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A BAYESIAN APPROACH IN THE PHILOSOPHY OF INFERENCE*

Philosophical disputes that are not dissolved by Joad's law, 'It depends what you mean', often yield to Good's law, 'It 's a matter of degree'. But Gilbert Ryle, one-time ruler of British philosophy, in a letter in 1947, said 'degrees of belief are meaningless because you either believe something or you don't. In that letter he did not suggest an alternative phrase to capture the concept, such as 'intensity of conviction' or 'personal probability' or 'subjective probability'. If Ryle were right in a substantive sense there would be no intermediate stage between belief and disbelief and a change from one state to the other could never occur gradually. Common law seems to agree with Ryle because an accused has to be 'found' guilty or innocent if a decision is reached at all, but, as Mr Bumble in Oliver Twist said " . . . the law is a ass, a idiot'. If a jury reached a unanimous verdict that the probability of guilt was between 0.8 and 0.95, it might be charged with contempt of court. Many statisticians during the last few decades have decided that degrees of belief are necessary for clarifying the foundations of inference, and in the last ten years some professional philosophers have come round to this view. Horwich is one of them regarding various puzzles and paradoxes in the philosophy of science. The formal use of degrees of belief is often described as 'Bayesian' because of a couple of obscurely written papers by the Rev Thomas Bayes, published posthumously in 1763, but as usual the eponymy is misleading because Laplace's later but still pioneering contributions were much greater. There are many interpretations of 'Bayesian', but the essential ingredient is the acceptance of certain axioms for rational degrees of belief, or epistemic probabilities. These axioms have received several justifications, beginning with lectures of Ramsey (1926). Some of us regard interval-valued probabilities (upper and lower probabilities) as more fundamental than sharp numerical ones, but, for the sake of simplicity, numerical values are usually assumed and will be assumed in this review. The current modern notation for a probability, in works on statistics and mathematical probability, is P(A IB), read 'the probability of A given B or relative to B', or the 'conditional probability of A given B'. In practice all probabilities are conditional, but B is often taken for granted and then the notation is abbreviated, with some risk of error, to P(A). Many philosophers of statistics use other notations because they don't read statistical texts.