Inductive Logic and Statistics

There are strong parallels between statistics and inductive logic. An inductive logic is a system of inference that describes the relation between propositions on data, and propositions that extend beyond the data, such as predictions over future data, and general conclusions on data. Statistics, on the other hand, is a mathematical discipline that describes procedures for deriving results about a population from sample data. These results include decisions on rejecting or accepting a hypothesis about the population, the determination of probability assignments over such hypotheses, predictions on future samples, and so on. Both inductive logic and statistics are calculi for getting from the given data to propositions or results that transcend the data. Despite this fact, inductive logic and statistics have evolved more or less separately. This is partly because there are objections to viewing statistics, especially classical statistical procedures, as inferential. A more important reason, to my mind, is that inductive logic has been dominated by the Carnapian programme, and that statisticians have perhaps not recognised Carnapian inductive logic as a discipline that is much like their own. Statistical hypotheses and models do not appear in the latter, but they are the start and finish of most statistical procedures. Much of the mutual misunderstanding stems from this difference between the roles of hypotheses in the two programmes, or so I believe. In this chapter I aim to show that Carnapian inductive logic can be developed to encompass inference over statistical hypotheses, and that the resulting inductive logic can, at least partly, capture statistical procedures. In doing so, I hope to bring the philosophical discipline of inductive logic and the mathematical discipline of statistics closer together. I believe both disciplines can benefit from such a rapprochement. First, framing statistical procedures as inferences in an inductive logic may help to clarify the presuppositions and foundations of these procedures. Second, by relating statistics to inductive logic, insights from inductive logic may be used to enrich and improve statistics. And finally, showing the parallels between inductive logic and statistics may show the relevance, also to inductive logicians themselves, of their discipline to the sciences, and thereby direct further philosophical research. The reader may wonder where in this chapter she can read about the history of inductive logic in relation to the historical development of statistics. Admittedly,