Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/147

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

Dem.

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

Dem.

$$\scriptstyle{\vdash:p.\equiv.p.p\or q\quad[*3\cdot26.*2\cdot2]}$$

The following formulae are due to De Morgan, or rather, are the propositional analogues of formulae given by De Morgan for classes. The first of them, it will be observed, merely embodies our definition of the logical product.

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

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

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

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

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

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

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

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

The following formulae are obtained immediately from the above. They are important as showing how to transform implications into sums or into denials of products, and vice versa. It will be observed that the first of them merely embodies the definition *1·01.