Probabilistic Puzzles

This paper generalizes the (epistemic) preface paradox beyond the principle of belief aggregation and constructs a similar paradox for deontic reasoning. The analysis of the deontic case yields a solution strategy---restricting belief/obligation aggregation rather than giving it up altogether---that can be transferred to the epistemic case. Our proposal amounts to a reasonable compromise between two goals: (i) sticking to bridge principles between evidence and belief, such as the Lockean Thesis, and (ii) obtaining a sufficiently strong logic of doxastic and deontic (...) reasoning. (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)

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.

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.

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)

How can different individuals' probability functions on a given sigma-algebra of events be aggregated into a collective probability function? Classic approaches to this problem often require 'event-wise independence': the collective probability for each event should depend only on the individuals' probabilities for that event. In practice, however, some events may be 'basic' and others 'derivative', so that it makes sense first to aggregate the probabilities for the former and then to let these constrain the probabilities for the latter. We formalize (...) this idea by introducing a 'premise-based' approach to probabilistic opinion pooling, and show that, under a variety of assumptions, it leads to linear or neutral opinion pooling on the 'premises'. This paper is the second of two self-contained, but technically related companion papers inspired by binary judgment-aggregation theory. (shrink)

How can different individuals' probability assignments to some events be aggregated into a collective probability assignment? Classic results on this problem assume that the set of relevant events -- the agenda -- is a sigma-algebra and is thus closed under disjunction (union) and conjunction (intersection). We drop this demanding assumption and explore probabilistic opinion pooling on general agendas. One might be interested in the probability of rain and that of an interest-rate increase, but not in the probability of rain or (...) an interest-rate increase. We characterize linear pooling and neutral pooling for general agendas, with classic results as special cases for agendas that are sigma-algebras. As an illustrative application, we also consider probabilistic preference aggregation. Finally, we compare our results with existing results on binary judgment aggregation and Arrovian preference aggregation. This paper is the first of two self-contained, but technically related companion papers inspired by binary judgment-aggregation theory. (shrink)

Igor Douven establishes several new intransitivity results concerning evidential support. I add to Douven’s very instructive discussion by establishing two further intransitivity results and a transitivity result.

Jeff Paris proves a generalized Dutch Book theorem. If a belief state is not a generalized probability then one faces ‘sure loss’ books of bets. In Williams I showed that Joyce’s accuracy-domination theorem applies to the same set of generalized probabilities. What is the relationship between these two results? This note shows that both results are easy corollaries of the core result that Paris appeals to in proving his dutch book theorem. We see that every point of accuracy-domination defines a (...) dutch book, but we only have a partial converse. (shrink)

This paper starts by indicating the analysis of Hempel's conditions of adequacy for any relation of confirmation (Hempel, 1945) as presented in Huber (submitted). There I argue contra Carnap (1962, Section 87) that Hempel felt the need for two concepts of confirmation: one aiming at plausible theories and another aiming at informative theories. However, he also realized that these two concepts are conflicting, and he gave up the concept of confirmation aiming at informative theories. The main part of the paper (...) consists in working out the claim that one can have Hempel's cake and eat it too - in the sense that there is a logic of theory assessment that takes into account both of the two conflicting aspects of plausibility and informativeness. According to the semantics of this logic, a is an acceptable theory for evidence β if and only if a is both sufficiently plausible given β and sufficiently informative about β. This is spelt out in terms of ranking functions (Spohn, 1988) and shown to represent the syntactically specified notion of an assessment relation. The paper then compares these acceptability relations to explanatory and confirmatory consequence relations (Flach, 2000) as well as to nonmonotonic consequence relations (Kraus et al., 1990). It concludes by relating the plausibility-informativeness approach to Carnap's positive relevance account, thereby shedding new light on Carnap's analysis as well as solving another problem of confirmation theory. (shrink)

Lewis’s Principal Principle says that one should usually align one’s credences with the known chances. In this paper I develop a version of the Principal Principle that deals well with some exceptional cases related to the distinction between metaphysical and epistemic modal­ity. I explain how this principle gives a unified account of the Sleeping Beauty problem and chance-­based principles of anthropic reasoning. In doing so, I defuse the Doomsday Argument that the end of the world is likely to be nigh. (...) GPI Working Paper No. 8-2021. (shrink)

Is the fact that our universe contains fine-tuned life evidence that we live in a multiverse? Ian Hacking and Roger White influentially argue that it is not. We approach this question through a systematic framework for self-locating epistemology. As it turns out, leading approaches to self-locating evidence agree that the fact that our own universe contains fine-tuned life indeed confirms the existence of a multiverse. This convergence is no accident: we present two theorems showing that, in this setting, any updating (...) rule that satisfies a few reasonable conditions will have the same feature. The conclusion that fine-tuned life provides evidence for a multiverse is hard to escape. (shrink)

This paper proposes a new explanation for the paradoxes related to anthropic reasoning. Solutions to the Sleeping Beauty Problem and the Doomsday argument are discussed in detail. The main argument can be summarized as follows: -/- Our thoughts, reasonings and narratives inherently comes from a certain perspective. With each perspective there is a center, or using the term broadly, a self. The natural first-person perspective is most primitive. However we can also think and express from others’ perspectives with a theory (...) of mind. A perspective’s center could be unrelated to the topic of discussion so its de se thoughts need not to be considered, e.g. the perspective of an outside observer. Let’s call these the third-person perspective. First-person reasoning allows primitive self identification as I am inherently unique as the center of the perspective. Whereas from third-person perspective I am not fundamentally special comparing to others so a reference class of observers including me can be defined. It is my contention that reasonings from different perspectives should not mix. Otherwise it could lead to paradoxes even independent of anthropic reasoning. The paradoxes surrounding anthropic reasoning are caused by the aforementioned perspective mix. Regarding the sleeping beauty problem the correct answer should be double halving. Lewisian halving and thirding uses unique reasonings from both first and third-person perspectives. Indexical probabilities such as “the probability that this is the first awakening” or “the probability of me being one of the first 100 billion human beings” also mixes first- and third-person reasonings. Therefore invalid. Readers against perspectivism may disagree with point 1 and suggest we could reason in objective terms without the limit of perspectives. My argument is compatible with this belief. Objective reasoning would be analytically identical to the third-person perspective. My argument would become that objective reasoning and perspective reasonings should not mix. In the following I would continue to use “third-person perspective” but readers can switch that to “objective reasoning” if they wish so. (shrink)

Abstract: Four main forms of Doomsday Argument (DA) exist—Gott’s DA, Carter’s DA, Grace’s DA and Universal DA. All four forms use different probabilistic logic to predict that the end of the human civilization will happen unexpectedly soon based on our early location in human history. There are hundreds of publications about the validity of the Doomsday argument. Most of the attempts to disprove the Doomsday Argument have some weak points. As a result, we are uncertain about the validity of DA (...) proofs and rebuttals. In this article, a meta-DA is introduced, which uses the idea of logical uncertainty over the DA’s validity estimated based on a virtual prediction market of the opinions of different scientists. The result is around 0.4 for the validity of some form of DA, and even smaller for “Strong DA”, which predicts the end of the world in the near term. We discuss many examples of the validity of the DA in real life as an instrument to prove it “experimentally”. We also show that DA becomes strongest if it is based on the idea of the “natural reference class” of observers, that is, the observers who know about the DA (i.e. a Self-Referenced DA). Such a DA predicts that there is a high probability of a global catastrophe with human extinction in the 21st century, which aligns with what we already know based on analysis of different technological risks. (shrink)

The physicist J. Richard Gott has given an argument which, if good, allows one to make accurate predictions for the future longevity of a process, based solely on its present age. We show that there are problems with some of the details of Gott's argument, but we defend the core thesis: in many circumstances, the greater the present age of a process, the more likely a longer future duration.

We develop a Bayesian framework for thinking about the way evidence about the here and now can bear on hypotheses about the qualitative character of the world as a whole, including hypotheses according to which the total population of the world is infinite. We show how this framework makes sense of the practice cosmologists have recently adopted in their reasoning about such hypotheses.

Bayesian conceptions of evidence have been invoked in recent arguments regarding the existence of God, the hypothesis of multiple physical universes, and the Doomsday Argument. Philosophers writing on these topics often claim that, given a Bayesian account of evidence, our existence or something entailed by our existence (perhaps in conjunction with some background knowledge or assumption) may serve as evidence for each of us. In this paper, I argue that this widespread view is mistaken. The mere fact of one's existence (...) qua conscious creature cannot serve as evidence on the standard Bayesian conception of evidence because knowledge of one's existence is a necessary part of the background knowledge relative to which all epistemic probabilities are defined. It follows that some formulations of the fine-tuning argument (for theism or a multiverse), the argument from consciousness (for theism) and a rejoinder to the Doomsday argument are mistaken. (shrink)

How should we update our beliefs when we learn new evidence? Bayesian confirmation theory provides a widely accepted and well understood answer – we should conditionalize. But this theory has a problem with self-locating beliefs, beliefs that tell you where you are in the world, as opposed to what the world is like. To see the problem, consider your current belief that it is January. You might be absolutely, 100%, sure that it is January. But you will soon believe it (...) is February. This type of belief change cannot be modelled by conditionalization. We need some new principles of belief change for this kind of case, which I call belief mutation. In part 1, I defend the Relevance-Limiting Thesis, which says that a change in a purely self-locating belief of the kind that results in belief mutation should not shift your degree of belief in a non-self-locating belief, which can only change by conditionalization. My method is to give detailed analyses of the puzzles which threaten this thesis: Duplication, Sleeping Beauty, and The Prisoner. This also requires giving my own theory of observation selection effects. In part 2, I argue that when self-locating evidence is learnt from a position of uncertainty, it should be conditionalized on in the normal way. I defend this position by applying it to various cases where such evidence is found. I defend the Halfer position in Sleeping Beauty, and I defend the Doomsday Argument and the Fine-Tuning Argument. (shrink)

This article defends the Doomsday Argument, the Halfer Position in Sleeping Beauty, the Fine-Tuning Argument, and the applicability of Bayesian confirmation theory to the Everett interpretation of quantum mechanics. It will argue that all four problems have the same structure, and it gives a unified treatment that uses simple models of the cases and no controversial assumptions about confirmation or self-locating evidence. The article will argue that the troublesome feature of all these cases is not self-location but selection effects.

We give an analysis of the Monty Hall problem purely in terms of confirmation, without making any lottery assumptions about priors. Along the way, we show the Monty Hall problem is structurally identical to the Doomsday Argument.

The Carter-Leslie Doomsday argument, as standardly presented, relies on the assumption that you have knowledge of your approximate birth rank. I demonstrate that the Doomsday argument can still be given in a situation where you have no knowledge of your birth rank. This allows one to reply to Bostrom's defense of the Doomsday argument against the refutation based on the idea that your existence makes it more likely that many observers exist.

A review of Matt Cook's book Sleight of Mind: 75 Ingenious Paradoxes in Mathematics, Physics, and Philosophy.

Peter Baumann uses the Monty Hall game to demonstrate that probabilities cannot be meaningfully applied to individual games. Baumann draws from this first conclusion a second: in a single game, it is not necessarily rational to switch from the door that I have initially chosen to the door that Monty Hall did not open. After challenging Baumann's particular arguments for these conclusions, I argue that there is a deeper problem with his position: it rests on the false assumption that what (...) justifies the switching strategy is its leading me to win a greater percentage of the time. In fact, what justifies the switching strategy is not any statistical result over the long run but rather the "causal structure" intrinsic to each individual game itself. Finally, I argue that an argument by Hilary Putnam will not help to save Baumann's second conclusion above. (shrink)

How should we update our beliefs when we learn new evidence? Bayesian confirmation theory provides a widely accepted and well understood answer – we should conditionalize. But this theory has a problem with self-locating beliefs, beliefs that tell you where you are in the world, as opposed to what the world is like. To see the problem, consider your current belief that it is January. You might be absolutely, 100%, sure that it is January. But you will soon believe it (...) is February. This type of belief change cannot be modelled by conditionalization. We need some new principles of belief change for this kind of case, which I call belief mutation. In part 1, I defend the Relevance-Limiting Thesis, which says that a change in a purely self-locating belief of the kind that results in belief mutation should not shift your degree of belief in a non-self-locating belief, which can only change by conditionalization. My method is to give detailed analyses of the puzzles which threaten this thesis: Duplication, Sleeping Beauty, and The Prisoner. This also requires giving my own theory of observation selection effects. In part 2, I argue that when self-locating evidence is learnt from a position of uncertainty, it should be conditionalized on in the normal way. I defend this position by applying it to various cases where such evidence is found. I defend the Halfer position in Sleeping Beauty, and I defend the Doomsday Argument and the Fine-Tuning Argument. (shrink)

We give an analysis of the Monty Hall problem purely in terms of confirmation, without making any lottery assumptions about priors. Along the way, we show the Monty Hall problem is structurally identical to the Doomsday Argument.

What degrees of belief does Sleeping Beauty's evidence support? That depends.

A new philosophical analysis is provided of the notorious Sleeping Beauty Problem. It is argued that the correct solution is one-third, but not in the way previous philosophers have typically meant this. A modified version of the Problem demonstrates that neither self-locating information nor amnesia is relevant to the core Problem, which is simply to evaluate the conditional chance of heads given an undated Monday-or-Tuesday awakening. Previous commentators have failed to appreciate the significance of the information that Beauty gains upon (...) waking, and which is relevant to the conditional chance of heads: de re acquaintance with the awakening itself and the non-locating knowledge that it is an experimental awakening. David Lewis and company are committed to several unjustifiable and unacceptable probability assessments. Previous commentators have in effect confused the information that Beauty undergoes this particular experimental awakening for the information that she undergoes some experimental awakening or other. Lewis in particular thereby illegitimately tips the scales both in favor of heads and in favor of Monday. The Sleeping Beauty Problem is equivalent to a ball-in-urn word problem in elementary probability theory. (shrink)

Lewis’s Principal Principle says that one should usually align one’s credences with the known chances. In this paper I develop a version of the Principal Principle that deals well with some exceptional cases related to the distinction between metaphysical and epistemic modal­ity. I explain how this principle gives a unified account of the Sleeping Beauty problem and chance-­based principles of anthropic reasoning. In doing so, I defuse the Doomsday Argument that the end of the world is likely to be nigh. (...) GPI Working Paper No. 8-2021. (shrink)

We argue against the halving response to Sleeping Beauty. First, we outline an appealing constraint on probability assignments: the Principle of Irrelevant Information. Roughly, this says: if you don’t know whether C, but you would assign probability p to X regardless of whether C or not-C, then you should assign p to X. This Principle is deeply plausible, but we show that it contradicts halving. Second, we show that halving either violates solid statistical reasoning or draws absurd distinctions.

The Sleeping Beauty problem remains controversial with disagreement between so-called Halfers and Thirders, although the Thirders appear to be leading these days. I analyze three popular arguments for the Thirder position, including the long-run frequency argument, Egla’s ‘symmetry’ argument, and new-information arguments, and find problems with each. The long-run frequency argument is almost unequivocally thought to strongly support Thirders, but in formalizing the argument for an arbitrary number of repetitions, I show that the expected proportion of Heads-Awakenings for a single-trial (...) experiment is unambiguously 1/2. My criticisms of Elga’s symmetry argument and the new-information arguments point to subtle misalignments between the narrative/causal description of thought-experiments and the mathematical probability expressions and theory we use to describe these narratives. I end with distinguishing two varieties of possibility—a dynamic forward type and static historical type—that help clarify the Sleeping Beauty problem, nullify the main criticism against Lewis’s Halfer argument, and have applicability to probability theory in general. (shrink)

The Sleeping Beauty problem has attracted considerable attention in the literature as a paradigmatic example of how self-locating uncertainty creates problems for the Bayesian principles of Conditionalization and Reflection. Furthermore, it is also thought to raise serious issues for diachronic Dutch Book arguments. I show that, contrary to what is commonly accepted, it is possible to represent the Sleeping Beauty problem within a standard Bayesian framework. Once the problem is correctly represented, the ‘thirder’ solution satisfies standard rationality principles, vindicating why (...) it is not vulnerable to diachronic Dutch Book arguments. Moreover, the diachronic Dutch Books against the ‘halfer’ solutions fail to undermine the standard arguments for Conditionalization. The main upshot that emerges from my discussion is that the disagreement between different solutions does not challenge the applicability of Bayesian reasoning to centered settings, nor the commitment to Conditionalization, but is instead an instance of the familiar problem of choosing the priors. (shrink)

“Double-halfers” think that throughout the Sleeping Beauty Scenario, Beauty ought to maintain a credence of 1/2 in the proposition that the fair coin toss governing the experimental protocol comes up heads. Titelbaum (2012) introduces a novel variation on the standard scenario, one involving an additional coin toss, and claims that the double-halfer is committed to the absurd and embarrassing result that Beauty’s credence in an indexical proposition concerning the outcome of a future fair coin toss is not 1/2. I argue (...) that there is no reason to regard the credence required by the double-halfer as any less acceptable than the one deemed required by Titelbaum. (shrink)

Principles of chance deference face two kinds of problems. In the first place, they face difficulties with a priori knowable contingencies. In the second place, they face difficulties in cases where you've lost track of the time. I provide a principle of chance deference which handles these problem cases. This principle has a surprising consequence for Adam Elga's Sleeping Beauty Puzzle.

The epistemology of self-locating belief concerns itself with how rational agents ought to respond to certain kinds of indexical information. I argue that those who endorse the thesis of Time-Slice Rationality ought to endorse a particular view about the epistemology of self-locating belief, according to which ‘essentially indexical’ information is never evidentially relevant to non-indexical matters. I close by offering some independent motivations for endorsing Time-Slice Rationality in the context of the epistemology of self-locating belief.

The strong law of large numbers and considerations concerning additional information strongly suggest that Beauty upon awakening has probability 1/3 to be in a heads-awakening but should still believe the probability that the coin landed heads in the Sunday toss to be 1/2. The problem is that she is in a heads-awakening if and only if the coin landed heads. So, how can she rationally assign different probabilities or credences to propositions she knows imply each other? This is the problem (...) I address in this article. I suggest that ‘p whenever q and vice versa’ may be consistent with p and q having different probabilities if one of them refers to a sample space containing ordinary possible worlds and the other to a sample space containing centred possible worlds, because such spaces may fail to combine into one composite probability space and, as a consequence, ‘whenever’ may not be well defined; such is the main contribution of this article. 1The Sleeping Beauty Game2Groisman’s and Peter Lewis’s Approaches3Discussing Beauty’s Credences4The Principle of Equivalence's Failure5Making Sense of the Principle of Equivalence's Failure6Elga’s and Lewis’s Approaches7ConclusionAppendix. (shrink)

This paper considers fair betting odds for certain bets that might be placed in the situation discussed in the so-called Sleeping Beauty Problem. This paper examines what Thirders, Halfers, and Double Halfers must say about the odds as determined by various decision theoretic approaches and argues that Thirders and Halfers have difficulties formulating plausible and coherent positions concerning the relevant betting odds. Double Halfers do not face this problem and that is an important consideration in favor of Double Halfers.

This paper proposes a new explanation for the paradoxes related to anthropic reasoning. Solutions to the Sleeping Beauty Problem and the Doomsday argument are discussed in detail. The main argument can be summarized as follows: -/- Our thoughts, reasonings and narratives inherently comes from a certain perspective. With each perspective there is a center, or using the term broadly, a self. The natural first-person perspective is most primitive. However we can also think and express from others’ perspectives with a theory (...) of mind. A perspective’s center could be unrelated to the topic of discussion so its de se thoughts need not to be considered, e.g. the perspective of an outside observer. Let’s call these the third-person perspective. First-person reasoning allows primitive self identification as I am inherently unique as the center of the perspective. Whereas from third-person perspective I am not fundamentally special comparing to others so a reference class of observers including me can be defined. It is my contention that reasonings from different perspectives should not mix. Otherwise it could lead to paradoxes even independent of anthropic reasoning. The paradoxes surrounding anthropic reasoning are caused by the aforementioned perspective mix. Regarding the sleeping beauty problem the correct answer should be double halving. Lewisian halving and thirding uses unique reasonings from both first and third-person perspectives. Indexical probabilities such as “the probability that this is the first awakening” or “the probability of me being one of the first 100 billion human beings” also mixes first- and third-person reasonings. Therefore invalid. Readers against perspectivism may disagree with point 1 and suggest we could reason in objective terms without the limit of perspectives. My argument is compatible with this belief. Objective reasoning would be analytically identical to the third-person perspective. My argument would become that objective reasoning and perspective reasonings should not mix. In the following I would continue to use “third-person perspective” but readers can switch that to “objective reasoning” if they wish so. (shrink)

In this paper, I argue that some plausible principles concerning which credences are rationally permissible for agents given information about one another’s epistemic and credal states have some surprising consequences for which credences an agent ought to have in light of self-locating information. I provide a framework that allows us to state these constraints and draw out these consequences precisely. I then consider and assess the prospects for rejecting these prima facie plausible principles.

A longstanding issue in attempts to understand the Everett (Many-Worlds) approach to quantum mechanics is the origin of the Born rule: why is the probability given by the square of the amplitude? Following Vaidman, we note that observers are in a position of self-locating uncertainty during the period between the branches of the wave function splitting via decoherence and the observer registering the outcome of the measurement. In this period it is tempting to regard each branch as equiprobable, but we (...) argue that the temptation should be resisted. Applying lessons from this analysis, we demonstrate (using methods similar to those of Zurek's envariance-based derivation) that the Born rule is the uniquely rational way of apportioning credence in Everettian quantum mechanics. In doing so, we rely on a single key principle: changes purely to the environment do not affect the probabilities one ought to assign to measurement outcomes in a local subsystem. We arrive at a method for assigning probabilities in cases that involve both classical and quantum self-locating uncertainty. This method provides unique answers to quantum Sleeping Beauty problems, as well as a well-defined procedure for calculating probabilities in quantum cosmological multiverses with multiple similar observers. (shrink)

If an agent believes that the probability of E being true is 1/2, should she accept a bet on E at even odds or better? Yes, but only given certain conditions. This paper is about what those conditions are. In particular, we think that there is a condition that has been overlooked so far in the literature. We discovered it in response to a paper by Hitchcock (2004) in which he argues for the 1/3 answer to the Sleeping Beauty problem. (...) Hitchcock argues that this credence follows from calculating her fair betting odds, plus the assumption that Sleeping Beauty’s credences should track her fair betting odds. We will show that this last assumption is false. Sleeping Beauty’s credences should not follow her fair betting odds due to a peculiar feature of her epistemic situation. (shrink)

In Bradley, I offered an analysis of Sleeping Beauty and the Everettian interpretation of quantum mechanics. I argued that one can avoid a kind of easy confirmation of EQM by paying attention to observation selection effects, that halfers are right about Sleeping Beauty, and that thirders cannot avoid easy confirmation for the truth of EQM. Wilson agrees with my analysis of observation selection effects in EQM, but goes on to, first, defend Elga’s thirder argument on Sleeping Beauty and, second, argue (...) that the analogy I draw between Sleeping Beauty and EQM fails. I will argue that neither point succeeds. 1 Introduction2 Background3 Wilson’s Argument for ⅓ in Sleeping Beauty4 Reply: Explaining Away the Crazy5 Wilson's Argument for the Breakdown of the Analogy6 Reply: The Irrelevance of Chance7 Conclusion. (shrink)

A large number of essays address the Sleeping Beauty problem, which undermines the validity of Bayesian inference and Bas Van Fraassen's 'Reflection Principle'. In this study a straightforward analysis of the problem based on probability theory is presented. The key difference from previous works is that apart from the random experiment imposed by the problem's description, a different one is also considered, in order to negate the confusion on the involved conditional probabilities. The results of the analysis indicate that no (...) inconsistency takes place, whereas both Bayesian inference and 'Reflection Principle' are valid. (shrink)

In a recent article, Joel Pust argued that direct inference based on reference properties of differing arity are incommensurable, and so direct inference cannot be used to resolve the Sleeping Beauty problem. After discussing the defects of Pust's argument, I offer reasons for thinking that direct inferences based on reference properties of differing arity are commensurable, and that we should prefer direct inferences based on logically stronger reference properties, regardless of arity.

There are three main points of the paper. 1. There are straightforward ways of manipulating expected gains and losses that result in a divergence between fair betting odds and credence. Such manipulations are familiar from tools of finance. One can easily see that the Sleeping Beauty case is structured in such a way as to result in a divergence between fair betting odds and credence. 2. The inspection of credences and betting odds in certain betting situations shows that the two (...) main extant positions, Elga's and Lewis's, are both mistaken. 3. My proposal may seem to require a revision of probability theory but it does not. The Sleeping Beauty case merely calls attention to a constraint on partitioning of possibility space that is usually satisfied as a matter of course but can in fact be violated. (shrink)

In two excellent recent papers, Jacob Ross has argued that the standard arguments for the ‘thirder’ answer to the Sleeping Beauty puzzle lead to violations of countable additivity. The problem is that most arguments for that answer generalise in awkward ways when he looks at the whole class of what he calls Sleeping Beauty problems. In this note I develop a new argument for the thirder answer that doesn't generalise in this way.

The Sleeping Beauty puzzle provides a nice illustration of the approach to self-locating belief defended by Robert Stalnaker in Our Knowledge of the Internal World (Stalnaker, 2008), as well as a test of the utility of that method. The setup of the Sleeping Beauty puzzle is by now fairly familiar. On Sunday Sleeping Beauty is told the rules of the game, and a (known to be) fair coin is flipped. On Monday, Sleeping Beauty is woken, and then put back to (...) sleep. If, and only if, the coin landed tails, she is woken again on Tuesday after having her memory of the Monday awakening erased.1 On Wednesday she is woken again and the game ends. There are a few questions we can ask about Beauty’s attitudes as the game progresses. We’d like to know what her credence that the coin landed heads should be (a) Before she goes to sleep Sunday; (b) When she wakes on Monday; (c) When she wakes on Tuesday; and (d) When she wakes on Wednesday? Standard treatments of the Sleeping Beauty puzzle ignore (d), run together (b) and (c) into one (somewhat ill-formed) question, and then divide theorists into ‘halfers’ or ‘thirders’ depending on how they answer it. Following Stalnaker, I’m going to focus on (b) here, though I’ll have a little to say about (c) and (d) as well. I’ll be following orthodoxy in taking 1 2 to be the clear answer to (a), and in taking the correct answers to (b) and (c) to be independent of how the coin lands, though I’ll briefly question that assumption at the end. An answer to these four questions should respect two different kinds of constraints. The answer for day n should make sense ‘statically’. It should be a sensible answer to the question of what Beauty should do given what information she then has. And the answer should make sense ‘dynamically’. It should be a sensible answer to the question of how Beauty should have updated her credences from some earlier day, given rational credences on the earlier day. As has been fairly clear since the discussion of the problem in Elga (2000), Sleeping Beauty is puzzling because static and dynamic considerations appear to push in different directions.. (shrink)

The best arguments for the 1/3 answer to the Sleeping Beauty problem all require that when Beauty awakes on Monday she should be uncertain what day it is. I argue that this claim should be rejected, thereby clearing the way to accept the 1/2 solution.

Sometimes we learn what the world is like, and sometimes we learn where in the world we are. Are there any interesting differences between the two kinds of cases? The main aim of this article is to argue that learning where we are in the world brings into view the same kind of observation selection effects that operate when sampling from a population. I will first explain what observation selection effects are ( Section 1 ) and how they are relevant (...) to learning where we are in the world ( Section 2 ). I will show how measurements in the Many Worlds Interpretation of quantum mechanics can be understood as learning where you are in the world via some observation selection effect ( Section 3 ). I will apply a similar argument to the Sleeping Beauty Problem ( Section 4 ) and explain what I take the significance of the analogy to be ( Section 5 ). Finally, I will defend the Restricted Principle of Indifference on which some of my arguments depend ( Section 6 ). (shrink)

A number of cases involving self-locating beliefs have been discussed in the Bayesian literature. I suggest that many of these cases, such as the sleeping beauty case, are entangled with issues that are independent of self-locating beliefs per se. In light of this, I propose a division of labor: we should address each of these issues separately before we try to provide a comprehensive account of belief updating. By way of example, I sketch some ways of extending Bayesianism in order (...) to accommodate these issues. Then, putting these other issues aside, I sketch some ways of extending Bayesianism in order to accommodate self-locating beliefs. Finally, I propose a constraint on updating rules, the "Learning Principle", which rules out certain kinds of troubling belief changes, and I use this principle to assess some of the available options. (shrink)