Types of Probability and their Measurement

Abstract

So far as measurement is concerned, the logical concept of probability presents difficulties. On the one hand, saying that probabilities deal with logical relations apparently militates against measurement, for in what sense might logical relations be quantified? It certainly suggests that probabilities are not per se numerical, and that measurability will require justification. But, on the other, saying that probabilities are concerned with degrees of rational belief conveys some sense of quantitativeness and even possible numericalisation. A resolution of these problems is effected by Keynes, but it takes up two closely argued chapters in the TP, one of these expounding his famous Principle of Indifference, the relevance of which also extends to his theory of practical reason (see chapter 6). Compared to other theories of probability where numerical measurement is relatively unproblematic, Keynes’s ideas here are complex and subtle. He was among the first writers to take non-numerical probabilities seriously and to contend that these were the major type. In fact, the solution offered to the problem of measurement is one of the ‘characteristic features’ (86) of his philosophy with far-reaching implications for his theory of rational behaviour. In addition, this side of his philosophy is crucial to understanding two further issues in Keynes’s thought — his attitude to mathematics in social science (discussed in chapter 9), and his rejection in the GT of purely probabilistic analyses of uncertainty (see chapter 12).

#x2026; the conclusion that all probabilities can be compared by means of numbers is more in need of proof than the contrary.… [The] possibility of the measurement of any quantity must be in doubt until good reason for accepting it has been produced.

J.M. Keynes 1908

It has been assumed hitherto as a matter of course that probability is, in the full and literal sense of the word, measurable. I shall have to limit, not extend, the popular doctrine.

J.M. Keynes 1921

But a main point to which I would call your attention is that, on my theory of probability, the probabilities themselves … are not numerical. So that … the substitution of a numerical measure needs discussion.

J.M. Keynes 1938¹