Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/148

126 $$\scriptstyle{\vdash:\qquad~p\supset q.\equiv.\sim p\or q}$$

$$\scriptstyle{\vdash:\quad\,\sim(p\supset q).\equiv.p.\sim q}$$

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

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

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

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

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

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

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

Dem.


 * 4·71.$$\scriptstyle{\vdash:.p\supset q.\equiv:p.\equiv.p.q}$$

Dem.

The above proposition is constantly used. It enables us to transform every implication into an equivalence, which is an advantage if we wish to assimilate symbolic logic as far as possible to ordinary algebra. But when symbolic logic is regarded as an instrument of proof, we need implications, and it is usually inconvenient to substitute equivalences. Similar remarks apply to the following proposition.

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

Dem.