Interpretations of Probability

by

Alan Hájek, Christopher Hitchcock

• LAST REVIEWED: 11 October 2023
• LAST MODIFIED: 24 July 2024
• DOI: 10.1093/obo/9780195396577-0416

Introduction

If you check the weather prediction on your phone, you might read that there is a 30 percent chance of rain at 4:00 in the afternoon. What does this mean? More precisely, what is the fraction 30/100 a measure of? Probability is a concept that is widespread both in everyday life and in science. Ordinary speakers of English utter and have some understanding of sentences such as “I will probably be late for the meeting,” or “it’s unlikely that Luxembourg will win the next World Cup.” Various sciences make explicit probabilistic claims: “the probability that a radium atom will decay in 1620 years is 0.5”; “the probability that a house mouse whose father is heterozygous for the t haplotype will inherit that trait is 0.9.” Other claims implicitly invoke probability: “the life expectancy of a child born in Japan today is 85.3 years.” Probability theory is also a major branch of mathematics, and it was given its modern formulation by Kolmogorov in 1933. Kolmogorov’s formalism presents a function P that satisfies a set of axioms: it is non-negative, normalized, and additive. These axioms entail a rich set of theorems concerning the behavior of P; together they make up the probability calculus. While the resulting theory is a formal theory in its own right, it is also natural to interpret P—to attach meanings, or truth conditions to claims involving it. “What is P?”, one may ask. This may be understood as a metaphysical question about what kinds of things are probabilities, or more generally as a question about what makes probability statements true or false. The various interpretations of probability attempt to answer this question, one way or another. This article surveys the literature on the interpretations of probability, due to mathematicians and especially philosophers. It divides the interpretations into two broad categories. Epistemological interpretations understand probability in terms of an agent’s beliefs, the strength of evidence in support of a statement, or other epistemological categories. Physical interpretations view probability as a feature of the world that would exist regardless of what evidence exists or what agents believe. This is a natural taxonomy, but others could be adopted, and its sub-categories are also somewhat pliable. The authors would like to thank Kim Border, Chris Bottomley, Kenny Easwaran, Hanti Lin, Michael Nielsen, Charles Sebens, Glenn Shafer, Julia Staffel, Jeremy Strasser, Michael Titelbaum, and an anonymous referee for many helpful suggestions.

General Overviews

There are a number of surveys of interpretations of probability. Hájek 2019 is a good place to start. It is a detailed overview and evaluation of the leading interpretations of probability. Childers 2013 and Galavotti 2005 are readable books with a minimum of technical detail. Diaconis and Skyrms 2017 is a wide-ranging book that explores a number of aspects of probability. Huber 2019 is an undergraduate-level textbook. Gillies 2000 and Salmon 2017 are also accessible book-length treatments, although they begin to touch on more technical issues. Howson 1995 is a relatively short survey that also introduces some lesser-known approaches. Weisberg 2023 is an excellent, lively textbook for introductory philosophy courses on probability and inductive logic. Fine 1973 is an excellent text for more mathematically sophisticated readers.

• Childers, Timothy. Philosophy and Probability. Oxford: Oxford University Press, 2013.

  A recent, non-technical, engaging survey of the main interpretations of probability, and their relationship to the problem of induction. One distinctive feature of this introduction is the final chapter, which discusses maximum entropy principles and other issues related to information theory.

• Diaconis, Persi, and Brian Skyrms. Ten Great Ideas about Chance. Princeton, NJ: Princeton University Press, 2017.

  DOI: 10.2307/j.ctvc77m33

  An engaging book that explores a number of dimensions of probability. Chapter 2 discusses issues related to subjective interpretations, including Dutch Book theorems. Chapter 4 discusses issues related to frequency interpretations. Chapters 6 and 10 engage with connections between probability and evidence, and chapter 9 touches on issues connected with physical interpretations of probability. The book explains all of the relevant mathematics as it arises.

• Fine, Terrence. Theories of Probability: An Examination of Foundations. New York and London: Academic Press, 1973.

  A philosophically rich, technically advanced survey of the major interpretations of probability. Also has sophisticated discussions of the set-theoretic underpinnings of the Kolmogorov axiomatization and of alternatives to standard probability theory.

• Galavotti, Maria Carla. Philosophical Introduction to Probability. Stanford, CA: CSLI, 2005.

  An excellent overview of the major approaches to interpreting probability, suitable for an undergraduate course Relatively non-technical, and includes a brief introduction to the mathematics of probability. Particularly thorough in its coverage of the history of attempts to interpret probability.

• Gillies, Donald. Philosophical Theories of Probability. London and New York: Routledge, 2000.

  Surveys the major interpretations of probability. Contains a bit more technical detail than Galavotti 2005, but it is still suitable for an advanced undergraduate course. Provides particularly good coverage of Keynes’s version of the logical interpretation, de Finetti’s subjective interpretation, and von Mises’s frequentism. The author also presents his own novel version of the propensity interpretation.

• Hájek, Alan. “Interpretations of Probability.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. Stanford, CA: Stanford University, 2019.

  This online, open access article is self-contained, beginning with an introduction to the formalism of probability theory. It canvasses the major interpretations of probability, providing both exposition and critical discussion. It is updated every few years.

• Howson, Colin. “Theories of Probability.” British Journal for the Philosophy of Science 46.1 (1995): 1–32.

  DOI: 10.1093/bjps/46.1.1

  This survey article is a lucid presentation of the leading interpretations of probability, and it also contains some discussion of less standard approaches. For example, it discusses alternative formal approaches to epistemological probabilities, such as Dempster-Shafer belief functions; and the section on frequency interpretations includes a discussion of Dawid’s “prequential theory.”

• Huber, Franz. A Logical Introduction to Probability and Induction. Oxford: Oxford University Press, 2019.

  A clear and rigorous textbook for an introductory undergraduate course. Self-contained, carefully explains all of the necessary technical material, and includes exercises for the student. Chapters 5–10 cover interpretations of probability. One of the few surveys that provides coverage of accuracy-based foundations for subjective probability and of subjective approaches to physical probability.

• Salmon, Wesley. The Foundations of Scientific Inference. Pittsburgh, PA: University of Pittsburgh Press, 2017.

  DOI: 10.2307/j.ctt1s475s3

  First published in 1967, this book provides a very readable introduction to probability and the problem of induction, albeit from the author’s distinctive point of view. Chapters 4–6 focus on the problem of interpreting probability. Provides especially good exposition and critical discussion of Carnap’s logical interpretation and Reichenbach’s frequency interpretation. The 50th Anniversary Edition contains an introductory essay by Christopher Hitchcock that also discusses more recent developments.

• Weisberg, Jonathan. Odds & Ends. 2023.

  An open access, open-source textbook available in both PDF and HTML. Provides an introduction to some of the leading interpretations of probability, with some historical background and points of curiosity. Its emphasis on visual representations is distinctive and pedagogically useful.