An interesting feature of a priori propositions is that we can sometimes know one without knowing of a single instance. For example, it is known that multiplying two numbers together yields a third number, their product. The multiplication table is a record of all products less than 100. It is also known that "the number of integers is infinite, and that only a finite number of pairs of integers ever have been or ever will be thought of by human beings." Given what is known and what is necessarily unknown, we can formulate the proposition: "All products of two integers, which never have been and never will be thought of by any human being, are over 100." We can never know any instance of the proposition because its terms exclude knowing.
Russell exposes the epistemological relevance of such propositions with a return gesture to his earlier concepts. Knowledge of physical objects has been shown to depend on inference; we have no immediate knowledge of them. We can only give instances of immediate sense-data, not the associated physical objects. Russell writes: "Our knowledge as to physical objects depends throughout upon this possibility of general knowledge where no instance can be given. And the same applies to our knowledge of other people's minds, or of any other class of things of which no instance is known to us by acquaintance."
This chapter offers a useful summary outline of sources of knowledge as Russell has developed them. We must first separate knowledge of things from knowledge of truths. Each kind of knowledge is further divisible into an immediate branch and a derivative branch. Up to chapter ten, we have mostly discussed immediate knowledge of things, which Russell calls knowledge by acquaintance. We can be acquainted with particulars, through sense-data, or with universals, things like "sensible qualities, relations of space and time, similarity, and certain abstract logical universals." The other branch, our derivative knowledge of things, Russell calls knowledge by description. This method requires an acquaintance with something and some knowledge of truths, which brings us to the other half of our knowledge not as yet much discussed.
Like our knowledge of things, there is an immediate branch and a derivative branch of our knowledge of truths. As Russell will show, our immediate knowledge is aptly called "intuitive knowledge, and the truths so known may be called self-evident truths." Self-evident truths will include that which we gather from the senses and "certain abstract logical and arithmetical principles." Whatever we can deduce from these self-evident truths will comprise our derivative knowledge of truths. Since knowledge by description depends on direct acquaintance and knowledge of truths, it is enlightening to return to chapter five (on description) after learning about knowledge of truths.
The problem of error arises with knowledge of truths; it does not with knowledge of things. If we stay within the scope of the immediate object, as in sensation, then we do not risk error. Our beliefs can be confirmed or revealed as mistaken, when "we regard the immediate (sense-data) as representative of a physical object." Next, Russell will analyze the nature of our intuitions since our intuitive knowledge is, in part, the basis for our knowledge of truths.