Saturday, September 1, 2012

Surprise Quiz Paradox

The surprise quiz paradox can be stated as follows: A professor emails his students on Sunday and informs them that they will have a surprise quiz on some day throughout the week (Monday through Friday). An astute student reads this email incredulously because he perceives a logical flaw in it. He knows that the professor can't give the quiz on Friday because then the day of the quiz will have become apparent on Thursday, thereby making it unsurprising. 

Since Friday has now been eliminated as a possible day, Thursday becomes the new Friday of the week. Moreover, the same argument used to exclude Friday can now be used to exclude Thursday. This logic continues all the way to the beginning of the week, with the final conclusion reached that it would be logically impossible to give a surprise quiz. The paradox lies in that it is obvious the professor can still give the pop quiz as he pleases and surprise the students, as was his original intention.

Before we begin to look for a resolution, it would be helpful to first clarify the definition of "surprise" we are using that leads to the paradox. I define surprise in this instance as meaning roughly: "You will be unable to predict at any point throughout the week, at least a day in advance, the day on which I will give the quiz". This is the definition that leads to the core problem of Friday not qualifying as a valid surprise quiz day.

It seems, given the definition of surprise we have provided, it is indeed logically impossible to give a "surprise" quiz, and this is true. However, the paradox lies not in that the statement is logically impossible, but in that the professor can still give a surprise quiz despite all this. It is in this very inconsistency that we shall begin to find our resolution. It seems to me that the definition of surprise that is utilized in contemporary conversational language is not equivalent to the definition of surprise we have employed to establish the fallacy. It is a somewhat weaker definition, which I will now attempt to provide.

The definition of surprise that I believe the professor employs is this: "The exact day of the quiz will be unpredictable for at least the same number of days that it will be predictable". In this sense, the definition of surprise is not a binary concept and there are degrees of strength inherent within it. One individual may be more surprised than another even provided that both experienced the exact same occurrence at the exact same time.  For example, if the day of the quiz is known to someone for three days, but unknown to him for four days, the quiz will be a surprise, though not as much of a surprise as for the individual for whom the day of the quiz was known for three days, but unknown for five.

The corner case of this definition that we must account for is the case in which an individual is aware of when an event will occur for the same duration of time as he was unaware of it. I believe that, in this case, we can say that the individual is in a state that is simultaneously both surprised and unsurprised, although this state would be the theoretically lowest degree of both surprise and lack of surprise conceivable to us.

Given this, and stretching my definition to its limits, one can provide the following example: Suppose somebody tells you on Sunday that they will throw a surprise party for you within the next two weeks. If on the Sunday of the second week, someone happens to clue you in that your surprise party will be that Saturday, will it still be a surprise for you? Technically, my definition is still satisfied in this instance, since you were unaware of the exact date of the party for seven days, and will be aware of it for also seven days. However, I think most people would agree that, despite this, the surprise party will be pretty well ruined by the time it comes around. 

While this may at first appear to be a valid counterargument, it is ultimately specious. As I mentioned before, I believe that, conversationally, humans innately think in terms of degrees of surprise. If I knew of the exact time of the occurrence of a certain event for a longer time than you did, and we both knew this information for a shorter or equal amount of time than we didn't know it, I think I would be justified in saying that I am surprised, but less surprised than you, when that event finally occurs. 

Now since the individual in the surprise party example knew the exact day of the party for just as long as he didn't know it, one can say (as stated before) that, in a sense, he/she is both surprised and unsurprised. This to me still qualifies as a state of surprise, although the lowest one possible in terms of degrees of vigor. Only once the number of days of awareness exceeds the number of days of ignorance does one truly leave the realm of surprise regarding a particular event.

We have seen how, in the paradox, the clause that begins the chain of reasoning that ultimately leads to the surprise quiz becoming a logical fallacy is: "A surprise quiz cannot be given on Friday". The logical definition that we gave at the beginning of this article most definitely satisfies this clause. Let us now, however, try it out with the alternative (and in my belief, more common and natural) definition that I have just provided. Suppose the professor indeed plans the quiz for Friday. On Sunday (the day the students first receive the email), the students are unable to predict the day of the quiz. The same goes for Monday, Tuesday, and Wednesday. Only on Thursday does it become readily apparent that the quiz will be on Friday. This means that for four days the students were unaware of what day the quiz will be on, as opposed to the two days during which they were.

Thus, the definition that I have provided is satisfied, and it is now possible to give a surprise quiz on Friday, thereby not creating the domino effect that leads to the paradox. Suppose that, for robustness, instead of a five day week, we have a two day week, with Monday and Friday being the only days. Then on Sunday, the students are unable to predict whether the quiz will be on Monday or Friday. On Monday, they can safely say the quiz will be on Friday. The definition I provided remains satisfied, as the students were both unable and able to predict the quiz for exactly one day. 

I anticipate that the main point of contention with this argument will be my supposedly commonplace definition of "surprise". Clearly none of us think about the definition I have provided when we use the actual word. I argue that, although this may be the case, subconsciously we more often than not still employ some version of my definition. To illustrate this point further, allow me to provide a simple thought experiment: Suppose that a doctor tells you that within the next ten years, you will die unexpectedly at any minute. Suppose further that you are 100 percent certain that he is correct. Finally, suppose that your death does not occur for the next nine years, 364 days, 23 hours, and 59 minutes. Then and only then will you be sure that your death must occur in the following (the 60th) minute.

According to the initial definition I provided for the justification of the paradox, and given this thought experiment, your death ceases to be a surprise following the end of the 59th minute. However, I believe that most people, myself included, would still attribute to the moment of your death a "surprising" quality, given the vast amount of time that you were unaware of the exact minute of your impending doom (as well as the equally vast amount of time you were anticipating it), relative to the tiny amount of time that you were. Although this is an extreme example, I believe it adequately demonstrates that there are indeed gradations to the state of surprise. 

Thus, I believe the paradox lies mainly within a discrepancy between the logical definition of the word "surprise", and the conversational definition (like, for example, with the word "or"). It is my strong belief that the concepts invoked here can be applied generally to all permutations of the surprise quiz paradox.

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