L17 The Surprise Examination Paradox: Some Attempted Solutions
0. The final quiz will
take place on May 4th (that
was the clear majority verdict of my survey).
So Thursday April 29th will be an (optional) revision class
1. A feature that is liable to distract us from a
view of what is essential to the Surprise Examination paradox, is that the
clever student's argument involves ruling out five possibilities, corresponding
to the five days of the teaching week.
But this may be an irrelevancy, since a similar pattern of argument is
iterated on each day. We could eliminate
this `noise' and restrict our attention to the single-day version of the
paradox, where the teacher announces on Thursday morning `There will be an
examination tomorrow, but it will be a surprise in that you don't believe right
up till it actually takes place that there will be such an examination despite
my here announcing that there will be'.
Naturally, the students are puzzled by this, since they believe their teacher
to be totally honest. (Note that there
are authors such as Timothy Williamson and Ned Hall (Mind, 1999) who
think that the essence of the paradox is not distilled into this
single-day version.
2. W.V.
Quine published a solution in 1953.
Consider the single-day version.
According to Quine, when the announcement is made, the students do not
know whether or not there will be an examination tomorrow, because it is
possible that the teacher is mistaken.
In other words, the students cannot know that the teacher’s announcement
is true. So the students cannot know that there will be an examination
tomorrow. But it does not follow from
this that there will not be an examination tomorrow, because `We cannot know
that p is the case, therefore p is not the case’ is a version of the fallacy argumentum
ad ignorantiam. (Compare: `We cannot
know that God exists, therefore God does not exist’.) Quine’s solution, then, is that the students
cannot know that the teacher’s statement is true, and so cannot guarantee that
any conclusion drawn from it will be true – the inference may be unsound.
3. R. Shaw thinks that the paradox can be
strengthened in such a way that it escapes Quine’s solution. Instead of the announcement being made by a
fallible teacher, we could think of the examination as being determined by
rules of the school. We could build into
the story the requirement that the school rules are not broken. Now, if the school had just these two rules:
(i)
An examination will take place on one day
of the last week of term
(ii)
The examination will be unexpected, in the
sense that it will take place on such a day that on the previous evening it
will not be possible for the pupils to deduce from Rule (i) that the
examination will take place on the next day.
then pupils would be able to deduce that no examination could take place on the last day of the week, for otherwise Rule (ii) would be violated. But all the other days of the week would be possible exam days. We could rule out all those other days by adding a different rule:
(ii*) The examination will take place on such a day that on the previous evening the pupils will not be able to deduce from Rules (i) and (ii*) that the examination will take place on the next day.
Notice that (ii*) is self-referential. However, to point out that (ii*) is self-referential does not amount to solving the paradox. It would be nice to think that the self-referentiality makes (ii*) similar to the Liar but, as Sorensen points out (p.299) it is really more closely similar to a paradox discussed by Pseudo-Scotus, one version of which might be
Laurence
is an extremely handsome man
This
argument is invalid.
Another
objection to the claim that the clue to solving the paradox is in detecting its
self-referentiality is that self-referentiality does not seem to be essential
to the paradox. Certainly when stated in
its original form, no self-referentiality is present.
4. In the
single-day version of the paradox, the teacher's announcement could be
shortened to `There will be an examination tomorrow, but you don't believe that
there will be', or, shorter still, to `p, but you don't believe that p'. As Robert Binkley was the first to point out,
this resembles a Moore-paradoxical proposition.
The similarity becomes identity if the student, embracing the teacher's
announcement, asserts `p but I don't believe that p'.
5. There
would be nothing much gained by reducing one paradox to another, if the latter
were no less puzzling than the first.
Fortunately, however,
6. At the
heart of Wittgenstein's solution to Moore's problem is the observation that,
while `He believes that p', `I believed that p' and `If I believe that p, then
...' are standardly used to describe someone's state of mind, `I believe that
p' is typically used by a speaker to assert (perhaps hesitantly, sometimes
emphatically) that p. The latter is an
expression (Ausserung) of belief, not a description of one. Similarly, a speaker who says `I don't
believe that p' is denying that p or is expressing his refusal to accept that
proposition. Thus a speaker who says `p,
but I don't believe that p' is, or purports to be, simultaneously putting a
proposition forward and taking it back.
The Mooronic utterance is not a formal contradiction in that it is not
of the form `p and not-p', but, in making such an utterance, a speaker is
contradicting (etymology: speaking against) himself. Wittgenstein says of a Mooronic utterance
that it is similar to a contradiction in that it `plays a similar rôle
in logic'. He credits
7. Let's
consider a student,
8. Notice that the teacher cannot be charged
with dishonesty, for she was not to know in advance that
9. If a
teacher were to announce `There will be an examination next week and there
won't be', then her students' natural reaction would be to conclude that she
couldn't possibly mean that.
Pending clarification of her meaning, some students (the pessimistic
ones) will, with no other clues at their disposal, expect an exam; the
optimistic ones will expect no exam; others will think that the teacher was
merely raving. There is no course of
action that it is rational to take based on her utterance, for she is sending
out contradictory signals. Now, when a
teacher says `There will be an examination tomorrow, but you don't believe
that', she is not uttering a contradiction but a contingent statement
which is the conjunction of a claim about an examination and a claim about her
hearer's state of mind. However, if a
student, on hearing this statement, adopts it as his own, in the form `There
will be an examination tomorrow but I don't believe that' then he is,
on the Wittgensteinian account, contradicting himself, and so, as we have seen,
there is no course of action that can rationally be based on that claim. In particular, the student would be foolhardy
to use that claim as a basis for an inference about a future examination.
10. The student, then, must resist the temptation to adopt the teacher's second-person contingent statement in the form of a first-person, non-contingent statement. The proper way for the student to adopt her statement is to take it in her intended sense, viz. as a prediction about his state of mind. So, for example, the beginning of his reasoning, in the five-day version, would see the student musing on the teacher's announcement as follows: `If, by next Thursday evening, no exam has taken place earlier in the week, then there will be an exam on the following day, but I and my fellow students won't be in the state of believing that there will be an exam on that day'. Since, as we have seen, the consequent of this conditional is contingent -- both parts could be true, their conjunction need not be false -- the antecedent need not be false to preserve the truth of the conditional. In other words, the student cannot conclude that the exam must be held earlier than Friday, so he is in no position to rule out any day as a possible exam day.
11.A novel solution to the paradox of the Surprise Examination has been provided by Timothy Williamson, in his recent book Knowledge and its Limits (hereafter, KL). Williamson starts out by devising a two-dimensional (4 x 3) array of paradoxes, one member of which, a version of the familiar Surprise Examination (or Prediction- or Hangman-) Paradox, sits at the bottom right hand corner and is labelled `###4’. At the top left corner is a paradox of Williamson’s own invention called The Glimpse, labelled `#1’. A large number of paths lead stepwise via the intermediate paradoxes, from #1 to ###4, but the steps are small. Thus Williamson wants to say that any satisfactory solution to the Surprise Examination should work for these other paradoxes too, and a proposed solution that fails for one will not be a satisfactory solution for any. Williamson attempts to show, by the use of this array, that many past attempts at solving the Surprise Examination do not succeed in getting to the heart of the matter. Further, he offers a solution of his own that works, he claims, for the whole array and also for a number of paradoxes in decision theory. I want to show that Williamson’s strategy, ingenious though it is, breaks down, but that observing how it breaks down gives us the essential clue to a satisfactory solution..
12.Each of the 12 paradoxes in Williamson’s array concerns pupils reasoning about when an examination is to be held during an n-day school term. The Glimpse (#1) runs as follows:
A teacher’s pupils know that she rings all and only examination dates on the calendar in her office. At the beginning of term, the only knowledge they have of examination dates this term comes from a distant glimpse of the calendar, enough to see that one and only one date is ringed and that it is not very near the end of term, but not enough to narrow it down much more than that. The pupils recognize their situation. They know now that for all numbers i, if the examination is i+1 days from the end of term then they do not know now that it will not be i days from the end (o≤i<n). In particular, they know now that if it is on the penultimate day then they do not know now that it will not be on the last day. But they also know now from their glimpse of the calendar that it will not be on the last day. They deduce that it will not be on the penultimate day. They also know now that if it is on the antepenultimate day then they do not know now that it will not be on the penultimate day. They rule out every day of term as possible date for the examination (KL, p.135).
13.The second of Williamson’s paradoxes, #2, is just like #1, except that it is the school caretaker who catches a glimpse of the calendar. He is a reliable person, tells the pupils that there is only one ringed date and that it is not very near the end of term, so they now know this on the basis of his testimony, and reason as in #1. The reasoning concerns their present knowledge. Paradox #3 is just like #2, except that it is the (trustworthy) teacher who tells the pupils that the examination will not be very near the end of term. In Paradox #4, the teacher spells out for the pupils what, in the other versions of the paradox, they had worked out for themselves: She tells them that, for all i, if the examination is i+1 days from the end of term then they do not know now that it will not be i days from the end (o≤i<n ). And what she says appears to be true, because the date she has fixed for the examination is not very near the end of term.
14.In Paradox ##1, the situation is the same as in The Glimpse, #1 — the pupils catch a glimpse of the calendar — but they reason slightly differently. They figure that, when they wake up on the morning of the day that the examination is to take place, they will not know that the examination will not take place on the next day. Thus their reasoning here concerns future knowledge — they are thinking about a cognitive standpoint that they will occupy at some time in the future. In particular, they realise, at the beginning of the week, that if the examination has been set for the penultimate day of term, then they will not know that it will not be on the last day of term. But, remembering their glimpse of the calendar, they do know that it will not be on the last day of term. Hence they conclude that the examination has not been set for the penultimate day of term. With this conclusion in place, they proceed, in similar fashion, to eliminate the antepenultimate day of term as being the date for which the examination has been set.
15.The reasoning in Paradox ###1 is slightly different again. Here they figure that they will not know on the morning of the examination that it will be on that day. For if, as per their reasoning in Paradox ##1, they do not know that the examination will not be on the next day then, for all they know, it might be on that next day and not on the day that we, but not they, know to be the day of the examination. In ###4, the teacher tells the pupils what, in ###1, they had worked out for themselves, namely, that they will not know on the morning of the examination that the examination will be held on that day.
16.It will not be necessary to fill in the details of all the other paradoxes in Williamson’s array. The reader can do that for him/herself when armed with the knowledge that the four columns in the array could be labelled `Pupils glimpse’, `Caretaker snitches’, `Teacher discloses’ and `Teacher spells it out’, while the rows could be labelled according to the styles of reasoning involved as illustrated in the sketch of ##1 and ###1, bearing in mind how these differ from #1 in which the reasoning concerns only the present cognitive standpoint of the pupils.
17.The Glimpse may, in a certain respect, be regarded as a simplified version of the Surprise Examination Paradox, for, in the former, the pupils do not have to reason about the testimony of another person (the teacher) because they have the testimony of their own eyes. And they don’t have to bother with reasoning about their future states of mind since that was a factor introduced by the teacher into the Surprise Examination situation (`You will not know on or before the morning of the examination that it will be on that day’) but is not present in The Glimpse. Any proposed solution that targets perimetric factors (inessential factors that are at the perimeter, not at the centre of the paradox) has to be mistaken. The comparison of the Surprise Examination to The Glimpse shows, for example, that any solution to the former that depends on its being self-referential must be incorrect, since there is no essential self-reference in the latter. Any other attempt to assimilate the Surprise Examination to the Liar Paradox would also be inapposite because, as Williamson has noted, the teacher’s claim can be true but the Liar’s utterance cannot be true.
18.Some writers have proposed, as a solution to the Surprise Examination, that the teacher’s announcement is false or lacking in truth-value. Williamson believes that comparison with the Glimpse shows this to be a non-viable option. He writes:
The teacher’s announcement [in the Surprise Examination] corresponds to the claim in the Glimpse that if the examination is i+1 days from the end, then the pupils do not know that it is not i days from the end. Thus to say that the announcement is false or truth-valueless corresponds to saying that that attempted expression of the content of the pupils’ knowledge in the Glimpse is false or truth-valueless. To say that in the Surprise Examination the pupils cannot know in advance that the announcement is true corresponds to saying that in the Glimpse the pupils cannot know the limitation on their knowledge by reflecting on the poverty of their perceptual knowledge in that case. The obvious implausibility of these claims for the Glimpse points up their implausibility for the Surprise Examination too. Advance knowledge that there will be a test, fire drill, or the like of which one will not know the time in advance is an everyday fact of social life, but one denied by a surprising proportion of early work on the Surprise Examination. Who has not waited for the telephone to ring, knowing that it will do so within a week and that one will not know a second before it rings that it will ring a second later? Any adequate diagnosis of the Surprise Examination should allow the pupils to know that there will be a surprise examination (KL, p.139).
19.But is this suggested constraint on the adequacy of a diagnosis really as compelling as it first seems? That telephone call we simply knew would be made within the week — after all, didn’t the manager assure us that he would let us know by ‘phone before the end of the week whether our future was with the Spurs — well, we’ve been waiting and waiting and it’s now just…one second before the end of the week. At this point, memory of its failure to have rung earlier in the week would undermine what, at the beginning of the week, we had supposed was knowledge of the truth of the proposition that the telephone would ring within the week. We thought we knew the ‘phone would ring, but we were wrong — we did not know it. So to accept the condition on adequacy of a solution that Williamson recommends is by no means compulsory. If, by Thursday night, no examination has taken place, so that Friday is the only day remaining for a promised surprise examination, then, if the pupils know that the examination will take place on Friday, its taking place on Friday cannot be a surprise to them.
20.Williamson says that the paradox of the Glimpse depends on a concealed use of the KK Principle (that if one knows p, then one knows that one knows p), a principle that, in an earlier chapter of his book (KL, p.114 ff.), he argued is false, even in a restricted version in which the proposition known to the knower has come to his attention. According to Williamson, the pupils in the Glimpse situation have two bits of current knowledge that we could write as follows:
K(p → ~K~l)
K~l
(where `p’ is `the examination will be held on the penultimate day’ and `l’ is `The examination will be on the last day’.) But, for the pupils to reach their conclusion `~p’ from the premise `p → ~K~l’ they need not `~l’ as an additional premise, but `K~l’ – for then they can make their deduction by modus tollens. The theorist is operating with the reasonable assumption that, if a pupil knows some premises and deduces a certain conclusion from those premises, then he knows the conclusion. But, as we have seen, one of the premises that the pupil needs, if his deduction is to be valid, is `K~l’, so the theorist must assume that the pupil knows this premise – i.e., KK~l. So the theorist illicitly (if the KK principle is false) makes use of `KK~l’, when his original premise was only `K~l’ (KL, p.140).
21.Now, as Williamson points out, this diagnosis of The Glimpse does not generalize to the Surprise Examination, since the reasoning within the latter paradox can be reconstructed in a way that does not invoke the KK principle:
22.A careful analysis shows that what the theorist really
needs is the assumption that the pupils know on the first morning of term that
they will know on the second morning ... that they will know on the penultimate
morning that they will know on the last morning the truth of the teacher's
announcement.... The Glimpse too can be reconstructed with premises attributing
the same number of iterations of knowledge to the pupils, but all concerning
present knowledge, in place of the KK principle (KL, p.140).
23.However this difference between the Surprise Examination
and The Glimpse in its standard form does not require any major adjustment to
Williamson’s solution as already sketched.
For if my knowledge of the truth of the teacher’s announcement does not
guarantee that I know that I know its truth, then a fortiori the multiply iterated knowledge claim that featured in
the aforementioned `careful analysis’ is susceptible to falsity. In other
words, the pupils’ argument goes wrong because the multiply iterated K premise
is false. Or, in Williamson’s words:
The iteration of knowledge operators leads sooner or later
to falsity through a process of erosion resulting from the need for margins for
error. This applies just as much when the knowledge operators refer to
different cognitive standpoints. If anything, it applies with more force:
knowledge of future knowledge is usually less exact than knowledge of present
knowledge, so wider margins for error are needed (KL, pp.140-1).
24.Clearly the `margins for error principle’ plays a central rôle in Williamson’s diagnosis of these paradoxes. And I shall argue that this principle is false. What this would mean is that The Glimpse, since it rests on one such principle, is solved, and Williamson’s own solution would be otiose. The Surprise Examination, however, can be stated, and standardly is, with no mention of this principle. In the next lecture, I shall try to show a mistake in Williamson’s reasoning and to argue that this paradox has a solution entirely different from his.