ABSTRACT
So far the generalisations we have accounted for have been isolated, or as they are sometimes not very happily called, empirical, laws. This would matter less if their certainty could be regarded as complete, though even in that case we might have to ask how we were to apply them to the complexities of nature. But they are not as they stand absolutely certain; in many cases the degree of uncertainty attaching to them is practically appreciable, and in all cases it is of theoretic interest. We have therefore to inquire whether, by connecting these isolated generalisations with one another, we can wholly or in part eliminate this element of doubt? Suppose we have a generalisation a − b grounded on an induction strong enough to make it probable but not strong enough to make it certain. Suppose, further, that we can connect this induction with another α − β (3 which is also probable. Suppose, first, the connection to be such that if α − β is true a − b must also be true, and assume that the evidence for α − β is entirely independent of that from which we inferred a − b. It is clear that the probability of a − b is increased. A fresh and independent consideration is adduced in its favour. The nature of the increase is best seen by putting it arithmetically. Let both the generalisations have an independent probability of 3/4. Then the probability that a is not b is 1/4 before we investigate it on its own merits. The result of our investigation is independently to reduce the probability to 1/4. The actual resultant probability that a is not b is therefore 1/4×1/4, i.e. the probability of a−b is https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203588833/3fe8d384-656e-4866-b10a-a0695306222a/content/ch28_page385-01_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>.
