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This article is a short version of:

 The Surprise Examination or Unexpected Hanging Paradox
 Timothy Y. Chow
 The American Mathematical Monthly
 Volume 105, Number 1, January 1998, pp. 41 - 51
 http://arxiv.org/abs/math.LO/9903160

Before plunging into an attempt to "resolve" the paradox, let us pause for
a moment and think for a moment about what "resolving" a paradox means.

There are two steps involved in resolving a paradox.

1. Establish unambiguously what the paradoxical argument is.  To do this
it often helps to write down all the assumptions and the logical steps
explicitly, possibly in a formal language of some kind.

2. Exhibit the fault in the reasoning.  The fault may be due to incorrect
reasoning (e.g., dividing by zero or confusing two distinct things by
using the same word for both) or faulty '''assumptions'''.  In the latter
case it is often easy to identify exactly which assumption should be
rejected, although sometimes what we discover is a set of mutually
incompatible assumptions such that rejecting any one of them suffices
to eliminate the paradoxical reasoning.  In such a situation, simply
exhibiting the incompatible axioms without passing judgment on which
one is wrong is usually sufficient to dispel the confusion caused by
the paradox.  (We will see an example of this below.)

In some paradoxes the argument is presented clearly from the outset and
we do not need to bother with step 1.  Such is the case with many 1 = 0
"proofs."  However, some paradoxical arguments can be interpreted in more
than one way.  The classic problem of two envelopes, one with twice as
much money as the other, is an example.  In such cases, to resolve the
paradox, one should list the different possibilities for formulating the
paradoxical argument precisely and apply step 2 to each case.

With these ideas in mind, let us now turn to the prediction paradox.
[Note: all names refer to the list of papers that appears at the end
of this article.]

The first observation to make is that it is not completely clear what the
paradoxical argument is.  Therefore step 1 is necessary.  We must decide
what the argument is exactly.

How do we do this?  One natural way of interpreting the paradoxical
argument is that it consists of a series of formal deductions from the
statements in the announcement.  To write down the paradoxical argument
precisely, then, we need to translate the announcement into a set of
axioms and then reproduce the paradoxical argument as a set of formal
logical inferences from these axioms.

If we do this, as many have done, we quickly discover that in order to
reproduce the paradoxical argument we must define "unexpected" in a
self-referential way, e.g., following Shaw,

1. The exam will take place next week.
2. It will take place on a day that cannot be deduced in advance from the
preceding axiom plus '''this''' axiom.

Without the clause "plus '''this''' axiom" it is clear that one cannot carry
out the paradoxical argument.  In their argument, the students use the
fact that the exam is unexpected, and therefore the claim that the date
is not deducible from the axioms must be included in the axioms themselves.
Once we have axioms 1 and 2 the contradiction follows readily: if the exam
takes place on Friday then on Thursday the date will be deducible from
axiom 1 and therefore will be deducible in advance, contradicting axiom 2;
hence the exam cannot take place on Friday.  If the exam takes place on
Thursday, then on Wednesday the date of the exam will be deducible from
axioms 1 and 2 using the argument just given, but this contradicts axiom 2,
etc.  This argument can be made purely formal in different ways, but in
each case the source of the contradictory conclusion can be traced to a
self-referential axiom.  (See Medlin, Fitch and Windt for variations on
this theme.)

Does this resolve the paradox?  Well, not quite.  As other authors such
as Olin have observed, the crux of the paradox does not appear to have
been explained yet: why does the teacher's announcement appear to be
vindicated?  The authors mentioned above, along with several others who
interpret the paradox in more or less the same way, generally fail to
explain this part of the paradoxical argument---perhaps because the
process of locating a faulty assumption is usually the last step in
resolving a paradox, and hence the discovery of the self-referential
axiom creates the illusion that the last step has just been completed.
In fact, the mystery has not yet been explained.

This has led some authors (e.g., Olin) to reject the self-referential
approach lock, stock and barrel.  But there is no need to be so drastic.
The analysis above is not incorrect, it is just incomplete.  In addition
to the aforementioned self-referential axiom, there is another error in
the logic---namely that of confusing two different meaning of the word
"unexpected."  Here is how we can address the confused students.  "When
you are arguing that the exam must be unexpected, what exactly do you
mean by 'unexpected'?  Are you trying to argue from axioms 1 and 2 above?
If so, then your argument is invalid because you haven't actually defined
'unexpected' in a legitimate way.  Furthermore, when the exam seems to be
'unexpected' after all, it is only 'unexpected' in the sense that you
were psychologically unprepared for the test.  It is not 'unexpected' in
the sense of axioms 1 and 2 above, because that notion of unexpected is
not even well-defined.  The teacher's announcement is vindicated, but
only if you interpret 'unexpected' in a way that does not allow you to
carry out the paradoxical argument."  If they resist the idea that the
word "unexpected" is unclear, we can point out to them that from the way
they are arguing it appears that as soon as a day has been proved to be
a day on which an "unexpected" exam cannot occur, the occurrence of an
exam on that day suddenly becomes "unexpected"====  Therefore the students
====
need to clarify what they mean by "unexpected," and once they do this it
is easy to find the problem.

In other words, there are two problems with the students' reasoning.
The first problem is that they are confusing two different meanings
of the word "unexpected"; one meaning leads to the vindication of the
announcement, and the other meaning is the one needed for the paradoxical
argument.  The second problem is that the meaning needed for the argument
is not well-defined.  (This approach is more or less the same as Edman's.)

It might seem that we are done now, but there is more to come.  The point
is that the paradoxical argument is capable of another interpretation
which does not make use of illegitimate self-reference.  So we must go
back to step 1 and analyze this alternative interpretation, which (for
reasons that will become apparent) we will call the "epistemological
interpretation."

There are several versions of the epistemological interpretation; we give
one here (similar to Sorenson's) to illustrate the idea.  Reduce the
number of days to two for simplicity; let "1" denote "the exam occurs on
the first day" and let "2" denote "the exam occurs on the second day."
Let "Ka" denote "on the eve of the first day the students will know" and
let "Kb" denote "on the eve of the second day the students will know."
The announcement can then be written

	(1 <code>> ~Ka 1) & (2 </code>> (~Kb 2 & Kb ~1)) & (1 v 2).

We then introduce certain assumptions about knowledge, i.e., about the
logical properties of the predicates Ka and Kb: that after hearing the
announcement the students know the content of the announcement; that it
is not possible to "know" a false statement; and that the students know
whatever they can deduce from what they already know.  It is then easy
to translate the paradoxical argument into a proof that these assumptions
about knowledge are inconsistent.  (See, for example, Binkley or Sorenson
for a full demonstration.)

Once this interpretation of the paradox has been made precise (step 1),
we then need to proceed to step 2.  There has been much discussion of
which axiom about knowledge should be rejected.  One popular candidate is
the assumption that after hearing the announcement the students "know" the
content of the announcement.  For those who maintain that we can never
"know" things by authority or that we can never "know" things about the
future, rejecting this assumption is clearly an appealing alternative.
However, even those who are not skeptics of this kind have good reason to
reject the assumption, because the statement that the students are
supposed to "know" is a statement that says something about the students'
inability to know certain things.  Consider the statement, "John Doe knows
nothing."  Clearly, John Doe cannot know the content of this statement
even if the statement is true and it is uttered in his hearing.  The
prediction paradox is (allegedly) nothing more than a more intricate
version of the John Doe situation.  (This approach is essentially the one
offered by Quine, Binkley, Olin, and O'Beirne, but it is notably rejected
by Sorenson; we refer the interested reader to his papers for details.)

Although this is a promising candidate for the mistaken axiom, we can,
as we noted before, actually take a conservative view and leave open the
question of just which axiom it is that is wrong.  We can simply make the
observation that the said list of axioms for knowledge is inconsistent.
To resolve the students' perplexity we simply show them the list of axioms
and ask them which ones they agree with.  If they accept them all, then we
show them that the problem arises because they are making contradictory
assumptions about what "knowledge" is.  If they reject one or more of the
axioms, then we simply show them the point at which their paradoxical
argument breaks.  For the purpose of resolving the paradox, it is not
necessary to reach consensus about exactly which axiom is wrong.

To summarize: to resolve the paradox we observe that there are two ways
of interpreting the paradox---the "epistemological" and "self-referential"
interpretations.  Under the first interpretation, we resolve the paradox
by exhibiting the list of inconsistent assumptions about knowledge that
lead to the paradoxical argument.  Under the second interpretation, we
distinguish between two meanings of the word "unexpected" and show how
the confusion is dispelled provided we choose one meaning consistently.
(Note that we have not eliminated---and obviously cannot eliminate---the
possibility that somebody will come up with a way of interpreting the
paradoxical argument that bears no resemblance to any formalization so
far constructed.  But we can deal with that when it happens.)

One remark about the frequently heard comment that there is no agreement
about how the prediction paradox ought to be resolved: this comment can
convey the impression that the paradox remains a great mystery to this day.
In fact, most of the disagreement falls into one of two categories: there
is disagreement about exactly how the paradox should be interpreted, and
there is disagreement (among those who interpret the paradox in terms of
knowledge) about exactly which axiom of knowledge should be rejected.  But
these disagreements are not really substantive: as we have indicated, one
should really consider both interpretations of the paradox, and one does
not need to take sides on which axiom of knowledge should be dropped.  The
"controversy" over the paradox has more to do with mistaken ideas about
what constitutes a "resolution" than with anything in the paradox itself.

Finally, a word to the skeptic who thinks that all this analysis is wasted
effort: as a dividend from this analysis one can, in addition to gaining
some insights about the nature of knowledge, obtain some theorems about the
inconsistency of certain sets of axioms whose inconsistency is not obvious
at first glance.  See Kaplan-Montague for a splendid example of this.

                           - ''' - ''' - ''' - ''' -

Let me give my personal opinions on the literature, for those who are
interested in reading some of the papers and want some guidance.  A
good review of the articles up to 1965 appears in Bennett-Cargile.
For the self-referential approach, Shaw, Medlin, and Fitch are the
most enlightening.  For the epistemological approach it is good to
start with Sorenson, since he points out some errors in previous
papers and presents ingenious new variations of the paradox.  You
might then want to read Binkley and Olin (and maybe Quine because
it is a short paper and everyone refers to it).  Finally, Edman and
Kaplan-Montague are excellent papers which offer a different slant
from the rest of the pack.

It is good to be critical when reading the literature because many of
the papers have serious flaws.  Bennett-Cargile points out an error in
Lyon's paper, which is essentially repeated in the most recent paper
on the list (Janaway).  Nerlich is also seriously flawed, as pointed
out by Medlin.  Several authors do not seem to have studied earlier
work carefully; Olin is apparently unaware that her ideas have been
anticipated in Chapman-Butler, and Chihara makes some criticisms of
Olin's paper that are explicitly answered in the very paper he
criticizes====  So take things with a grain of salt.
====

                           - ''' - ''' - ''' - ''' -

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-- Timothy Y. Chow (tycchow@math.mit.edu)  Tue Dec  7 17:13:24 1993
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