Returning now to the problem of a priori knowledge, which we left unsolved when we began the consideration of universals, we find ourselves in a position to deal with it in a much more satisfactory manner than was possible before. Let us revert to the proposition “two and two are four.” It is fairly obvious, in view of what has been said, that this proposition states a relation between the universal “two” and the universal “four.” This suggests a proposition which we shall now endeavour to establish: namely, “All a priori knowledge deals exclusively with the relations of universals.” This proposition is of great importance, and goes a long way towards solving our previous difficulties concerning a priori knowledge.
The only case in which it might seem, at first sight, as if our proposition were untrue, is the case in which an a priori proposition states that all of one class of particulars belong to some other class, or (what comes to the same thing) that all particulars having some one property also have some other. In this case it might seem as though we were dealing with the particulars that have the property rather than with the property. The proposition “two and two are four” is really a case in point, for this may be stated in the form “any two and any other two are four,” or “any collection formed of two twos is a collection of four.” If we can show that such statements as this really deal only with universals, our proposition may be regarded as proved.
One way of discovering what a proposition deals with is to ask ourselves what words we must understand—in other words, what objects we must be acquainted with—in order to see what the proposition means. As soon as we see what the proposition means, even if we do not yet know whether it is true or false, it is evident that we must have acquaintance with whatever is really dealt with by the proposition. By applying this test, it appears that many propositions which might seem to be concerned with particulars are really concerned only with universals. In the special case of “two and two are four,” even when we interpret it as