Analogy and Exchangeability in Predictive Inferences

Within the research programme in probabilistic confirmation theory initiated by Carnap — also called inductive probability theory — several kinds of inductive methods have been designed. In particular, some inductive methods, which can be called analogical (inductive) methods, take into account certain considerations of analogy by similarity for predictive inferences. In many situations it seems desirable to use exchangeable analogical methods, i. e., analogical methods that, besides taking into account ‘predictive analogy', have an attractive property known as exchangeability. Unfortunately, the design of exchangeable analogical methods is much more difficult than one might think. In this paper, a precise reformulation of the problem of predictive analogy is given and a new family of exchangeable analogical methods is introduced. Our research on this subject has been stimulated by a paper by Skyrms (1993) where a simple exchangeable method taking predictive analogy into account is introduced. Unfortunately, Skyrms's method (Sk) cannot be considered as a completely satisfying solution to the problem of predictive analogy, since the predictive analogy properties exhibited by Sk are very weak, while the best known general principles of predictive analogy are violated by Sk (see Section 4). On the other hand, we think that the strategy used by Skyrms — consisting of the usage of particular "hyper-Carnapian methods", i. e., mixtures of Carnapian inductive methods — is a significant step in the right direction. Indeed, we will show that certain hyper-Carnapian methods satisfy at least an interesting general principle of predictive analogy. After introducing, in Section 2, some basic concepts concerning Carnapian inductive methods, in Section 3 we will discuss a number of predictive analogy properties. In Section 4 we will analyze some basic features of Sk and, more generally, of Skyrms's approach to predictive analogy. In Section 5, a new family of exchangeable analogical methods, consisting of a particular set of hyper-Carnapian methods, is introduced and it is proved that, under certain conditions, such methods satisfy an interesting general principle of predictive analogy.