Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/155

elementary propositions, is called an elementary function. For reasons explained in Chapter II of the Introduction, it would seem that negation and disjunction and their derivatives must have a different meaning when applied to elementary propositions from that which they have when applied to such propositions as $\scriptstyle {(x).\phi x}$ or $\scriptstyle {(\exists x).\phi x}$. If $\scriptstyle {\phi {\hat {x}}}$ is an elementary function, we will in this number call $\scriptstyle {(x).\phi x}$ and $\scriptstyle {(\exists x).\phi x}$ "first-order propositions." Then in virtue of the fact that disjunction and negation do not have the same meanings as applied to elementary or to first-order propositions, it follows that, in asserting the primitive propositions of *1, we must either confine them, in their application, to propositions of a single type, or we must regard them as the simultaneous assertion of a number of different primitive propositions, corresponding to the different meanings of "disjunction" and "negation." Likewise in regard to the primitive ideas of disjunction and negation, we must either, in the primitive propositions of *1, confine them to disjunctions and negations of elementary propositions, or we must regard them as really each multiple, so that in regard to each type of propositions we shall need a new primitive idea of negation and a new primitive idea of disjunction. In the present number, we shall show how, when the primitive ideas of negation and disjunction are restricted to elementary propositions, and the $\scriptstyle {p}$, $\scriptstyle {q}$, $\scriptstyle {r}$ of *1–*5 are therefore necessarily elementary propositions, it is possible to obtain definitions of the negation and disjunction of first-order propositions, and proofs of the analogues, for first-order propositions, of the primitive propositions *1·2–·6. (*1·1 and *1·11 have to be assumed afresh for first-order propositions, and the analogues of *1·7·71·72 require a fresh treatment.) It follows that the analogues of the propositions of *2–*5 follow by merely repeating previous proofs. It follows also that the theory of deduction can be extended from first-order propositions to such as contain two apparent variables, by merely repeating the process which extends the theory of deduction from elementary to first-order propositions. Thus by merely repeating the process set forth in the present number, propositions of any order can be reached. Hence negation and disjunction may be treated in practice as if there were no difference in these ideas as applied to different types; that is to say, when "$\scriptstyle {\sim p}$" or "$\scriptstyle {p\lor q}$" occurs, it is unnecessary in practice to know what is the type of $\scriptstyle {p}$ or $\scriptstyle {q}$, since the properties of negation and disjunction assumed in *1 (which are alone used in proving other properties) can be asserted, without formal change, of propositions of any order or, in the case of $\scriptstyle {p\lor q}$, of any two orders. The limitation, in practice, to the treatment of negation or disjunction as single ideas, the same in all types, would only arise if we ever wished to assume that there is some one function of $\scriptstyle {p}$ whose value is always $\scriptstyle {\sim p}$, whatever may be the order of $\scriptstyle {p}$, or that there is some one function of $\scriptstyle {p}$ and $\scriptstyle {q}$ whose value is always $\scriptstyle {p\lor q}$, whatever may be the orders of $\scriptstyle {p}$ and $\scriptstyle {q}$. Such an assumption is not involved so long as $\scriptstyle {p}$ (and $\scriptstyle {q}$) remain real variables,