Interpretations of Probability

Por • 29 jul, 2020 • Sección: Leyes

First published Mon Oct 21, 2002; substantive revision Wed Aug 28, 2019

Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.—Bertrand Russell, 1929 Lecture (cited in Bell 1945, 587)

The Democrats will probably win the next election.’

‘The coin is just as likely to land heads as tails.’

‘There’s a 30% chance of rain tomorrow.’

‘The probability that a radium atom decays in one year is roughly 0.0004.’

One regularly reads and hears probabilistic claims like these. But what do they mean? 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. At a first pass, various interpretations of probability answer this question, one way or another.

However, there is also a stricter usage: an ‘interpretation’ of a formal theory provides meanings for its primitive symbols or terms, with an eye to turning its axioms and theorems into true statements about some subject. In the case of probability, Kolmogorov’s axiomatization (which we will see shortly) is the usual formal theory, and the so-called ‘interpretations of probability’ usually interpret it. That axiomatization introduces a function ‘PP’ that has certain formal properties. We may then ask ‘What is PP?’. Several of the views that we will discuss also answer this question, one way or another.

Our topic is complicated by the fact that there are various alternative formalizations of probability. Moreover, as we will see, some of the leading ‘interpretations of probability’ do not obey all of Kolmogorov’s axioms, yet they have not lost their title for that. And various other quantities that have nothing to do with probability do satisfy Kolmogorov’s axioms, and thus are ‘interpretations’ of it in the strict sense: normalized mass, length, area, volume, and other quantities that fall under the scope of measure theory, the abstract mathematical theory that generalizes such quantities. Nobody seriously considers these to be ‘interpretations of probability’, however, because they do not play the right role in our conceptual apparatus.

Perhaps we would do better, then, to think of the interpretations as analyses of various concepts of probability. Or perhaps better still, we might regard them as explications of such concepts, refining them to be fruitful for philosophical and scientific theorizing (à la Carnap 1950).

However we think of it, the project of finding such interpretations is an important one. Probability is virtually ubiquitous. It plays a role in almost all the sciences. It underpins much of the social sciences — witness the prevalent use of statistical testing, confidence intervals, regression methods, and so on. It finds its way, moreover, into much of philosophy. In epistemology, the philosophy of mind, and cognitive science, we see states of opinion being modeled by subjective probability functions, and learning being modeled by the updating of such functions. Since probability theory is central to decision theory and game theory, it has ramifications for ethics and political philosophy. It figures prominently in such staples of metaphysics as causation and laws of nature. It appears again in the philosophy of science in the analysis of confirmation of theories, scientific explanation, and in the philosophy of specific scientific theories, such as quantum mechanics, statistical mechanics, evolutionary biology, and genetics. It can even take center stage in the philosophy of logic, the philosophy of language, and the philosophy of religion. Thus, problems in the foundations of probability bear at least indirectly, and sometimes directly, upon central scientific, social scientific, and philosophical concerns. The interpretation of probability is one of the most important such foundational problems.

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