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  1. Geometric Pooling: A User's Guide.Richard Pettigrew & Jonathan Weisberg - forthcoming - British Journal for the Philosophy of Science.
    Much of our information comes to us indirectly, in the form of conclusions others have drawn from evidence they gathered. When we hear these conclusions, how can we modify our own opinions so as to gain the benefit of their evidence? In this paper we study the method known as geometric pooling. We consider two arguments in its favour, raising several objections to one, and proposing an amendment to the other.
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  2. On the pragmatic and epistemic virtues of inference to the best explanation.Richard Pettigrew - 2021 - Synthese 199 (5-6):12407-12438.
    In a series of papers over the past twenty years, and in a new book, Igor Douven has argued that Bayesians are too quick to reject versions of inference to the best explanation that cannot be accommodated within their framework. In this paper, I survey their worries and attempt to answer them using a series of pragmatic and purely epistemic arguments that I take to show that Bayes’ Rule really is the only rational way to respond to your evidence.
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  3. Trivalent Conditionals: Stalnaker's Thesis and Bayesian Inference.Paul Égré, Lorenzo Rossi & Jan Sprenger - manuscript
    This paper develops a trivalent semantics for indicative conditionals and extends it to a probabilistic theory of valid inference and inductive learning with conditionals. On this account, (i) all complex conditionals can be rephrased as simple conditionals, connecting our account to Adams's theory of p-valid inference; (ii) we obtain Stalnaker's Thesis as a theorem while avoiding the well-known triviality results; (iii) we generalize Bayesian conditionalization to an updating principle for conditional sentences. The final result is a unified semantic and probabilistic (...)
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  4. How to Analyse Retrodictive Probabilities in Inference to the Best Explanation.Andrew Holster - manuscript
    IBE ('Inference to the best explanation' or abduction) is a popular and highly plausible theory of how we should judge the evidence for claims of past events based on present evidence. It has been notably developed and supported recently by Meyer following Lipton. I believe this theory is essentially correct. This paper supports IBE from a probability perspective, and argues that the retrodictive probabilities involved in such inferences should be analysed in terms of predictive probabilities and a priori probability ratios (...)
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  5. On Being a Random Sample.David Manley - manuscript
    It is well known that de se (or ‘self-locating’) propositions complicate the standard picture of how we should respond to evidence. This has given rise to a substantial literature centered around puzzles like Sleeping Beauty, Dr. Evil, and Doomsday—and it has also sparked controversy over a style of argument that has recently been adopted by theoretical cosmologists. These discussions often dwell on intuitions about a single kind of case, but it’s worth seeking a rule that can unify our treatment of (...)
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  6. Revealing the Beauty behind the Sleeping Beauty Problem.Ioannis Mariolis - manuscript
    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 (...)
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  7. What the "Equal Weight View" is.Randall G. McCutcheon - manuscript
    Dawid, DeGroot and Mortera showed, a quarter century ago, that any agent who regards a fellow agent as a peer--in particular, defers to the fellow agent's prior credences in the same way that she defers to her own--and updates by split-the-difference is prone to diachronic incoherence. On the other hand one may show that there are special scenarios in which Bayesian updating approximates difference splitting, so it remains an important question whether it remains a viable response to ``generic" peer update. (...)
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  8. Self-locating belief and the goal of accuracy.Richard Pettigrew - manuscript
    The goal of a partial belief is to be accurate, or close to the truth. By appealing to this norm, I seek norms for partial beliefs in self-locating and non-self-locating propositions. My aim is to find norms that are analogous to the Bayesian norms, which, I argue, only apply unproblematically to partial beliefs in non-self-locating propositions. I argue that the goal of a set of partial beliefs is to minimize the expected inaccuracy of those beliefs. However, in the self-locating framework, (...)
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  9. A non-pragmatic dominance argument for conditionalization.Robert Williams - manuscript
    In this paper, I provide an accuracy-based argument for conditionalization (via reflection) that does not rely on norms of maximizing expected accuracy. -/- (This is a draft of a paper that I wrote in 2013. It stalled for no very good reason. I still believe the content is right).
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  10. Pooling, Products, and Priors.Richard Pettigrew & Jonathan Weisberg -
    We often learn the opinions of others without hearing the evidence on which they're based. The orthodox Bayesian response is to treat the reported opinion as evidence itself and update on it by conditionalizing. But sometimes this isn't feasible. In these situations, a simpler way of combining one's existing opinion with opinions reported by others would be useful, especially if it yields the same results as conditionalization. We will show that one method---upco, also known as multiplicative pooling---is specially suited to (...)
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  11. My way or her way: A conundrum in bayesian epistemology of disagreement.Tomoji Shogenji - manuscript
    The proportional weight view in epistemology of disagreement generalizes the equal weight view and proposes that we assign to judgments of different people weights that are proportional to their epistemic qualifications. It is shown that if the resulting degrees of confidence are to constitute a probability function, they must be the weighted arithmetic means of individual degrees of confidence, while if the resulting degrees of confidence are to obey the Bayesian rule of conditionalization, they must be the weighted geometric means (...)
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  12. Conditionalization.Lisa Cassell - forthcoming - In Matthias Steup Kurt Sylvan (ed.), Blackwell Companion to Epistemology, Third Edition. Wiley-Blackwell.
  13. Time-Slice Epistemology for Bayesians.Lisa Cassell - forthcoming - Inquiry: An Interdisciplinary Journal of Philosophy.
    Recently, some have challenged the idea that there are genuine norms of diachronic rationality. Part of this challenge has involved offering replacements for diachronic principles. Skeptics about diachronic rationality believe that we can provide an error theory for it by appealing to synchronic updating rules that, over time, mimic the behavior of diachronic norms. In this paper, I argue that the most promising attempts to develop this position within the Bayesian framework are unsuccessful. I sketch a new synchronic surrogate that (...)
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  14. Knowledge-First Evidentialism about Rationality.Julien Dutant - forthcoming - In Fabian Dorsch & Julien Dutant (eds.), The New Evil Demon Problem. Oxford University Press.
    Knowledge-first evidentialism combines the view that it is rational to believe what is supported by one's evidence with the view that one's evidence is what one knows. While there is much to be said for the view, it is widely perceived to fail in the face of cases of reasonable error—particularly extreme ones like new Evil Demon scenarios (Wedgwood, 2002). One reply has been to say that even in such cases what one knows supports the target rational belief (Lord, 201x, (...)
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  15. Know Your Way Out of St. Petersburg: An Exploration of "Knowledge-First" Decision Theory.Frank Hong - forthcoming - Erkenntnis:1-20.
    This paper explores the consequences of applying two natural ideas from epistemology to decision theory: (1) that knowledge should guide our actions, and (2) that we know a lot of non-trivial things. In particular, we explore the consequences of these ideas as they are applied to standard decision theoretic puzzles such as the St. Petersburg Paradox. In doing so, we develop a “knowledge-first” decision theory and we will see how it can help us avoid fanaticism with regard to the St. (...)
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  16. Generalized Conditionalization and the Sleeping Beauty Problem, II.Terence Horgan - forthcoming - Erkenntnis.
    In “Generalized Conditionalization and the Sleeping Beauty Problem,” Anna Mahtani and I offer a new argument for thirdism that relies on what we call “generalized conditionalization.” Generalized conditionalization goes beyond conventional conditionalization in two respects: first, by sometimes deploying a space of synchronic, essentially temporal, candidate-possibilities that are not “prior” possibilities; and second, by allowing for the use of preliminary probabilities that arise by first bracketing, and then conditionalizing upon, “old evidence.” In “Beauty and Conditionalization: Reply to Horgan and Mahtani,” (...)
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  17. Jeffrey Conditionalization Permits Undermining.Marc Lange - forthcoming - Journal of Philosophical Logic:1-7.
    It has frequently been argued recently that Jeffrey Conditionalization (JC) does not permit undermining. For JC to be inapplicable in cases where the evidence could be undermined would severely compromise JC’s range. However, this paper contends that the argument fails to show that JC cannot accommodate undermining. This response turns on using the proper partition to capture the direct impact of our evidence in redistributing our credences.
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  18. Not So Phenomenal!Maria Lasonen-Aarnio & John Hawthorne - forthcoming - The Philosophical Review.
    Our main aims in this paper is to discuss and criticise the core thesis of a position that has become known as phenomenal conservatism. According to this thesis, its seeming to one that p provides enough justification for a belief in p to be prima facie justified (a thesis we label Standard Phenomenal Conservatism). This thesis captures the special kind of epistemic import that seemings are claimed to have. To get clearer on this thesis, we embed it, first, in a (...)
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  19. An accuracy-based approach to quantum conditionalization.Alexander Meehan & Jer Alex Steeger - forthcoming - British Journal for the Philosophy of Science.
  20. 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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  21. Rational Aversion to Information.Sven Neth - forthcoming - British Journal for the Philosophy of Science.
    Is more information always better? Or are there some situations in which more information can make us worse off? Good (1967) argues that expected utility maximizers should always accept more information if the information is cost-free and relevant. But Good's argument presupposes that you are certain you will update by conditionalization. If we relax this assumption and allow agents to be uncertain about updating, these agents can be rationally required to reject free and relevant information. Since there are good reasons (...)
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  22. How should your beliefs change when your awareness grows?Richard Pettigrew - forthcoming - Episteme:1-25.
    Epistemologists who study credences have a well-developed account of how you should change them when you learn new evidence; that is, when your body of evidence grows. What's more, they boast a diverse range of epistemic and pragmatic arguments that support that account. But they do not have a satisfactory account of when and how you should change your credences when you become aware of possibilities and propositions you have not entertained before; that is, when your awareness grows. In this (...)
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  23. Non-Factive Kolmogorov Conditionalization.Michael Rescorla - forthcoming - Review of Symbolic Logic:1-46.
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  24. An Improved Argument for Superconditionalization.Julia Staffel & Glauber De Bona - forthcoming - Erkenntnis:1-27.
    Standard arguments for Bayesian conditionalizing rely on assumptions that many epistemologists have criticized as being too strong: (i) that conditionalizers must be logically infallible, which rules out the possibility of rational logical learning, and (ii) that what is learned with certainty must be true (factivity). In this paper, we give a new factivity-free argument for the superconditionalization norm in a personal possibility framework that allows agents to learn empirical and logical falsehoods. We then discuss how the resulting framework should be (...)
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  25. Best Laid Plans: Idealization and the Rationality–Accuracy Bridge.Brett Topey - forthcoming - British Journal for the Philosophy of Science.
    Hilary Greaves and David Wallace argue that conditionalization maximizes expected accuracy and so is a rational requirement, but their argument presupposes a particular picture of the bridge between rationality and accuracy: the Best-Plan-to-Follow picture. And theorists such as Miriam Schoenfield and Robert Steel argue that it's possible to motivate an alternative picture—the Best-Plan-to-Make picture—that does not vindicate conditionalization. I show that these theorists are mistaken: it turns out that, if an update procedure maximizes expected accuracy on the Best-Plan-to-Follow picture, it's (...)
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  26. Higher-Order Evidence and the Dynamics of Self-Location: An Accuracy-Based Argument for Calibrationism.Brett Topey - forthcoming - Erkenntnis:1-27.
    The thesis that agents should calibrate their beliefs in the face of higher-order evidence—i.e., should adjust their first-order beliefs in response to evidence suggesting that the reasoning underlying those beliefs is faulty—is sometimes thought to be in tension with Bayesian approaches to belief update: in order to obey Bayesian norms, it's claimed, agents must remain steadfast in the face of higher-order evidence. But I argue that this claim is incorrect. In particular, I motivate a minimal constraint on a reasonable treatment (...)
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  27. On the Ecological and Internal Rationality of Bayesian Conditionalization and Other Belief Updating Strategies.Olav Benjamin Vassend - forthcoming - British Journal for the Philosophy of Science.
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  28. Epistemic Probabilities are Degrees of Support, not Degrees of (Rational) Belief.Nevin Climenhaga - 2024 - Philosophy and Phenomenological Research 108 (1):153-176.
    I argue that when we use ‘probability’ language in epistemic contexts—e.g., when we ask how probable some hypothesis is, given the evidence available to us—we are talking about degrees of support, rather than degrees of belief. The epistemic probability of A given B is the mind-independent degree to which B supports A, not the degree to which someone with B as their evidence believes A, or the degree to which someone would or should believe A if they had B as (...)
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  29. Bayesian defeat of certainties.Michael Rescorla - 2024 - Synthese 203 (2):1-38.
    When P(E) > 0, conditional probabilities P(H|E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(H|E)$$\end{document} are given by the ratio formula. An agent engages in ratio conditionalization when she updates her credences using conditional probabilities dictated by the ratio formula. Ratio conditionalization cannot eradicate certainties, including certainties gained through prior exercises of ratio conditionalization. An agent who updates her credences only through ratio conditionalization risks permanent certainty in propositions against which she has overwhelming evidence. To avoid this undesirable consequence, (...)
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  30. Support for Geometric Pooling.Jean Baccelli & Rush T. Stewart - 2023 - Review of Symbolic Logic 16 (1):298-337.
    Supra-Bayesianism is the Bayesian response to learning the opinions of others. Probability pooling constitutes an alternative response. One natural question is whether there are cases where probability pooling gives the supra-Bayesian result. This has been called the problem of Bayes-compatibility for pooling functions. It is known that in a common prior setting, under standard assumptions, linear pooling cannot be nontrivially Bayes-compatible. We show by contrast that geometric pooling can be nontrivially Bayes-compatible. Indeed, we show that, under certain assumptions, geometric and (...)
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  31. 'Logic Will Get You From A to B, Imagination Will Take You Anywhere'.Francesco Berto - 2023 - Noûs.
    There is some consensus on the claim that imagination as suppositional thinking can have epistemic value insofar as it’s constrained by a principle of minimal alteration of how we know or believe reality to be – compatibly with the need to accommodate the supposition initiating the imaginative exercise. But in the philosophy of imagination there is no formally precise account of how exactly such minimal alteration is to work. I propose one. I focus on counterfactual imagination, arguing that this can (...)
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  32. Learning from experience and conditionalization.Peter Brössel - 2023 - Philosophical Studies 180 (9):2797-2823.
    Bayesianism can be characterized as the following twofold position: (i) rational credences obey the probability calculus; (ii) rational learning, i.e., the updating of credences, is regulated by some form of conditionalization. While the formal aspect of various forms of conditionalization has been explored in detail, the philosophical application to learning from experience is still deeply problematic. Some philosophers have proposed to revise the epistemology of perception; others have provided new formal accounts of conditionalization that are more in line with how (...)
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  33. Why Aren’t I Part of a Whale?David Builes & Caspar Hare - 2023 - Analysis 83 (2):227-234.
    We start by presenting three different views that jointly imply that every person has many conscious beings in their immediate vicinity, and that the number greatly varies from person to person. We then present and assess an argument to the conclusion that how confident someone should be in these views should sensitively depend on how massive they happen to be. According to the argument, sometimes irreducibly de se observations can be powerful evidence for or against believing in metaphysical theories.
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  34. When the (Bayesian) ideal is not ideal.Danilo Fraga Dantas - 2023 - Logos and Episteme 15 (3):271-298.
    Bayesian epistemologists support the norms of probabilism and conditionalization using Dutch book and accuracy arguments. These arguments assume that rationality requires agents to maximize practical or epistemic value in every doxastic state, which is evaluated from a subjective point of view (e.g., the agent’s expectancy of value). The accuracy arguments also presuppose that agents are opinionated. The goal of this paper is to discuss the assumptions of these arguments, including the measure of epistemic value. I have designed AI agents based (...)
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  35. Credal imprecision and the value of evidence.Nilanjan Das - 2023 - Noûs 57 (3):684-721.
    This paper is about a tension between two theses. The first is Value of Evidence: roughly, the thesis that it is always rational for an agent to gather and use cost‐free evidence for making decisions. The second is Rationality of Imprecision: the thesis that an agent can be rationally required to adopt doxastic states that are imprecise, i.e., not representable by a single credence function. While others have noticed this tension, I offer a new diagnosis of it. I show that (...)
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  36. The Value of Biased Information.Nilanjan Das - 2023 - British Journal for the Philosophy of Science 74 (1):25-55.
    In this article, I cast doubt on an apparent truism, namely, that if evidence is available for gathering and use at a negligible cost, then it’s always instrumentally rational for us to gather that evidence and use it for making decisions. Call this ‘value of information’ (VOI). I show that VOI conflicts with two other plausible theses. The first is the view that an agent’s evidence can entail non-trivial propositions about the external world. The second is the view that epistemic (...)
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  37. Expert deference and Adams conditionalization.Patryk Dziurosz-Serafinowicz - 2023 - Synthese 201 (5):1-21.
    The standard principle of expert deference says that conditional on the expert’s credence in a proposition _A_ being _x_, your credence in _A_ ought to be _x_. The so-called Adams conditionalization is an attractive update rule in situations when learning experience prompts a shift in your conditional credences. In this paper, I show that, except in some trivial situations, when your prior conditional credence in _A_ obeys the standard principle of expert deference and then is revised by Adams conditionalization in (...)
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  38. Updating without evidence.Yoaav Isaacs & Jeffrey Sanford Russell - 2023 - Noûs 57 (3):576-599.
    Sometimes you are unreliable at fulfilling your doxastic plans: for example, if you plan to be fully confident in all truths, probably you will end up being fully confident in some falsehoods by mistake. In some cases, there is information that plays the classical role of evidence—your beliefs are perfectly discriminating with respect to some possible facts about the world—and there is a standard expected‐accuracy‐based justification for planning to conditionalize on this evidence. This planning‐oriented justification extends to some cases where (...)
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  39. Only CDT values knowledge.Michael Nielsen - 2023 - Analysis 84 (1):67-82.
    Causal decision theory is often motivated as a ‘revision of [decision theory] intended to solve Newcomb’s problem’ (Bacon 2022). In this paper, I give a direct argument for CDT by deriving it from a venerable decision-theoretic principle: the value of knowledge principle. The general framework that I use to deliver this result also supports a new argument for conditionalization.
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  40. The Supremacy of IBE over Bayesian Conditionalization.Seungbae Park - 2023 - Problemos 103:66-76.
    Van Fraassen does not merely perform Bayesian conditionalization on his pragmatic theory of scientific explanation; he uses inference to the best explanation (IBE) to justify it, contrary to what Prasetya thinks. Without first using IBE, we cannot carry out Bayesian conditionalization, contrary to what van Fraassen thinks. The argument from a bad lot, which van Fraassen constructs to criticize IBE, backfires on both the pragmatic theory and Bayesian conditionalization, pace van Fraassen and Prasetya.
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  41. Bayesian updating when what you learn might be false.Richard Pettigrew - 2023 - Erkenntnis 88 (1):309-324.
    Rescorla (Erkenntnis, 2020) has recently pointed out that the standard arguments for Bayesian Conditionalization assume that whenever I become certain of something, it is true. Most people would reject this assumption. In response, Rescorla offers an improved Dutch Book argument for Bayesian Conditionalization that does not make this assumption. My purpose in this paper is two-fold. First, I want to illuminate Rescorla’s new argument by giving a very general Dutch Book argument that applies to many cases of updating beyond those (...)
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  42. Reflecting on diachronic Dutch books.Michael Rescorla - 2023 - Noûs 57 (3):511-538.
    Conditionalization governs how to reallocate credence in light of new evidence. One prominent argument in favor of Conditionalization holds that an agent who violates it is vulnerable to a diachronic Dutch book: a series of acceptable bets offered at multiple times that inflict a sure loss. van Fraassen argues that an agent who violates the Principle of Reflection is likewise vulnerable to a diachronic Dutch book. He concludes that agents should conform to both Conditionalization and Reflection. Some authors reply that (...)
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  43. Essential materials for Bayesian Mindsponge Framework analytics.Aisdl Team - 2023 - Sm3D Science Portal.
    Acknowledging that many members of the SM3D Portal need reference documents related to Bayesian Mindsponge Framework (BMF) analytics to conduct research projects effectively, we present the essential materials and most up-to-date studies employing the method in this post. By summarizing all the publications and preprints associated with BMF analytics, we also aim to help researchers reduce the time and effort for information seeking, enhance proactive self-learning, and facilitate knowledge exchange and community dialogue through transparency.
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  44. Reflection and conditionalization: Comments on Michael Rescorla.Bas C. van Fraassen - 2023 - Noûs 57 (3):539-552.
    Rescorla explores the relation between Reflection, Conditionalization, and Dutch book arguments in the presence of a weakened concept of sure loss and weakened conditions of self‐transparency for doxastic agents. The literature about Reflection and about Dutch Book arguments, though overlapping, are distinct, and its history illuminates the import of Rescorla's investigation. With examples from a previous debate in the 70s and results about Reflection and Conditionalization in the 80s, I propose a way of seeing the epistemic enterprise in the light (...)
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  45. Just As Planned: Bayesianism, Externalism, and Plan Coherence.Pablo Zendejas Medina - 2023 - Philosophers' Imprint 23.
    Two of the most influential arguments for Bayesian updating ("Conditionalization") -- Hilary Greaves' and David Wallace's Accuracy Argument and David Lewis' Diachronic Dutch Book Argument-- turn out to impose a strong and surprising limitation on rational uncertainty: that one can never be rationally uncertain of what one's evidence is. Many philosophers ("externalists") reject that claim, and now seem to face a difficult choice: either to endorse the arguments and give up Externalism, or to reject the arguments and lose some of (...)
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  46. Comparative Opinion Loss.Benjamin Eva & Reuben Stern - 2022 - Philosophy and Phenomenological Research 107 (3):613-637.
    It is a consequence of the theory of imprecise credences that there exist situations in which rational agents inevitably become less opinionated toward some propositions as they gather more evidence. The fact that an agent's imprecise credal state can dilate in this way is often treated as a strike against the imprecise approach to inductive inference. Here, we show that dilation is not a mere artifact of this approach by demonstrating that opinion loss is countenanced as rational by a substantially (...)
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  47. Dutch‐booking indicative conditionals.Melissa Fusco - 2022 - Philosophy and Phenomenological Research 107 (1):208-231.
    Recent literature on Stalnaker's Thesis, which seeks to vindicate it from Lewis (1976)'s triviality results, has featured linguistic data that is prima facie incompatible with Conditionalization in iterated cases (McGee 1989, 2000; Kaufmann 2015; Khoo & Santorio, 2018). In a recent paper (2021), Goldstein & Santorio make a bold claim: they hold that these departures light the way to a new, non‐conditionalizing theory of rational update.Here, I consider whether this new form of update is subject to a Dutch book. On (...)
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  48. (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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  49. Multiple Universes and Self-Locating Evidence.Yoaav Isaacs, John Hawthorne & Jeffrey Sanford Russell - 2022 - Philosophical Review 131 (3):241-294.
    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 (...)
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  50. Accuracy-First Epistemology Without Additivity.Richard Pettigrew - 2022 - Philosophy of Science 89 (1):128-151.
    Accuracy arguments for the core tenets of Bayesian epistemology differ mainly in the conditions they place on the legitimate ways of measuring the inaccuracy of our credences. The best existing arguments rely on three conditions: Continuity, Additivity, and Strict Propriety. In this paper, I show how to strengthen the arguments based on these conditions by showing that the central mathematical theorem on which each depends goes through without assuming Additivity.
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