Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/144

122 $$\scriptstyle{\vdash:.p.q\or r.\equiv:p.q.\or.p.r}$$

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

The second of these forms has no analogue in ordinary algebra.

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

I.e. $$\scriptstyle{p}$$ implies $$\scriptstyle{q}$$ when, and only when, $$\scriptstyle{p}$$ is equivalent to $\scriptstyle{p.q}$. This proposition is used constantly; it enables us to replace any implication by an equivalence.

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

I.e. a true factor may be dropped from or added to a proposition without altering the truth-value of the proposition.


 * 4·01.$$\scriptstyle{p\equiv q.=.p\supset q.q\supset p\quad\text{Df}}$$

$$\scriptstyle{p\equiv q\equiv r.=.p\equiv q.q\equiv r\quad\text{Df}}$$

This definition serves merely to provide a convenient abbreviation.


 * 4·1.$$\scriptstyle{\vdash:p\supset q.\equiv.\sim q\supset\sim p}$$


 * 4·11.$$\scriptstyle{\vdash:p\equiv q.\equiv.\sim p\equiv\sim q}$$

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


 * 4·13.$$\scriptstyle{\vdash.p\equiv\sim(\sim p)}$$

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

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


 * 4·2.$$\scriptstyle{\vdash.p\equiv p}$$


 * 4·21.$$\scriptstyle{\vdash:p\equiv q.\equiv.q\equiv p}$$


 * 4·22.$$\scriptstyle{\vdash:p\equiv q.q\equiv r.\supset.p\equiv r}$$

Dem.