philosophy paper 3
Valerielee
12.1UnderstandingthePredictionParadox.pdf
8/27/2015
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Phil 2: Puzzles and Paradoxes
Prof. Sven Bernecker
University of California, Irvine
Lecture 12.1
Understanding the
Prediction Paradox
The Unexpected Exam Paradox
Imagine a schoolteacher who announces:
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Announcement
Class, there will be an exam next week. I will not tell you on
which day I will give you the exam; it‘s timing will be a surprise.
You will not be able to justifiably believe (know), prior to the day
of the exam, on which day the exam will be held.
One version involves a warden
who schedules an unexpected
hanging for a prisoner. The
umbrella term for the different
versions is Prediction Paradox.
• The exam will not be totally unexpected, since the class knows
that it will occur sometime during the next week (month, year …)
• But the exam is unexpected in this sense: on the evening prior to
the day on which the exam is held, the class will not be able to
justifiably believe (know) that it will occur on the following day,
even though it knows (has evidence to believe) the teacher’s
announcement.
• Intuitively, there is no reason why the teacher should not be able
to give a surprise exam.
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A student who is very good at logic claims that the teacher cannot give a surprise
exam. She argues as follows:
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Student’s Reply
The teacher‘s announcement has two parts: that she will give an exam, and that it
will be unexpected. The teacher cannot give the exam on Friday (assuming this to
be the last possible day of the week); for, by the time Thursday evening arrives, and
we know that all the previous says have been exam-free, we would have every
reason to expect the exam to occur on Friday. So leaving the exam until Friday is
inconsistent with giving an unexpected exam. So Friday is out. For similar reasons,
the exam cannot be held on Thursday. Given the previous conclusion that it cannot
be delayed until Friday, we would know, when Wednesday evening came, and the
previous days had been exam-free, that it would have to be held on Thursday. So if
it were held on Thursday, it would not be unexpected. So Thursday is out. Similar
reasoning shows that there is no day of the week on which it can be held. So the
teacher cannot do what she announced she would do.
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Possible day for
exam
Reasons to exclude day from being the day of the unexpected exam
Friday Would be last possible day to hold exam, and if the exam were held on Friday it
would not be a surprise.
Thursday Given that Friday has been eliminated, Thursday is now the last possible day to
hold the exam. But if the exam is given on Thursday, it would not be unexpected
because Friday has been eliminated.
Wednesday Given that Friday and Thursday have now been eliminated, Wednesday is now
the last possible day to hold the exam. But if the exam were given on Wednesday,
it would not be unexpected because Friday and Thursday have already been
eliminated.
Tuesday Given that Friday, Thursday, and Wednesday now have been eliminated, Tuesday
is the last possible day to hold the exam. But if the exam were given on Tuesday
it wouldn‘t be unexpected because Friday, Thursday and Wednesday have
already been eliminated.
Monday Monday is now the only day left on which to hold the exam. But given that this is
so, if the exam were to be held on Monday, it would not be unexpected.
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• The “Student‘s Reply“ is a reductio ad absurdum: the student assumes
that the teacher‘s announcement is true and then shows that this leads to
a contradiction, and hence that the assumption needs to be rejected.
• This is a paradox because there is an apparently impeccable argument
for a conclusion that seems clearly false. Clearly the teacher can do what
she announced she would do. A surprise exam is possible.
• There is widespread agreement that the conclusion of the argument is
false. But what is wrong with the reductio argument of the
“Announcement“?
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Presuppositions of the paradox:
• The teacher is speaking the truth when she announces an
unexpected exam.
• Neither the teacher nor the student forget about the exam during
the week.
• The teacher and the student keep track of the days of the week
and remember whether an exam has been held.
• During the week the student gets no additional sources of
evidence relevant to the teacher‘s announcement.
• The student is an ideal reasoner.
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History of Prediction Paradox
• The paradox has its origin in a historical event. In 1943/44,
it was announced on Swedish radio that a civil defense
exercise would take place one day of the following week
but that no one would be able to predict the day of the test
in advance.
• The Swedish mathematician Lennart Ekbom was the first to
detect the paradox lurking behind the radio announcement.
Ekbom taught at Östermalms College in Stockholm.
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Doris Olin
A Philosopher at the
University of York,
Canada
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Proposed Solutions
• Reinterpreting the teacher‘s announcement (Olin, “Believing in
Surprises“, pp. 42-3)
• Disallowing reasoning backwards in time (Olin, “Believing in Surprises,“
pp. 41-2)
• Denying knowledge of future contingents
• Rejecting the KK-thesis (Olin, “Believing in Surprises,“ pp. 46-51)
• Denying knowledge of the teacher‘s announcement (Olin, “Believing in
Surprises,“ pp. 51-7)
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