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The surprise examination or unexpected hanging paradox
 American Mathematical Monthly
, 1998
"... 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 socalled “s ..."
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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 socalled “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 Gödel’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].
Hempel’s raven paradox: A lacuna in the standard Bayesian solution. British Journal for the Philosophy of Science 55:545–560
, 2004
"... Abstract. According to Hempel’s paradox, evidence (E) that an object is a nonblack nonraven confirms the hypothesis (H) that every raven is black. According to the standard Bayesian solution, E does confirm H but only to a minute degree. This solution relies on the almost never explicitly defended a ..."
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Abstract. According to Hempel’s paradox, evidence (E) that an object is a nonblack nonraven confirms the hypothesis (H) that every raven is black. According to the standard Bayesian solution, E does confirm H but only to a minute degree. This solution relies on the almost never explicitly defended assumption that the probability of H should not be affected by evidence that an object is nonblack. I argue that this assumption is implausible, and I propose a way out for Bayesians.
From Hypocomputation to Hypercomputation
, 2008
"... Hypercomputational formal theories will, clearly, be both structurally and foundationally different from the formal theories underpinning computational theories. However, many of the maps that might guide us into this strange realm have been lost. So little work has been done recently in the area of ..."
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Hypercomputational formal theories will, clearly, be both structurally and foundationally different from the formal theories underpinning computational theories. However, many of the maps that might guide us into this strange realm have been lost. So little work has been done recently in the area of metamathematics, and so many of the previous results have been folded into other theories, that we are in danger of loosing an appreciation of the broader structure of formal theories. As an aid to those looking to develop hypercomputational theories, we will briefly survey the known landmarks both inside and outside the borders of computational theory. We will not focus in this paper on why the structure of formal theory looks the way it does. Instead we will focus on what this structure looks like, moving from hypocomputational, through traditional computational theories, and then beyond to hypercomputational theories.
Vexing Expectations
"... We introduce a St. Petersburglike game, which we call the ‘Pasadena game’, in which we toss a coin until it lands heads for the first time. Your payoffs grow without bound, and alternate in sign (rewards alternate with penalties). The expectation of the game is a conditionally convergent series. A ..."
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We introduce a St. Petersburglike game, which we call the ‘Pasadena game’, in which we toss a coin until it lands heads for the first time. Your payoffs grow without bound, and alternate in sign (rewards alternate with penalties). The expectation of the game is a conditionally convergent series. As such, its terms can be rearranged to yield any sum whatsoever, including positive infinity and negative infinity. Thus, we can apparently make the game seem as desirable or undesirable as we want, simply by reordering the payoff table, yet the game remains unchanged throughout. Formally speaking, the expectation does not exist; but we contend that this presents a serious problem for decision theory, since it goes silent when we want it to speak. We argue that the Pasadena game is more paradoxical than the St. Petersburg game in several respects. We give a brief review of the relevant mathematics of infinite series. We then consider and rebut a number of replies to our paradox: that there is a privileged ordering to the expectation series; that decision theory should be restricted to finite state spaces; and that it should be restricted to bounded utility functions. We conclude that the paradox remains live. 1. The Pasadena paradox It’s your lucky day. We offer you at no charge the following game, which with winking homage to a more famous game that inspired it, we will call the Pasadena game. We toss a fair coin until it lands heads for the first time. We have written on consecutive cards your payoff for each possible outcome. The cards read as follows: (Top card) If the first heads is on toss #1, we pay you $2. (2nd top card) If the first heads is on toss #2, you pay us $2. (3rd top card) If the first heads is on toss #3, we pay you $8/3. (4th top card) If the first heads is on toss #4, you pay us $4.