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.
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."
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.