The central work of this chapter is explaining general principles that function like the principle of induction. Knowledge on these principles cannot be proved or disproved yet can attain the same degree of certainty as knowledge by direct experience. When we practice induction, "we realize some particular application of the principle, and then we realize that the particularity is irrelevant and that there is a generality which may equally truly be affirmed." One plain example of this realization takes place with the arithmetic operation: "two plus two are four." First, we grasp one instance of the statement's truth, then we see that it applies in some other particular case. Then, sooner or later we are able to see the general truth that the statement is true for any particular case. Russell continues that the same practice occurs with logical principles. It is familiar to us that if the premises in an argument are true, then the conclusion is also true.
Take the example of a dialogue between two men, disputing a date. One says: "You will admit that if yesterday was the 15th to-day must be the 16th," to which the other assents. Then, the first continues, that in fact "yesterday was the 15th, because you dined with Jones, and your diary will tell you that was on the 15th," to which the other agrees. Thus, since both premises are true, then the conclusion "to-day is the 16th," follows. In such a case of reasoning, the principle in use may be stated: "Suppose it known that if this is true, then that is true. (And) suppose it also known that this is true, then it follows that that is true." What follows from a proposition that is known to be true is a conclusion that must also be true. The validity of this principle is obvious yet important to examine because the principle allows us to gain positive knowledge without appealing to our senses. It is a self-evident principle exercised by thought, not experience.
There are a number of logical principles like the one described above. Some must be granted before others can be proved, although these last proved seem to have the same kind of obvious certainty intrinsic to those first granted. Russell lists three essential, though arbitrary, such principles, collectively called "Laws of Thought." The first is the law of identity, which states that: "whatever is, is." The second, the law of contradiction, holds that "nothing can both be and not be." And the third, the law of the excluded middle, means that "everything must either be or not be." Calling these principles "laws" is misleading because our thinking does not have to conform to them in any way. Calling them laws serves to recognize their authority; things we observe "behave in accordance with them," and when we think in such accordance, "we think truly."
After preparing the groundwork of general principles, Russell begins a comparative discussion between two schools of thought. The controversy between the empiricists and the rationalists is over the issue of how we come by our knowledge. The British empiricists, Locke, Berkeley, and Hume, believe that our knowledge comes from experience while the rationalists, mainly in the seventeenth century, Descartes and Leibniz, held that we learn from experience and that we also have knowledge of "innate principles" independent of all our experience.
We have already established that we have logical principles that cannot be proven through experience, which are logically independent, in agreement with the rationalists. However, the relation the principles have with experience is not thoroughly independent, for we must have experience first in order to bring forth our knowledge. We must start from particular instances to develop general principles. Russell admits a modification with present-day philosophy, that the rationalist belief in "innate principles" is now more accurately known as "a priori" knowledge. So, although we admit all knowledge to be caused by experience, we can understand a priori knowledge as independent to the degree that experience does not prove it but merely directs us to see the truth of the a priori in itself.
Another way in which our understanding, with Russell, agrees with the empiricist theory is in the position that "nothing can be known to exist" except through experience. To prove that something beyond our experience exists, we must appeal to something else of which we have experience. We have already seen this case through the theory of knowledge by description being dependent on knowledge by acquaintance. Something we know directly must be in the premise of the argument adduced for something we do not know directly. For example, knowing that the Bismarck existed depends on sense-data gained through acquaintance with testimony.