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.

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)

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

_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)

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)

Quantum mechanical entangled configurations of particles that do not satisfy Bell’s inequalities, or equivalently, do not have a joint probability distribution, are familiar in the foundational literature of quantum mechanics. Nonexistence of a joint probability measure for the correlations predicted by quantum mechanics is itself equivalent to the nonexistence of local hidden variables that account for the correlations (for a proof of this equivalence, see Suppes and Zanotti, 1981). From a philosophical standpoint it is natural to ask what sort of (...) concept can be used to provide a “joint” analysis of such quantum correlations. In other areas of application of probability, similar but different problems arise. A typical example is the introduction of upper and lower probabilities in the theory of belief. A person may feel uncomfortable assigning a precise probability to the occurrence of rain tomorrow, but feel comfortable saying the probability should be greater than ½ and less than ⅞. Rather extensive statistical developments have occurred for this framework. A thorough treatment can be found in Walley (1991) and an earlier measurement-oriented development in Suppes (1974). It is important to note that this focus on beliefs, or related Bayesian ideas, is not concerned, as we are here, with the nonexistence of joint probability distributions. Yet earlier work with no relation to quantum mechanics, but focused on conditions for existence has been published by many people. For some of our own work on this topic, see Suppes and Zanotti (1989). Still, this earlier work naturally suggested the question of whether or not upper and lower measures could be used in quantum mechanics, as a generalization of.. (shrink)

This book explores important topics in Darwin’s theory of evolution and in the twentieth- and twenty-first-century theories that grew out of it. These topics include fitness, adaptation, altruism, intragenomic conflict, units of selection, genetic drift, the randomness of mutation, gradualism, common ancestry, taxonomy, race, phylogenetic inference, and optimality models. The book brings these and other biological topics into contact with numerous philosophical ideas − operationalism, reductionism, conventionalism, null hypotheses and default reasoning, instrumentalism versus realism, likelihoods versus probabilities, model selection, (...) essentialism, falsifiability, parsimony, the principle of the common cause, comparisons of causal power, interpretations of probaiblity, indeterminism, sensitivity to initial conditions, and the knowability of the past. (shrink)

It is well known that a degree-of-belief function P is coherent if and only if it satisfies the probability calculus. In this paper, we show that the notion of coherence can be extended to higher order probabilities such as P(P(h)=p)=q, and that a higher order degree-of-belief function P is coherent if and only if it satisfies the probability calculus plus the following axiom: P(h)=p iff P(P(h)=p)=1. Also, a number of lemmata which extend an incomplete probability function to a complete (...) one are established. (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)

La première partie de Particle rend compte des conditions philosophiques et du contexte historique dans lequel s'est développée la conception du probable chez Wolff. La seconde met trois aspects de cette conception en relief: 1) la distinction entre la probalité d'un évènement et la probalité d'une proposition, 2) l'interprétation du principe d'égale possibilité en tant que principe de cause non-suffisante, 3) les quelques restrictions conservés par Wolff à l'égard de la validité et des domaines d'application du principe d'égale possibilité. La (...) troisième partie traite de ce que l'on nomme les „experientiae indubitatae". Elles se fondent sur une notion de la certitude qui peut être interprétée comme cas limite de la probabilité. Les certitudes de cette sorte sont acceptées par Wolff comme les premisses d'une preuve. De cette façon Wolff utilise sa conception de la probabilité pour unir les exigences de la raison et celles de l'expérience et débouche sur une conception de la science qui dépasse celle en cours jusque-là. A partir d'elle, il essaye de réaliser le programme rationaliste de l'édification des sciences empiriques. (shrink)

Abstract In Section 1 I examine the use of probabilistic data to establish causal conclusions in non?experimental research. In Section 2 I show that the probabilities involved in such research are inhomogeneous ?mixed? probabilities. Section 3 then argues that such mixed probabilities are responsible for the way common causes screen off correlations between their joint effects. Section 4 concludes that mixed probabilities are therefore crucial for the nature of the causal relation itself.

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 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)

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)

This paper defends the view that the asymmetry of causation can be explained in terms of probabilistic relationships between event types. Papineau first explores three different versions of the "fork asymmetry", namely David Lewis' asymmetry of overdetermination, the screening-off property of common causes, and Spirtes', Glymour's and Scheines' analysis of probabilistic graphs. He then argues that this fork asymmetry is both a genuine phenomenon and a satisfactory metaphysical reduction of causal asymmetry. In his final section he shows how this reduction (...) can account for the relevance of causal direction to human agency, and in particular for the fact that we can manipulate causes to influence their effects, but not vice versa. (shrink)

The present paper deals with the tools that can be used to represent causation and to reason about it and, specifically, with their diversity. It focuses on so-called “causal probabilities”—that is, probabilities of effects given one of their causes—and critically surveys a recent paper in which Joyce argues that the values of these probabilities do not depend on one’s conception of causation. I first establish a stronger independence claim: I show that the very definition of causal (...) class='Hi'>probabilities is independent of one’s conception of causation. Second, I investigate whether causal probabilities indeed take the same values under their different possible definitions. (shrink)

I argue that Nancy Cartwright's largely methodological arguments for capacities and against Hume's regularity account of causation are only partially successful. They are especially problematic in establishing the primacy of singular causation and the reality of mixed-dual capacities. Therefore, her arguments need to be supported by ontological ones, and I propose the propensity interpretation of causal probabilities as a natural way of doing this.

The modal interpretation of quantum theory ascribes at each instant physical magnitudes with definite values to quantum systems. Starting from certain natural requirements, I determine unique solutions for the evolution of these possessed magnitudes in free systems and in special cases of interacting systems. The evolution is given in terms of transition probabilities that relate the values of the possessed magnitudes at one instant to the values at a second instant. I also determine a joint property ascription to a (...) composite system and its separate subsystems. Finally, I give a proof that the predictions of the modal interpretation with respect to measurement outcomes agree with the predictions of the standard interpretation. (shrink)

In this paper, we look at Richard Dawkins’ “Spectrum of Theistic Probabilities” from his book “The God Delusion”. The spectrum is edited to account for a mathematical error by Richard Dawkins. Correcting this oversight leads to 2 new theological positions being discovered. One of these positions is defended in detail. In its defense, we argue against a potential flaw of the position and also list a merit for the position among other arguments for the position. The merit involves a (...) rebuttal for Pascal’s wager found in no other piece of literature. (shrink)

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.

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)

The non-random mixing of biomembrane components, especially saturated phospholipids, exhibits important consequences in molecular biology. Particularly, the distribution of lipids within natural and model membranes is strongly determined by the selective association processes. These processes of phospholipids take place due to the cooperative modes in multiparticle systems as well as the specific lipid-lipid interactions both in the hydrophobic core and in the region of the polar headgroups. We demonstrated that the investigation of the selective association processes of saturated phospholipids might (...) contribute to the insight of the lipid domains appearance inside the bilayer membranes. The association probabilities of like- pairs and cross- pairs from a binary mixture of saturated phospholipids were tested for both parallel and anti-parallel alignments of the polar headgroups. The present model confirms the experimental evidence for saturated phospholipids to have a high tendency for association in parallel configuration of the electric dipole moments of the polar headgroups whether the cross-sectional area of the polar headgroup is in an usual range of 25-55 2. There are three major lipid domains in a binary mixture of saturated phospholipids: lipid domains in non-mixed phase of the first mixture component, in parallel alignment of the polar headgroups; lipid domains in non-mixed phase of the second mixture component, in anti-parallel alignment of the polar headgroups; lipid domains in mixed phase. We think that the selective association processes of phospholipids are neither exclusively, nor only involved in promoting the lipid domains appearance through bilayer phospholipid membranes. (shrink)

Why is understanding causation so important in philosophy and the sciences? Should causation be defined in terms of probability? Whilst causation plays a major role in theories and concepts of medicine, little attempt has been made to connect causation and probability with medicine itself. Causality, Probability, and Medicine is one of the first books to apply philosophical reasoning about causality to important topics and debates in medicine. Donald Gillies provides a thorough introduction to and assessment of competing theories of causality (...) in philosophy, including action-related theories, causality and mechanisms, and causality and probability. Throughout the book he applies them to important discoveries and theories within medicine, such as germ theory; tuberculosis and cholera; smoking and heart disease; the first ever randomized controlled trial designed to test the treatment of tuberculosis; the growing area of philosophy of evidence-based medicine; and philosophy of epidemiology. This book will be of great interest to students and researchers in philosophy of science and philosophy of medicine, as well as those working in medicine, nursing and related health disciplines where a working knowledge of causality and probability is required. (shrink)

The term probability can be used in two main senses. In the frequency interpretation it is a limiting ratio in a sequence of repeatable events. In the Bayesian view, probability is a mental construct representing uncertainty. This 2002 book is about these two types of probability and investigates how, despite being adopted by scientists and statisticians in the eighteenth and nineteenth centuries, Bayesianism was discredited as a theory of scientific inference during the 1920s and 1930s. Through the examination of a (...) dispute between two British scientists, the author argues that a choice between the two interpretations is not forced by pure logic or the mathematics of the situation, but depends on the experiences and aims of the individuals involved. The book should be of interest to students and scientists interested in statistics and probability theories and to general readers with an interest in the history, sociology and philosophy of science. (shrink)

A few purported counterexamples to the Adams thesis have cropped up in the literature in the last few decades. I propose a theory that accounts for them, in a way that makes the connections between indicative conditionals and counterfactuals clearer.

De Finetti is one of the founding fathers of the subjective school of probability. He held that probabilities are subjective, coherent degrees of expectation, and he argued that none of the objective interpretations of probability make sense. While his theory has been influential in science and philosophy, it has encountered various objections. I argue that these objections overlook central aspects of de Finetti’s philosophy of probability and are largely unfounded. I propose a new interpretation of de Finetti’s theory that (...) highlights these aspects and explains how they are an integral part of de Finetti’s instrumentalist philosophy of probability. I conclude by drawing an analogy between misconceptions about de Finetti’s philosophy of probability and common misconceptions about instrumentalism. (shrink)

This pair of articles provides a critical commentary on contemporary approaches to statistical mechanical probabilities. These articles focus on the two ways of understanding these probabilities that have received the most attention in the recent literature: the epistemic indifference approach, and the Lewis-style regularity approach. These articles describe these approaches, highlight the main points of contention, and make some attempts to advance the discussion. The first of these articles provides a brief sketch of statistical mechanics, and discusses the (...) indifference approach to statistical mechanical probabilities. (shrink)

I vindicate Hume’s argument against belief in miracle reports against a prevalent objection. Hume has us balance the probability of a miracle’s occurrence against the probability of its being falsely attested to, and argues that the latter must inevitably be the greater; thus, reason requires us to reject any miracle report. The "flaw" in this reasoning, according to Butler and many others, is that it proves too much--it counsels us to never believe historians, newspaper reports of lottery results, and so (...) on; and this is clearly absurd. I show that this objection is misguided: far from providing counterexamples to Hume’s "balancing principle", as I call it, these cases actually confirm it, as some simple calculations of probabilities show. (shrink)

Cohen and Meskin 2006 recently offered a counterfactual theory of information to replace the standard probabilistic theory of information. They claim that the counterfactual theory fares better than the standard account on three grounds: first, it provides a better framework for explaining information flow properties; second, it requires a less expensive ontology; and third, because it does not refer to doxastic states of the information-receiving organism, it provides an objective basis. In this paper, I show that none of these is (...) really an advantage. Moreover, the counterfactual theory fails to satisfy one of the basic properties of information flow, namely the Conjunction principle. Thus, I conclude, there is no reason to give up the standard probabilistic theory for the counterfactual theory of information. (shrink)

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.

This pair of articles provides a critical commentary on contemporary approaches to statistical mechanical probabilities. These articles focus on the two ways of understanding these probabilities that have received the most attention in the recent literature: the epistemic indifference approach, and the Lewis-style regularity approach. These articles describe these approaches, highlight the main points of contention, and make some attempts to advance the discussion. The second of these articles discusses the regularity approach to statistical mechanical probabilities, and (...) describes some areas where further research is needed. (shrink)

The foundation of the mathematical theory of probabilities is still a controversial subject. There are schools of insufficient reasoning and of cogent reasoning, of a priori determination and of frequency determination, of subjective and of objective probability. Two main difficulties exist. The first is the definition of equally like events. The second difficulty is the relation between the laws of causal natural science and the laws of statistical regularity. Is it really necessary to add to the laws of mechanics (...) one or more independent “laws of large numbers” to explain the regularities in dice throwing? In all these cases different answers exist, and the existence of a mathematical theory of probabilities remains a kind of miracle, as esoteric to the further domain of natural science as the resurrection of the dead. (shrink)

Let R(X, B) denote the class of probability functions that are defined on algebra X and that represent rationally permissible degrees of certainty for a person whose total relevant background evidence is B. This paper is concerned with characterizing R(X, B) for the case in whichX is an algebra of propositions involving two properties and B is empty. It proposes necessary conditions for a probability function to be in R(X, B), some of which involve the notion of statistical dependence. The (...) class of probability functions that satisfy these conditions, here denoted PI, includes a class that Carnap once proposed for the same situation. Probability functions in PI violate Carnap's axiom of analogy but, it is argued, that axiom should be rejected. A derivation of Carnap's model by Hesse has limitations that are not present in the derivation of PI given here. Various alternative probability models are considered and rejected. (shrink)