surprise examination or unexpected hanging paradox, The

Many mathematicians have a dismissive attitude towards paradoxes. This is unfortunate, because many paradoxes are rich in easy in mind having connections with serious mathematical ideas as well as having pedagogical value in teaching elementary logical reasoning. An superior example is the so-called "surprise examination paradox" (described below), which is an argument that assumes at first to be too silly to merit much attention. However, it has inspired an amazing variety of philosophical and mathematical investigations, that have in cast uncovered links to Godel's incompleteness theorems, game theory, and several other logical paradoxes (eg the liar paradox and the sorites paradox). Unfortunately, greatest in number mathematicians are unaware of this because greatest in quantity of the literature has been published in philosophy journals.

In this article, I describe a certain quantity of of this work, emphasizing the ideas that are particularly interesting mathematically. I also prove by experiment to dispel some of the confusion that environs the paradox and plagues equal the published literature. However, I do not attempt to correct every error or explain each idea that has ever appeared in print. Readers who want more comprehensive examines should see [30, chapters 7 and 8] [201 and [16]

At times I assume about knowledge of mathematical logic (such as may be rest in Enderton [10]), but the reader who lacks this background may safely skim these sections.

1 THE PARADOX AND THE META-PARADOX. suffer us begin by recalling the paradox. It has many variants, the earliest probably being Lennart Ekbom's surprise drill, and the best known to mathematicians (thanks to Quine and Gardner) being an unexpect hanging. We shall give the surprise examination version.

A teacher announces in class that an examination will be held forward some day during the following week, and moreover that the examination will be a surprise. The learners argue that a surprise exam cannot happen For suppose the exam were forward the last day of the week. Then forward the previous night, the scholars would be able to predict that the exam would come into one's head on the following day, and the exam would not be a surprise. with equal reason it is impossible for a surprise exam to present itself on the last day. yet then a surprise exam cannot come into one's head on the penultimate day, either, for in that case the observers knowing that the last day is an impossible day for a surprise exam, would be able to predict forward the night before the exam that the exam would arise on the following day. Similarly, the pupils argue that a surprise exam cannot befall on any other day of the week either. Confident in this conclusion, they are of course totally surprised when the exam come to passs (on Wednesday, say). The announcement is vindicated after all. Where did the students' reasoning advance wrong?

The natural reaction to a paradox like this is to examine to resolve it. Indeed, if you have not seen this paradox before, I encourage you to prove to resolve it now before reading onward However, I do not want to discuss the resolution of the paradox right away. Instead, for reasons that should become apparent, I discuss what I call the "meta-paradox" first.

The meta-paradox consists of pair seemingly incompatible facts. The first is that the surprise exam paradox be seens easy to resolve. Those seeing it for the first time typically have the instinctive reaction that the flaw in the students' reasoning is obvious. Furthermore, mostly readers who have tried to think it in consequence of have had little difficulty resolving it to their allow satisfaction.

The second (astonishing) fact is that to date nearly a hundr papers forward the paradox have been published, and still no consensus onward its correct resolution has been reached. The paradox has equal been called a "significant problem" for philosophy [30 chapter 7 section VII]. to what degree can this be? Can as it is a ridiculous argument really be a major unsolv mystery? If not, to what end does paper after paper begin through brusquely dismissing all previous work and claiming that it alone currents the long-awaited simple solution that lays the paradox to stillness once and for all?

Some other paradoxes put up with from a similar meta-paradox, on the other hand the problem is especially acute in the case of the surprise examination paradox. For principally other trivial-sounding paradoxes there is broad consensus in succession the proper resolution, whereas for the surprise exam paradox there is not flat agreement on its proper formulation. Since one's view of the meta-paradox influences the way individual views the paradox itself, I must put to the test to clear up the former before discussing the latter.

In my view, greatest in number of the confusion has been caused from authors who have plunged into the proces of "resolving" the paradox without first having a clear idea of what it means to "resolve" a paradox. The goal is poorly understood, in such a manner controversy over whether the goal has been attained is inevitable. give leave to me now suggest a way of thinking about the proces of "resolving a paradox" that I believe dispels the meta-paradox.

In general, there are couple steps involved in resolving a paradox. First, common establishes precisely what the paradoxical argument is. Any unclear space of times are defined carefully and all assumptions and logical grades are stated clearly and explicitly, possibly in a formal language of a certain kind. Second, one finds the fault in the argument. Sometimes, simply performing pace one reveals the flaw, eg when the paradox hinges in succession confusing two different meanings of the same word, to such a degree that pointing out the ambiguity suffices to dispel the confusion. In other cases, however, something more requires to be done; one must locate the bad assumptions, the bad reasoning, or (in desperate circumstances) the flaw in the edifice of logic itself.