EFQ and the Unexpected Examination Paradox GABRIELE RASTELLO to Kiririn-shi June 4, 2017 Abstract Analysis of the surprise examination paradox supporting the idea that the students' conclusion is not wrong but simply contradictory. Keywords: Ex Falso Quodlibet, Principle of explosion, Scoto's law, Surprise Examination Paradox Introduction Let's start by introducing the paradox; there are many variants of this paradox and of course the core of the paradox is the same in all of them. I personally love the Surprise Examination version so we're gonna work on that. Let's first introducing the paradox. Here's the paradox exactly as it is presented in [1]: A teacher announces in class that an examination will be held on some day during the following week, and more over that the examination will be a surprise. The students argue that a surprise exam cannot occur. For suppose the exam were on the last day of the week. Then on the previous night, the students would be able to predict that the exam would occur on the following day, and the exam would not be a surprise. So it is impossible for a surprise exam to occur on the last day. But then a surprise exam cannot occur on the penultimate day, either, for in that case the students, knowing that the last day is an impossible day for a surprise exam, would be able to predict on the night before the exam that the exam would occur on the following 1 day. Similarly, the students argue that a surprise exam cannot occur on any other day of the week either. Confident in this conclusion, they are of course totally surprised when the exam occurs (on Wednesday, say). The announcement is vindicated after all. Where did the students' reasoning go wrong? The tool I'm gonna use to explain why the students' reasoning is wrong it's a law of classical logic called Ex Falso Quodlibet1 according to witch we can prove any statement from a single contradiction. If φ and ψ are statements we can write EFQ formally as: pφqpψq : φ. „ φ Ą ψ. The reader can easily convince itself that EFQ hold for every possible statement φ and ψ by reading the proof you can find at [2]. Paradox breakdown Let's then start by considering a very simple case: let's suppose that instead of "an examination will be held on some day during the following week" the teacher had said "an examination will be held on some day during the following two days". Let's also suppose, as many others have done, that surprise mean "not predictable". Let's try to apply the students' reasoning to this simple case. First let's suppose the exam were on the second day, then we are able to predict it the previous night and it cannot be a surprise. So we can infer that no examination can be hold on the second day. But now, since there are just two possible days for the surprise examination to occur and we already proved that it can't be on the second day the examination must be held on the first one and again it cannot be a surprise. We've just proved that: α: No surprise examination can occur on the second day. β: No surprise examination can occur on the first day. Since we're considering just two days we can now conjugate these two proposition to obtain the same result the students obtained: φ: α.β: no surprise examination can occur2. We're now in the same situation the students in the paradox are: we are sure that no surprise examination will occur but let's say that the second day the teacher decides to hold an examination: we would be really surprised. 1Sometimes called Principle of explosion or Principle of Pseudo-Scotus. From now on EFQ. 2In the following two days. 2 This seems to be against our conclusion φ that no surprise examination will occur. Where did our reasoning go wrong? Many thinkers focus on the process that bring us to φ, arguing that there's something fallacious there, but I think that the real problem is more φ rather than how we obtain it. We have to recognize that we're assuming the teacher statement "a surprise examination will be held on some day during the following two days" as an axiom, let' call it σ. Without assuming σ as an axiom we can't obtain φ. Let's see why. σ is basically the conjunction of two statements: (σ1) that state "an examination will occur" and (σ2) that state "the examination will be a surprise". When we supposed the exam will be held on the second day we argued that it would be impossible because the examination must be a surprise; in other words the exam can't occur on the second day because this contradict σ (particularly σ2). Then we said that since an examination must occur (this is σ1) and it can't be hold on the second day (via α, that follows from σ2) then the examination must occur on the first day; but this is against σ2. Now it's not hard to see that φ is the negation of σ since one state "a surprise examination will occur" and the other "no surprise examination will occur". This mean that we can actually construct, from σ, a statement like σ. „ σ and now, via EFQ we can prove everything. Of course every line of this section still apply if we pick 3, 4, 5, n days instead of 2 in account. Conclusions Since we can prove almost everything from the teacher's statement we now have to ask ourselfs: is σ (the teacher's statement) a valid statement? The answer can be yes or not and I kinda support both. If we say σ is worthless then we're basically saying there is some problem with it. The problem can be hard to find but since every word in σ seems pretty clear except surprise I think the problem is there. We had assumed surprise to mean "not predictable" but since this lead us to contradictions maybe we should have gone with a different meaning; like "at random" or "the day I like but I'm not going to tell you". Statements like "an examination will occur during the following week at a day chosen at random" or "an examination will occur during the following week at a day I like and I'm not going to tell you" does not permit to infer statement like α and β and are safe from contradictions. But if we chose to maintain surprise to mean "not predictable" and accept σ as invalid statement we still don't have very much problems. We know that from σ we can infer anything thanks to EFQ so we know that no 3 examination will occur, that an examination will occur on the first day, that an examination will not occur on the first day, that an examination will occur on the second day and that it will not occur on the second day and so on. We know everything but nothing at all so on any day3 an examination would be a surprise. To conclude: there's nothing wrong in the students' reasoning; they just did not confronted they're conclusion with the statement they derived it from. References 1. Timothy Y. Chow. The surprise examination or unexpected hanging paradox. The American Mathematical Monthly, 105:41–51, 1998. 2. Wikipedia. Principle of explosion - wikipedia, the free encyclopedia, 2017. [Online; accessed 3-June-2017]. 3except maybe the very last.