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In this chapter, Russell continues his discussion of knowledge of truths. He has just established a criterion for what we mean by truth and now turns to the more interesting question concerning how we can know what is true from what is false. Since it is plain that some of our beliefs are erroneous, it becomes difficult to regard any unexamined belief with certainty. What we must ask ourselves now is: "can we ever know anything at all"? So, Russell sets out first to define "knowing" and "knowledge."

He begins by positing "true belief" as a definition for knowledge. Although it sometimes happens that we believe something that happens to be true, we engage the word "know" in everyday language in a way that prohibits us from saying matter-of-factly that we have knowledge of this belief. In one instance a man might claim that he knows that the last Prime Minister's last name started with 'B'. He might believe correctly since the last prime minister (in Russell's 1912 example) was Sir Henry Campbell Bannerman. However, if this particular man holds his belief because he believes the minister's name was Mr. Balfour, then his belief could not be granted as proper knowledge. Russell states "a true belief is not knowledge when it is deduced from a false belief." Analogously, a true belief does not constitute knowledge when one deduces it by a "fallacious process of reasoning." The premises "All Greeks are men; Socrates was a man" are true. The inferred conclusion that "Socrates was a Greek" is true in itself but does not follow from the premises. Thus, this process of inference cannot be said to lead to knowledge.

The remaining alternative seems to be that "nothing is knowledge except what is validly deduced from true premises." Russell cannot accept this because it is not enough that premises are true; they must be known as well. However, if we change the alternative from "true premises" to "known premises," the definition becomes circular, assuming one has knowledge before the act of deducing knowledge. Russell allows that this definition at best defines "derivative knowledge," that which is "validly deduced from premisses known intuitively." Russell briefly postpones his discussion of intuitive knowledge to consider this definition.

One objection to the definition is that "it unduly limits knowledge." Russell claims that it frequently comes about that a person will hold a true belief, not because she has validly inferred it, but because she has been familiar with some piece of intuitive knowledge. Consider the beliefs created in the act of reading. If the newspapers announce that a king has died, then upon reading it our belief is justified, as the papers are usually correct when making such statements. However, our belief is based on knowledge that a sense-data exists, that of print which delivers news. Comprehension of meaning occurs, but not realization from direct experience. Although the reader could theoretically draw an inference from printed letters to meanings, she does not perform that act; she reads and associates an act of inference. Still, we would say that she "know(s) that the newspaper announces the King's death." Therefore, Russell admits derivative knowledge to be "the result of intuitive knowledge even if by mere association." Logical processes of reasoning are not required for such knowledge though there must be such a connection possible. Reading print is just one example of a "psychological inference," a process by which we often pass from one belief to another.

At this point Russell declares that the major difficulty that arises with respect to knowledge does not involve the derivative kind, rather the intuitive. One may use intuitive knowledge to test the derivative, but there is no known criterion for testing the intuitive. Russell maintains that "all our knowledge of truths (are) infected with some degree of doubt." However, the earlier established notion of self-evidence does something to diminish this difficulty.

The possibility for self-evidence in our truths contains a sense in which a truth may be judged infallible. "When a belief is true," Russell reminds from the previous chapter, "there is a corresponding fact, in which the several objects of the belief form a single complex." The belief then constitutes "knowledge of this fact." Besides knowledge from corresponding fact, we may also entertain knowledge of facts "constituted by perception." This method, by way of knowledge of things, allows for a case where one looks west, sees the setting sun, and knows a fact that the sun is setting. The same fact, that the sun is setting, can be known by way of knowledge of truths, a belief corresponding to fact. If the hour of sunset is known, then wherever one is at that hour, one can know that the sun is setting. There are thus two theoretical ways in which the same complex fact can be known, by acquaintance or by judgment.