The first point is of course the most important characteristic of a priori theories and the one which gives them their name. It is what distinguishes them most strongly from the relative frequency or ‘empirical’ school.¹ It is also what occasions the single greatest objection to them: How can a priori principles do probability theory’s practical work of predicting the future in the real world? Again, it is the source of their greatest single advantage – they can establish probabilities without the necessity of waiting for a very (infinitely?) long sequence of repetitive empirical events.

The second of our basic ideas, like the first, is so central to AP theories that it has generated a name for them: ‘Logical Theories of Probability.’ As interest has grown in the inductive logics of Carnap, Kemeny, Hintikka, et al, this term has been more and more commonly used for such theories. I agree that it is a suitable, descriptive name for this sub-class of a priori theories. I object, however, to its extension to theories of the Keynesian variety, because there the initial probabilities are not obtained from quantitative logistic systems but from a priori intuition aided by the Principle of Indifference.² My objection is not fervid, however, and if the philosophical tide continues to flow towards empiricism in general but away from the dogmas of logical positivism, I foresee a time when ‘a priori’ will become a general pejorative and I too shall abandon it. Of course the important point here is that these theories do define probability as a logical relation – not whether that fact is sufficient to warrant a label.

This concern with logic was motivated in Carnap’s case, apparently, by a conscious desire for an inductive logic (more than by a need to account for the probability calculus, for example). He had earlier accepted RF theories as adequate for scientific probability –not until he engaged in logical research did he shift his allegiance to AP theories (or, rather, divide it between the two). Keynes, on the other hand, seems to have set out looking for the nature of probability and then ‘discovered’ that it is a matter of logic.

The two also disagreed on the relation between deductive logic and probability. Carnap took the (normal) position that probability theory results from adding probability rules (c-functions) to the ordinary deductive logic, thereby increasing its power and range. Keynes, however, thought probability theory was the basic theory of inference – deductive logic is merely that degenerate case where all probabilities are 1 or 0.

Our third basic idea is that AP theories recognize probabilities relative to given evidence only. Many RF theorists consider this to be a grave defect in AP theories, rendering a probability subordinate to the state of our knowledge and therefore odiously ‘subjective’. Their theories, they maintain, deal with the real probability, which is objectively determined and not relative to anything. There is a sense in which this claim for RF theories is true and a sense in which it is false – we shall discuss this problem in the chapter on Relative Frequency theories below. Our present concern is with the AP theories, and in this context evidence-dependence is seen as a virtue rather than a defect. It is a necessary consequence of the fact that probability is defined as a relation between a proposition and some evidence for that proposition.

John Maynard Keynes (1883-1946), British economist and man of letters, was ‘one of the creators of the modern world.’³ His theoretical contributions to political economy are familiar to everyone as an important part of the rationale for the ever-increasing government intervention in non-communist economies. His practical contributions to the British Treasury and to international monetary conferences such as Bretton Woods are well known to at least economists and historians. What is less well known is that philosophy was Keynes’s earliest love. He studied it under G. E. Moore and Bertrand Russell as an undergraduate at Cambridge while earning his degree in mathematics. During his subsequent employment with the India Company, Keynes combined these fields in a thorough review of probability and induction. This resulted in a dissertation which earned Keynes a fellowship in King’s College and allowed him to return to Cambridge as a philosopher. He was persuaded to teach in the Economics faculty instead, and subsequently made that field his chief intellectual interest, so his brief career as a philosopher culminated in the publication of an expanded version of his dissertation as A Treatise on Probability.⁴ This one work established Keynes as the authority of his time on probability theory, and maintains his reputation today as a leading spokesman for the a priori (AP) interpretation of probability.

Keynes was the first self-conscious apriorist in probability theory. He asserted emphatically that probability is a logical relation between propositions and, according to Carnap, was the first to perceive and emphasize the fact that such probabilities are inherently relative to given evidence and to nothing else.⁵

Such assertions, and the philosophical argumentation which supports them, constitute Keynes’s most famous and significant contributions to the development of probability theory. His derivation of theorems in the probability calculus represented at best a mild improvement over his predecessors and is not historically important. His attempted revision of the Principle of Indifference is frequently referred to and generally considered to be an improvement, but it does not attack the fundamental question whether any such principle can be a legitimate source of initial probabilities. Thus Keynes is seldom cited for these achievements. Instead, he finds his place as the pre-eminent representative of a priori probability theory, and when he is discussed by philosophers it is usually in this role as the embodiment and spokesman of a priori probability rather than as the source of some particular argument or theorem (in contrast to Bernoulli or Bayes, for example).

Rudolf Carnap (1891-1970) was ‘the most prominent representative of the logical empiricist, or logical positivist, school in the philosophy of science and logic.’⁶ His early training in physics led him to accept the relative frequency (RF) theory of probability,⁷ but his name is now more associated with his later development of AP probability theory as a form of quantitative inductive logic.

Carnap’s AP theory is of a piece with his notable work in the related fields of logic, syntax, semantics, and formal languages. It is impossible to convey fully the nature of his system of a priori probability without requiring or providing some grounding in the concepts and formalizations he employs. At least a hundred pages of his Logical Foundations of Probability⁸ are devoted to just this preliminary spadework – the task is ...