On the Alleged Impossibility of Inductive Probability*

Karl Popper and David Miller [1983] have argued that there is no such thing as probabilistic inductive support, as conceived of, for example, in the Bayesian theory of evidence. A little more specifically, they argue that 'all probabilistic support is purely deductive' (p. 688), and that only probabilistic countersupport (disconfirmation) could be inductive in nature. Their interesting and striking argument, which they say is 'completely devastating to the inductive interpretation of probability' (p. 688), has received considerable attention, both in the way of criticism and in the way of defense: Richard Jeffrey [1984] and I. J. Good [1984] have criticised the argument, Isaac Levi [1984] attempts to improve it, Popper and Miller [1984] defend it from Jeffrey's and Good's criticisms, M. L. G. Redhead [1985] offers a new kind of criticism, and Donald Gillies [1986] defends the argument from Redhead's objection. I will not discuss these responses here. After a brief rehearsal of the Popper-Miller argument, I will criticize it in a way different from the criticisms just cited. I will argue that the Popper-Miller argument fails to establish its conclusion on any interesting way of understanding its conclusion. This will involve a little clarification of the structures of inductive and deductive support. According to the Bayesian theory of probabilistic inductive support, the degree to which evidence e supports a hypothesis h is given by the measure: