=22. Negation.= When the characteristics a, b, c, d of a group have been determined, then the aggregate of all things existing can be divided into two parts, namely, the things which belong to the group A and those which do not belong to it. This second aggregate may then be regarded as a group by itself. If we call this group "not-A," it follows from the definition of this group that the two groups, A and not-A, together form the aggregate of all things. This is the meaning and the significance of the linguistic form of _negation_. It excludes the thing negated from any group given in a proposition, and this relegates it to the second or complementary group. The characteristic of such a group is the common absence of the characteristics of the positive group. We must note here that the absence of even _one_ of the characteristics a, b, c, d excludes the incorporation of the thing into the group A, while the mere absence of this characteristic suffices to include it in the group not-A. We can therefore by no means predicate of group not-A that each one of its members must lack _all_ the characteristics a, b, c, d. We can only say that each of its members lacks at least one of the characteristics, but that one or some may be present, and several or all may be absent. From this follows a certain asymmetry of the two groups, which we must bear in mind. The consideration of this subject is especially important in the treatment of negation in the conclusions of formal logic. As we shall make no special use of formal logic, we need not enter into it in detail. =23. Artificial and Natural Groups.= The combination of the characteristics which are to serve for the definition of a group is at first purely arbitrary. Thus, when we have chosen such an arbitrary combination, a, b, c, d, we can eliminate one of the characteristics, as, for example, c, and form a group with the characteristics a, b, d. Such a group, which is _poorer in characteristics_, will, in general, be _richer in members_, for to it belong, in the first place, all the things with the characteristics a, b, c, d, of which the first group consisted, and in addition all the things which, though not possessing c, possess a, b, and d. If we call such groups related as contain common characteristics, though containing them in different members and combinations, so that the definition of the one group can be derived from the other by the elimination or incorporation of individual characteristics, then we can postulate the general thesis _that in related groups those must be richer in members which are poorer in characteristics, and inversely_. This is the precise statement of the proposition of the less definite thesis stated above.