3. When, in the book, Williamson first discusses the `margins for error’ principle, it is in relation to an example in which Mr. Magoo can see a tree some distance away and estimates its height, knowing full well that his estimate may be off by at least a few centimetres either way. Mr. Magoo, reflecting on the limitations of his eyesight and his ability to judge heights, arrives at the principle that, for each relevant natural number i:
4. I know that if the tree is i + 1 inches tall, then I do not know that the tree is not i inches tall.
5. Clearly, this principle does not hold for i = 0 just in virtue of the fact that it is given that Mr. Magoo can see a tree. It may be a bonsai tree that Magoo sees; he may not know that it is a bonsai, but he knows that it is a tree. Now, in the case of the Glimpse, we are given that one and only one date on the calendar is ringed by the teacher as being the date of the examination, and that it is not very near the end of term (KL, p.135). Let us suppose that the ringed date is in fact 24 days before the end of term. Then, certainly, the pupils cannot tell, from their glimpse of the calendar, that the examination will not be held 23 days before the end of term. But such considerations, which concern examinations held not very near the end of term, do not licence the generalised margin of error principle
(MEP) A pupil knows now that if the examination is i+1 days from the end of term, he does not know now that it will not be held i days from the end of term.
We can accept
that for large i,
(MEP) holds, but we should question whether it holds for small i too. To suppose that it does
is to fall prey to a Sorites
fallacy.
6. In order to make it clear why, so far as the holding of (MEP) goes, size matters, I shall use, as an analogy, an example of Descartes’ (though my use of the example will be somewhat different from his). We cannot discriminate, in the mind’s eye, between a chiliagon and a chili-plus-one-agon. It does not follow that we cannot see, in the mind’s eye, the difference between a hexagon and a heptagon. Let us call a polygon Cartesian if one cannot tell, at a glance, that it has a number of sides different from that of a polygon with one side more or one side less than it. The polygons in question, we shall assume, are of a suitable size to be taken in at a glance. (Also, in order not to be too Cartesian, forget about the introspective mind’s eye, and consider a glance to be a ½-second coup of the outer eye.) Now, it may be the case that there is a sharp cut-off point between Cartesian and non-Cartesian polygons, or it may be that there is an imperceptible slide from one kind to the other, in rather the same way as a heap of sand transforms imperceptibly into a non-heap by surreptitious removal, one by one, of its grains. Either way, it would be wrong to infer, from the fact that polygons with a large number of sides are Cartesian, that polygons with a small number of sides must be Cartesian too. Equally, it is wrong to infer from the fact that (MEP) holds for large i that the principle holds for small i too.
7. (MEP23) — the instance of (MEP) in which i = 23 — looks fine: If a pupil now knows that if the examination is 24 days from the end of term, he does not know, from his glimpse of the calendar, that it will not be held 23 days from the end of term. And, since there is imperceptibly less margin of error for the pupils if the ringed date is 23 days from the end of term, we can accept that if (MEP23) is fine, then (MEP22) is too. So we conclude that (MEP22) is fine. Fine. But it is easy enough to see where this line of reasoning is heading — right down the slippery slope, by a series of modus ponenses, to the false conclusion that (MEP0) is also fine. But (MEP0), which features crucially in Williamson’s argument, is false. A conclusion that we draw about a margin of error when the ringed date is 24 days before the end of term will not necessarily hold when the number of days remaining is much fewer. (MEP23) is clearly true, but, as i decreases, we enter a `penumbral’ region where (MEPi) is neither clearly true nor clearly false. But, as i decreases further, we emerge from the other side of the penumbra and into a region of clearly false margin for error principles. There is no difference relevant to present concerns between a glimpse of a calendar and a glance at a polygon. And nobody thinks that a square is Cartesian. The `polygon’ version of (MEP):
(PMEP) For all i, a person knows that if a polygon has i+1 sides, he cannot tell at a glance that it does not have i sides
is clearly FALSE.
8. If a pupil can see that the ringed date on a calendar is not very near the end of term then he has perceptual knowledge and cannot close his eyes to the evidence of his own eyesight. This is why it is in reality impossible for any but the most stupid pupil who glimpses a circled date which is 24 days before the end of term to convince himself that no examination will take place. By contrast, a pupil in the Surprise Examination situation really could come to believe that the examination promised by the teacher simply could not occur. Here is prima facie evidence that The Glimpse and The Surprise Examination are rather different in nature.
9. Michael Clark (in personal correspondence) grants that MEP does not hold for small i., but believes that Williamson can successfully develop a genuinely paradoxical version of The Glimpse, one which invokes only large i. In this version, the pupils' glimpse does not enable them to distinguish between an exam 15 days from the end of term and one 16 days from the end, or between 16 days from the end and 17, etc.. They do not make a sharp distinction between 15 days and 16 days before the end. They can tell from their glimpse that the exam will be at least 16 days before the end of term, and they think it will probably not be during the last 20 days. You could probably press them to say they knew it would not be 16 or even 17 days before the end of term. Now replace the unrestricted MEP in Williamson's argument by 'for all numbers i ≥ 15, if the exam is i+1 days from the end of term they do not know it will not be i days from the end'. They can now rule out every day, since for i ≤ 15 they know from their glimpse that the exam won't take place i days from the end, and the backwards induction rules out the remaining cases (where i > 15).
10.
This may seem to be a `strengthened’ version of The
Glimpse, but in fact, the only effect is to transform our original concerns
with the small i cases into new concerns with small j
cases, where j = i + 15. Let us say that the date 15
days before the end of term is March 19 (I have designated it this way for our convenience,
the pupils themselves might not be calendar-literate and might not know that
the day is so-designated). In
11. The `strengthened’ version of The Glimpse is emasculated once we observe that it spreads fudge at a critical juncture. It wants there to be both a sharp cut-off (all i ≤ 15 definitely ruled out as possible examination days), yet it also wants fuzziness in the area 15 < i < 20. But one cannot have it both ways – both that the pupils do not make a sharp distinction between 15 days and 16 days before the end of term and that they know, from their glimpse, that the examination cannot take place 15 days before the end. The premise crucial for the setting up of this version of the paradox is `If the exam is 16 days from the end then the pupils do not know that it is not 15 days from the end.' However, this is a premise to which we are not entitled. For, if the examination is 16 days from the end of term, then pupils do know that it is not 15 days from the end because they can see that it's not 15 days from the end -- which is precisely why they can rule out that day.
12. This contrast between The Glimpse and The Surprise Examination can be observed most clearly if we take the `extreme’ case where it is now the last day of term and no examination has, so far, taken place. In The Glimpse, with a single date showing on the calendar, the pupils could, it is true, be uncertain whether it is ringed or not (say, from one angle, there seemed to be a ring, from another not). But they could not derive a conclusion Williamson-style, for there is no margin for error in this case, no other date on which they might not know the examination not to be. It is quite possible, however, to generate a Surprise Examination paradox by making an announcement on the last day of term. If you are a teacher, you can actually try this out. I shall assume that you are a scrupulously honest teacher and known to be such. Now follow these simple instructions:
Go into the
classroom at
When (as will inevitably happen) a pupil immediately says `But please, Miss, you just said that there will be an examination, so we do know it’, answer `Yes, it’s confusing, isn’t it, but I don’t have time to stand here arguing with you’ and, without further ado, leave the room.
Return at
13.
Pupils who reasoned that you could not hold the promised examination were wrong. And, convinced that
there could not be any such examination, they were surprised when it actually
happened. Others, who reasoned that there would be an examination, because you
are honest and you said that there would be an examination, would also be
inclined to believe that the examination would come as a surprise, because you
said that too. That inclination would, of course, be tempered by recognition
that your announcement alerted them to an impending examination. Put yourself
in the position of a pupil: You hear the
teacher’s apparently crazy announcement. You just don’t know what to believe.
In this uncomfortable agnostic state, you don’t rule out an examination at
14. I have argued that the principle (MEP) that is used to set up The Glimpse is false. But The Surprise Examination is a genuine paradox and it can be set up without any reference to (MEP). The pupils first go through the usual argument to prove that there can be no surprise examination: On Monday, when the announcement of a surprise examination is made, they reason that if, by Thursday evening, no examination has taken place then, with Friday remaining as the only available day for the examination, its taking place on that day could not be a surprise. So Friday is ruled out. And so forth, backwards, ruling out each other day of the week. But then they reflect that, if past school experience has taught them anything, it is that there can be a surprise examination. So they are perplexed. On the Monday of the announcement, they can envisage their perplexity reaching a peak of acuity by Thursday evening if, by that time, no examination has taken place. The teacher’s announcement of a surprise examination would, by then, amount to an assurance (a) that there will be an examination on Friday, but (b) that pupils do not know this — ~K(a). Given that the teacher is honest, and that pupils know to be true what she tells them, we also have K(a) and K(b). But `(b) & K(a)’ is a contradiction.
15. Faced with this contradiction the sensible thing for the pupils to do is not (pace classical logic) to infer everything, but to refrain from inferring anything. However, a pupil who on Thursday, after four consecutive examination-free days, wanted to bet on the result, should bet that an examination will take place on the next day. For the honest teacher is committed to both (a) and (b). Now (a) is a proposition about the examination, while (b) is a proposition about the pupils’ psychological states. And the teacher is more likely to have made an honest mistake about the latter – although the pupils can by no means be sure that that is the mistake she has made. This would mean that, in the standard (5-day) version of the Surprise Examination paradox, the last day of the week should not be ruled out as a possible examination date, and hence the backward induction, ruling out the other days in reverse order, cannot get started.
16. Some authors are too hasty in dismissing the 1-day version of the paradox, and, to illustrate the point, in the previous lecture, I gave instructions for actually getting pupils into the relevant 1-day bind. The 4-day version of the paradox seems just like the 5-day version with a slight reduction of `noise’, so here is prima facie evidence that there is no essential difference between the 1-day version and the 5-day version, in the absence of any Sorites-type effect. And if that were so, then a solution to the 5-day version would fall to a solution of the 1-day version. A further point of interest about the 1-day version is that here there is no scope for appeal to any `margins of error’ principle. And another important point about the 1-day version is that it is connected with two other paradoxes in a surprising and mutually illuminating way, as we saw in a previous lecture.
17. The Newcomb paradox can be made vivid by embedding it in a TV game show in which a contestant is presented with two boxes at the front of a stage. Box A is transparent – everyone can see the $10,000 that has been placed inside it. Box B is opaque – whatever, if anything, it contains is not visible. Contestants have to choose EITHER to open Box B only OR to open both boxes. The prize is the content of the box or boxes they choose to open. Since it is a rule of this game that the content of the boxes cannot be changed after the contestant has made his/her choice, it seems obvious that the infallible strategy for winning most money is to open both boxes. But wait ….. The presenter is an expert psychologist who runs an examination on contestants before the show begins, in order to determine their psychological profiles. And it is another rule of the show, known to everyone, that if the presenter predicts that a contestant genuinely wants to one-box, then, just before the contestant walks onto the stage from behind a screen, the presenter will put $1000,000 into opaque Box B and close the lid; if, on the other hand, the presenter predicts that a contestant is a two-boxer, then she leaves Box B empty. The show is very popular. It has aired each day for several years, and everyone knows that the presenter is terrifically accurate in her predictions. Over the course of many, many shows, 99% of the contestants who have two-boxed have left the stage sheepishly, with only $10,000, their tails between their legs to the derisory hoots of the audience. Whereas 99% who have one-boxed have left the stage triumphantly, waving $1 million. Jim is the next contestant up. Should he one-box or two?
18. Jim is avaricious. Some (a very few) contestants in the past have walked off with the maximum $1010,000 jackpot, and this is the outcome that Jim wants for himself. Now, the situation that must obtain for Jim to stand a good chance of getting what he wants is that he really wants to one-box, so that the presenter, having read this aspect of his mind, has put $1M in Box B. (It’s no good Jim pretending to be a one-boxer, for the presenter almost invariably detects such faking.) But, when it comes to actually making the choice, Jim must two-box, thus securing the additional $10,000. So, writing `one-box’ as an abbreviation for `Jim one-boxes’, and `W’ for `wants that’, then the winning situation S for Jim is
S: Wj(one-box) & ~one-box
(Since one-boxing and two-boxing are exhaustive and mutually exclusive alternatives, two-boxing is equivalent to not one-boxing.)
19. So Jim can win simply by his being in situation S. The catch is, though, that Jim wants that situation to obtain, WjS, i.e.
Wj(Wj(one-box) & ~one-box)
But, given that `W’ distributes over conjunction, it is obvious that this state of Jim is incompatible with his wanting to one-box, Wj(one-box). Therefore, since one-boxing was a necessary condition for winning, Jim, if he wants to win, cannot win. He can only win if, irrationally, he doesn’t want to. It’s just a fool and his money that are never parted in this game.
20. Next, consider Greg Kavka’s Toxin puzzle. A perverse millionairess hands Jim a vial of toxin and pledges to give him $10 million if he forms the intention to drink it. The poison is particularly nasty -- although it will not kill Jim, it will, as he is perfectly well aware, leave him writhing in dreadful agony for a week. Jim would drink the toxin and suffer the consequences to get the $10 million. But remember the terms of the deal – he only has to intend to drink it; once that intention is formed he doesn’t actually have to drink the stuff, since he has already done what the millionaires bad him do. So the real `winning’ situation S* for Jim is his intending (I) to drink the toxin but not drinking it:
S*: Ij(drink) &~drink
Therefore, if Jim is rational, he’ll try to get himself into this situation; he’ll intend to bring it about; I S*, i.e.
Ij(Ij(drink) &~drink)
21. But, given that `I’ distributes over conjunction, it is obvious that this state of Jim is incompatible with his intending to drink the toxin, Ij(drink). Since his intending to drink the toxin is a necessary condition for his winning the $10 million, he cannot, under these circumstances, win the money.
22. Back, now, to the 1-day version of the Surprise Examination paradox. For the sake of simplicity, we shall assume that Jim has a private tutor and, for the sake of variety, we shall cash surprise not as not knowing that something will occur, but as not believing that it will. The tutor announces the surprise examination by saying to Jim `You will not believe that there will be an examination today, but there will be one’. The situation S** described by the tutor is
S**: ~Bj(exam) & exam
Jim knows the teacher to be honest, so he believes to obtain the situation described by her, BS**, i.e.
Bj(~Bj(exam) & exam)
23. But, given that `B’ distributes over conjunction, it is obvious that this state of Jim is incompatible with his not believing that an examination will take place. Since Jim knows the tutor to be truthful, it would be irrational of him not to believe what she says, but his believing what she says — Bj(~Bj(exam) & exam) — is incompatible with his not believing that there will be an examination, and, since the tutor has said that he will not believe that there will be an examination, he has to conclude that his respected tutor is not truthful. His believing that she is truthful entails that she is not truthful.
24. In the Newcomb case, having a rational want undermined the possibility of Jim getting something he wanted. Would that he could suppress that rational want! In the Kavka case, having a rational intention undermined the possibility of his making a fortune from forming a certain intention. Would that he could suppress the rational intention! In the Surprise Examination case, having a rational belief (that the teacher was speaking the truth when she told him that there would be a surprise examination that day) undermined the possibility of his believing something she said. Now, suppressing a rational want is difficult, especially when what you want is a large sum of money and it appears to be there for the taking. Suppressing a rational intention is difficult, especially if that intention is to avoid writhing in agony for a week. But suppressing a rational belief is not so difficult, especially when there is an equal and opposite rational belief in the offing. If Jim’s honest girlfriend tells him that she is going to wear an all-pink dress to the prom and that she is going to wear a an all-black dress to the same prom, then he would be well advised, pending getting the situation sorted out, not to buy a bow tie matching the colour of her dress. Likewise, his honest tutor has told him that there will be an examination that day, so he acquires the belief that there will be an examination that day. But, in the same breath, she tells him that he will not believe that there will be an examination that day. He is rational to endorse that too, so he also rationally does not believe that there will be an examination that day. Just as in the prom dilemma, where he found himself rationally subscribing to two opposing beliefs, the best thing for him to do in the present situation is to do nothing. In particular, to refrain from inferring anything — either that there will be an examination today or that there won’t be.
25. Now, as we noted earlier, the 1-day situation is just what the student in the standard (5-day) paradox envisages himself being in by Thursday evening / Friday morning if no examination has, up to that point, occurred. What he ought to do, as we have just seen, is to draw no inference about whether an examination will or will not take place on the Friday. Thus, on the Monday, when he has heard the teacher’s announcement, he will not rule out the possibility of a Friday examination, and so will not be able to proceed to the conclusion that there will be no examination at all that week. This is the correct result — for surprise examinations certainly do take place.