Russell's topic in this chapter is knowledge by induction; he addresses its validity and our capacity to understand it. The principle of induction is the cornerstone in Russell's discussion of knowledge of things beyond acquaintance. He has established so far that we are acquainted with our sense-data and our memories of past sense-data (and probably also with ourselves). To extend our understanding beyond the range of immediate experience, we draw inferences. In this way we approach things outside our realm of acquaintance, like physical objects, matter, other people, a past before individual consciousness, things we could not know otherwise. Inferences depend on general principles. In order to draw an inference, it must be known that "some one sort of thing A, is a sign of the existence of some other sort of thing, B." The existence of thunder usually signifies that lightning has come just before. Russell believes that inferential judgments happen every day and, though they cannot be proven to be accurate, provide a useful extension of knowledge beyond our private experience.

Our expectation that the sun will rise tomorrow is an essential case for Russell. Such an expectation is a usual one, one which never seems to come under suspicion or doubt. Now, Russell asks whether or not this belief is a reasonable one. Though there is no simple test, he undertakes to find a source of general belief that would justify our expectation. If asked why we believe that the sun will rise tomorrow, one could openly answer, "Because it has always risen every day." We expect the future based on the past. Or, when asked, one might appeal to laws of motion. Unless something interferes with the orbit of earth, a rotating body, then it will continue the same as it always has. To this, Russell rephrases the initial question: what reason do we have to suppose that a law of motion will be sustained from this day to the next?

We believe in the laws of motion, just as we believe in the rising sun, because to our knowledge, there has never been a break in this repetition, this constancy. However, is this reason enough for our belief? "Do any number of cases of a law being fulfilled in the past afford evidence that it will be fulfilled in the future?" Uncertainty about the expectations by which we live our daily lives, such as the expectation that we will not be poisoned by the bread at our next meal, is an unattractive possibility. Russell tries to show next that it is of the essence to our daily life that our expectations seem probable, not certain. He sets out to find a reason in support of the view that our expectations will probably be fulfilled.

Experience shows that "uniform succession or coexistence has been a cause of our expecting the same succession or coexistence on the next occasion." We associate repeated sensations with a certain outcome by habit. Our instincts cause us to anticipate the sun each morning, and they seem valid. Still, the question as to whether there is "reasonable ground" for following such instincts persists. Should we believe in these patterns that are merely consistent as far as we know? Russell proposes that we instinctually assume "the uniformity of nature." We believe that "everything that has happened or will happen is an instance of some general law to which there are no exceptions." We also find this attitude (and perhaps mimic it) in the province of scientific investigation. Science frequently assumes that "general rules that have exceptions can be replaced by general rules which have no exceptions." Laws of motion and laws of gravitation came to account for balloons and airplanes replacing the old rule, "unsupported bodies in air fall," which failed and counted balloons and airplanes as exceptions. Science isolates uniformities that hold as uniform as far as our experience extends. Yet, the uniformity of nature is an assumption that cannot be proven. It holds for all instances in the past, but there is no way of knowing if it will remain constant in the future. Despite many repetitions, an outcome could change even at the last instance and thus "probability is all we ought to seek."

The most stringent degree of certainty about future expectations that we can secure is that the more often that A signifies the occurrence of B, the more probable it is that the instance will also be the case in the future. We may also hope that if A indicates B very frequently, then we may estimate the frequency tantamount to an almost certainty. Russell formulates these observations into two parts, outlining the principle of induction.

First, when a thing of a certain sort A has been found to be associated with a thing of a certain other sort B and has never been found dissociated from a thing of the sort B, the greater the number of cases in which A and B have been associated, the greater is the probability that they will be associated in a fresh case in which one of them is known to be present. Second, under the same circumstances, a sufficient number of cases of association will make the probability of a fresh association nearly a certainty and will make it approach certainty without limit.