Principia mathematica, vol. 1 (of 3)

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:

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 .

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.

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.

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 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.