Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/137

SECTION A] of the type of the argument $$\scriptstyle{x}$$ in $\scriptstyle{\chi(x,y,z,\ldots)}$, and therefore both $$\scriptstyle{\phi x}$$ and $$\scriptstyle{\psi x}$$ take arguments of the same type. Hence, in such a case, if both $$\scriptstyle{\phi x}$$ and $$\scriptstyle{\psi x}$$ can be asserted, so can $\scriptstyle{\phi x.\psi x}$.

As an example of the use of this proposition, take the proof of *3·47. We there prove and what we wish to prove is  which is *3·47. Now in (1) and (2), $\scriptstyle{p}$, $\scriptstyle{q}$, $\scriptstyle{r}$, $$\scriptstyle{s}$$ are elementary propositions (as everywhere in Section A); hence by *1·7·71, applied repeatedly, "$\scriptstyle{p\supset r.q\supset r.\supset:p.q.\supset.q.r}$" and "$\scriptstyle{p\supset r.q\supset s.\supset:q.r.\supset.r.s}$" are elementary propositional functions. Hence by *3·03, we have whence the result follows by *3·43 and *3·33.

The principal propositions of the present number are the following:

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

I.e. "$\scriptstyle{p}$ implies that $$\scriptstyle{q}$$ implies $\scriptstyle{p.q}$," i.e. if each of two propositions is true, so is their logical product.

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

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

I.e. if the logical product of two propositions is true, then each of the two propositions severally is true.

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

I.e. if $$\scriptstyle{p}$$ and $$\scriptstyle{q}$$ jointly imply $\scriptstyle{r}$, then $$\scriptstyle{p}$$ implies that $$\scriptstyle{q}$$ implies $\scriptstyle{r}$. This principle (following Peano) will be called "exportation," because $$\scriptstyle{q}$$ is "exported" from the hypothesis. It will be referred to as "Exp."

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

This is the correlative of the above, and will be called (following Peano) "importation" (referred to as "Imp").

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

I.e. "if $$\scriptstyle{p}$$ is true, and $$\scriptstyle{q}$$ follows from it, then $$\scriptstyle{q}$$ is true." This will be called the "principle of assertion" (referred to as "Ass"). It differs from *1·1 by the fact that it does not apply only when $$\scriptstyle{p}$$ really is true, but requires merely the hypothesis that $$\scriptstyle{p}$$ is true.

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

I.e. if a proposition implies each of two propositions, then it implies their logical product. This is called by Peano the "principle of composition." It will be referred to as "Comp."