Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/146

124 $$\scriptstyle{p.q.r.=.(p.q).r\quad\text{Df}}$$

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

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

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

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


 * 4·4.$$\scriptstyle{\vdash:.p.q\or r.\equiv:p.q.\or.p.r}$$

This is the first form of the distributive law.

Dem.


 * 4·41.$$\scriptstyle{\vdash:.p.\or.q.r:\equiv.p\or q.p\or r}$$

This is the second form of the distributive law—a form to which there is nothing analogous in ordinary algebra. By the conventions as to dots, "$\scriptstyle{p.\or.q.r}$" means "$\scriptstyle{p\or(q.r)}$."

Dem.

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

Dem.