Volume 1 / Part 1 — *20. GENERAL THEORY OF CLASSES.

*20. GENERAL THEORY OF CLASSES.

Summary of *20.

The following theory of classes, although it provides a notation to represent them, avoids the assumption that there are such things as classes. This it does by merely defining propositions in whose expression the symbols representing classes occur, just as, in *14, we defined propositions containing descriptions.

The characteristics of a class are that it consists of all the terms satisfying some propositional function, so that every propositional function determines a class, and two functions which are formally equivalent (i.e. such that whenever either is true, the other is true also) determine the same class, while conversely two functions which determine the same class are formally equivalent. When two functions are formally equivalent, we shall say that they have the same extension. The incomplete symbols which take the place of classes serve the purpose of technically providing something identical in the case of two functions having the same extension; without something to represent classes, we cannot, for example, count the combinations that can be formed out of a given set of objects.

Propositions in which a function occurs may depend, for their truth-value, upon the particular function , or they may depend only upon the extension of . In the former case, we will call the proposition concerned an intensional function of ; in the latter case, an extensional function of . Thus, for example, ( or ( is an extensional function of , because, if is formally equivalent to , i.e. if , we have ( and (. But on the other hand "I believe (" is an intensional function, because, even if , it by no means follows that I believe ( provided I believe (. The mark of an extensional function of a function is (We write "" when we wish to speak of the function itself as opposed to its argument.) The functions of functions with which mathematics is specially concerned are all extensional.

When a function of is extensional, it may be regarded as being about the class determined by , since its truth-value remains unchanged so long as the class is unchanged. Hence we require, for the theory of classes, a method of obtaining an extensional function from any given function of a function. This is effected by the following definition:

*20·01.

Here is in reality a function of , which is defined whenever is significant for predicative functions . But it is convenient to regard as though it had an argument ), which we will call "the class determined by the function ." It will be proved shortly that is always an extensional function of , and that, applying the definition of identity (*13·01) to the fictitious objects ) and ), we have This last is the distinguishing characteristic of classes, and justifies us in treating ) as the class determined by .

With regard to the scope of ), and to the order of elimination of two such expressions, we shall adopt the same conventions as were explained in *14 for (. The condition corresponding to which is always satisfied because of *12·1.

Following Peano, we shall use the notation to express " is a member of the class determined by ." We therefore introduce the following definition:

*20·02.

In this form, the definition is never used; it is introduced for the sake of the proposition which results from *20·02 and *20·01, and leads to by the help of *12·1.

We shall use small Greek letters (other than , , , , , , ) to represent classes, i.e. to stand for symbols of the form or . When a small Greek letter occurs as apparent variable, it is to be understood to stand for a symbol of the form , where is properly the apparent variable concerned. The use of single letters in place of such symbols as or is practically almost indispensable, since otherwise the notation rapidly becomes intolerably cumbrous. Thus "" will mean " is a member of the class ," and may be used wherever no special defining function of the class is in question.

The following definition defines what is meant by a class.

*20·03.

Note that the expression "" has no meaning in isolation: we have merely defined (in *20·01) certain uses of such expressions. What the above definition decides is that the symbol "Cls" may replace the symbol "," wherever the latter occurs, and that the meaning of the combination of symbols concerned is to be unchanged thereby. Thus "Cls," also, has no meaning in isolation, but merely in certain uses.

The above definition, like many future definitions, is ambiguous as to type. The Latin letter , according to our conventions, is to represent the lowest type concerned; thus is of the type next above this. It is convenient to speak of a class as being of the same type as its defining function; thus is of the type next above that of , and "Cls" is of the type next above that of . Thus the type of "Cls" is fixed relatively to the lowest type concerned; but if, in two different contexts, different types are the lowest concerned, the meaning of "Cls" will be different in these two contexts. The meaning of "Cls" only becomes definite when the lowest type concerned is specified.

Equality between classes is defined by applying *13·01, symbolically unchanged, to their defining functions, and then using *20·01.

The propositions of the present number may be divided into three sets. First, we have those that deal with the fundamental properties of classes; these end with *20·43. Then we have a set of propositions dealing with both classes and descriptions; these extend from *20·5 to *20·59 (with the exception of *20·53·54). Lastly, we have a set of propositions designed to prove that classes of classes have all the same formal properties as classes of individuals.

In the first set, the principal propositions are the following.

*20·15.

I.e. two classes are identical when, and only when, their defining functions are formally equivalent. This is the principal property of classes.

*20·31.

I.e. two classes are identical when, and only when, they have the same members.

*20·43.

This is the same proposition as *20·31, merely employing Greek letters in place of and .

*20·18.

I.e. if two classes are identical, any property of either belongs also to the other. This is the analogue of *13·12.

*20·2·21·22, which prove that identity between classes is reflexive, symmetrical and transitive.

*20·3.

I.e. a term belongs to a class when, and only when, it satisfies the defining function of the class.

In the second set of propositions (*20·3—·59), we show that, under suitable circumstances, expressions such as ( may be substituted for in *20·3 and various other propositions of the first set, and we prove a few properties of such expressions as "(," i.e. "the class which satisfies the function ." Here it is to be remembered that "" stands for "," and that "" therefore stands for "." This is, in reality, a function of , namely the extensional function associated with by means of *20·01. Thus an expression containing a variable class is always an abbreviation for an expression containing a variable function.

In the third set of propositions, we prove that variable classes satisfy all the primitive propositions assumed for variable individuals or functions, whence it follows, by merely repeating the proofs of the first set of propositions (*20·1—·43), that classes of classes have all the formal properties of classes of individuals or functions. We shall never have occasion explicitly to consider classes of functions, but classes of classes will occur constantly—for example, every cardinal number will be defined as a class of classes. Classes of relations, which will also frequently occur, will be considered in *21.

*20·01.

*20·02.

*20·03.

The three following definitions serve merely for purposes of abbreviation.

*20·04.

*20·05.

*20·06.

The following definitions merely extend to symbols representing classes the definitions which have already been given for other symbols, with the smallest possible modifications.

*20·07. (

*20·071. (

*20·072.

*20·08.

*20·081.

The propositions which follow give the most general properties of classes.

*20·1.

*20·11.

Dem.

This proves that every proposition about a class expresses an extensional property of the determining function of the class, and therefore does not depend for its truth or falsehood upon the particular function selected for determining the class, but only upon the extension of the determining function.

*20·111.

Dem.

*20·112.

Dem.

Thus the axiom of reducibility still holds for classes as arguments.

*20·12.

*20·13.

The meaning of "" is obtained by a double application of *20·01 to *13·01, remembering the convention that is to have a larger scope than because it occurs first.

Dem.

*20·14.

Dem.

This proposition is the converse of *20·13.

*20·15.

This proposition states that two functions determine the same class when, and only when, they are formally equivalent, i.e. are satisfied by the same set of values. This is the essential property of classes, and gives the justification of the definition *20·01.

*20·151.

Dem.

In virtue of this proposition, all classes can be obtained from predicative functions. This fact is especially important when classes are used as apparent variables. For in that case, according to the definitions *20·07·071, the apparent variable really involved is a predicative function. In virtue of *20·151, this places no limitation upon the classes concerned, except the limitation which inevitably results from the nature of their membership.

A class, therefore, unlike a function, has its order completely determined by the order of its possible members, i.e. of the arguments which render its defining function significant.

*20·16.

*20·17.

*20·18.

*20·19.

Dem.

*20·191.

*20·2.

Dem.

*20·21.

*20·22.

The above propositions are not immediate consequences of *13·15·16·17, for a reason analogous to that explained in the note to *14·13, namely because is not a value of , and therefore in particular "" is not a value of "."

*20·23.

*20·24.

*20·25.

Dem.

*20·3.

Dem.

This proposition shows that is a member of the class determined by when, and only when, satisfies .

*20·31.

*20·32.

*20·33.

Dem.

Here is written in place of some expression of the form . The use of the single Greek letter is more convenient whenever the determining function is irrelevant.

*20·34.

Dem.

The above proposition and *20·25 illustrate the use of Greek letters as apparent variables.

*20·35.

*20·4.

*20·41.

*20·42.

A Greek letter, such as , is merely an abbreviation for an expression of the form , thus this proposition is *20·32 repeated.

Dem.

*20·43.

The following propositions deal with cases in which both classes and descriptions occur. In such cases, we shall, in the absence of any indication to the contrary, adopt the convention that the descriptions are to have a larger scope than the classes, in applying the definitions *14·01 and *20·01.

*20·5.

Dem.

*20·51.

Dem.

*20·52.

Dem.

*20·53.

This is the analogue of *13·191.

Dem.

*20·54.

This proposition is the analogue of *13·195.

Dem.

*20·55.

Dem.

*20·56.

*20·57.

Dem.

*20·58.

Dem.

*20·59.

Dem.

In the following propositions, we shall prove that classes have all the formal properties of individuals, and have the same relations to classes of classes as individuals have to classes of individuals. It is only necessary to prove the analogues of our primitive propositions, and of our definitions in cases where their analogues are not themselves definitions. We shall take the propositions *10·1·11·12·121·122, rather than those of *9, and we shall prove the analogue of *10·01. As was pointed out in *10, we shall thus have proved everything upon which subsequent proofs depend. The analogues of *20·01·02 and of *14·01 remain definitions, but those of *10·01 and *13·01 become propositions to be proved. *9·131 must be extended by the definition: Two classes are "of the same type" when they have predicative defining functions of the same type. In addition to these, we have to prove the analogues of *10·1·11·12·121·122, *11·07 and *12·1·11. When these have been proved, the analogues of other propositions follow by merely repeating previous proofs. These analogues will, therefore, be quoted by the numbers of the original propositions whose analogues they are.

*20·6.

Dem.

This is the analogue of *10·01.

*20·61.

Dem.

This is the analogue of *10·1.

In practice we also need This is *20·17.

We need further

This is *20·41.

*20·62. When is true, whatever possible argument of the form may be, then ( is true.

This is the analogue of *10·11.

Dem.

when is true, whatever possible argument may be, then ( is true, i.e. (by *20·07), ( is true.

*20·63.

This is the analogue of *10·12.

Dem.

*20·631. If "" is significant, then if is of the same type as , "" is significant, and vice versa.

This is the analogue of *10·121.

Dem.

By *20·151, is of the form , and therefore, by *20·01, is a function of . Similarly is of the form , and is a function of . Hence by applying *10·121 to and the result follows.

*20·632. If, for some , there is a proposition , then there is a function , and vice versa.

Dem.

By the definition in *20·01, is a function of . Hence the proposition follows from *10·122.

*20·633. "Whatever possible class may be, is true whatever possible class may be" implies the corresponding statement with and interchanged except in "." (The corresponding exception is to be understood in *11·07.)

This is the analogue of *11·07, and follows at once from *11·07 because is a function of the defining functions of and .

*20·64.

Dem.

Observe that "" is merely an abbreviation for any symbol of the form . This is why nothing further is required in the above proof.

The above proposition is the analogue of *10·14. Like that proposition, it requires, for the significance of the conclusion, that and should be functions which take arguments of the same type. This is not required for the significance of the hypothesis. Hence, though the above proposition is true whenever it is significant, it is not true whenever its hypothesis is significant.

*20·7.

This is the analogue of *12·1.

*20·701.

[The proof proceeds as in *20·112, using *12·11 instead of *12·1.]

*20·702.

[Proof as in *20·701.]

*20·703.

Dem.

*20·701·702·703 give the analogues, for classes, of *12·11.

*20·71.

This is the analogue of *13·01.

This completes the proof that all propositions hitherto given apply to classes as well as to individuals. Precisely similar reasoning extends this result to classes of classes, classes of classes of classes, etc.

From the above propositions it appears that, although expressions such as have no meaning in isolation, yet those of their formal properties with which we have been hitherto concerned are the same as the corresponding properties of symbols which have a meaning in isolation. Hence nothing in the apparatus hitherto introduced requires us to determine whether a given symbol stands for a class or not, unless the symbol occurs in a way in which only a class can occur significantly. This is an important result, which enables us to give much greater generality to our propositions than would otherwise be possible.

The two following propositions (*20·8·81) are consequences of *13·3. The "type" of any object will be defined in *63 as the class of terms either identical with or not identical with . We may define the "type of the arguments to " as the class of arguments for which "" is significant, i.e. the class . Then the first of the following propositions shows that if "" is significant, the type of the arguments to is the type of ; the second proposition shows that, if "" and "" are both significant, the type of the arguments to is the same as the type of the arguments to , because each is the type of . *20·8 will be used in *63·11, which is a fundamental proposition in the theory of relative types.

*20·8.

Dem.

*20·81.

Dem.

In the third line of the above proof, the use of *10·121 depends upon the fact that the "" in both (1) and (2) must be such as to render the hypothesis significant, i.e. such as to render significant. Hence the "" in (1) and the "a" in (2) must be of the same type, by *10·121, and hence by *10·13 we can assert the product of (1) and (2), identifying the two "'s."

Since a type is the range of significance of a function, if is a function which is always true, must be a type. For if a function is always true, the arguments for which it is true are the same as the arguments for which it is significant; hence is the range of significance of , if ( holds. Thus any class is a type if (. It follows that, whatever function may be, is a type; and in particular, is a type. Since is a member of this class, this class is the type to which a belongs. In virtue of *20·8, if is significant, the type to which a belongs is the class of arguments for which is significant, i.e. . And if there is any argument a for which and are both significant, then and have the same range of significance, in virtue of *20·81.