Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/149

Section A] *4·73. $$\scriptstyle{\vdash:.q.\supset:p.\equiv.p.q\quad[\text{Simp}.*4\cdot71]}$$

This proposition is very useful, since it shows that a true factor may be omitted from a product without altering its truth or falsehood, just as a true hypothesis may be omitted from an implication.

$$\scriptstyle{\vdash:.\sim p.\supset:q.\equiv.p\or q}$$

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

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

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

Dem.

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

Dem.

Note. The analogues, for classes, of *4·78·79 are false. Take, e.g. *4·78, and put $\scriptstyle{p=}$English people, $\scriptstyle{q=}$men, $\scriptstyle{r=}$women. Then $$\scriptstyle{p}$$ is contained in $$\scriptstyle{q}$$ or $\scriptstyle{r}$, but is not contained in $$\scriptstyle{q}$$ and is not contained in $\scriptstyle{r}$.

$$\scriptstyle{\vdash:p\supset\sim p.\equiv.\sim p}$$

$$\scriptstyle{\vdash:\sim p\supset p.\equiv.p}$$

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

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

Note. *4·82·83 may also be obtained from *4·43, of which they are virtually other forms.

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

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

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

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

$\scriptstyle{[\text{Exp}.\text{Comm}.\text{Imp}]}$


 * 4·87 embodies in one proposition the principles of exportation and importation and the commutative principle.