Probabilistic Puzzles

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 (...) related symmetry principle to the failure of probabilistic regularity. We contend that these arguments fail, because they rely on auxiliary assumptions about probability which are inconsistent with symmetry to begin with. We argue, moreover, that symmetry should be rejected in light of this inconsistency, and because it has implausible decision- theoretic implications. -/- The weaker principle of probabilistic invariance says that the probabilistic comparison of any two regions is unchanged by rotations of the sphere. This principle supports a more compelling argument for imprecise probability. We show, however, that invariance is incompatible with mundane judgments about what is probable. Ultimately, we find reason to be suspicious of the application of principles like symmetry and invariance to nonmeasurable regions. (shrink)

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 (...) to competing proposals; in particular, we can formulate the rules of correct doxastic reasoning without reference to probabilistic features of the involved propositions. (shrink)

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 (...) to A to enumeration oracles and hence enumeration degrees. We isolate several classes of enumeration degrees based on their behavior with respect to this relation: the PA bounded degrees, the degrees that have a universal class, the low for PA degrees, and the ${\left \langle {\text {self}\kern1pt}\right \rangle }$ -PA degrees. We study the relationship between these classes and other known classes of enumeration degrees. We also investigate a group of classes of enumeration degrees that were introduced by Kalimullin and Puzarenko [14] based on properties that are commonly studied in descriptive set theory. As part of this investigation, we give characterizations of three of their classes in terms of a special sub-collection of relativized $\Pi ^0_1$ classes—the separating classes. These three can then be seen to be direct analogues of three of our classes. We completely determine the relative position of all classes in question. (shrink)

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 (...) or incoherent credences. The main idea behind this extension is that the agent should avoid updating plans that increase the possible sure loss from Dutch strategies. This happens to be equivalent to avoiding updating plans that increase incoherence according to a distance-based incoherence measure. (shrink)

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.

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.

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 (...) entropy. In situations of information loss, or whenever the agent relies on a qualitative description of her information state - such as a plausibility ranking over hypotheses, or a belief set - the dynamics of AGM belief revision are compatible with Bayesian conditioning; indeed, through the maximum entropy principle, conditioning naturally generated AGM revision operators. This mitigates an impossibility theorem of Lin and Kelly for tracking Bayesian conditioning with AGM revision, and suggests an approach to the compatibility problem that highlights the information loss incurred by acceptance rules in passing from probabilistic to qualitative representations of beliefs. (shrink)

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.

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. (...) In this paper, we propose a revised norm, Kolmogorov-Blackwell conditionalization, which avoids this problem. We prove a Dutch book theorem and converse Dutch book theorem for this revised norm, and relate our results to those of Rescorla (2018). (shrink)

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 (...) de AM, a probabilidade de que a teoria T seja aproximadamente verdadeira é grande, apenas se segue se assumirmos para T um valor não-negligenciável. Tal crítica de Howson despertou duas categorias de reação na literatura especializada: supondo que AM comete a falácia da taxa base, alguns autores propuseram abandonar AM enquanto um argumento para a tese epistemológica do realismo científico, ao passo que outros defenderam que apenas uma versão de AM comete a falácia da taxa-base; defendendo que AM não comete a falácia da taxa base, certos autores alegaram alguns problemas na formalização probabilística de AM, enquanto outros buscaram refinar tal formalização de modo que a nova evite a falácia em questão. Tendo apresentado esse estado da arte, investigaremos se a formalização probabilística do argumento do milagre é razoável. Discutiremos se ela acomoda a natureza de inferência à melhor explicação pressuposta no AM. Por fim, indicaremos, brevemente, algumas dificuldades para a crítica de Howson ao assumir-se que AM instancia, na verdade, uma abdução Peirceana. (shrink)

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 (...) these questions that have been defended in the literature. In addition, my account complements fruitful probabilistic theories of assertion and knowledge. (shrink)

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 (...) raise a diachronic case which illustrates that neither satisfies both Lin and Kelly’s constraint that the update on orderings track the update on credence functions, and the intuitive constraint that credence of at least 0.5 is necessary for rational belief. I conclude by suggesting that we reformulate these proposals in terms of orderings over entire algebras based on partitions rather than orderings just over the partitions themselves. Reformulating both rules in this way yields acceptance rules which avoid the Lottery Paradox while satisfying both the tracking and likeliness constraints. (shrink)

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 (...) of the solutions. The discussion of this improved version suggests that the Goodman paradox actually embodies three different problems; yet, one of them is not Goodman’s but Hume’s problem. The discussion also suggests that the naturalistic approach is probably the best for basing on it a theory of confirmation. Finally, I analyze one of Hume’s insights that seems to have been largely ignored. This insight shows a surprising similarity to a central feature of the naturalistic solutions. (shrink)

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 (...) I find his defense unsatisfactory for two reasons: it appeals to false premises, and it saves Poi only at the expense of something that Poi’s believers likely find just as important. So in this essay I defend Poi by arguing that its antecedent does hold in the relevant thought experiments, and that the further propositions needed to reject Poi are false. I play only defense in this essay; I don’t argue that Poi is true (even though I think it is), but rather that one popular refutation is faulty. In showing this, I also note something that has to my knowledge gone unnoticed: given some innocuous-looking assumptions the denial of Poi is equivalent to a version of epistemic permissivism , and Poi itself is equivalent to a version of epistemic uniqueness. (shrink)

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 (...) minimizing expected predictive error. In contrast, formal learning theory appeals to Ockham's razor, which it justifies by reference to the goal of enhancing efficient convergence to the truth. In this essay, I show that, despite their differences, statistical and formal learning theory yield precisely the same result for a class of inductive problems that I call strongly VC ordered, of which Goodman's riddle is just one example. (shrink)

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 (...) the "problem of old evidence". Several other novel aspects of Goodman's argument are also discussed. (shrink)

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 (...) in which the latter principle and the principle of Countable Additivity cannot be jointly satisfied. The most plausible response to this tension, I argue, is to accept both of these principles, and to maintain that when an agent cannot satisfy them both, she is faced with a rational dilemma. (shrink)

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 (...) testable consequences,and so the sampling assumptions themselves must be regarded as empirical claims.The result of testing some of these consequences is that both doomsday argumentsare empirically disconfirmed. (shrink)