# Probabilistic Puzzles

Edited by Darren Bradley (University of Leeds)
 Summary In this category belong a range of puzzles that are analysed using probabililty and have philosophical implications. Perhaps the best known is Goodman's New Riddle of Induction (Grue), which can be seen as a strengthened version of Hume's problem of induction. The Paradox of the Ravens (the paradox of confirmation) is one of the central problems for theories of confirmation. It seems to show that obvious principles of confirmation generate the result that a white sneaker confirms that all ravens are black. The Sleeping Beauty problem concerns an agent who is woken on either one day or two, and faces the question of whether the current waking is part of the single waking or the double waking. This raises the issue of incorporating self-locating beliefs into the Bayesian framework. The Doomsday Argument purports to show that humans will die out sooner than we previously thought, based merely on our own birth rank among humans. The Monty Hall Problem is about whether you should swap doors, after tentatively choosing one of the three doors, one of which contains a prize, and finding that the door you selected does not have the prize.
 Key works The New Riddle of Induction was introduced in Goodman 1954. The Paradox of the Ravens was introduced by Hosiasson-Lindenbaum 1940 and influentially discussed by Hempel 1945 I and Hempel 1945 II. Sleeping Beauty was introduced by Elga 2000, shortly followed by Lewis 2001. The Doomsday Argument was popularized largely by Leslie 1989.
 Introductions The new riddle of induction and the paradox of the ravens are explained in section 5 of Vickers 2008. This Bostrom manuscript explains the Doomsday Argument and Titelbaum 2016 gives a summary of the responses to Sleeping Beauty
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1. It is tempting to think that a process of choosing a point at random from the surface of a sphere can be probabilistically symmetric, in the sense that any two regions of the sphere which differ by a rotation are equally likely to include the chosen point. Isaacs, Hájek, and Hawthorne (2022) argue from such symmetry principles and the mathematical paradoxes of measure to the existence of imprecise chances and the rationality of imprecise credences. Williamson (2007) has argued from a (...)

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2. The Epistemic and the Deontic Preface Paradox.Lina M. Lissia & Jan Sprenger - forthcoming - Philosophical Quarterly.
This paper generalizes the preface paradox beyond the conjunctive aggregation of beliefs and constructs an analogous paradox for deontic reasoning. The analysis of the deontic case suggests a systematic restriction of intuitive rules for reasoning with obligations. This proposal can be transferred to the epistemic case: it avoids the preface and the lottery paradox and saves one of the two directions of the Lockean Thesis (i.e., high credence is sufficient, but not necessary for rational belief). The resulting account compares favorably (...)

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3. Pa Relative to an Enumeration Oracle.G. O. H. Jun Le, Iskander Sh Kalimullin, Joseph S. Miller & Mariya I. Soskova - 2023 - Journal of Symbolic Logic 88 (4):1497-1525.
Recall that B is PA relative to A if B computes a member of every nonempty $\Pi ^0_1(A)$ class. This two-place relation is invariant under Turing equivalence and so can be thought of as a binary relation on Turing degrees. Miller and Soskova [23] introduced the notion of a $\Pi ^0_1$ class relative to an enumeration oracle A, which they called a $\Pi ^0_1{\left \langle {A}\right \rangle }$ class. We study the induced extension of the relation B is PA relative (...)

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4. Updating incoherent credences ‐ Extending the Dutch strategy argument for conditionalization.Glauber De Bona & Julia Staffel - 2021 - Philosophy and Phenomenological Research 105 (2):435-460.
In this paper, we ask: how should an agent who has incoherent credences update when they learn new evidence? The standard Bayesian answer for coherent agents is that they should conditionalize; however, this updating rule is not defined for incoherent starting credences. We show how one of the main arguments for conditionalization, the Dutch strategy argument, can be extended to devise a target property for updating plans that can apply to them regardless of whether the agent starts out with coherent (...)

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5. An Infinite Lottery Paradox.John D. Norton & Matthew W. Parker - 2022 - Axiomathes 32 (1):1-6.
In a fair, infinite lottery, it is possible to conclude that drawing a number divisible by four is strictly less likely than drawing an even number; and, with apparently equal cogency, that drawing a number divisible by four is equally as likely as drawing an even number.

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6. (Almost) all evidence is higher-order evidence.Brian Hedden & Kevin Dorst - 2022 - Analysis 82 (3):417-425.
Higher-order evidence is evidence about what is rational to think in light of your evidence. Many have argued that it is special – falling into its own evidential category, or leading to deviations from standard rational norms. But it is not. Given standard assumptions, almost all evidence is higher-order evidence.

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7. Probabilistic stability, agm revision operators and maximum entropy.Krzysztof Mierzewski - 2020 - Review of Symbolic Logic:1-38.
Several authors have investigated the question of whether canonical logic-based accounts of belief revision, and especially the theory of AGM revision operators, are compatible with the dynamics of Bayesian conditioning. Here we show that Leitgeb's stability rule for acceptance, which has been offered as a possible solution to the Lottery paradox, allows to bridge AGM revision and Bayesian update: using the stability rule, we prove that AGM revision operators emerge from Bayesian conditioning by an application of the principle of maximum (...)

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8. The Prisoner's Dilemma Paradox: Rationality, Morality, and Reciprocity.Rory W. Collins - 2022 - Think 21 (61):45-55.
This article examines the prisoner's dilemma paradox and argues that confessing is the rational choice, despite this probably entailing a less-than-ideal outcome.

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9. Kolmogorov Conditionalizers Can Be Dutch Booked.Alexander Meehan & Snow Zhang - forthcoming - Review of Symbolic Logic:1-36.
A vexing question in Bayesian epistemology is how an agent should update on evidence which she assigned zero prior credence. Some theorists have suggested that, in such cases, the agent should update by Kolmogorov conditionalization, a norm based on Kolmogorov’s theory of regular conditional distributions. However, it turns out that in some situations, a Kolmogorov conditionalizer will plan to always assign a posterior credence of zero to the evidence she learns. Intuitively, such a plan is irrational and easily Dutch bookable. (...)

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10. Objetivamos discutir a crítica, avançada por Colin Howson em Hume's problem, segundo a qual o argumento do milagre (doravante, AM) comete a falácia da taxa-base. Por falácia da taxa-base, entende-se a negligência do valor da probabilidade prévia de determinada hipótese ou teoria T, P(T). Por sua vez, em uma de suas versões, AM assere que apenas assumindo que uma teoria científica madura T é aproximadamente verdadeira não faz de seu sucesso preditivo um milagre. Formalizado probabilisticamente, Howson argumenta que a conclusão (...)

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11. Full Belief and Loose Speech.Sarah Moss - 2019 - Philosophical Review 128 (3):255-291.
This paper defends an account of full belief, including an account of its relationship to credence. Along the way, I address several familiar and difficult questions about belief. Does fully believing a proposition require having maximal confidence in it? Are rational beliefs closed under entailment, or does the preface paradox show that rational agents can believe inconsistent propositions? Does whether you believe a proposition depend partly on your practical interests? My account of belief resolves the tension between conflicting answers to (...)

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12. R. A. Sharpe. Validity and the paradox of confirmation. The philosophical quarterly , vol. 14 , pp. 170–173. [REVIEW]David Kaplan - 1967 - Journal of Symbolic Logic 32 (2):251.

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13. (1 other version)Samuel Goldberg. Probability. An introduction. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1960, xiv + 322 pp. [REVIEW]Stefan Bauer-Mengelberg - 1971 - Journal of Symbolic Logic 36 (3):543-544.

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14. (2 other versions)

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15. The World, the Deceiver, and The Face in the Frost.Lydia McGrew - 2018 - Quaestiones Disputatae 8 (2):112-146.

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16. Rigged lotteries: a diachronic problem for reducing belief to credence.Jonathan Wright - 2018 - Synthese 195 (3):1355-1373.
Lin and Kelly :957–981, 2012) and Leitgeb :1338–1389, 2013, Philos Rev 123:131–171, 2014), offer similar solutions to the Lottery Paradox, defining acceptance rules which determine a rational agent’s beliefs in terms of broader features of her credal state than just her isolated credences in individual propositions. I express each proposal as a method for obtaining an ordering over a partition from a credence function, and then a belief set from the ordering. Although these proposals avoid the original Lottery Paradox, I (...)

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17. (1 other version)Leblanc Hugues. Évidence logique et degré de confirmation. Revue philosophique de Louvain, vol. 52 , pp. 619–625.Richard Montague - 1960 - Journal of Symbolic Logic 25 (1):86-86.

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18. (2 other versions)

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19. (1 other version)

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20. (1 other version)Hempel Carl G.. A logical appraisal of operationism. The scientific monthly, vol. 79 , pp. 215–220.Henry Mehlberg - 1958 - Journal of Symbolic Logic 23 (3):354-356.

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21. (1 other version)Freund John E.. On the problem of confirmation. Methodos, vol. 3 , pp. 33–42Somenzi Vittorio. Discussion. Methodos, vol. 3 , p. 42. [REVIEW]Hilary Putnam - 1958 - Journal of Symbolic Logic 23 (1):76-77.

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22. (1 other version)Crawshay-Williams Rupert. Equivocal confirmation. Analysis , vol. 11 no. 4 , pp. 73–79.Hilary Putnam - 1957 - Journal of Symbolic Logic 22 (4):406-407.

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23. (1 other version)Goodman Nelson. New notes on simplicity.Frederic B. Fitch - 1953 - Journal of Symbolic Logic 18 (2):179-179.

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24. (1 other version)Goodman Nelson. Some reflections of the theory of systems. English with Spanish abstract. Philosophy and phenomenological research, vol. 9 no. 3 , pp. 620–626. [REVIEW]Frederic B. Fitch - 1950 - Journal of Symbolic Logic 15 (3):218-218.

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25. (1 other version)Goodman Nelson. On likeness of meaning. Analysis , vol. 10 no. 1 , pp. 1–7.Alonzo Church - 1950 - Journal of Symbolic Logic 15 (2):150-151.

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26. (1 other version)Miller David L.. Comments on “Studies in the logic of explanation.” Philosophy of science, vol. 15 , pp. 348–349.Thomas Storer - 1949 - Journal of Symbolic Logic 14 (2):133-133.

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27. (2 other versions)Morris Charles. Signs, language, and behavior. Prentice-Hall, Inc., New York 1946, v + 356 pp. [REVIEW]Arthur Francis Smullyan - 1947 - Journal of Symbolic Logic 12 (2):49-51.

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28. (2 other versions)Hempel Carl G.. A note on the paradoxes of confirmation. Mind, n. s. vol. 55 , pp. 79–82.Max Black - 1946 - Journal of Symbolic Logic 11 (4):124-124.

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29. (2 other versions)Goodman Nelson. A query on confirmation. The journal of philosophy, vol. 43 , pp. 383–385.Max Black - 1946 - Journal of Symbolic Logic 11 (3):81-81.

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30. (1 other version)Hempel C. G.. Geometry and empirical science. The American mathematical monthly, vol. 52 , pp. 7–17.Alonzo Church - 1946 - Journal of Symbolic Logic 11 (3):100-100.

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31. (1 other version)Hempel Carl G. and Oppenheim Paul. A definition of “degree of confirmation.” Philosophy of science, vol. 12 , pp. 98–115. [REVIEW]Max Black - 1946 - Journal of Symbolic Logic 11 (1):18-19.

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32. (3 other versions)Whiteley C. H.. Hempel's paradoxes of confirmation. Mind, n. s. vol. 54 , pp. 156–158.Max Black - 1945 - Journal of Symbolic Logic 10 (3):104-104.

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33. (1 other version)Hempel. Carl G. A purely syntactical definition of confirmation.Max Black - 1944 - Journal of Symbolic Logic 9 (2):47-47.

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34. (1 other version)Goodman Nelson. On the simplicity of ideas.George D. W. Berry - 1944 - Journal of Symbolic Logic 9 (2):52-53.

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35. (1 other version)Goodman Nelson. Sequences.J. C. C. McKinsey - 1942 - Journal of Symbolic Logic 7 (3):120-120.

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36. (1 other version)Leonard Henry S. and Goodman Nelson. The calculus of individuals and its uses. [REVIEW]Laurence J. Lafleur - 1940 - Journal of Symbolic Logic 5 (3):113-114.

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37. (1 other version)Gonseth F.. Qu'est-ce que la logique? Actualités scientifiques et industrielles 524, Hermann & Cie, Paris 1937, 89 pp. [REVIEW]Paul Henle - 1938 - Journal of Symbolic Logic 3 (4):164-164.

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38. (1 other version)Hempel C. G.. Le problème de la vérité. Theoria, vol. 3 , pp. 206–246.C. H. Langford - 1937 - Journal of Symbolic Logic 2 (4):170-171.

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39. Updating a progic.Eric Raidl - 2016 - Journal of Applied Logic 14:65-94.

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40. The Goodman Paradox: Three Different Problems and a Naturalistic Solution to Two of Them.Nathan Stemmer - 2004 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 35 (2):351-370.
It is now more than 50 years that the Goodman paradox has been discussed, and many different solutions have been proposed. But so far no agreement has been reached about which is the correct solution to the paradox. In this paper, I present the naturalistic solutions to the paradox that were proposed in Quine (1969, 1974), Quine and Ullian (1970/1978), and Stemmer (1971). At the same time, I introduce a number of modifications and improvements that are needed for overcoming shortcomings (...)

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41. A Defense of the Principle of Indifference.Greg Novack - 2010 - Journal of Philosophical Logic 39 (6):655-678.
The principle of indifference (hereafter ‘Poi’) says that if one has no more reason to believe A than B (and vice versa ), then one ought not to believe A more than B (nor vice versa ). Many think it’s demonstrably false despite its intuitive plausibility, because of a particular style of thought experiment that generates counterexamples. Roger White ( 2008 ) defends Poi by arguing that its antecedent is false in these thought experiments. Like White I believe Poi, but (...)

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42. (1 other version)Testability and Ockham’s Razor: How Formal and Statistical Learning Theory Converge in the New Riddle of Induction.Daniel Steel - 2009 - Journal of Philosophical Logic 38 (5):471-489.
Nelson Goodman's new riddle of induction forcefully illustrates a challenge that must be confronted by any adequate theory of inductive inference: provide some basis for choosing among alternative hypotheses that fit past data but make divergent predictions. One response to this challenge is to distinguish among alternatives by means of some epistemically significant characteristic beyond fit with the data. Statistical learning theory takes this approach by showing how a concept similar to Popper's notion of degrees of testability is linked to (...)

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43. Goodman’s “New Riddle‘.Branden Fitelson - 2008 - Journal of Philosophical Logic 37 (6):613-643.
First, a brief historical trace of the developments in confirmation theory leading up to Goodman's infamous "grue" paradox is presented. Then, Goodman's argument is analyzed from both Hempelian and Bayesian perspectives. A guiding analogy is drawn between certain arguments against classical deductive logic, and Goodman's "grue" argument against classical inductive logic. The upshot of this analogy is that the "New Riddle" is not as vexing as many commentators have claimed. Specifically, the analogy reveals an intimate connection between Goodman's problem, and (...)

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44. Sleeping Beauty, Countable Additivity, and Rational Dilemmas.Jacob Ross - 2010 - Philosophical Review 119 (4):411-447.
Currently, the most popular views about how to update de se or self-locating beliefs entail the one-third solution to the Sleeping Beauty problem.2 Another widely held view is that an agent‘s credences should be countably additive.3 In what follows, I will argue that there is a deep tension between these two positions. For the assumptions that underlie the one-third solution to the Sleeping Beauty problem entail a more general principle, which I call the Generalized Thirder Principle, and there are situations (...)

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45. An Empirical Critique of Two Versions of the Doomsday Argument – Gott's Line and Leslie's Wedge.E. Sober - 2003 - Synthese 135 (3):415-430.
I discuss two versions of the doomsday argument. According to Gott's Line'',the fact that the human race has existed for 200,000 years licences the predictionthat it will last between 5100 and 7.8 million more years. According to Leslie'sWedge'', the fact that I currently exist is evidence that increases the plausibilityof the hypothesis that the human race will come to an end sooner rather than later.Both arguments rest on substantive assumptions about the sampling process thatunderlies our observations. These sampling assumptions have (...)

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46. (1 other version)‘Yes:—no:—I have been sleeping—and now—now—I am dead’: undeath, the body and medicine.Megan Stern - 2008 - Studies in History and Philosophy of Science Part C: Studies in History and Philosophy of Biological and Biomedical Sciences 39 (3):347-354.

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47. Incoherence and inconsistency.Michael Schippers - 2014 - Review of Symbolic Logic 7 (3):511-528.

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48. 34. The Problem of the Enjoyment of Beauty.Guy Sircello - 1975 - In New Theory of Beauty. Princeton, N.J.: Princeton University Press. pp. 126-129.

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49. 25. The Problem of Moral Beauty.Guy Sircello - 1975 - In New Theory of Beauty. Princeton, N.J.: Princeton University Press. pp. 81-84.