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Logic
— the Art of Reasoning
Mathematics
— the Art of Studying Patterns Using Logic
The Surprise Paradox, Is It a Paradox?
Uri Geva
MathVentures, a Division of Ten
Ninety
Document Version 0.3 (draft)
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Abstract
Surprisingly, there is a good reason why the scenario
set forth by the surprise paradox results with a surprise Thus, it is no
paradox.
Background
The Surprise
Examination Paradox, also known as
the Prediction
Paradox, "first appeared in print
in D.J. O'Connor (1948). The paradox originated earlier when a Swedish
mathematician, Lennart Ekbom, discussed at Ostermalms College a difficulty
he had noticed with an announcement by the Swedish Broadcasting Company
during World War II. The announcement said that a civil defense exercise
was to be held during a particular week. In order to ensure preparedness,
no one was to know in advance which particular day of the week the exercise
would be conducted. Ekbom noticed that the unexpectedness of the exercise
was problematic, which forms the core of the Surprise Examination Paradox.
This paradox appears in many guises and under many names, including among
others the Prediction and Hangman paradoxes. All have essentially the same
form as that represented by the surprise examination, on which we focus
here." [Jonathan L. Kvanvig, Department of
Philosophy, Texas A&M University, College Station, TX 778434237, (409)8455679;
(409)6906263, Source: http://www.missouri.edu/~kvanvigj/papers/epistemicparadoxes.htm]

"During World War II the Swedish Broadcasting Company
made the following announcement on the radio:
"A CIVIL DEFENSE
EXERCISE WILL BE HELD THIS WEEK. IN ORDER TO MAKE SURE THAT THE CIVILDEFENSE
UNITS ARE PROPERLY PREPARED, NO ONE WILL KNOW IN ADVANCE ON WHAT DAY THIS
EXERCISE WILL TAKE PLACE." [Note that a
minor misspelling, the source of which is not known, was corrected.]
"A Swedish mathematician, Lennart Ekbom, immediately
recognized something odd about this announcement. He discussed the situation
with his class at Ostermalms College. From there is spread around the world.
By 1948 it had reached print in the British magazine mind.
In 1951, Michael Scriven announced ‘A new and powerful paradox has come
to light.’ " [Bryan Bunch, Mathematical
Fallacies and Paradoxes, (1982, Dover Publications, Inc.), p, 34.]
The Paradox
This well known Paradox is usually stated like so:
Concise paradox Statement. A teacher
tells her students that next week she will give the class a surprise exam.
Some versions of this paradox include that the teacher adds something like
"you will not know in advance when I’ll give the exam."
Question. Is that possible?
Student Analysis. If the exam is not given
by Friday, then the student would know it is about to be given on Friday
and therefore it would not be a surprise. Hence, the surprise exam cannot
be given on Friday of next week. Then, if the exam is not given by Thursday,
then the student would know that it must be given on Thursday, since it
cannot be given on Friday. But again it would not be a surprise. Hence,
the surprise exam cannot be given on Thursday of next week. Repeating the
same argument we can exclude each day of the week. Since the teacher is
assumed to be truthful, he either gives a surprise exam or no exam at all.
Answer. A surprise exam cannot be announced
in advance for a given period of time.
The Paradox. However, we know from experience
that a surprise exam can be announced in advance for a given period of
time and the students will be surprised on the day it is given. How
come?
Martin Gardner’s Version of the
Paradox
[Source: Bryan Bunch,
Mathematical
Fallacies and Paradoxes, (1982, Dover Publications, Inc.), p, 35.]
"One of the clearest versions come from Martin
Gardner, columnist for Scientific
American. A loving husband tells his
wife that she will receive an unexpected gift for her birthday. It will
be a gold watch. The husband is the person who has set the conditions.
"Now the wife uses logic. Her husband would not
lie to her. Since he has said the gift would be unexpected, it will be
unexpected. But she now expects a gold watch. Therefore, it cannot be a
gold watch.
"But, of course it is.
"And it is unexpected, for she had used logic
to show that it could not be a gold watch."
Others’ Analysis
There are various opinions (see references below.)
Willard Van Orman Quinn and some others argue essentially that it is not
possible for one person to know what it is in the mind of another. This
view is summarized thus:
"…Logic
does not apply to another person’s thoughts. The wife cannot reason about
her husband’s peculiar statement that the gold watch will be unexpected.
This is as forbidden as division by zero." [Source: Bryan Bunch, Mathematical
Fallacies and Paradoxes, (1982, Dover Publications, Inc.), p, 35.]
My Analysis — Not a Paradox
This is no paradox. That is, it is not, or, at least,
should not be surprising (pun intended) that the students are surprised
when the exam is given. Whether the surprise is supposed to be derived
from the timing of the event (the Swedish exercise or the surprise exam)
or from the action itself (Gardner’s gift paradox), my view is the same.
First we can simplify it by stating it in a simpler
manner.
We start by clarifying the notion of surprise.
I agree with Gardner that a surprise event is an unexpected
event. Or, a surprise
is when the unexpected
happens. The purpose of the additional part of the statement that is some
times included is to clarify this point. This does not effect my analysis.
The original statement can now be revised and
stated in a simpler form:
I am about
to say your name at noon but you will not expect me to say your name at
noon.
It is important that I give you sufficient time between
the time I tell you what I am about to do and the time I state I will do
it. Also, it is important that both of us are logical persons and we both
know it about each other. That is, I know that you will execute a logical
reasoning to evaluate my statement. And you have no doubt about my reasoning
and intention to be truthful.
Your Analysis. Since you know the specifics
of the event and since you trust that my intention is to surprise you without
lying, you conclude that the event cannot happen as I state it. (For simplicity,
you can assume that I have full control over my ability to carry out the
action at the time I say I would.) Either the timing or the content of
the event must be beyond my control and therefore it will not happen as
I predicted. For you know only about my truthfulness but
you don’t know what will happen.
The paradox. I do say your name exactly
at noon, a fact which you did not expect (it surprises you).
Reasoning. The initial expectation to be sorprised
is base on lack
of knowledge. That
is, because ordinarily if one does not know when an event will happen,
when it does happen one is surprised. This is true in the case of the surprise
exam, before the students carry out the complete reasoning analysis of
the situation.
What really happens is the surprise that results
from having
too much knowledge. You
do not expect me to say your name at noon and the students
do not expect the exam at any time because, based on a careful analysis,
you and they reached the affirmative conclusion that the predicted event
cannot take place under the stated conditions. This certainty is then contradicted
when the exam is given, whenever it is (within the specified period),
and when I say your name at noon.
The element of surprise is always present. If
the students do not do their logical reasoning, surprise is the result
of not knowing the timing of the exam. If they dom then it is the result
of reaching a conclusion that the exam cannot be given at all.
So the teacher outsmarted the students by creating
a situation in which she can indeed surprise her students.
When people consider this "paradox" the goal is
to claim that a truthful teacher cannot make such a statement. My analysis
shows that the teacher successfully surprises the student not by
lying to them but by forcing them to reach a conclusion that will necessarily
fail to predict her action.
Note that an essential component of the teacher’s
success is the fact that surprise is an ambiguous even — it can
be caused by unexpected events and by seemingly impossible event.
If we distinguish between two different type of surprises then a truthful
teacher cannot make this statement.
The Importance of Some Elements of the Apparent Paradox
Without these elements the paradoxical situation
cannot be developed. At the same time, these elements are also necessary
for the argument that the situation is not paradoxical.

Knowledge vs. conclusion based on logical analysis.
If my statement referred to an event that you can know whether or not it
will happen, there will be no surprise following your analysis. For example,
if I said "I will say your name when the sun is at the Zenith over our
heads yet it will surprise you." Or, "The sun will rise tomorrow and you
will not expect it." Depending on our location at the time and our ability
to be relocated by the deadline, you can know whether the
even I predict will happen or not. If we are not at the North Pole during
its dead winter, you know that the sun will rise tomorrow. And If we are
in the US and the time now is 11:00 AM, you know that the sun will not
be in its zenith over our heads. But if we are in Colombia during the spring
or fall equinox (especially if we are near the equator), you can conclude
that it will happen.
It is necessary that you should not be able
to know whether or not the even I predict will happen or not and the only
way you can reach a conclusion with respect to the future of this event
is by logical reasoning.

Complete Logical Reasoning. I must be sure
that you will carry out the logical reasoning, of which I know the conclusion
that you will reach, if you make no mistake. And to facilitate your complete
reasoning, I must give you the all time that you need to complete it. Most
people, who are not logicians, will see no paradox in this case if they
are not directed to follow the detailed reasoning. For example, ordinarily,
when a class is told about a surprise exam it accepts the statement and
the conditions it implies, including the surprise, and is surprises whenever
the exam happens without a second thought. On the other hand, an
ordinary woman will accept the gold watch and ask her husband, "why did
you ruin the surprise?"
With regard to the argument that one cannot depend
(as oppose to use) on logical reasoning when evaluating the state of mind
of another person [See Logic
Does Not Apply To Another Person’s Thoughts.], it should be noted
that, there is no need for this exclusionprohibition since the simple
argument I introduced above resolves the paradox.
Military Practical Application
The same tactic is a common practice for military
planners or strategygame players like chess players. In general, one opponent
attack the other in a manner the second opponent has concluded to be impossible.
More specifically, attackers create a pattern of attacks that leads the
defender to conclude the way the next attack will be conducted. Then the
defender reasons that, because the attack pattern is so predictable, the
attacker would assume that defender is about to take the appropriate defense
against it. Therefore, the defender concludes that the attacker would not
make this attack and therefore the defender plans his defense against another
attack. The defender is then surprised when the attack is conducted exactly
according to the predictable pattern. For example, during several consecutive
attacks, a strike force has always flanked the defending force from the
left. Each time the defending was able to repel the attack. Then the defending
commander concludes that on the next attack, the offensive force will strike
from the right. So he shifts all of his forces to the right flank in order
to destroy the attacking forces once and for all. This is his mistake.
The attacking commander created this pattern, in order to get the defending
commander to reach this very conclusion. All along he planned to keep striking
from the same direction knowing that once the defending commander reaches
the erroneous conclusion, the left flank will be defended with little force
and his attack will succeed.
Timing Consideration — Cannot Be a Surprise In the First
Place
When a teacher announces to the class
"Put your books
and notebooks away and have only a pencil and a clean sheet of paper on
your desks. We are going to have a surprise exam."
Or, when civil defense authorities announce to the
public
"We are about
to activate the civildefense alarm and start a surprise civildefense
drill."
Some time, perhaps a few minutes, elapses before
the exam actually commences. The length of this time gap, the duration
from the announcement and the actual commencement of the exam, should have
no bearing on whether the students are or the public is surprised. So,
say a teacher announces
"Next week
we will have a surprise exam."
or, when civil defense authorities announce to the
public
"Tomorrow we
will have a surprise civildefense drill."
Then, at every moment during the specified period
(a week for the class and a day for the public) every person should think,
"Since
we did not have a test so far, we are about to have one right now."
A person, who keeps this in mind, will not be
surprised at all when the exam or drill actually commences.
After all, the purpose of a surprise exam or surprise
drill is readiness. And if one is ready, then there is no surprise.
The only way to have a surprise is to actually
do it without any prior announcement whatsoever. In the case of the civildefense
drill, the alarm is sounded all of a sudden. In the case of the class exam,
the teacher poses a problem to the students and then says:
"You have ten
minutes to answer this question without using your books or notes. This
is a surprise exam."
References

[Bryan Bunch, Mathematical Fallacies and Paradoxes,
(1982, Dover Publications, Inc.), p, 34–36.]

Jonathan L. Kvanvig, Department of Philosophy, Texas
A&M University, College Station, TX 778434237, (409)8455679; (409)6906263,
Source:
http://www.missouri.edu/~kvanvigj/papers/epistemicparadoxes.htm,
Last Modified: Wednesday, January 30, 2002 6:40:39 PM GMT]
Surprise or Prediction Paradoxes
Below are links to related web sites and some
of my comments concerning the discussion therein.
Swedish civil defense authorities announced that
a civil defense drill would be held one day the following week, but the
actual day would be a surprise. However, we can prove by induction that
the drill cannot be held. Clearly, they cannot wait until Friday, since
everyone will know it will be held that day. But if it cannot be held on
Friday, then by induction it cannot be held on Thursday, Wednesday, or
indeed on any day.
What is wrong with this proof?
Solution. 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
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