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# Logical probability

Keynes pays specific attention to the concept of probability, since we cannot understand the scientific method if we do not investigate the meaning of probability. In his A Treatise on Probability (1973a [1921]) he searches for rational objective principles to justify inductive judgements in the awareness that science needs a concept of probability that is not merely dependent on a valuation that may be different from subject to subject.

In this volume, Kyburg (Chapter 2), Levi (Chapter 3) and Fano (Chapter 4) deal extensively with logical probability. For our purposes we note only that it is a logical relation that represents a degree of rational belief established objectively in the sense that every agent in the same circumstances establishes the same probability. More specifically, probability is subjective in the sense that

it is without significance to call a proposition probable unless we specify the knowledge to which we are relating it. ... But in the sense important to logic, probability is not subjective. ... A proposition is not probable because we think it so. When once the facts are given which determine our knowledge, what is probable or improbable in these circumstances has been fixed objectively, and is independent on our opinion. The theory of probability is logical, therefore, because it is concerned with the degree of belief which is rational to entertain in given conditions, and not merely with the actual beliefs of particular individuals, which may or may not be rational.

(Keynes, 1973a [1921],p. 4)

The appeal is to reason as source of knowledge. In particular, 'where our experience is incomplete, we cannot hope to derive from it judgments of probability without the aid either of intuition or of some further a priori principle' (1973a [1921], p. 94). In this way, Keynes (1973a [1921], ch. 3) claims to deal with probability in its widest sense, and this means that probabilities may be numerically non-measurable and non-com- parable.9

An economic example of this measurability question is given by Russell (1948, Part V, ch.1). Consider the probability that a given insurance policy is good business for an individual. The problem of this individual is different from that of the insurance company, which is not interested in the individual case, since it offers insurance to all the members of a certain class of individuals, and so needs to know only the statistical mean. An individual, in contrast, may have personal reasons to expect a more or less long life. His or her health and way of life are important, and some of the details of these may be so rare that statistics cannot give reliable help. In addition, a doctor may not be able to give a scientific judgement about personal health conditions. In this case, the probability that taking out insurance is a good business for the individual is very vague and absolutely impossible to measure numerically.

In this volume Costantini and Garibaldi (Chapter 8), by considering events and not propositions, argue that 'nowadays the theory of probability is very different from the classical theory criticized by Keynes' and give a stochastic representation of an economic system where agents behave as organicistic and not as atomistic agents. Since long-time expectations change as time goes by, probabilities change with evidence. Therefore, changes in agents' strategy are governed by probability conditions which determine 'the statistical behavior of the economic system expressed by a probability distribution, hence its average behavior too' (s. 8.7).

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