# Introduction

## Abstract and Keywords

This is an exciting time for the philosophy of probability, and probability theory’s value to philosophy has never been as appreciated as it is nowadays. The introduction to this Handbook sets out the context of the current debate in this area and provides a primer on those parts of probability theory that are most important for philosophers to know. It then goes on to introduce the seven main sections of the handbook: 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.

Keywords: probability, formalism, standard probability theory, probabilistic judgment, applications of probability, philosophy

Probability theory has long played a central role in statistics, the sciences, and the social sciences, and it is an important branch of mathematics in its own right. It has also been playing an increasingly significant role in philosophy—in epistemology, philosophy of science, ethics, social philosophy, philosophy of religion, and elsewhere. A case can be made that probability is as vital a part of the philosopher’s toolkit as logic. Moreover, there is a fruitful two-way street between probability theory and philosophy: the theory informs much of the work of philosophers, and philosophical inquiry, in turn, has shed considerable light on the theory.

This volume encapsulates and furthers the influence of philosophy on probability, and of probability on philosophy. Nearly forty chapters summarize the state of play and present new insights in various areas of research at the intersection of these two fields. The chapters should be of special interest to practitioners of probability who seek a greater understanding of its mathematical and conceptual foundations, and to philosophers of probability who want to get up to speed on the cutting edge of research in this area. There is also plenty here to entice philosophical readers who don’t work especially *on* probability but who want to learn more *about* it and its applications. Indeed, this volume should appeal to the intellectually curious generally; after all, there is much here to be curious about.

We do not expect all of this volume’s audience to have a thorough training in probability theory. And while probability is relevant to the work of many philosophers, they often do not have much of a background in its formalism. With this in mind, we begin with “Probability for Everyone—Even Philosophers”, a primer on those parts of probability theory that we believe are most important for philosophers to know. The rest of the volume is divided into seven main sections:

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*History*•

*Formalism*•

*Alternatives to Standard Probability Theory*•

*Interpretations*•

*Probabilistic Judgment and Its Applications*•

*Applications of Probability: Science*•

*Applications of Probability: Philosophy*

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Some historians of probability, notably Hacking,^{1} regard probability as having arrived surprisingly late on the intellectual scene, given its relative simplicity and practical value. Specifically, the birth of probability is usually dated to 1654, when Blaise Pascal and Pierre de Fermat began to correspond about a problem inspired by gambling on dice. (By contrast, Descartes’ groundbreaking work in analytic geometry—a much more complex and abstract topic—appeared in 1637.) James Franklin’s chapter in this volume traces the origins of probabilistic thinking much further back, even to antiquity. To be sure, the mid-to-late 17th century represents something of a watershed in the study of probability, and Edith Sylla takes up its history at that point, focusing on the work of Jacob Bernoulli and his influence in Continental Europe. Meanwhile, in Britain in the 17th and 18th centuries, probability theory was appropriated in various applications, as David Bellhouse details in his chapter. Hans Fischer continues the theory’s history from early in the 19th century until around the middle of the 20th century, as probability increasingly became an autonomous branch of pure mathematics. During this period, statistics became an important field in England especially; this is the topic of John Aldrich’s contribution to this volume. Finally, Maria Carla Galavotti canvases the work and legacy of some of the leading philosophers of probability in the 20th century, which created the field of philosophy of probability in its own right, and which set the stage for research in that field right up to today.

Two of the chief areas of research in the foundations of probability concern its *formalism*, and its *interpretation*. We begin with some suitable *formal theory* of probability, some codification of how probabilities are to be represented and how they behave. We then *interpret* that theory, bringing the formalism to life with an account of what probabilities are and of what grounds them. Regarding the formalism, Kolmogorov’s axiomatization of 1933 remains orthodoxy. However, it has also found its share of critics. The chapters by Aidan Lyon and Kenny Easwaran discuss Kolmogorov’s formalism and some of the sources of discontent with his approach. Richard Neapolitan and Xia Jiang’s chapter on causal Bayes nets describes a newer formalism for efficiently representing and computing probabilities. The chapters by Terrence Fine, James Hawthorne, and Fabio Cozman describe formalisms intended as alternatives to standard probability theory, such as imprecise probabilities and qualitative analogues of probability. J. Robert G. Williams discusses how Kolmogorov’s approach may be generalized to accommodate various nonclassical logics.

Regarding the interpretation of probability, we are pulled in multiple directions. Probability apparently begins in uncertainty, but it arguably does not end there. We are irremediably ignorant of various aspects of the world; probability theory has been our chief tool for systematizing and managing this ignorance. Our evidence is impoverished, and it typically fails to settle various matters of interest to us. But even if Hume was right that there are no necessary connections between distinct existences, still it seems that there are *probabilistic* connections between the evidence that we have and the hypotheses that we entertain. Moreover, many authors believe that modern physics gives us reason to think that the world itself has not settled various matters either: probability is part of the fabric of reality. If this is right, then *pace* Einstein, God *does* play dice.

Accordingly, philosophers have homed in on three leading kinds of probability: evidential, subjective, and physical. (Note that one can consistently adhere to more than
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one interpretation of probability. Indeed, a case can be made for embracing evidential, subjective, and physical probabilities for different purposes.) *Evidential probability* is meant to capture the degree to which available or hypothetical evidence supports various hypotheses, where typically such support falls short of entailment. One might think of this as how objectively plausible the hypotheses are in light of the evidence, irrespective of what anyone actually thinks. The earliest incarnation of this idea was enshrined in the *classical* interpretation of probability, in which the probability of an event is regarded as the ratio of the number of live possibilities favourable to the event divided by the total number of live possibilities. (The *live* possibilities are those that have not been ruled out.) This interpretation is founded on the *indifference* principle: when there is no relevant evidence, or when the relevant evidence bears symmetrically on the alternative possibilities, the possibilities should be given equal weight. Sandy Zabell’s chapter on symmetry arguments in probability traces the history of the indifference principle, and more recent heirs of the classical interpretation that aim to ground probability values in symmetries.

The *logical* interpretation of probability generalizes this approach to evidential probability, seeking to measure the degree of support that a body of evidence gives a particular hypothesis, whatever the evidence and hypothesis. The possibilities can be assigned *unequal* weights, and probabilities can be computed whatever the evidence may be, symmetrically balanced with respect to the hypothesis or not. The result, to the extent that it succeeds, is a comprehensive *inductive logic* or *confirmation* theory.^{2} Several chapters in this volume discuss at least to some extent these themes of evidential probability, the classical and logical interpretations of probability, and confirmation theory. Maria Carla Galavotti’s chapter discusses the logical interpretation of probability in the early 20th century, with particular attention to the work of Harold Jeffreys. Vincenzo Crupi and Katya Tentori focus on confirmation theory and inductive logic in their chapter. Matthew Kotzen also addresses evidential probability in his contribution, paying special attention to the work of Kyburg and Williamson. Sandy Zabell’s discussion of symmetry arguments additionally covers many ideas that have been associated with the logical interpretation of probability.

Some authors in philosophy of probability’s pantheon, however, were sceptical of any notion of logical probability—notably, Ramsey and de Finetti. They advocated a more permissive *subjectivism* about probability, which interprets probabilities as degrees of confidence of suitably rational agents. This interpretation is addressed especially by Lyle Zynda’s chapter, but it also takes centre stage in the chapters by Fabio Cozman, Franz Dietrich and Christian List, Stephen Hora, Michael Smithson, and Jan Sprenger.

Meanwhile, a number of authors hold that probabilities reside in the world itself, mind-independently—these are *physical* probabilities, often called *chances*.^{3} Frequentist interpretations identify such probabilities with appropriate relative frequencies in some sequence of events; see the chapter by Adam La Caze. These interpretations seem to fare better when there are many trials of the relevant event type—flips of a coin, throws of a die,
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and the like. Frequentist interpretations fare worse when there are few trials, and especially poorly when there is just one—this is ‘the problem of the single case’. Partly as a response to this problem, propensity interpretations regard probabilities as graded dispositions or tendencies—applicable even in the single case (at least on some versions). Donald Gillies surveys these interpretations. More recently, *best systems* approaches to physical probability have become popular. On this type of view, chances fall out of the theory of the universe that best balances certain theoretical virtues. This is the topic of Wolfgang Schwarz’s chapter.

There are further interpretive issues regarding probabilities that are not specifically interpretations *of* probability, although they interact with these interpretations. Probability has been thought by many authors to bear interesting connections to *randomness*, the subject of Antony Eagle’s chapter. There is also considerable debate about whether chance is compatible with *determinism*, an issue that Roman Frigg takes up. Issues connected to the interpretation of probability also underlie the debate between champions of Bayesian and frequentist approaches to statistical inference —see Jan Sprenger’s contribution.

We then turn to probabilistic judgment and its applications. Michael Smithson explores some of the psychological literature on how people reason with probabilities. While it is often useful to represent the opinion of someone, especially an expert, in probabilistic form, many of us cannot simply assign a number to the degree of our conviction on the basis of introspection. Stephen Hora describes a number of strategies for eliciting probabilities from subjects. Franz Dietrich and Christian List consider how the probabilistic judgments of a number of individuals can be pooled in various ways.

Next, our authors discuss applications of probability, beginning with science. Physics explicitly traffics in probabilities, especially in quantum mechanics and statistical mechanics. Guido Bacciagaluppi and Wayne Myrvold, respectively, examine the place of probabilities in each of these theories. Furthermore, probabilities make their way both explicitly and implicitly into biology, especially in connection with the concept of fitness in evolutionary biology—see Roberta Millstein’s chapter.

While the earlier sections on formalism and interpretations fall under the *philosophy of probability*, the final section concerns the myriad applications of *probability in philosophy*. Many areas of philosophy have benefited from probability theory. Several chapters display this: Matthew Kotzen’s on epistemology; Hannes Leitgeb’s on logic; David McCarthy’s on ethics; Paul Bartha’s on philosophy of religion; and Eric Swanson’s on philosophy of language. A number of chapters survey more targeted applications of probability in philosophy: Vincenzo Crupi and Katya Tentori’s on confirmation theory; Michael Titelbaum’s on self-locating credences; Lara Buchak’s on decision theory; and Christopher Hitchcock’s on probabilistic causation.

Stepping back, we see just how fertile the interaction of probability and philosophy can be. This is an exciting time for the philosophy of probability, and probability theory’s value to philosophy has never been as appreciated as it is nowadays. We thus thought it was especially timely when Peter Momtchiloff of Oxford University Press approached us with the idea of this Handbook. He has been a pleasure to work with, and we thank him for his ongoing encouragement and advice. Many thanks are also due to John Cusbert and Edward Elliott for their incisive comments on various drafts from the authors. Above all, we thank the authors themselves for their fine work.

## Notes:

(^{1})
*The Emergence of Probability, A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference.* 1975. Cambridge: Cambridge University Press.

(^{2})
Terminological caution: in ordinary English, “confirms” usually means *establishes* or *verifies*, but confirmation theory’s relations are *probabilistic*. Moreover, these relations include those of evidential *counter*-support.

(^{3})
They are sometimes also called *objective* probabilities. However, logical probabilities are often regarded as objective also (much as logic itself is often regarded as objective). We thus prefer to speak of *physical* probabilities.