This chapter is a shining example of Russell's ability to tell the story of how modern philosophy developed into what it is today, through the rationalists and empiricists. Russell first explicitly appreciated the British empiricists in this work. One recognizes between the rationalist school and the empiricists a composite picture that ultimately merges into view as Russell's philosophy. His work with induction and deduction foreshadow his later associations with a constructive realism, a view that held many parts of reality to be logically constructed out of other, more basic parts.
Three key points to take from this chapter are the following observations. All of our knowledge rests, in part, on experience. We understand this through the illustration that in order to grasp the a priori necessity of "two and two are four," we must first experience at least one instance. Another point is that the a priori quality of necessity is meaningfully distinct from the empirical generalization, which has the quality of mere fact and can be imagined to not be the case. The essential point, however, is the hypothesis that we have knowledge of general principles, a priori knowledge, about which we can have the same degree of certainty that we grant to our direct knowledge by acquaintance.
Another kind of the a priori, besides the logical form and pure math, is knowledge on "ethical value." Something is desirable or useful if it obtains some end, an end that is intrinsically "valuable on its own account." Through experience, we learn that "happiness is more desirable than misery, knowledge than ignorance." These value judgments are elicited through experience but cannot be proven by it (just because something exists and has been experienced cannot indicate if it is good or bad). These ethical judgments are a priori in the sense that they are immediate and logically independent of experience.
At the end of this chapter, Russell gestures toward Immanuel Kant, German philosopher (1724–1804). Kant's discussion of a priori knowledge is fundamentally significant to an understanding of Russell's or any other modern thinker's philosophy. The next chapter is concerned exclusively with Kant's distinctions.