It was earlier established that particulars are divisible into those known by acquaintance and those known by description. Universals are also divisible along these lines. The qualities of sense-data, like "white, red, black, sweet, sour, loud," are universals with which we are acquainted. We become acquainted with these universals by seeing similar instances of one thing, for example several white things, and abstracting the essential whiteness. These universals, called "sensible qualities" by Russell, are the least removed from our knowledge of particulars; they seem more immediate than universals like relations. Another readily apprehensible universal is the relation between parts of a "single complex sense-datum." We are acquainted, for instance, with seeing one page of writing at a time and at the same time understanding that different parts of the page are in specific relations to other parts. We have knowledge by acquaintance with the universal relation "being to the left of" when we abstract from a page of writing that what certain spaces of the page have in common is their position on the left. Another good example of abstracting a universal relation from sense-data can be seen in the event of hearing a sequence of chimes. One hears the progression of chimes and at the end can recall the entire sequence while at once knowing which chime came before or after others. In other words, this process of abstraction allows us to gain knowledge of space and time relations.

Another relation, resemblance, becomes clear when we see two shades of green compared simultaneously with a shade of red. It is clear which colors are like and which unlike. One more step of abstraction grants another relation, that of "greater than." In the case of two greens and a red, it is easy to see that the similarity between the greens is "greater than" the degree of similarity between a red and a green. We have immediate knowledge of these universal relations, just as we have immediate knowledge of particulars via sense-data.

The topic of universals arose from Russell's earlier consideration of a priori knowledge. Returning now to the problem of how a priori knowledge is possible, Russell points out that the proposition "two and two are four" involves a relation between the universal "two" and the universal "four." From this, he formulates a tentative proposition, that "All a priori knowledge deals exclusively with the relations of universals." Russell sets out to prove the truth of this proposition by countering the only objection he can think of—"all of one class of particulars belong to some other class" or, in other words, "that all particulars having some one property might also have some other." The objection suggests that what we are really dealing with are particulars that have some property, instead of the universal property itself. On this objecting view, the a priori proposition above could be restated "any two and any other two are four."

Russell maintains the view that a priori propositions deal with universals. He tests the proposition at hand by familiarizing himself with the words in it, for they substantiate the task of understanding the statement. We comprehend the case of "two and two are four," as soon as we grasp "two" and "four." It is unnecessary (and impossible) to know all of the couples in the world in order to grasp the statement. "Thus," Russell writes, "although our general statement implies statements about particular couples, as soon as we know that there are such particular couples," it does not assert any particular couple "and thus fails to make any statement whatever about any actual particular couple." Thus, the a priori proposition contains universals, not particulars.

Our power of abstracting universals grants us knowledge of a priori logical and arithmetic truths. Earlier consideration of the a priori was troubling with respect to experience. But our knowledge of the a priori is general and all of its applications involve particulars, which must be known empirically, through experience. Facts about the world like which two and which two make a collection of four depend on experience and thereby the confusion about the role of experience in a priori knowledge evaporates.

Russell contrasts the empirical generalization "all men are mortals" with our previous a priori judgment. The difference between these statements rests with the kind of evidence involved. We can grasp the generalization as soon as we grasp the constituent universals, "man" and "mortal." It is unnecessary to be acquainted with the entire human race to understand the statement. Still, the generalization is based in experience because we know of many instances of men dying and no instances of immortality. We infer that all men are mortal; we do not perceive an a priori connection between the words.