*4. EQUIVALENCE AND FORMAL RULES.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
*4·51.
*4·52.
*4·53.
*4·54.
*4·55.
*4·56.
*4·57.
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.
*4·6.
*4·61.
*4·62.
*4·63.
*4·64.
*4·65.
*4·66.
*4·67.
*4·7.
Dem.
*4·71.
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.
*4·72.
Dem.
*4·73.
This proposition is very useful, since it shows that a true factor may be omitted from a product without altering its truth or falsehood, just as a true hypothesis may be omitted from an implication.
*4·74.
*4·76.
*4·77.
*4·78.
Dem.
*4·79.
Dem.
Note. The analogues, for classes, of *4·78·79 are false. Take, e.g. *4·78, and put = English people, = men, = women. Then is contained in or , but is not contained in and is not contained in .
*4·8.
*4·81.
*4·82.
*4·83.
Note. *4·82·83 may also be obtained from *4·43, of which they are virtually other forms.
*4·84.
*4·85.