Natural Philosophy

by Wilhelm Ostwald

From these observations it becomes further apparent that logic, whether it is the superfluous classic logic or modern effective inductive logic, is nothing but a part of the group theory, or science of manifoldness, which appears as the first, because it is the most general member of the mathematical sciences (this word taken in its widest significance). But according to the hierarchic system in harmony with which the scheme of all the sciences had been consciously projected, we cannot expect anything else than that those sciences which are needful for the pursuit of all other sciences (and logic has always been regarded as such an indispensable science, or, at least, art) should be found collected and classified in the first science.

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.

For the purposes of systematization we have assumed that we can arbitrarily eliminate one or another characteristic of a group. In experience, however, this often proves inadmissible. As a rule we find that the things which lack one of the characteristics of a group will also lack a number of other characteristics; in other words, that the characteristics are not all independent of one another, but that a certain number of them go together, so that they are present in a thing either in common or not at all.