Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/156

since, in that case, there is no need to give the same meaning to negation and disjunction for different values of $\scriptstyle {p}$ (and $\scriptstyle {q}$), when these different values are of different types. But if $\scriptstyle {p}$ (or $\scriptstyle {q}$) is going to be turned into an apparent variable, then, since our two primitive ideas $\scriptstyle {(x).\phi x}$ and $\scriptstyle {(\exists x).\phi x}$ both demand some definite function $\scriptstyle {\phi }$, and restrict the apparent variable to possible arguments for $\scriptstyle {\phi }$, it follows that negation and disjunction must, wherever they occur in the expression in which $\scriptstyle {p}$ (or $\scriptstyle {q}$) is an apparent variable, be restricted to the kind of negation or disjunction appropriate to a given type or pair of types. Thus, to take an instance, if we assert the law of excluded middle in the form

“$\scriptstyle {\vdash .p\lor \sim p}$”

there is no need to place any restriction upon $\scriptstyle {p}$: we may give to $\scriptstyle {p}$ a value of any order, and then give to the negation and disjunction involved those meanings which are appropriate to that order. But if we assert

“$\scriptstyle {\vdash .(p).p\lor \sim p}$”

it is necessary, if our symbol is to be significant, that “$\scriptstyle {p\lor \sim p}$” should be the value, for the argument $\scriptstyle {p}$, of a function $\scriptstyle {\phi p}$; and this is only possible if the negation and disjunction involved have meanings fixed in advance, and if, therefore, $\scriptstyle {p}$ is limited to one type. Thus the assertion of the law of excluded middle in the form involving a real variable is more general than in the form involving an apparent variable. Similar remarks apply generally where the variable is the argument to a typically ambiguous function.

In what follows the single letters $\scriptstyle {p}$ and $\scriptstyle {q}$ will represent elementary propositions, and so will “$\scriptstyle {\phi x}$,” “$\scriptstyle {\psi x}$,” etc. We shall show how, assuming the primitive ideas and propositions of *1 as applied to elementary propositions, we can define and prove analogous ideas and propositions as applied to propositions of the forms $\scriptstyle {(x).\phi x}$ and $\scriptstyle {(\exists x).\phi x}$. By mere repetition of the analogous process, it will then follow that analogous ideas and propositions can be defined and proved for propositions of any order; whence, further, it follows that, in all that concerns disjunction and negation, so long as propositions do not appear as apparent variables, we may wholly ignore the distinction between different types of propositions and between different meanings of negation and disjunction. Since we never have occasion, in practice, to consider propositions as apparent variables, it follows that the hierarchy of propositions (as opposed to the hierarchy of functions) will never be relevant in practice after the present number.

The purpose and interest of the present number are purely philosophical, namely to show how, by means of certain primitive propositions, we can deduce the theory of deduction for propositions containing apparent variables from the theory of deduction for elementary propositions. From the purely technical point of view, the distinction between elementary and other propositions may be ignored, so long as propositions do not appear as apparent variables; we may then regard the primitive propositions of *1 as applying