Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/47

a class and $\scriptstyle {\phi {\hat {x}}}$ one of its determining functions [so that $\scriptstyle {\alpha ={\hat {z}}(\phi z)}$], it is not sufficient that $\scriptstyle {\phi \alpha }$ be a false proposition, it must be nonsense. Thus a certain classification of what appear to be objects into things of essentially different types seems to be rendered necessary. This whole question is discussed in Chapter II, on the theory of types, and the formal treatment in the systematic exposition, which forms the main body of this work, is guided by this discussion. The part of the systematic exposition which is specially concerned with the theory of classes is *20, and in this Introduction it is discussed in Chapter III. It is sufficient to note here that, in the complete treatment of *20, we have avoided the decision as to whether a class of things has in any sense an existence as one object. A decision of this question in either way is indifferent to our logic, though perhaps, if we had regarded some solution which held classes and relations to be in some real sense objects as both true and likely to be universally received, we might have simplified one or two definitions and a few preliminary propositions. Our symbols, such as "$\scriptstyle {{\hat {x}}(\phi x)}$" and $\scriptstyle {\alpha }$ and others, which represent classes and relations, are merely defined in their use, just as $\scriptstyle {\nabla ^{2}}$, standing for

$\scriptstyle {{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}}$,

has no meaning apart from a suitable function of $\scriptstyle {x}$, $\scriptstyle {y}$, $\scriptstyle {z}$ on which to operate. The result of our definitions is that the way in which we use classes corresponds in general to their use in ordinary thought and speech; and whatever may be the ultimate interpretation of the one is also the interpretation of the other. Thus in fact our classification of types in Chapter II really performs the single, though essential, service of justifying us in refraining from entering on trains of reasoning which lead to contradictory conclusions. The justification is that what seem to be propositions are really nonsense.

The definitions which occur in the theory of classes, by which the idea of a class (at least in use) is based on the other ideas assumed as primitive, cannot be understood without a fuller discussion than can be given now (cf. Chapter II of this Introduction and also *20). Accordingly, in this preliminary survey, we proceed to state the more important simple propositions which result from those definitions, leaving the reader to employ in his mind the ordinary unanalysed idea of a class of things. Our symbols in their usage conform to the ordinary usage of this idea in language. It is to be noticed that in the systematic exposition our treatment of classes and relations requires no new primitive ideas and only two new primitive propositions, namely the two forms of the "Axiom of Reducibility" (cf. next Chapter) for one and two variables respectively.

The propositional function "$\scriptstyle {x}$ is a member of the class $\scriptstyle {\alpha }$" will be expressed, following Peano, by the notation

$\scriptstyle {x\in \alpha }$.