Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/122

primitive proposition *1·11, since it states a practically important consequence of this fact, is called the "axiom of identification of type."

Another consequence of the principle that, if there is an argument $\scriptstyle {a}$ for which both $\scriptstyle {\phi a}$ and $\scriptstyle {\psi a}$ are significant, then $\scriptstyle {\phi x}$ is significant whenever $\scriptstyle {\psi x}$ is significant, and vice versa, will be given in the "axiom of identification of real variables," introduced in *1·72. These two propositions, *1·11 and *1·72, give what is symbolically essential to the conduct of demonstrations in accordance with the theory of types.

The above proposition *1·11 is used in every inference from one asserted propositional function to another. We will illustrate the use of this proposition by setting forth at length the way in which it is first used, in the proof of *2·06. That proposition is

"$\scriptstyle {\vdash :.p\supset q.\supset :q\supset r.\supset .p\supset r.}$"

We have already proved, in *2·05, the proposition

$\scriptstyle {\vdash :.q\supset r.\supset :p\supset q.\supset .p\supset r.}$

It is obvious that *2·06 results from *2·05 by means of *2·04, which is

$\scriptstyle {\vdash :.p.\supset .q\supset r:\supset :q.\supset .p\supset r.}$

For if, in this proposition, we replace $\scriptstyle {p}$ by $\scriptstyle {q\supset r}$, $\scriptstyle {q}$ by $\scriptstyle {p\supset q}$, and $\scriptstyle {r}$ by $\scriptstyle {p\supset r}$, we obtain, as an instance of *2·04, the proposition

$\scriptstyle {\vdash ::q\supset r.\supset :p\supset q.\supset .p\supset r:.\supset :.p\supset q.\supset :q\supset r.\supset .p\supset r\qquad {\text{(1)}}}$,

and here the hypothesis is asserted by *2·05. Thus our primitive proposition *1·11 enables us to assert the conclusion.

*1·2. $\scriptstyle {\vdash :p\lor p.\supset .p\quad {\text{Pp.}}}$

This proposition states: "If either $\scriptstyle {p}$ is true or $\scriptstyle {p}$ is true, then $\scriptstyle {p}$ is true." It is called the "principle of tautology," and will be quoted by the abbreviated title of "Taut." It is convenient, for purposes of reference, to give names to a few of the more important propositions; in general, propositions will be referred to by their numbers.

*1·3. $\scriptstyle {\vdash :q.\supset .p\lor q\quad {\text{Pp.}}}$

This principle states: "If $\scriptstyle {q}$ is true, then '$\scriptstyle {p}$ or $\scriptstyle {q}$' is true." Thus e.g. if $\scriptstyle {q}$ is "to-day is Wednesday" and $\scriptstyle {p}$ is "to-day is Tuesday," the principle states: "If to-day is Wednesday, then to-day is either Tuesday or Wednesday." It is called the "principle of addition," because it states that if a proposition is true, any alternative may be added without making it false. The principle will be referred to as "Add."

*1·4. $\scriptstyle {\vdash :p\lor q.\supset .q\lor p\quad {\text{Pp.}}}$

This principle sates that "$\scriptstyle {p}$ or $\scriptstyle {q}$" implies "$\scriptstyle {q}$ or $\scriptstyle {p}$." It states the permutative law for logical addition of propositions, and will be called the "principle of permutation." It will be referred to as "Perm."