These various kinds of correlations have importance in various connections, some for one purpose, some for another. The general notion of one-one correlations has boundless importance in the philosophy of mathematics, as we have partly seen already, but shall see much more fully as we proceed. One of its uses will occupy us in our next chapter.
CHAPTER VI
SIMILARITY OF RELATIONS
WE saw in Chapter II. that two classes have the same number of terms when they are "similar," i.e. when there is a one-one relation whose domain is the one class and whose converse domain is the other. In such a case we say that there is a "one-one correlation" between the two classes.
In the present chapter we have to define a relation between relations, which will play the same part for them that similarity of classes plays for classes. We will call this relation "similarity of relations," or "likeness" when it seems desirable to use a different word from that which we use for classes. How is likeness to be defined?
We shall employ still the notion of correlation: we shall assume that the domain of the one relation can be correlated with the domain of the other, and the converse domain with the converse domain; but that is not enough for the sort of resemblance which we desire to have between our two relations. What we desire is that, whenever either relation holds between two terms, the other relation shall hold between the correlates of these two terms. The easiest example of the sort of thing we desire is a map. When one place is north of another, the place on the map corresponding to the one is above the place on the map corresponding to the other; when one place is west of another, the place on the map corresponding to the one is to the left of the place on the map corresponding to the other; and so on. The structure of the map corresponds with that of the country of which it is a map. The space-relations in the map have "likeness" to the space-relations in the country mapped. It is this kind of connection between relations that we wish to define.
We may, in the first place, profitably introduce a certain restriction. We will confine ourselves, in defining likeness, to such relations as have "fields," i.e. to such as permit of the formation of a single class out of the domain and the converse domain. This is not always the case. Take, for example, the relation "domain," i.e. the relation which the domain of a relation has to the relation. This relation has all classes for its domain, since every class is the domain of some relation; and it has all relations for its converse domain, since every relation has a domain. But classes and relations cannot be added together to form a new single class, because they are of different logical "types." We do not need to enter upon the difficult doctrine of types, but it is well to know when we are abstaining from entering upon it. We may say, without entering upon the grounds for the assertion, that a relation only has a "field" when it is what we call "homogeneous," i.e. when its domain and converse domain are of the same logical type; and as a rough-and-ready indication of what we mean by a "type," we may say that individuals, classes of individuals, relations between individuals, relations between classes, relations of classes to individuals, and so on, are different types. Now the notion of likeness is not very useful as applied to relations that are not homogeneous; we shall, therefore, in defining likeness, simplify our problem by speaking of the "field" of one of the relations concerned. This somewhat limits the generality of our definition, but the limitation is not of any practical importance. And having been stated, it need no longer be remembered. We may define two relations
and
as "similar," or as having "likeness," when there is a one-one relation