Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/46

 used in the ordinary way express some identity, and thus use the sign of equality in the above sense.

If $$\scriptstyle{x}$$ and $$\scriptstyle{y}$$ are identical, either can replace the other in any proposition without altering the truth-value of the proposition; thus we have This is a fundamental property of identity, from which the remaining properties mostly follow.

It might be thought that identity would not have much importance, since it can only hold between $$\scriptstyle{x}$$ and $$\scriptstyle{y}$$ if $$\scriptstyle{x}$$ and $$\scriptstyle{y}$$ are different symbols for the same object. This view, however, does not apply to what we shall call "descriptive phrases," i.e. "the so-and-so." It is in regard to such phrases that identity is important, as we shall shortly explain. A proposition such as "Scott was the author of Waverley" expresses an identity in which there is a descriptive phrase (namely "the author of Waverley"); this illustrates how, in such cases, the assertion of identity may be important. It is essentially the same case when the newspapers say "the identity of the criminal has not transpired." In such a case, the criminal is known by a descriptive phrase, namely "the man who did the deed," and we wish to find an $$\scriptstyle{x}$$ of whom it is true that "$\scriptstyle{x=}$the man who did the deed." When such an $$\scriptstyle{x}$$ has been found, the identity of the criminal has transpired.

Classes and relations. A class (which is the same as a manifold or aggregate) is all the objects satisfying some propositional function. If $$\scriptstyle{\alpha}$$ is the class composed of the objects satisfying $\scriptstyle{\phi\hat x}$, we shall say that $$\scriptstyle{\alpha}$$ is the class determined by $\scriptstyle{\phi\hat x}$. Every propositional function thus determines a class, though if the propositional function is one which is always false, the class will be null, i.e. will have no members. The class determined by the function $$\scriptstyle{\phi\hat z}$$ will be represented by $\scriptstyle{\hat z~(\phi z)}$. Thus for example if $$\scriptstyle{\phi x}$$ is an equation, $$\scriptstyle{\hat z~(\phi z)}$$ will be the class of its roots; if $$\scriptstyle{\phi x}$$ is "$\scriptstyle{x}$ has two legs and no feathers," $$\scriptstyle{\hat z~(\phi z)}$$ will be the class of men; if $$\scriptstyle{\phi x}$$ is "$\scriptstyle{0<x< 1}$," $$\scriptstyle{\hat z~(\phi z)}$$ will be the class of proper fractions, and so on.

It is obvious that the same class of objects will have many determining functions. When it is not necessary to specify a determining function of a class, the class may be conveniently represented by a single Greek letter. Thus Greek letters, other than those to which some constant meaning is assigned, will be exclusively used for classes.

There are two kinds of difficulties which arise in formal logic; one kind arises in connection with classes and relations and the other in connection with descriptive functions. The point of the difficulty for classes and relations, so far as it concerns classes, is that a class cannot be an object suitable as an argument to any of its determining functions. If $$\scriptstyle{\alpha}$$ represents