Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/102

The above definition of "the class defined by the function $\scriptstyle {\phi {\hat {z}}}$," or rather, of any proposition in which this phrase occurs, is, in symbols, as follows:

$\scriptstyle {f\{{\hat {z}}(\phi z)\}.=:(\exists \psi ):\phi x.\equiv _{x}.\psi !x:f\{\psi !{\hat {z}}\}\quad {\text{Df.}}}$

In order to recommend this definition, we shall enumerate five requisites which a definition of classes must satisfy, and we shall then show that the above definition satisfies these five requisites. We require of classes, if they are to serve the purposes for which they are commonly employed, that they shall have certain properties, which may be enumerated as follows. (1) Every propositional function must determine a class, which may be regarded as the collection of all the arguments satisfying the function in question. This principle must hold when the function is satisfied by an infinite number of arguments as well as when it is satisfied by a finite number. It must hold also when no arguments satisfy the function; i.e. the "null-class" must be just as good a class as any other. (2) Two propositional functions which are formally equivalent, i.e. such that any argument which satisfies either satisfies the other, must determine the same class; that is to say, a class must be something wholly determined by its membership, so that e.g. the class "featherless bipeds" is identical with the class "men," and the class "even primes" is identical with the class "numbers identical with 2." (3) Conversely, two propositional functions which determine the same class must be formally equivalent; in other words, when the class is given, the membership is determinate: two different sets of objects cannot yield the same class. (4) In the same sense in which there are classes (whatever this sense may be), or in some closely analogous sense, there must also be classes of classes. Thus for example "the combinations of $\scriptstyle {n}$ things $\scriptstyle {m}$ at a time," where the $\scriptstyle {n}$ things form a given class, is a class of classes; each combination of $\scriptstyle {m}$ things is a class, and each such class is a member of the specified set of combinations, which set is therefore a class whose members are classes. Again, the class of unit classes, or of couples, is absolutely indispensable; the former is the number 1, the latter the number 2. Thus without classes of classes, arithmetic becomes impossible. (5) It must under all circumstances be meaningless to suppose a class identical with one of its own members. For if such a supposition had any meaning, "$\scriptstyle {\alpha \in \alpha }$" would be a significant propositional function^[1], and so would "$\scriptstyle {\alpha \sim \in \alpha }$." Hence, by (1) and (4), there would be a class of all classes satisfying the function "$\scriptstyle {\alpha \sim \in \alpha }$." If we call this class $\scriptstyle {\kappa }$, we shall have

$\scriptstyle {\alpha \in \kappa .\equiv _{\alpha }.\alpha \sim \in \alpha }$.

Since, by our hypothesis, "$\scriptstyle {\kappa \in \kappa }$" is supposed significant, the above equivalence, which holds with all possible values of $\scriptstyle {\alpha }$, holds with the value $\scriptstyle {\kappa }$, i.e.

$\scriptstyle {\kappa \in \kappa .\equiv .\kappa \sim \in \kappa }$.

1. As explained in Chapter I (pp. 25, 26), "$\scriptstyle {x\in \alpha }$" means "$\scriptstyle {x}$ is a member of the class $\scriptstyle {\alpha }$," or, more shortly, "$\scriptstyle {x}$ is an $\scriptstyle {\alpha }$." The definition of this expression in terms of our theory of classes will be given shortly.