Knowledge by acquaintance with perception is only possible "when there really is such a fact," when the parts of a complex whole really are present in the appropriate relation to form the whole. By comparison, the knowledge of truths by judgment only demands the "reality of the parts and the relation: the relation may not relate those parts in that way, and yet the judgment may (erroneously) occur."

The double-standard of self-evidence, discussed in chapter 11, suggested two kinds of evidence, one which gave "an absolute guarantee of truth," the other truth in degrees. Russell further distinguishes the two. The first absolute sense occurs when we "have acquaintance with the fact which corresponds to the truth," knowledge of a truth of perception. The fact involved in "Othello believes that Desdemona loves Cassio" is "Desdemona's love," a fact with which only Desdemona could have direct acquaintance. Thus, she is the only one who could regard this truth (if it were true) as self-evident. This is an example of a mental fact; the same privacy holds for facts known through sense-data. Each fact about particular sense-data can only be self-evident in this absolute sense to one person. (It is important to note that although our knowledge of the truth of a complex fact can be absolutely self-evident, we do not have the guarantee that a certain judgment concerning that fact is true. For we analyze the complex fact in passing from perception to judgment. "We have to separate out 'the sun' and 'shining' as constituents of the fact." We might make a judgment that does not correspond to fact.)

The second sense of self-evidence accompanies judgments not based in perception. This kind has degrees, from a high degree of certainty to "a bare inclination in favor of the belief." Consider cases of gradation, not in the sense-data themselves, but in the self-evidence of our judgments based on them. When a horse trots away from us, our certainty of hearing the rap of hooves is first clear, then "there comes a moment when we think perhaps it was imagination…then we think we no longer hear anything, and at last we know we no longer hear anything." Russell offers other illustrations of degree-valued phenomena, resolving that we can trust the higher degrees more than the lower ones.

In our deductions from derivative knowledge, the premises must have some clear degree of self-evidence and this degree must be present at each stage of reasoning. As with derivative knowledge, intuitive knowledge is reliable in a proportion to its degree of certainty. Sense-data and truths of logic and arithmetic may be taken as examples of the high certainty end of the gradation, while judgments "only just more probable than their opposites" exemplify the other end.

When we firmly believe in something intuitive or something inferred from the intuitive, and it is true, then we have knowledge. When we firmly believe in the above and it is false, we are in error. And when we believe something "that is neither knowledge or error" hesitatingly because it has a lower degree of self- evidence, then what we believe "may be called probable opinion." Most of what would pass for knowledge, before Russell's enquiry, ends up being describable as probable opinion. The test of coherence (which failed as a definition of truth) is useful with respect to probable opinions in that a body of coherent opinions are more probable than one probable opinion in isolation. Some scientific hypotheses acquire recognition in this way. Russell notably cites the distinction between waking life and dreaming; "if our dreams, night after night, were as coherent one with another as our days, we should hardly know whether to believe the dreams or the waking life." But the test of coherence "condemns the dreams and confirms the waking life."

Analysis

The content of this chapter, concerning derivative and intuitive knowledge of truths, is the height of Russell's technical outline of knowledge. It contains an echo of the Platonic dialogue "Protagoras," which also asks the question: How can we know anything at all? Russell's answer may be abridged as follows: Derivative knowledge is knowledge from known premises where the known premises are known intuitively. Psychological inference is an unclearly developed middle factor that partially explains our capacity for derivative knowledge. The only qualification for intuitive knowledge is a degree of self-evidence. We have highly self-evident intuitions from our knowledge of perception, our acquaintances with sense-data (fact). As we saw in the previous chapter, belief corresponding to fact is an ideal criterion for truth. We can make judgments, without being acquainted with fact, which may be true and leave room for error. These judgments are removed from our direct perception and may have a low degree of self-evidence. Probable opinion is the last category of self-evident truths, which have the lowest degree of self-evidence.