Introduction to Mathematical Philosophy

relative product of one of them and its converse implies identity, where the "relative product" of two relations and is that relation which holds between and when there is an intermediate term , such that has the relation to and has the relation to . Thus, for example, if is the relation of father to son, the relative product of and its converse will be the relation which holds between and a man when there is a person , such that is the father of and is the son of . It is obvious that and must be [Pg 47] the same person. If, on the other hand, we take the relation of parent and child, which is not one-many, we can no longer argue that, if is a parent of and is a child of , and must be the same person, because one may be the father of and the other the mother. This illustrates that it is characteristic of one-many relations when the relative product of a relation and its converse implies identity. In the case of one-one relations this happens, and also the relative product of the converse and the relation implies identity. Given a relation , it is convenient, if has the relation to , to think of as being reached from by an "-step" or an "-vector." In the same case will be reached from by a "backward -step." Thus we may state the characteristic of one-many relations with which we have been dealing by saying that an -step followed by a backward -step must bring us back to our starting-point. With other relations, this is by no means the case; for example, if is the relation of child to parent, the relative product of and its converse is the relation "self or brother or sister," and if is the relation of grandchild to grandparent, the relative product of and its converse is "self or brother or sister or first cousin." It will be observed that the relative product of two relations is not in general commutative, i.e. the relative product of and is not in general the same relation as the relative product of and . E.g. the relative product of parent and brother is uncle, but the relative product of brother and parent is parent. One-one relations give a correlation of two classes, term for term, so that each term in either class has its correlate in the other. Such correlations are simplest to grasp when the two classes have no members in common, like the class of husbands and the class of wives; for in that case we know at once whether a term is to be considered as one from which the correlating relation goes, or as one to which it goes. It is convenient to use the word referent for the term from which the relation goes, and the term relatum for the term to which it goes. Thus if and are husband and wife, then, with respect to the relation [Pg 48] "husband," is referent and relatum, but with respect to the relation "wife," is referent and relatum. We say that a relation and its converse have opposite "senses"; thus the "sense" of a relation that goes from to is the opposite of that of the corresponding relation from to . The fact that a relation has a "sense" is fundamental, and is part of the reason why order can be generated by suitable relations. It will be observed that the class of all possible referents to a given relation is its domain, and the class of all possible relata is its converse domain. But it very often happens that the domain and converse domain of a one-one relation overlap. Take, for example, the first ten integers (excluding 0), and add 1 to each; thus instead of the first ten integers we now have the integers These are the same as those we had before, except that 1 has been cut off at the beginning and 11 has been joined on at the end. There are still ten integers: they are correlated with the previous ten by the relation of to , which is a one-one relation. Or, again, instead of adding 1 to each of our original ten integers, we could have doubled each of them, thus obtaining the integers Here we still have five of our previous set of integers, namely, 2, 4, 6, 8, 10. The correlating relation in this case is the relation of a number to its double, which is again a one-one relation. Or we might have replaced each number by its square, thus obtaining the set On this occasion only three of our original set are left, namely, 1, 4, 9. Such processes of correlation may be varied endlessly. The most interesting case of the above kind is the case where our one-one relation has a converse domain which is part, but [Pg 49] not the whole, of the domain. If, instead of confining the domain to the first ten integers, we had considered the whole of the inductive numbers, the above instances would have illustrated this case. We may place the numbers concerned in two rows, putting the correlate directly under the number whose correlate it is. Thus when the correlator is the relation of to , we have the two rows: When the correlator is the relation of a number to its double, we have the two rows: When the correlator is the relation of a number to its square, the rows are: In all these cases, all inductive numbers occur in the top row, and only some in the bottom row. Cases of this sort, where the converse domain is a "proper part" of the domain (i.e. a part not the whole), will occupy us again when we come to deal with infinity. For the present, we wish only to note that they exist and demand consideration. Another class of correlations which are often important is the class called "permutations," where the domain and converse domain are identical. Consider, for example, the six possible arrangements of three letters: [Pg 50] Each of these can be obtained from any one of the others by means of a correlation. Take, for example, the first and last, and . Here is correlated with , with itself, and with . It is obvious that the combination of two permutations is again a permutation, i.e. the permutations of a given class form what is called a "group." 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. [Pg 51]