Principia mathematica, vol. 1 (of 3)

General Observations on Relations. The notion of "relation" is so general that it is important to realize the different sorts of relations to which the notations defined in the preceding section may be applied. It often happens that a proposition which holds for any relation is only important for relations of certain kinds; hence it is desirable that the reader should have in mind some of the principal kinds of relations. Of the various uses to which different sorts of relations may be put, there are three which are specially important, namely (1) to give rise to descriptive functions, (2) to establish correlations between different classes, (3) to generate series. Let us consider these in succession.

(1) In order that a relation may give rise to a descriptive function, it must be such that the referent is unique when the relatum is given. Thus, for example, the relations , , , , , , , defined above, all give rise to descriptive functions. In general, if gives rise to a descriptive function, there will be a certain class, namely , to which the argument of the function must belong in order that the function may have a value for that argument. For example, taking the sine as an illustration, and writing "" instead of "," must be a number in order that may exist. Then is the relation of to when . If we put numbers between and , both included, will be the relation of to when and . The converse of this relation, which is , will also give rise to a descriptive function; thus that value of which lies between and . This illustrates a case which arises very frequently, namely, that a relation does not, as it stands, give rise to a descriptive function, but does do so when its domain or converse domain is suitably limited. Thus for example the relation "parent" does not give rise to a descriptive function, but does do so when its domain is limited to males or limited to females. The relation "square root," similarly, gives rise to a descriptive function when its domain is limited to positive numbers, or limited to negative numbers. The relation "wife" gives rise to a descriptive function when its converse domain is limited to Christian men, but not when Mohammedans are included.

The domain of a relation which gives rise to a descriptive function without limiting its domain or converse domain consists of all possible values of the function; the converse domain consists of all possible arguments to the function. Again, if gives rise to a descriptive function, will be the class of those arguments for which the value of the function is . Thus consists of all numbers whose sine is , i.e. all values of . Again, will be the sines of the various members of . If is a class of numbers, then, by the notation of *38, will be the doubles of those numbers, the trebles of them, and so on. To take another illustration, let be a pencil of lines, and let be the intersection of a line with a given transversal. Then will be the intersections of lines belonging to the pencil with the transversal.

(2) Relations which establish a correlation between two classes are really a particular case of relations giving rise to descriptive functions, namely the case in which the converse relation also gives rise to a descriptive function. In this case, the relation is "one-one," i.e. given the referent, the relatum is determinate, and vice versa. A relation which is to be conceived as a correlation will generally be denoted by or . In such cases, we are as a rule less interested in the particular terms and for which , than in classes of such terms. We generally, in such cases, have some class contained in the converse domain of our relation , and we have a class such that . In this case, the relation correlates the members of and the members of . We shall have also , so that, for such a relation, the correlation is reciprocal. Such relations are fundamental in arithmetic, since they are used in defining what is meant by saying that two classes (or series) have the same cardinal (or ordinal) number of terms.

(3) Relations which give rise to series will in general be denoted by or , and in propositions whose chief importance lies in their application to series we shall also, as a rule, denote a variable relation by or . When is used, it may be read as "precedes." Then may be read "follows," may be read "predecessors of ," may be read "followers of ." will be all members of the series generated by except the last (if any), will be all members of the series except the first (if any), will be all the members of the series. will consist of all terms preceding some member of . Suppose, for example, that our series is the series of real numbers, and that is the class of members of an ascending series , , , ... , ... Then will be the segment of the real numbers defined by this series, i.e. it will be all the predecessors of the limit of the series. (In the event of the series , , , ... , ... growing without limit, will be the whole series of real numbers.)

It very often happens that a relation has more or less of a serial character, without having all the characteristics necessary for generating series. Take, for example, the relation of son to father. It is obvious that by means of this relation series can be generated which start from any man and end with Adam. But these series are not the field of the relation in question; moreover this relation is not transitive, i.e. a son of a son of is not a son of . If, however, we substitute for "son" the relation "descendant in the direct male line" (which can be defined in terms of "son" by the method explained in *90 and *91), and if we limit the converse domain of this relation to ancestors of in the direct male line, we obtain a new relation which is serial, and has for its field and all his ancestors in the direct male line. Again, one relation may generate a number of series, as for example the relation " is east of ." If and are points on the earth's surface, and in the eastern hemisphere, this relation generates one series for every parallel of latitude. By confining the field of the relation further to one parallel of latitude, we obtain a relation which generates a series. (The reason for confining and to one hemisphere is to insure that the relation shall be transitive, since otherwise we might have east of and east of , but west of .)

A relation may have the characteristics of all the three kinds of relations, provided we include in the third kind all those which lead to series by some such limitations as those just described. For example, the relation , i.e. (in virtue of the notation of *38) the relation of to , where is supposed to be a finite cardinal integer, has the characteristics of all three kinds of relations. In the first place, it leads to the descriptive function (, i.e. . In the second place, it correlates with any class of numbers the class obtained by adding 1 to each member of , i.e. (. This correlation may be used to prove that the number of finite integers is infinite (in one of the two senses of the word "infinite"); for if we take as our class all the natural numbers including 0, the class ( consists of all the natural numbers except 0, so that the natural numbers can be correlated with a proper part[58] of themselves. Again the relation may be used, like that of father to son, to generate a series, namely the usual series of the natural numbers in order of magnitude, in which each has to its immediate predecessor the relation . Thus this relation partakes of the characteristics of all three kinds of relations.