Examination or Unexpected Hanging Paradox

Many mathematicians have a dismissive attitude towards paradoxes. This is unfortunate , because many paradoxes are rich in content, having connections with serious mathematical ideas as well as having pedagogical value in teaching elementary logical reasoning. An excellent example is the so-called " surprise examination paradox " (described below), which is an argument that seems at first to be too silly to deserve much attention. However, it has inspired an amazing variety of philosophical and mathematical investigations that have in turn uncovered links to G ¨ odel's incompleteness theorems, game theory, and several other logical paradoxes (e.g., the liar paradox and the sorites paradox). Unfortunately, most mathematicians are unaware of this because most of the literature has been published in philosophy journals. In this article, I describe some of this work, emphasizing the ideas that are particularly interesting mathematically. I also try to dispel some of the confusion that surrounds the paradox and plagues even the published literature. However, I do not try to correct every error or explain every idea that has ever appeared in print. Readers who want more comprehensive surveys should see [30, chapters 7 and 8], [20], and [16]. At times I assume some knowledge of mathematical logic (such as may be found in Enderton [10]), but the reader who lacks this background may safely skim these sections. 1. The paradox and the meta-paradox Let 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 unexpected hanging. We shall give the surprise examination version. A teacher announces in class that an examination will be held on some day during the following week, and moreover 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 …