Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/152

130 $$\scriptstyle{\vdash:p\equiv q.\equiv.\sim(p\equiv\sim q)}$$

$$\scriptstyle{\vdash.\sim(p\equiv\sim p)}$$

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

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

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

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

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

From *5·25 it appears that we might have taken implication, instead of disjunction, as a primitive idea, and have defined "$\scriptstyle{p\or q}$" as meaning "$\scriptstyle{p\supset q.\supset.q}$." This course, however, requires more primitive propositions than are required by the method we have adopted.

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

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

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

This proposition is constantly required in subsequent proofs.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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