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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

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Surprisingly, there is a good reason why the scenario set forth by the surprise paradox results with a surprise Thus, it is no paradox.


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 77843-4237, (409)845-5679; (409)690-6263, Source: http://www.missouri.edu/~kvanvigj/papers/epistemicparadoxes.htm] "A CIVIL DEFENSE EXERCISE WILL BE HELD THIS WEEK. IN ORDER TO MAKE SURE THAT THE CIVIL-DEFENSE 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.]
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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?

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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.]

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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.

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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. 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. 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 exclusion-prohibition since the simple argument I introduced above resolves the paradox.
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Military Practical Application

The same tactic is a common practice for military planners or strategy-game 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.
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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 civil-defense alarm and start a surprise civil-defense 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 civil-defense 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 civil-defense 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."  
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References 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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