A solution to the unexpected day paradox
By Albert Frank
Several versions of the unexpected day paradox can be find. Here is one:
On a Monday morning, a teacher says to his class, "I will give you a surprise
examination someday this week. It may be today, tomorrow, Wednesday, Thursday,
or Friday at the latest. On the morning of the examination, when you come to class,
you will not know that this is the day of the examination."
A student reasoned as follows: "Obviously I can't get the exam on the last day,
Friday, because if I haven't got the exam by the end of Thursday's class, then on
Friday morning I'll know that this is the day, and the exam won't be a surprise. This
rules out Friday, so I now know that Thursday is the last possible day. And, if I don't
get the exam by the end of Wednesday, then I'll know on Thursday morning that this
must be the day (because I have already ruled out Friday), hence it won't be a
surprise. So Thursday is also ruled out."
The student then ruled out Wednesday by the same argument, then Tuesday, and
finally Monday, the day on which the professor was speaking. He concluded:
"Therefore I cannot get the exam at all; the professor cannot possibly fulfil his
statement." Just then, the professor said: "Now I will give you your exam." The
student was most surprised!
++++++++++++++++++++++++++++++++++++++++++++
There are a lot of studies about this old paradox. The "classical solutions" tries to
proof that the recurrence is valid only for the last day. This looks very dubious.
Several logicians, among which Raymond Smullyan had other ideas. Mainly:
The teacher said two things: (1) You will get an exam someday this week; (2) The
exact day will be a surprise, you won't know on the morning of the exam that
this is the day. It is important that these two statements should be separated. It could
be that the professor was right in the first statement and wrong in the second …
Suppose Friday morning comes and I haven't got the exam yet. What would I then
believe? …
+++++++++++++++++++++++++++++++++++++++++++
After reading a lot about this paradox, I came to the following conclusion: This
paradox is a good example of semantic paradox. In the context, the word surprise
does not have a constant meaning, and the use of it (like if it would have a constant
same value) is an abuse! In his first sentence, the teacher uses the word surprise
with its "Monday's value", in his second sentence with its "Friday's value", and in his
third sequence with its "Thursday value".
+++++++++++++++++++++++++++++++++++++++++++
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