Theory of Probability

11Inductive Reasoning; Statistical Inference

11.1 Introduction

11.1.1. Within the ambit of the logic of certainty, that is to say ordinary logic, valid arguments are deductive arguments. Conclusions which are certain can only be arrived at by establishing that they are implicit in something already known. In other words, we arrive at the particular through the general. In doing so, however, it is clear that we can never enlarge our field of knowledge (except in the sense that certain features of our previously acquired knowledge, of which, perhaps, we were previously unaware, are now made more explicit).

The form of argument leading to conclusions that go beyond what is already known, or what has previously been ascertained, is different; this is the so‐called inductive form of argument. We have used ‘so‐called’ because, in fact, we must first of all discuss whether, and in what sense, it is legitimate to refer to it as a form of ‘argument’ at all (see Sections 11.1.3–11.1.4).

The problem of induction arises in every field and at every level: from the examination of arguments for and against various scientific theories, to those concerning the guilt of someone suspected of a crime; from methods for establishing, on the basis of some given data, the conditions for a specific kind of insurance policy, to methods of estimating some quality or other to the required degree of accuracy on the basis of measurements which are inherently imprecise.

It is particularly instructive to ...