you can figure it out” adds a new piece of information, and perhaps only after such N “repeated” pieces of information are added does the situation become clear—there are quite a few such math problems, so I won’t elaborate.
That is to say, the force of these newly added pieces of information is “one-time only”: they can be used only for one round of inference, that is, for analysis of the situation before this information was announced; yet at the very moment people begin to analyze on this basis, the situation has already changed somewhat.
In the Hollis paradox, is the problem also like this? Based on the information given by C, A does indeed have the right to infer that B did not choose 1; however, is it still legitimate to continue using this assertion from C in subsequent reasoning?
But the situation in the Hollis paradox is not that simple. What if C’s announcement were, “You will ‘never’ be able to figure out whose number is larger”? In other words, this requires C to state his claim more strictly—namely A=“Not only can you not figure out right now whose number is larger; even if you know ‘this information,’ you still cannot figure out whose number is larger!”
However, the above 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 of the sentence X, namely, “you cannot figure out right now whose number is larger,” then since we regard the whole piece of information A provided by C, its effect is always “one-time only,” because the situation changes at the very 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.” It is something like saying, “None of you can figure it out” — “Still none of you can figure it out.” Under this condition, A indeed has the right to infer that B could not have chosen 2! But if the information C gives contains only one layer of “nesting,” then the inference A can make on that basis can only go as far as 2! If one wants A to be entitled to reason indefinitely on the basis of the conditions given by C, then C’s information must contain infinitely many “nestings”; that is to say, the “this information” in A must refer to A itself!
In other words, the paradox arises only when A itself contains “self-reference.” At this point, we connect the Hollis paradox with set-theoretic paradoxes and the liar paradox. The common problem underlying these paradoxes is the appearance of “self-reference” in the proposition.
2006年5月27日
We know that the making of an assertion may change the truth value of that assertion itself. This conclusion is by no means unfamiliar in everyday life. For example, I declare: “You certainly don’t know that I am a Peking University student!”—this assertion may well be true; you really did not know it beforehand. But after hearing what I said, you do know it. Or take the famous stock analyst who declares: “This stock will rise”—if this analyst is merely talking to himself at home, then perhaps his judgment will be wrong. However, if he publicly makes such a prediction on television in the capacity of an expert forecaster, then more believers will go buy that stock, and the stock will indeed rise. So is the analyst’s prediction accurate or not?
In short, speaking an assertion aloud can quite possibly make a previously correct assertion false, or turn a previously false assertion into a true one, because the announcement of the assertion makes the assertion itself a new part of the total situation.
In order to better contrast this with the egg problem, let us first further rewrite the previous math problem into a new-style paradox:
A and B each randomly take one egg from among eight eggs. The eggs are numbered from 1 to 8, and the two of them can only see the number on the egg in their own hand. A’s number is 3, and B’s is 5. C asks them, “Whose number is larger?” The two answer in unison, “Don’t know!” After a while, C asks again, “Whose number is larger?” and the two again answer in unison, “Don’t know!” At this point, A already knows the answer.
One can change the pattern of C asking and A and B answering in unison into a direct assertion by C, because, by correct reasoning, one can fully anticipate A and B’s answers. Then the situation is—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 instead D were to say to E, “A and B still don’t know whose number is larger!”, then E would think D was wrong, since A has clearly already arrived at the correct judgment. However, if it is still C saying to A and B, “After hearing my two previous statements, neither of you still knows whose number is larger!”, then A will probably start to get confused—“Clearly I have already inferred from the first two statements that my number is smaller than B’s; how can you still 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 not wrong, then the third statement must be wrong; but I cannot determine which of them is wrong, so… am I being forced to admit that all three of C’s statements are correct?! Because I have no way of determining which one is wrong, I really have never truly known whose number 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, and 6. Since B’s own number is 4, he infers that A’s number must be 5. Unfortunately, his reasoning is wrong.
Then C says a fourth sentence: “After hearing my previous three statements, still none of you knows whose number is larger!” For A, who is already completely bewildered, and B, who has drawn the wrong conclusion, C’s fourth statement is indeed correct, and B begins to get confused as well……
Here, C’s announcement does not need infinitely many nestings; merely by saying these four sentences in succession, he has already completely baffled the two of them.
……I’m already dizzy too, so let me rest for two days before I think about it again. I feel this new-style paradox is somewhat enlightening, but I’m afraid it still won’t be easy to figure out clearly. Paradox papers are really hard to write……
2006年5月28日
Abstract form of the sudden rehearsal paradox:
1、 Everything below is true
2、 Everything below is true
3、 Everything below is true
4、 Everything below is true
5、 Everything below is true
6、 Everything below is true
7、 Everything below is true
8、 Everything below is true
9、 At least one of the above is a lie
Simplified variant:
1、 There is a lie in this box
2、 There is a lie in this box
3、 There is a lie in this box
……
The simplest one: “This sentence is a lie”
Bibliography
Translated from the Chinese original with AI assistance. The original text is authoritative.
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