This chapter presents an overview of the major interpretations of probability followed by an outline of the objective Bayesian interpretation and a discussion of the key challenges it faces. I discuss the ramifications of interpretations of probability and objective Bayesianism for the philosophy of mathematics in general.

Probability is increasingly important for our understanding of the world. What is probability? How do we model it, and how do we use it? Timothy Childers presents a lively introduction to the foundations of probability and to philosophical issues it raises. He keeps technicalities to a minimum, and assumes no prior knowledge of the subject.

_Philosophy of Probability: Contemporary Readings_ is the first anthology to collect essential readings in this important area of philosophy. Featuring the work of leading philosophers in the field such as Carnap, Hájek, Jeffrey, Joyce, Lewis, Loewer, Popper, Ramsey, van Fraassen, von Mises, and many others, the book looks in depth at the following key topics: subjective probability and credence probability updating: conditionalization and reflection Bayesian confirmation theory classical, logical, and evidential probability frequentism physical probability: propensities and objective chances. The book (...) features a useful primer on the mathematics of probability, and each section includes an introduction by the editor, as well as a guide to further reading. A broad-ranging and highly accessible exploration of the subject, _Philosophy of Probability_ is ideal for any student of formal epistemology, philosophy of science, metaphysics, or philosophy of mathematics. (shrink)

Two new philosophical problems surrounding the gradation of certainty began to emerge in the 17th century and are still very much alive today. One is concerned with the evaluation of inductive reasoning, whether in science, jurisprudence, or elsewhere; the other with the interpretation of the mathematical calculus of change. This book, aimed at non-specialists, investigates both problems and the extent to which they are connected. Cohen demonstrates the diversity of logical structures that are available for judgements of probability, and explores (...) the rationale for their appropriateness in different contexts of application. Thus his study deals with the complexity of the underlying philosophical issues without simply cataloging alternative conceptions or espousing a particular "favorite" theory. (shrink)

Probability theory promises to deliver an exact and unified foundation for inquiry in epistemology and philosophy of science. But philosophy of religion is also fertile ground for the application of probabilistic thinking. This volume presents original contributions from twelve contemporary researchers, both established and emerging, to offer a representative sample of the work currently being carried out in this potentially rich field of inquiry. Grouped into five parts, the chapters span a broad range of traditional issues in religious epistemology. The (...) first three parts discuss the evidential impact of various considerations that have been brought to bear on the question of the existence of God. These include witness reports of the occurrence of miraculous events, the existence of complex biological adaptations, the apparent 'fine-tuning' for life of various physical constants and the existence of seemingly unnecessary evil. The fourth part addresses a number of issues raised by Pascal's famous pragmatic argument for theistic belief. A final part offers probabilistic perspectives on the rationality of faith and the epistemic significance of religious disagreement. (shrink)

This Element has two main aims. The first one is an historically informed review of the philosophy of probability. It describes recent historiography, lays out the distinction between subjective and objective notions, and concludes by applying the historical lessons to the main interpretations of probability. The second aim focuses entirely on objective probability, and advances a number of novel theses regarding its role in scientific practice. A distinction is drawn between traditional attempts to interpret chance, and a novel methodological study (...) of its application. A radical form of pluralism is then introduced, advocating a tripartite distinction between propensities, probabilities and frequencies. Finally, a distinction is drawn between two different applications of chance in statistical modelling which, it is argued, vindicates the overall methodological approach. The ensuing conception of objective probability in practice is the 'complex nexus of chance'. (shrink)

This dissertation is a contribution to formal and computational philosophy. -/- In the first part, we show that by exploiting the parallels between large, yet finite lotteries on the one hand and countably infinite lotteries on the other, we gain insights in the foundations of probability theory as well as in epistemology. Case 1: Infinite lotteries. We discuss how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. The solution boils down to the introduction (...) of infinitesimal probability values, which can be achieved using non-standard analysis. Our solution can be generalized to uncountable sample spaces, giving rise to a Non-Archimedean Probability (NAP) theory. Case 2: Large but finite lotteries. We propose application of the language of relative analysis (a type of non-standard analysis) to formulate a new model for rational belief, called Stratified Belief. This contextualist model seems well-suited to deal with a concept of beliefs based on probabilities ‘sufficiently close to unity’. -/- The second part presents a case study in social epistemology. We model a group of agents who update their opinions by averaging the opinions of other agents. Our main goal is to calculate the probability for an agent to end up in an inconsistent belief state due to updating. To that end, an analytical expression is given and evaluated numerically, both exactly and using statistical sampling. The probability of ending up in an inconsistent belief state turns out to be always smaller than 2%. (shrink)

This chapter contains sections titled: Introduction The Mathematical Theory of Probability The Philosophical Theory of Probability Conclusion References.

This chapter presents an overview of the major interpretations of probability followed by an outline of the objective Bayesian interpretation and a discussion of the key challenges it faces. I discuss the ramifications of interpretations of probability and objective Bayesianism for the philosophy of mathematics in general.

Probability theory is a key tool of the physical, mathematical, and social sciences. It has also been playing an increasingly significant role in philosophy: in epistemology, philosophy of science, ethics, social philosophy, philosophy of religion, and elsewhere. This Handbook encapsulates and furthers the influence of philosophy on probability, and of probability on philosophy. Nearly forty articles summarise the state of play and present new insights in various areas of research at the intersection of these two fields. The articles will be (...) of special interest to practitioners of probability who seek a greater understanding of its mathematical and conceptual foundations, and to philosophers who want to get up to speed on the cutting edge of research in this area. The volume begins with a primer on those parts of probability theory that we believe are most important for philosophers to know, and the rest is divided into seven main sections: History; Formalism; Alternatives to Standard Probability Theory; Interpretations and Interpretive Issues; Probabilistic Judgment and Its Applications; Applications of Probability: Science; and Applications of Probability: Philosophy. (shrink)

This volume defends a novel approach to the philosophy of physics: it is the first book devoted to a comparative study of probability, causality, and propensity, and their various interrelations, within the context of contemporary physics -- particularly quantum and statistical physics. The philosophical debates and distinctions are firmly grounded upon examples from actual physics, thus exemplifying a robustly empiricist approach. The essays, by both prominent scholars in the field and promising young researchers, constitute a pioneer effort in bringing out (...) the connections between probabilistic, causal and dispositional aspects of the quantum domain. The book will appeal to specialists in philosophy and foundations of physics, philosophy of science in general, metaphysics, ontology of physics theories, and philosophy of probability. (shrink)

A. J. Ayer was one of the foremost analytical philosophers of the twentieth century, and was known as a brilliant and engaging speaker. In essays based on his influential Dewey Lectures, Ayer addresses some of the most critical and controversial questions in epistemology and the philosophy of science, examining the nature of inductive reasoning and grappling with the issues that most concerned him as a philosopher. This edition contains revised and expanded versions of the lectures and two additional essays. Ayer (...) begins by considering Hume's formulation of the problem of induction and then explores the inferences on which we base our beliefs in factual matters. In other essays, he defines the three kinds of probability that inform inductive reasoning and examines the various criteria for verifiability and falsifiability. In his extensive introduction, Graham Macdonald discusses the arguments in _Probability and Evidence_, how they relate to Ayer's other works, and their influence in contemporary philosophy. He also provides a brief biographical sketch of Ayer, and includes a bibliography of works about and in response to _Probability and Evidence_. (shrink)

Philosophy of Probability provides a comprehensive introduction to theoretical issues that occupy a central position in disciplines ranging from philosophy of ...

This book explores a question central to philosophy--namely, what does it take for a belief to be justified or rational? According to a widespread view, whether one has justification for believing a proposition is determined by how probable that proposition is, given one's evidence. In this book this view is rejected and replaced with another: in order for one to have justification for believing a proposition, one's evidence must normically support it--roughly, one's evidence must make the falsity of that proposition (...) abnormal in the sense of calling for special, independent explanation. This conception of justification bears upon a range of topics in epistemology and beyond. Ultimately, this way of looking at justification guides us to a new, unfamiliar picture of how we should respond to our evidence and manage our own fallibility. This picture is developed here. (shrink)

Table of Contents: Preface.- 1. Introduction; Mauricio Suárez.- PART I: PROBABILITIES.- 2. Probability and time symmetry in classical Markov processes; Guido Bacciagaluppi.- 3. Probability assignments and the principle of indifference: An examination of two eliminative strategies; Sorin Bangu.- 4. Why typicality does not explain the approach to equilibrium; Roman Frigg; PART II: CAUSES.- 5. From metaphysics to physics and back: The example of causation; Federico Laudisa.- 6. On explanation in retro-causal interpretations of quantum mechanics; Joseph Berkovitz.- 7. Causal completeness (...) in general probability theories; Balasz Gyenis, Miklós Rédei.- 8. Causal Markov, robustness and the quantum correlations; Mauricio Suárez, Iñaki San Pedro.- PART III: PROPENSITIES.- 9. Do dispositions and propensities have a role in the ontology of quantum mechanics? Some critical remarks; Mauro Dorato.- 10. Is the quantum world composed of propensitons?; Nicholas Maxwell.- 11. Derivative dispositions and multiple derivative levels; Ian Thompson. (shrink)

When a doctor tells you there’s a one percent chance that an operation will result in your death, or a scientist claims that his theory is probably true, what exactly does that mean? Understanding probability is clearly very important, if we are to make good theoretical and practical choices. In this engaging and highly accessible introduction to the philosophy of probability, Darrell Rowbottom takes the reader on a journey through all the major interpretations of probability, with reference to real–world situations. (...) In lucid prose, he explores the many fallacies of probabilistic reasoning, such as the ‘gambler’s fallacy’ and the ‘inverse fallacy’, and shows how we can avoid falling into these traps by using the interpretations presented. He also illustrates the relevance of the interpretation of probability across disciplinary boundaries, by examining which interpretations of probability are appropriate in diverse areas such as quantum mechanics, game theory, and genetics. Using entertaining dialogues to draw out the key issues at stake, this unique book will appeal to students and scholars across philosophy, the social sciences, and the natural sciences. (shrink)

How do we ascribe subjective probability? In decision theory, this question is often addressed by representation theorems, going back to Ramsey (1926), which tell us how to define or measure subjective probability by observable preferences. However, standard representation theorems make strong rationality assumptions, in particular expected utility maximization. How do we ascribe subjective probability to agents which do not satisfy these strong rationality assumptions? I present a representation theorem with weak rationality assumptions which can be used to define or measure (...) subjective probability for partly irrational agents. (shrink)

The use of probability theory is widespread in our daily life as well as in scientific theories. In virtually all cases, calculations can be carried out within the framework of classical probability theory. A special exception is given by quantum mechanics, which gives rise to a new probability theory: quantum probability theory. This dissertation deals with the question of how this formalism can be understood from a philosophical and physical perspective. The dissertation is divided into three parts. In the first (...) part, the formalism of quantum probability theory and its relation to quantum mechanics is presented. The second part considers the possibility of reformulating quantum probability theory to embed it within classical probability. Mathematically, this possibility overlaps with the possibility of the existence of hidden variables theories of quantum mechanics: theories that attempt to solve fundamental problems in quantum mechanics by introducing additional physical properties of systems. The conclusion of this part is that, although classical reformulations of quantum probability theory are formally possible, they offer little insight into the formalism of quantum probability theory itself. The third part regards a more direct investigation of quantum probability theory. A reformulation of quantum probability theory is obtained by constructing a quantum logic on the basis of empirical non-probabilistic predictions of quantum mechanics. In this reformulation quantum probability functions are conditional probability functions on an algebra of propositions about measurements and measurement outcomes. (shrink)

I’m not sure how much knowledge everyone already has, so I’d like to start with a little questionnaire. On a card, say for each of the following topics whether you’re familiar with the topic, have heard of it but aren’t familiar with it, or have never heard of it.

This book sheds light on some recent discussions of the problems in probability theory and their history, analysing their philosophical and mathematical significance, and the role pf mathematical probability theory in other sciences.

Review of Maria Carla Galavotti (ed), Notes on Philosophy, Probability and Mathematics, 1991, Bibliopolis. Notes are selected from manuscripts by Frank Plumpton Ramsey at the University of Pittsburgh's Hillman Library.

At the beginning of his section “Of Miracles,” Hume mentions an argument of Dr. Tillotson. The doctrine of “the real presence” seems contradicted by our senses. We see a piece of bread, but are asked to believe it consists in the substance of the body of Christ.

In _Probability and Evidence_, one of Britain's foremost twentieth-century philosophers addresses central questions in the theory of knowledge and the philosophy of science. This book contains A.J. Ayer's John Dewey Lectures delivered at Columbia University, together with two additional essays, "Has Harrod Answered Hume?" and "The Problem of Conditionals.".

A Selection of Papers Contributed to Sections IV, VI, and XI of the Fourth International Congress for Logic, Methodology, and Philosophy of Science, Bucharest, September 1971.

Recognizing that probability (the Greek doxa) was understood in pre-modern theories as the polar opposite of certainty (episteme), the author of this study elaborates the forms which these polar opposites have taken in some twentieth century writers and then, in greater detail, in the writings of Thomas Aquinas. Profiting from subsequent more sophisticated theories of probability, he examines how Aquinas’s judgments about everything from God to gossip depend on schematizations of the polarity between the systematic and the non-systematic: revelation/reason, science/opinion, (...) saints/philosophers, Aristotle/others. Post-medieval developments have provided the means to discern within Thomas’s thought both a logical theory and a kind of relative frequency theory of probability. The former depends on Aristotle’s theory of demonstration, the latter on his theory of an orderly cosmos; but each as used by Thomas lead to convictions that are sometimes naive, sometimes amusing or even terrifying. (shrink)

This is the only book to chart the history and development of modern probability theory. It shows how in the first thirty years of this century probability theory became a mathematical science. The author also traces the development of probabilistic concepts and theories in statistical and quantum physics. There are chapters dealing with chance phenomena, as well as the main mathematical theories of today, together with their foundational and philosophical problems. Among the theorists whose work is treated at some length (...) are Kolmogorov, von Mises and de Finetti. The principal audience for the book comprises philosophers and historians of science, mathematicians concerned with probability and statistics, and physicists. The book will also interest anyone fascinated by twentieth-century scientific developments because the birth of modern probability is closely tied to the change from a determinist to an indeterminist world-view. (shrink)

This is the only book to chart the history and development of modern probability theory. It shows how in the first thirty years of this century probability theory became a mathematical science. The author also traces the development of probabilistic concepts and theories in statistical and quantum physics. There are chapters dealing with chance phenomena, as well as the main mathematical theories of today, together with their foundational and philosophical problems. Among the theorists whose work is treated at some length (...) are Kolmogorov, von Mises and de Finetti. The principal audience for the book comprises philosophers and historians of science, mathematicians concerned with probability and statistics, and physicists. The book will also interest anyone fascinated by twentieth-century scientific developments because the birth of modern probability is closely tied to the change from a determinist to an indeterminist world-view. (shrink)

The book was planned and written as a single, sustained argument. But earlier versions of a few parts of it have appeared separately. The object of this book is both to establish the existence of the paradoxes, and also to describe a non-Pascalian concept of probability in terms of which one can analyse the structure of forensic proof without giving rise to such typical signs of theoretical misfit. Neither the complementational principle for negation nor the multiplicative principle for conjunction applies (...) to the central core of any forensic proof in the Anglo-American legal system. There are four parts included in this book. Accordingly, these parts have been written in such a way that they may be read in different orders by different kinds of reader. (shrink)

This book provides a systematic treatment of Locke's theory of probable assent. It shows how the theory applies to Locke's philosophy of science, moral epistemology, and religious epistemology. There is a powerful case to be made that the most important dimension of Locke's philosophy is his theory of rational probable assent, rather than his theory of knowledge. According to Locke, we largely live our lives in the "twilight of probability" rather than in "the sunshine of certain knowledge". Locke's theory of (...) probable assent is towering insofar as it contains a wealth of novel, independently interesting, and prescient elements that precede the modern field of formal epistemology. In this book, the author argues for the central role of probable assent in Locke's philosophy. Locke's theory of probable assent is based on an epistemic modesty that claims, roughly, that our cognitive abilities are quite limited and that we ought to carry ourselves in believing with due caution. This modesty motivates the author's discussion of other aspects of Locke's epistemology, notably his principle of proportionality, his doxastic involuntarism, his epistemological pragmatism, and his theory of testimony. The book concludes by connecting the theory of probable assent with Locke's views on the limits of science, moral epistemology, and the rationality of faith. Locke's Twilight of Probability will appeal to scholars and advanced students working on Locke and the history of early modern philosophy. (shrink)