Volume 1 / Part 1 — *21. GENERAL THEORY OF RELATIONS.

*21. GENERAL THEORY OF RELATIONS.

Summary of *21.

The definitions and propositions of this number are exactly analogous to those of *20, from which they differ by being concerned with functions of two variables instead of one. A relation, as we shall use the word, will be understood in extension: it may be regarded as the class of couples ( for which some given function is true. Its relation to the function is just like that of the class to its determining function. We put

*21·01.

Here "" has no meaning in isolation, but only in certain of its uses. In *21·01 the alphabetical order of and corresponds to the typographical order of and in , so that This is important in relation to the substitution-convention below.

It will be shown that i.e. that two relations, as above defined, are identical when, and only when, they are satisfied by the same pairs of arguments.

For substitution in and , we adopt the convention that when a function (as opposed to its values) is represented in a form involving and , or any other two letters of the alphabet, the value of this function for the arguments and is to be found by substituting for and for , while the value for the arguments and is to be found by substituting for and for . That is, the argument mentioned first is to be substituted for the letter which comes first in the alphabet, and the argument mentioned second for the later letter; thus the mode of substitution depends upon the alphabetical order of the letters which have circumflexes and the typographical order of the other letters.

The above convention as to order is presupposed in the following definition, where is the first argument mentioned and the second:

*21·02.

Hence, following the convention,

This definition is not used as it stands, but is introduced for the sake of which results from *21·01·02. We shall use capital Latin letters to represent variable expressions of the form , just as we used Greek letters for variable expressions of the form . If a capital Latin letter, say , is used as an apparent variable, it is supposed that the which occurs in the form "(" or "(" is to be replaced by "(" or "(," while the which occurs later is to be replaced by "." In fact we put The use of single letters for such expressions as is a practically indispensable convenience.

The following is the definition of the class of relations:

*21·03.

Similar remarks apply to it as to the definition of "Cls" (*20·03).

In virtue of the definitions *21·01·02 and the convention as to capital Latin letters, the notation "" will mean " has the relation to ." This notation is practically convenient, and will, after the preliminaries, wholly replace the cumbrous notation .

The proofs of the propositions of this number are usually omitted, since they are exactly analogous to those of *20, merely substituting *12·11 for *12·1, and propositions in *11 for propositions in *10.

The propositions of this number, like those of *20, fall into three sections. Those of the second section are seldom referred to. Those of the third section, extending to relations the formal properties hitherto assumed or proved for individuals and functions, are not explicitly referred to in the sequel, but are constantly relevant, namely whenever a proposition which has been assumed or proved for individuals and functions is applied to relations. The principal propositions of the first section are the following.

*21·15.

I.e. two relations are identical when, and only when, their defining functions are formally equivalent.

*21·31.

I.e. two relations are identical when, and only when, they hold between the same pairs of terms. The same fact is expressed by the following proposition:

*21·43.

*21·2·21·22 show that identity of relations is reflexive, symmetrical and transitive.

*21·3.

I.e. two terms have a given relation when, and only when, they satisfy its defining function.

*21·151.

I.e. every relation can be defined by a predicative function. Hence when, using *21·07 or *21·071, we have a relation as apparent variable, and are therefore confined to predicative defining functions, there is no loss of generality.

*21·01.

On the convention as to order in *21·01·02, cf. p. 211, and thus relate , to , so that

*21·02.

*21·03.

The following definitions merely extend to relations, with as little modification as possible, the definitions already given for other symbols.

*21·07. (

*21·071. (

*21·072.

*21·08.

*21·081.

The convention as to typographic and alphabetic order is here retained.

*21·082.

*21·083.

*21·1.

*21·11.

This proposition proves that every proposition about a relation expresses an extensional property of the determining function.

*21·111.

*21·112.

It is *12·1, not *12·11, which is required in this proposition, because we are concerned with a function ( of one variable, namely , although that one variable is itself a function of two variables.

*21·12.

This is the first use of the primitive proposition *12·11, except in *20·701·702·703.

*21·13.

*21·14.

*21·15.

This proposition states that two double functions determine the same relation when, and only when, they are formally equivalent, i.e. are satisfied by the same pairs of arguments. This is a fundamental property of relations as defined above (*21·01).

*21·151.

*21·16.

*21·17.

*21·18.

*21·19.

*21·191.

*21·2.

*21·21.

*21·22.

*21·23.

*21·24.

*21·3.

This shows that has to y the relation determined by when, and only when, and satisfy .

Note that the primitive proposition *12·11 is again required here.

*21·31.

*21·32.

*21·33.

Here is written for some expression of the form . The use of a single capital letter for a relation is convenient whenever the determining function is irrelevant.

*21·4.

*21·41.

*21·42.

*21·43.

*20·5·51·52 have no analogues in the theory of relations.

*21·53.

*21·54.

*21·55.

*21·56.

*21·57.

*21·58.

The following propositions are the analogues of *20·6 ff., and have a similar purpose.

*21·6.

*21·61.

*21·62. When is true, whatever possible argument of the form may be, ( is true. [Proof as in *20·62]

*21·63.

*21·631. If "" is significant, then if is of the same type as , "" is significant, and vice versa.

[Proof as in *20·631]

*21·632. If, for some , there is a proposition , then there is a function , and vice versa.

[Proof as in *20·632]

*21·633. "Whatever possible relation may be, is true whatever possible relation may be" implies "whatever possible relation may be, is true whatever possible relation may be."

[Proof as in *20·633]

*21·64.

*21·7.

*21·701.

*21·702.

*21·703.

*21·704.

*21·705.

*21·71.

From the above propositions it appears that relations, like classes, have all the formal properties which they would have if they were symbols having a meaning in isolation. Hence unless a symbol occurs in a way in which only a relation can occur significantly, we do not need to decide whether it stands for a relation or not. This result, like the corresponding result for classes mentioned at the end of *20, is important as giving greater generality to our propositions than they would otherwise possess. The results obtained in *20 and *21 for classes and relations whose members or terms are neither classes nor relations can be extended, by mere repetition of the proofs, to classes of classes, classes of relations, relations of classes, relations of relations, and so on.