Phil 2511: Paradoxes
L16 The Surprise Examination Paradox
0.
Please note that we now have an `Easter break’, and
that the next lecture will be on April 20.
During this time, you should study the rather extensive notes I handed
out at Tuesday’s lecture, and also catch up on any reading on the Liar paradox
that interests you. Also, you can
prepare yourself for the last section of the course by looking at some work on
the Paradox of the Surprise Examination.
1. There is
a very thorough account of pre-1988 solutions to the Surprise Examination
Paradox in Roy Sorensen, Blindspots, Chaps. 7-9. I’ll draw quite heavily on Sorensen in this
part of the course. Sorensen gives lots
of useful references; some additional ones are R.L. Kirkham, `The two paradoxes
of the unexpected examination’, Philosophical Studies 49 (1986),
pp.19-26; Christopher Janaway, `Knowing about surprises: a supposed antinomy
revisited’, Mind 98 (1989), pp.391-409; Don S. Levi, `Surprise!’, The
Southern Journal of Philosophy 38 (2000), pp.447-464. In a subsequent lecture, I shall say
something about Timothy Williamson’s solution in his new book Knowledge and
its Limits (Oxford University Press, 2000).
2. Sorensen
reports that the paradox (sometimes also called the Prediction Paradox or the
Hangman Paradox’ first appeared in the philosophical literature in D.J.
O’Connor, `Pragmatic Paradoxes’, Mind 57 (1948). (Those writing a term paper on the pragmatic
paradoxes may find this paper useful, as well as a reply to it by L.J. Cohen in
Mind 1950.) Both O’Connor and
Cohen assimilate the teacher’s announcement to a Moore-paradoxical utterance,
such as `I believe that Uranus has three moons, but it doesn’t’. The point is that it might be true
both that Uranus does not have three moons and that I believe that Uranus has
three moons. However, if I myself make
that announcement, it is bound to be false. This week’s lecture, which
introduces the subject of the Paradox of the Surprise Examination largely
extracted from two papers of mine: `Inescapable Surprises and Acquirable
Intentions', which you can find in Analysis
53 (1993), pp.93-99 and `Examining boxing and toxin’, Analysis 63 (2003), pp.242-244.
You should also, of course, consult the relevant sections of your two
course texts (it’s called the Paradox of the Unexpected Examination in Clark,
pp.206-208).
3. A
teacher, at school's end on Friday, says to her class `I'm going to give you a
surprise examination one day next week'.
A bright pupil figures out that next Friday can't be that day since, if
no exam is given by close of school on Thursday, there would be only one day
left for the exam; students would know this and therefore they would expect the
exam that day -- it wouldn't be a surprise.
But, with Friday ruled out, we can go through the same kind of reasoning
to rule out Thursday: by Wednesday
evening, with Friday already ruled out as surprise day, only Thursday would
remain as possible, so the exam would have to be given on that day; no
surprise. A similar line of argument
rules out Wednesday, Tuesday and Monday as possible exam days, hence the
argument shows that the surprise exam promised by the teacher cannot take
place. But then, on, say, Tuesday, the
teacher walks into the class and gives out the exam papers. This comes as a great surprise to the
students, especially those who regarded the clever student's argument as
impeccable. Clearly it's not
impeccable. But where does it go wrong?
4. Some
writers have suggested that the paradox is essentially about time, and
shows that some contingent statements made about the future cannot be
true. But Roy Sorensen has presented a
number of entertaining variants in which temporality is not an issue (see his Blindspots,
Chap.8).
5. Sorensen
usefully discusses Greg Kavka’s toxin puzzle for the purpose of comparison. In this, a guy (let’s call him `Tox’) is
promised $1M if, by noon
tomorrow, he intends to drink a vial of toxin that afternoon. The toxin will, he knows, only make him
violently ill for a day. The catch is
that Tox is not to be paid the money for drinking the stuff, but for
performing the apparently simpler task of having the intention to drink it. As soon as Tox has the intention to drink it
he can collect the money; there's no penalty if he subsequently doesn't drink
it. But that's the trouble. After he's earned the money by intending to
drink the toxin, there's no need for him to actually drink it, and every reason
for him not to drink it, since it would leave him writhing in agony for a day. So he can be quite confident that he wouldn't
drink it. But then, if he knows he won't
drink it, he can't form the intention to drink it!
6. The
person who makes the offer to Tox is generous but nasty. He certainly thinks he may have to pay up,
but only after reaping the sadistic pleasure of seeing a squirming Tox
resolving to cause himself great suffering.
This malebenefactor has a taste for mental, rather than physical
cruelty; that's why he made the payout conditional on the intention rather than
the implementation. So the
malebenefactor is prepared to bet that, by midday
tomorrow, Tox can intend to drink the toxin even though he may never actually
drink it. Tox, let's suppose, is
impressed by the fact that his malebenefactor has a history of safe bets, but
if he endorses the malebenefactor's hunch by asserting `I intend to drink the
toxin and I shall not drink it', his utterance is Mooronic, a Wittgensteinian
contradiction. Therefore he can infer
anything from it (if he believes in the principle ex contradictione
quodlibet) or, more prudently, he can refuse to infer anything from
it. But the correct thing for him
to do is to discount those options and read his malebenefactor's claim in the
way it was intended, viz., as a prediction about what he, Tox, would do. And, as a matter of fact, this prediction is
a reasonable one that can be made quite honestly if the malebenefactor knows
Tox to be like you and me.
7. Put yourself
in the position of Tox, seconds before noon
tomorrow. The loaded vial is in your
hand and your malebenefactor is taunting you by waving a large wad of hundred
dollar bills right under your nose. You
fight off the idea that you won't drink the poison, you steel yourself and, as
12 o' clock strikes, you swiftly raise the vial to your lips with the intention
of downing it in one gulp. So you get
your $1M and, if you're extremely lucky, in your haste to get the vile liquid
drunk, you miss your mouth and spill it down your shirt, thereby vindicating
your malebenefactor's hunch. Of course,
you can't count on that, and the odds are that you will suffer agonies, but
just for a day.
8. That you can form the requisite intention is
easier to see by considering a five-offer version in which, if you win the $1M,
the offer gets renewed up to a maximum of four more times. By the time the fifth offer comes round, you
would be back in the single-offer situation, with the difference that you now
have enough money in your pocket for a modest apartment on the Lower
East Side and may be having recurrent nightmares of
four days in hell. You can be almost dead
certain in advance that you wouldn't drink the fifth draft. But, if you don't intend to drink the fifth,
you would have no motive for qualifying for the fifth offer and that
removes an incentive to drink the fourth, something you will need to try to do
in order to have had the intention to do it.
And so on. So in this case the
stakes are higher. Failing to drink the
first draft could cost you up to $5M.
But, looking dispassionately at your future intentions, you cannot be
dead certain what the state of your mind will be after up to four doses of
poison; you can't predict what intentions you will have at that time. So you cheerfully form the intention to drink
the first draft. Who knows but that, in
a couple of weeks' time, you will be a rich person with five toxin-stained
shirts as testimony to your serendipitous clumsiness.
9. There is
a phenomenological connection between Kavka’s puzzle, the Surprise
Examination and Newcomb’s Paradox. This
may be a hint that there is a common solution………
The teacher says to Calvin, `There will be an exam later today
but you don’t believe that there will be, so the exam will come as a surprise’.
Calvin cannot stop the teacher doing what she likes with regard to giving the
exam, nevertheless he wants to win, albeit a small victory, by bringing it
about that what she says is false. He has no control over the truth or falsity
of the first conjunct of her statement, and so can only win by believing that
there will be an exam. But, since he knows that the teacher tells the truth at
least 95% of the time, in the absence of any relevant evidence to the contrary,
Calvin cannot rationally believe that she is not telling the truth on this
occasion. And she said that he does not believe that there will be an exam that
day so, if he is rational, he will accept it as true that he does not believe
that there will be an examination that day. He can only believe that there will
be an exam that day (and hence win) if he is irrational. He has to believe that
there will be an exam that day while somehow refusing to accept what the
smallest amount of reflection would force him to accept — namely, that he does
not believe it.
The Predictor in the eponymous TV game show says to Calvin,
`As you can see, there is $10,000 in transparent Box A. In Box B, which is
opaque and closed, and whose content cannot now be changed, I have put nothing
if I have predicted that you will opt to take the content of both boxes. You
can either take the content of both boxes or take just the content of Box B,
and if I have predicted that you will do the latter, I will have placed
$1000,000 in Box B’. Calvin cannot stop the Predictor doing what she likes with
regard to the content of Box B, nevertheless he wants to win the maximum of
$1,010,000 by bringing it about that he two-boxes, while the Predictor predicts
that he is a one-boxer (and so has deposited $1,000,000 in Box B). But, since
he knows that the Predictor is an excellent psychologist who has played this
game with hundreds of different contestants and, on the basis of detailed
studies of their psychological profiles, has predicted right at least 95% of
the time, in the absence of any relevant evidence to the contrary, Calvin
cannot rationally believe that she will be wrong on this occasion. And so he
cannot rationally choose to two-box, and so, if rational, will one-box, thereby
depriving himself of the chance of achieving his goal of winning $1010,000. He
can only achieve that winning goal if he is irrational. He has to actually be a
one-boxer (not just pretend to be one — the Predictor is wise to that ploy)
while somehow refusing to accept that, when the crunch comes and he has to
choose, he will be a two-boxer.
The president of Philip Morris says to Calvin, `Here is a vial
of toxin. I’ll give you $10M if you intend to drink it — just for having that
intention. Whether or not you subsequently actually drink it, you keep the
$10M. The toxin will not kill you, but
it is really nasty — it will have you writhing in agony for five days’. Calvin
cannot alter the terms of the deal that the president has set out,
nevertheless, he wants the best outcome for himself, which is obviously that he
forms the intention to drink the toxin but doesn’t actually drink it. However,
he is aware that, on at least 95% of occasions, refraining from performing a
voluntary action is preceded by the intention to refrain from performing it,
which is obviously incompatible with forming the intention to perform it. And
so he cannot, on this occasion, rationally intend to drink the toxin while
harbouring the plan to not drink it after that intention has been formed. He
can only achieve the goal of intending to drink the toxin while avoiding the
agonized writhing if he is irrational. He has to really intend to drink the
toxin (the president does not pay out to fakers) while somehow refusing to
accept that, when the crunch comes and he is to carry out that intention, he
will acquire the intention to refrain from drinking it.
Despite apparently very different
set-ups, the catch is common to all these challenges with which Calvin is
presented, as is the resulting bind where the possibility of winning is
reserved only for the truly irrational. Calvin is currently busy trying to
become irrational but, given the huge prizes he knows to be available for
success in that venture, the more he tries, the more he shows himself incapable
of succeeding.