Volume 1 / Part 1 — *32. REFERENTS AND RELATA OF A GIVEN TERM WITH RESPECT TO A GIVEN RELATION.
Reading edition
*32. REFERENTS AND RELATA OF A GIVEN TERM WITH RESPECT TO A GIVEN RELATION.
Summary of *32.
Given any relation , the class of terms which have the relation to a given term are called the referents of , and the class of terms to which a given term has the relation are called the relata of . We shall denote by the relation of the class of referents of to , and by the relation of the class of relata of to . It is convenient also to have a notation for the relations of and to . We shall denote the relation of to by "," where "" stands for "sagitta." Similarly we shall denote by "" the relation of to , to suggest an arrow running from right to left instead of from left to right. and are chiefly useful for the sake of the descriptive functions to which they give rise; thus and . Thus e.g. if is the relation of parent to son, the parents of , the sons of . If is the relation of less to greater among numbers of any kind, numbers less than , and numbers greater than . When exists, is the class whose only member is . But when there are many terms having the relation to , , which is the class of those terms, supplies a notation which cannot be supplied by . And similarly if there are many terms to which has the relation , supplies the notation for these terms. Thus for example let be the relation "," i.e. the relation which has to when . Then "" represents all values of such that , i.e. all values of or . Unlike the usual symbol, it is not ambiguous, since instead of representing some one of these values, it represents the class of them.
The definitions of , , , are as follows:
*32·01.
*32·02.
*32·03.
*32·04.
In virtue of the above definitions, we shall have , . This gives an alternative notation which is convenient in dealing with a relation not represented by a single letter.
It should be observed that if is a homogeneous relation (i.e. one in which referents and relata are of the same type), then and are not homogeneous, but relate a class to objects of the type of its members.
In virtue of the definitions of and , we shall have
*32·13.
*32·131.
Thus by *14·21, we always have and . Thus whatever relation may be, we have and . We do not in general have or (. Thus taking to be the relation of parent and child, and . Thus , i.e. when is childless, and , i.e. , when is Adam or Eve. The two sorts of existence, and , can both be significantly predicated of , because "" is a descriptive function whose value is a class; and the same applies to . It will be seen that (by *14·21) , but the converse implication does not hold in general.
We have
*32·16.
Also by *32·18·181,
Thus by the use of or , every statement of the form "" can be reduced to a statement asserting membership of a class. Since, however, the class in question is given by a descriptive function, and descriptive functions are defined by means of relations, we do not thus obtain a method of reducing the theory of relations to the theory of classes.
*32·01.
*32·02.
*32·03.
*32·04.
*32·1.
*32·101.
*32·11.
*32·111.
*32·12.
*32·121.
"" must not be confounded with "." The former means that there is such a class as , which, as we have just seen, is always true; the latter means that is not null, which is only true if is a term to which some other term has the relation . Note that, by *14·21, both and imply . The contradictory of is not , but . This last would not imply , but for the fact that is always true.
*32·13.
*32·131.
*32·132.
*32·133.
The use of *20·57 will in general be tacit. It happens constantly that we have propositions such as *32·13, in which a descriptive expression is shown to be identical with a class. In such cases, whenever the properties of the class are asserted of the descriptive expression, *20·57 is relevant.
*32·14.
Dem.
*32·15.
*32·16.
*32·18.
*32·181.
*32·182.
The transformation from "" to "" is one commonly effected in language. E.g. suppose "" is " loves ," then "" is " is a lover of ."
*32·19.
Dem.
*32·2.
*32·201.
*32·21.
*32·211.
*32·22.
*32·221.
*32·23.
*32·231.
*32·24.
Dem.
*32·241.
*32·25.
*32·251.
*32·3.
Notice that we do not have
Dem.
*32·31.
*32·32.
*32·33.
*32·34.
*32·35.
The proofs of the above propositions are similar to that of *32·3.
*32·4.
*32·41.
Dem.
*32·42.