In contrast to the empiricists, rationalists believed themselves capable of deducing the existence of something in the world just from "general consideration as to what must be." A priori knowledge, which comes the closest to resembling the kind of independent truth the rationalists had in mind, depends on something being the case first. There is a conditional "If" which precedes each statement and tells us that if "one thing exists," then "another must exist." A priori propositions are purely hypothetical, "giving connexions among things that exist or may not exist, but not giving actual existence." They require the knowledge that a first thing exists, that the first premise is indeed the case. When this condition is satisfied, as it can only be through experience (because "all knowledge that something exists must be in part dependent on experience"), then the a priori principle assumes the authority of truth. Both experience and an a priori hypothesis are required to prove that something exists. All our knowledge that asserts that something exists is based, at least in part, on experience. It is therefore aptly describable as empirical knowledge.

Pure math is another kind of a priori knowledge, besides the logical form. Empiricists denied this possibility, claiming that experience was an essential source of our mathematical knowledge. By repeated experience of finding two and two to be four, they argued, we conclude by induction that two and two will always be four. However, Russell states that the way that our mathematical knowledge works is based on a number of instances which allow us to "think of two abstractly, rather than of two coins or two books." Then, "as soon as we are able to divest our thoughts of irrelevant particularity, we become able to see the general principle." We do not, after this, feel more certain about our knowledge after seeing new instances. Each further instance is merely "typical." We identify some "quality of necessity" about the 'two and two' proposition.

The empirical generalization differs as having obtained a mere quality of fact. As a fact, we are able to imagine another world where the generalization might not be fact, where it is not the case. And in our actual world, it just happens to be the case. As against fact, the necessity of "two and two are four" demands that "everything actual and possible" abide by it.

Considering the empirical generalization, "All men are mortal." We can admit that we share this belief because there is no known case of a man living to be older than a certain age. That is our experience with men and death. However, we would probably not draw this conclusion after observing only one case of a man being mortal. Yet, in the case of "two and two are four," one case is adequate to convince us of its truth and necessity. Russell illustrates with the example of Jonathan Swift's imaginary "race of Struldbugs who never die," which we can imagine easily, much more easily than "a world where two and two make five." This latter world would diminish the "whole fabric of our knowledge," casting everything into doubt.

Mathematical and logical judgments are apparent to us without the use of inference, provided some instance indicates a first meaning. The processes that facilitate these judgments are deduction, which progresses from the general to the particular, and induction, which as we have seen usually goes from the particular to the general.

In order to illustrate these processes, Russell takes up the classic example of deduction: "All men are mortal; Socrates is a man, therefore Socrates is mortal." Russell suggests that the best knowledge that we have about men being mortal is really that some certain men, "A, B, C," were mortals. We know this because they have died. He asserts that if we know that Socrates was a member of this certain set, then it is unnecessary to go the obtuse route through deduction in order to prove that "Socrates is mortal." The argument is more certain if induction is applied rather than deduction, because there is a greater probability that Socrates, one man, is mortal than the probability that all men are mortal. Russell holds that this "illustrates the difference between general propositions known a priori, such as "two and two are four," and empirical generalizations such as "all men are mortal." In regard to the former, deduction is the right mode of argument," because we can easily see that this general proposition will apply in future instances; whereas in regard to empirical generalizations, "induction is always theoretically preferable, and warrants greater confidence in the truth of our conclusion, because all empirical generalizations are more uncertain than the instances of them."