Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/53

 There are no classes which contain objects of more than one type. Accordingly there is a universal class and a null-class proper to each type of object. But these symbols need not be distinguished, since it will be found that there is no possibility of confusion. Similar remarks apply to relations.

Descriptions. By a "description" we mean a phrase of the form "the so-and-so" or of some equivalent form. For the present, we confine our attention to the in the singular. We shall use this word strictly, so as to imply uniqueness; e.g. we should not say "$\scriptstyle{A}$ is the son of $\scriptstyle{B}$" if $$\scriptstyle{B}$$ had other sons besides $\scriptstyle{A}$. Thus a description of the form "the so-and-so" will only have an application in the event of there being one so-and-so and no more. Hence a description requires some propositional function $$\scriptstyle{\phi\hat x}$$ which is satisfied by one value of $$\scriptstyle{x}$$ and by no other values; then "the $$\scriptstyle{x}$$ which satisfies $\scriptstyle{\phi\hat x}$" is a description which definitely describes a certain object, though we may not know what object it describes. For example, if $$\scriptstyle{y}$$ is a man, "$\scriptstyle{x}$ is the father of $\scriptstyle{y}$" must be true for one, and only one, value of $\scriptstyle{x}$. Hence "the father of $\scriptstyle{y}$" is a description of a certain man, though we may not know what man it describes. A phrase containing "the" always presupposes some initial propositional function not containing "the"; thus instead of "$\scriptstyle{x}$ is the father of $\scriptstyle{y}$" we ought to take as our initial function "$\scriptstyle{x}$ begot $\scriptstyle{y}$"; then "the father of $\scriptstyle{y}$" means the one value of $$\scriptstyle{x}$$ which satisfies this propositional function.

If $$\scriptstyle{\phi\hat x}$$ is a propositional function, the symbol "$\scriptstyle{(}$$\scriptstyle{x)(\phi x)}$" is used in our symbolism in such a way that it can always be read as "the $$\scriptstyle{x}$$ which satisfies $\scriptstyle{\phi\hat x}$." But we do not define "$\scriptstyle{(}$$\scriptstyle{x)(\phi x)}$" as standing for "the $$\scriptstyle{x}$$ which satisfies $\scriptstyle{\phi\hat x}$," thus treating this last phrase as embodying a primitive idea. Every use of "$\scriptstyle{(}$$\scriptstyle{x)(\phi x)}$," where it apparently occurs as a constituent of a proposition in the place of an object, is defined in terms of the primitive ideas already on hand. An example of this definition in use is given by the proposition "$\scriptstyle{\mathbf{E!}(}$$\scriptstyle{x)(\phi x)}$" which is considered immediately. The whole subject is treated more fully in Chapter III.

The symbol should be compared and contrasted with "$\scriptstyle{\hat x(\phi x)}$" which in use can always be read as "the $\scriptstyle{x}$'s which satisfy $\scriptstyle{\phi\hat x}$." Both symbols are incomplete symbols defined only in use, and as such are discussed in Chapter III. The symbol "$\scriptstyle{\hat x(\phi x)}$" always has an application, namely to the class determined by $\scriptstyle{\phi x}$; but "$\scriptstyle{(}$$\scriptstyle{x)(\phi x)}$"" only has an application when $$\scriptstyle{\phi\hat x}$$ is only satisfied by one value of $\scriptstyle{x}$, neither more nor less. It should also be observed that the meaning given to the symbol by the definition, given immediately below, of $\scriptstyle{\mathbf{E!}(}$$\scriptstyle{x)(\phi x)}$ does not presuppose that we know the meaning of "one." This is also characteristic of the definition of any other use of "$\scriptstyle{(}$$\scriptstyle{x)(\phi x)}$.