A plea: If you're going to propose a Bayesian framework for updating self-locating degrees of belief, please read this piece first. I've tried to survey all the extant formalisms, group them by their general approach, then describe challenges faced by every formalism employing a given approach. Hopefully this survey will prevent further instances of authors' re-inventing updating rules already proposed elsewhere in the literature.

The Twentieth Century has seen a dramatic rise in the use of probability and statistics in almost all fields of research. This has stimulated many new philosophical ideas on probability. _Philosophical Theories of Probability_ is the first book to present a clear, comprehensive and systematic account of these various theories and to explain how they relate to one another. Gillies also offers a distinctive version of the propensity theory of probability, and the intersubjective interpretation, which develops the subjective theory.

Normative requirements are often overlooked, but they are central features of the normative world. Rationality is often thought to consist in acting for reasons, but following normative requirements is also a major part of rationality. In particular, correct reasoning – both theoretical and practical – is governed by normative requirements rather than by reasons. This article explains the nature of normative requirements, and gives examples of their importance. It also describes mistakes that philosophers have made as a result of confusing (...) normative requirements with reasons. (shrink)

Though the revised edition of A Theory of Justice, published in 1999, is the definitive statement of Rawls's view, so much of the extensive literature on Rawls's theory refers to the first edition.

Confirmational holism is at odds with Jeffrey conditioning --- the orthodox Bayesian policy for accommodating uncertain learning experiences. Two of the great insights of holist epistemology are that (i) the effects of experience ought to be mediated by one's background beliefs, and (ii) the support provided by one's learning experience can and often is undercut by subsequent learning. Jeffrey conditioning fails to vindicate either of these insights. My aim is to describe and defend a new updating policy that does better. (...) In addition to showing that this new policy is more holism-friendly than Jeffrey conditioning, I will also show that it has an accuracy-centered justification. (shrink)

The chapter argues that a model‐building methodology like that widespread in contemporary natural and social science already plays a significant role in philosophy. One neglected form of progress in philosophy over the past fifty years has been the development of better and better formal models of significant phenomena. Examples are given from both philosophy of language and epistemology. Philosophy can do still better in the future by applying model‐building methods more systematically and self‐consciously, with consequent readjustments to its methodology. Although (...) it is neither likely nor desirable for model‐building to become the sole or even main philosophical method, its use enhances the power and reliability of philosophical thinking. (shrink)

I argue that its appearing to you that P does not provide justification for believing that P unless you have independent justification for the denial of skeptical alternatives – hypotheses incompatible with P but such that if they were true, it would still appear to you that P. Thus I challenge the popular view of ‘dogmatism,’ according to which for some contents P, you need only lack reason to suspect that skeptical alternatives are true, in order for an experience as (...) of P to justify belief that P. I pursue three lines of objection to dogmatism, having to do with probabilistic reasoning, considerations of future or hypothetically available justification, and epistemic circularity. I briefly sketch a fall-back position which avoids the problems raised. (shrink)

I have been supposing that for the theory of reasoning, explicit belief is an all-or-nothing matter, I have assumed that, as far as principles of reasoning are concerned, one either believes something explicitly or one does not; in other words an appropriate "representation" is either in one's "memory" or not. The principles of reasoning are principles for modifying such all-or-nothing representations. This is not to deny that in some ways belief is a matter of degree. For one thing implicit belief (...) is certainly a matter of degree, since it is a matter of how easily and automatically one can infer something from what one believes explicitly. Furthermore, explicit belief is a matter of degree in the sense that one believes some things more strongly than others. Sometimes one is only somewhat inclined to believe something, sometimes one is not sure what to believe, sometimes one is inclined to disbelieve something, sometimes one is quite confident something is not so, and so forth. (shrink)

With this treatise, an insightful exploration of the probabilistic connection between philosophy and the history of science, the famous economist breathed new life into studies of both disciplines. Originally published in 1921, this important mathematical work represented a significant contribution to the theory regarding the logical probability of propositions. Keynes effectively dismantled the classical theory of probability, launching what has since been termed the “logical-relationist” theory. In so doing, he explored the logical relationships between classifying a proposition as “highly probable” (...) and as a “justifiable induction.” Unabridged republication of the classic 1921 edition. (shrink)

The principle of indifference states that in the absence of any relevant evidence, a rational agent will distribute their credence equally among all the possible outcomes under consideration. Despite its intuitive plausibility, PI famously falls prey to paradox, and so is widely rejected as a principle of ideal rationality. In this article, I present a novel rehabilitation of PI in terms of the epistemology of comparative confidence judgments. In particular, I consider two natural comparative reformulations of PI and argue that (...) while one of them prescribes the adoption of patently irrational epistemic states, the other provides a consistent formulation of PI that overcomes the most salient limitations of existing formulations. (shrink)

Objective Bayesianism is a methodological theory that is currently applied in statistics, philosophy, artificial intelligence, physics and other sciences. This book develops the formal and philosophical foundations of the theory, at a level accessible to a graduate student with some familiarity with mathematical notation.

An examination of topics involved in statistical reasoning with imprecise probabilities. The book discusses assessment and elicitation, extensions, envelopes and decisions, the importance of imprecision, conditional previsions and coherent statistical models.

Richard Pettigrew offers an extended investigation into a particular way of justifying the rational principles that govern our credences. The main principles that he justifies are the central tenets of Bayesian epistemology, though many other related principles are discussed along the way. Pettigrew looks to decision theory in order to ground his argument. He treats an agent's credences as if they were a choice she makes between different options, gives an account of the purely epistemic utility enjoyed by different sets (...) of credences, and then appeals to the principles of decision theory to show that, when epistemic utility is measured in this way, the credences that violate the principles listed above are ruled out as irrational. The account of epistemic utility set out here is the veritist's: the sole fundamental source of epistemic utility for credences is their accuracy. Thus, Pettigrew conducts an investigation in the version of epistemic utility theory known as accuracy-first epistemology. (shrink)

Scientific reasoning is—and ought to be—conducted in accordance with the axioms of probability. This Bayesian view—so called because of the central role it accords to a theorem first proved by Thomas Bayes in the late eighteenth ...

While there are several arguments on either side, it is far from clear as to whether or not countable additivity is an acceptable axiom of subjective probability. I focus here on de Finetti's central argument against countable additivity and provide a new Dutch book proof of the principle, To argue that if we accept the Dutch book foundations of subjective probability, countable additivity is an unavoidable constraint.

Dogmatism is sometimes thought to be incompatible with Bayesian models of rational learning. I show that the best model for updating imprecise credences is compatible with dogmatism.

Kolmogorov''s axiomatization of probability includes the familiarratio formula for conditional probability: 0).$$ " align="middle" border="0">.

I advocate Time-Slice Rationality, the thesis that the relationship between two time-slices of the same person is not importantly different, for purposes of rational evaluation, from the relationship between time-slices of distinct persons. The locus of rationality, so to speak, is the time-slice rather than the temporally extended agent. This claim is motivated by consideration of puzzle cases for personal identity over time and by a very moderate form of internalism about rationality. Time-Slice Rationality conflicts with two proposed principles of (...) rationality, Conditionalization and Reflection. Conditionalization is a diachronic norm saying how your current degrees of belief should fit with your old ones, while Reflection is a norm enjoining you to defer to the degrees of belief that you expect to have in the future. But they are independently problematic and should be replaced by improved, time-slice-centric principles. Conditionalization should be replaced by a synchronic norm saying what degrees of belief you ought to have given your current evidence and Reflection should be replaced by a norm which instructs you to defer to the degrees of belief of agents you take to be experts. These replacement principles do all the work that the old principles were supposed to do while avoiding their problems. In this way, Time-Slice Rationality puts the theory of rationality on firmer foundations and yields better norms than alternative, non-time-slice-centric approaches. (shrink)

'Fundamentals of Bayesian Epistemology' provides an accessible introduction to the key concepts and principles of the Bayesian formalism. This volume introduces degrees of belief as a concept in epistemology and the rules for updating degrees of belief derived from Bayesian principles.--.

The holistic thesis developed by Pierre Duhem challenges the idea that our evidence can conclusively falsify a theory. Given that no scientific theory is tested in isolation, a negative experiment can always be attributed to components other than the theory we test – to the auxiliary hypotheses and background assumptions. How do scientists decide whether the experimental result undermines the theory or points at an error in the underlying assumptions? Duhem argues that we cannot offer a rule that directs when (...) the scientist should employ a radical or conservative strategy in light of a negative result, and ultimately they will appeal to their intuition. More recently philosophers have offered a number of strategies of how to locate error and justify the abandonment of a theory or an auxiliary hypothesis. This Element analyses Duhem's response to holism and subsequent accounts of how the problem can be resolved. (shrink)

How should we reason in science? Jan Sprenger and Stephan Hartmann offer a refreshing take on classical topics in philosophy of science, using a single key concept to explain and to elucidate manifold aspects of scientific reasoning. They present good arguments and good inferences as being characterized by their effect on our rational degrees of belief. Refuting the view that there is no place for subjective attitudes in 'objective science', Sprenger and Hartmann explain the value of convincing evidence in terms (...) of a cycle of variations on the theme of representing rational degrees of belief by means of subjective probabilities (and changing them by Bayesian conditionalization). In doing so, they integrate Bayesian inference—the leading theory of rationality in social science—with the practice of 21st century science. Bayesian Philosophy of Science thereby shows how modeling such attitudes improves our understanding of causes, explanations, confirming evidence, and scientific models in general. It combines a scientifically minded and mathematically sophisticated approach with conceptual analysis and attention to methodological problems of modern science, especially in statistical inference, and is therefore a valuable resource for philosophers and scientific practitioners. (shrink)

In formal epistemology, we use mathematical methods to explore the questions of epistemology and rational choice. What can we know? What should we believe and how strongly? How should we act based on our beliefs and values? We begin by modelling phenomena like knowledge, belief, and desire using mathematical machinery, just as a biologist might model the fluctuations of a pair of competing populations, or a physicist might model the turbulence of a fluid passing through a small aperture. Then, we (...) explore, discover, and justify the laws governing those phenomena, using the precision that mathematical machinery affords. For example, we might represent a person by the strengths of their beliefs, and we might measure these using real numbers, which we call credences. Having done this, we might ask what the norms are that govern that person when we represent them in that way. How should those credences hang together? How should the credences change in response to evidence? And how should those credences guide the person’s actions? This is the approach of the first six chapters of this handbook. In the second half, we consider different representations—the set of propositions a person believes; their ranking of propositions by their plausibility. And in each case we ask again what the norms are that govern a person so represented. Or, we might represent them as having both credences and full beliefs, and then ask how those two representations should interact with one another. This handbook is incomplete, as such ventures often are. Formal epistemology is a much wider topic than we present here. One omission, for instance, is social epistemology, where we consider not only individual believers but also the epistemic aspects of their place in a social world. Michael Caie’s entry on doxastic logic touches on one part of this topic, but there is much more. Relatedly, there is no entry on epistemic logic, nor any on knowledge more generally. There are still more gaps. These omissions should not be taken as ideological choices. This material is missing, not because it is any less valuable or interesting, but because we v failed to secure it in time. Rather than delay publication further, we chose to go ahead with what is already a substantial collection. We anticipate a further volume in the future that will cover more ground. Why an open access handbook on this topic? A number of reasons. The topics covered here are large and complex and need the space allowed by the sort of 50 page treatment that many of the authors give. We also wanted to show that, using free and open software, one can overcome a major hurdle facing open access publishing, even on topics with complex typesetting needs. With the right software, one can produce attractive, clear publications at reasonably low cost. Indeed this handbook was created on a budget of exactly £0 (≈ $0). Our thanks to PhilPapers for serving as publisher, and to the authors: we are enormously grateful for the effort they put into their entries. (shrink)

A Logical Introduction to Probability and Induction starts with elementary logic and uses it as basis for a philosophical discussion of probability and induction. Throughout the book results are carefully proved using the inference rules introduced at the beginning. The textbook is suitable for undergraduate courses in philosophy and logic.

In his new foreword to this edition, Hilary Putnam forcefully rejects these nativist claims.

How should thinkers cope with uncertainty? Julia Staffel breaks new ground in the study of rationality by answering this question and many others. She also explains how it is better to be less irrational, because less irrational degrees of belief are generally more accurate and better at guiding our actions.

Traditional philosophical discussions of knowledge have focused on the epistemic status of full beliefs. In this book, Moss argues that in addition to full beliefs, credences can constitute knowledge. For instance, your .4 credence that it is raining outside can constitute knowledge, in just the same way that your full beliefs can. In addition, you can know that it might be raining, and that if it is raining then it is probably cloudy, where this knowledge is not knowledge of propositions, (...) but of probabilistic contents. -/- The notion of probabilistic content introduced in this book plays a central role not only in epistemology, but in the philosophy of mind and language as well. Just as tradition holds that you believe and assert propositions, you can believe and assert probabilistic contents. Accepting that we can believe, assert, and know probabilistic contents has significant consequences for many philosophical debates, including debates about the relationship between full belief and credence, the semantics of epistemic modals and conditionals, the contents of perceptual experience, peer disagreement, pragmatic encroachment, perceptual dogmatism, and transformative experience. In addition, accepting probabilistic knowledge can help us discredit negative evaluations of female speech, explain why merely statistical evidence is insufficient for legal proof, and identify epistemic norms violated by acts of racial profiling. Hence the central theses of this book not only help us better understand the nature of our own mental states, but also help us better understand the nature of our responsibilities to each other. (shrink)

This book develops new techniques in formal epistemology and applies them to the challenge of Cartesian skepticism. It introduces two formats of epistemic evaluation that should be of interest to epistemologists and philosophers of science: the dual-component format, which evaluates a statement on the basis of its safety and informativeness, and the relative-divergence format, which evaluates a probabilistic model on the basis of its complexity and goodness of fit with data. Tomoji Shogenji shows that the former lends support to Cartesian (...) skepticism, but the latter allows us to defeat Cartesian skepticism. Along the way, Shogenji addresses a number of related issues in epistemology and philosophy of science, including epistemic circularity, epistemic closure, and inductive skepticism. (shrink)

According to Bayesian epistemology, rational learning from experience is consistent learning, that is learning should incorporate new information consistently into one's old system of beliefs. Simon M. Huttegger argues that this core idea can be transferred to situations where the learner's informational inputs are much more limited than Bayesianism assumes, thereby significantly expanding the reach of a Bayesian type of epistemology. What results from this is a unified account of probabilistic learning in the tradition of Richard Jeffrey's 'radical probabilism'. Along (...) the way, Huttegger addresses a number of debates in epistemology and the philosophy of science, including the status of prior probabilities, whether Bayes' rule is the only legitimate form of learning from experience, and whether rational agents can have sustained disagreements. His book will be of interest to students and scholars of epistemology, of game and decision theory, and of cognitive, economic, and computer sciences. (shrink)

Probabilistic coherence is not an absolute requirement of rationality; nevertheless, it is an ideal of rationality with substantive normative import. An idealized rational agent who avoided making implicit logical errors in forming his preferences would be coherent. In response to the challenge, recently made by epistemologists such as Foley and Plantinga, that appeals to ideal rationality render probabilism either irrelevant or implausible, I argue that idealized requirements can be normatively relevant even when the ideals are unattainable, so long as they (...) define a structure that links imperfect and perfect rationality in a way that enables us to make sense of the notion of better approximations to the ideal. I then analyze the notion of approximation to the ideal of coherence by developing a generalized theory of belief functions that allows for incoherence, and showing how such belief functions can be ordered with regard to greater or lesser coherence. (shrink)

I develop a conception of expressivism according to which it is chiefly a pragmatic thesis about some fragment of discourse, one imposing certain constraints on semantics. The first half of the paper uses credal expressivism about the language of probability as a stalking-horse for this purpose. The second half turns to the question of how one might frame an analogous form of expressivism about the language of deontic modality. Here I offer a preliminary comparison of two expressivist lines. The first, (...) expectation expressivism, looks again to Bayesian modelling for inspiration: it glosses deontically modal language as characteristically serving to express decision-theoretic expectation (expected utility). The second, plan expressivism, develops the idea (due to Gibbard 2003) that this language serves to express 'plan-laden' states of belief. In the process of comparing the views, I show how to incorporate Gibbard's modelling ideas into a compositional semantics for attitudes and modals, filling a lacuna in the account. I close with the question whether and how plan expressivism might be developed with expectation-like structure. (shrink)

Isn't probability 1 certainty? If the probability is objective, so is the certainty: whatever has chance 1 of occurring is certain to occur. Equivalently, whatever has chance 0 of occurring is certain not to occur. If the probability is subjective, so is the certainty: if you give credence 1 to an event, you are certain that it will occur. Equivalently, if you give credence 0 to an event, you are certain that it will not occur. And so on for other (...) kinds of probability, such as evidential probability. The formal analogue of this picture is the regularity constraint: a probability distribution over sets of possibilities is regular just in case it assigns probability 0 only to the null set, and therefore probability 1 only to the set of all possibilities. (shrink)

We present a conservative extension of a Bayesian account of confirmation that can deal with the problem of old evidence and new theories. So-called open-minded Bayesianism challenges the assumption—implicit in standard Bayesianism—that the correct empirical hypothesis is among the ones currently under consideration. It requires the inclusion of a catch-all hypothesis, which is characterized by means of sets of probability assignments. Upon the introduction of a new theory, the former catch-all is decomposed into a new empirical hypothesis and a new (...) catch-all. As will be seen, this motivates a second update rule, besides Bayes’ rule, for updating probabilities in light of a new theory. This rule conserves probability ratios among the old hypotheses. This framework allows for old evidence to confirm a new hypothesis due to a shift in the theoretical context. The result is a version of Bayesianism that, in the words of Earman, “keep[s] an open mind, but not so open that your brain falls out”. (shrink)

Inference to the Best Explanation (IBE) and Bayesianism are our two most prominent theories of scientific inference. Are they compatible? Van Fraassen famously argued that they are not, concluding that IBE must be wrong since Bayesianism is right. Writers since then, from both the Bayesian and explanationist camps, have usually considered van Fraassen’s argument to be misguided, and have plumped for the view that Bayesianism and IBE are actually compatible. I argue that van Fraassen’s argument is actually not so misguided, (...) and that it causes more trouble for compatibilists than is typically thought. Bayesianism in its dominant, subjectivist form, can only be made compatible with IBE if IBE is made subservient to conditionalization in a way that robs IBE of much of its substance and interest. If Bayesianism and IBE are to be fit together, I argue, a strongly objective Bayesianism is the preferred option. I go on to sketch this objectivist, IBE-based Bayesianism, and offer some preliminary suggestions for its development. (shrink)

Van Fraassen famously endorses the Principle of Reflection as a constraint on rational credence, and argues that Reflection is entailed by the more traditional principle of Conditionalization. He draws two morals from this alleged entailment. First, that Reflection can be regarded as an alternative to Conditionalization – a more lenient standard of rationality. And second, that commitment to Conditionalization can be turned into support for Reflection. Van Fraassen also argues that Reflection implies Conditionalization, thus offering a new justification for Conditionalization. (...) I argue that neither principle entails the other, and thus neither can be used to motivate the other in the way van Fraassen says. There are ways to connect Conditionalization to Reflection, but these connections depend on poor assumptions about our introspective access, and are not tight enough to draw the sorts of conclusions van Fraassen wants. Upon close examination, the two principles seem to be getting at two quite independent epistemic norms. (shrink)

Conditionalization and Jeffrey Conditionalization cannot simultaneously satisfy two widely held desiderata on rules for empirical learning. The first desideratum is confirmational holism, which says that the evidential import of an experience is always sensitive to our background assumptions. The second desideratum is commutativity, which says that the order in which one acquires evidence shouldn't affect what conclusions one draws, provided the same total evidence is gathered in the end. (Jeffrey) Conditionalization cannot satisfy either of these desiderata without violating the other. (...) This is a surprising problem, and I offer a diagnosis of its source. I argue that (Jeffrey) Conditionalization is inherently anti-holistic in a way that is just exacerbated by the requirement of commutativity. The dilemma is thus a superficial manifestation of (Jeffrey) Conditionalization's fundamentally anti-holistic nature. (shrink)