*80. ELEMENTARY PROPERTIES OF SELECTIONS.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
Summary of *80.
In this number, we shall give such properties of as follow most directly from the definition, without any restrictive hypothesis as to .
If , selects one member of , whenever , as the selected referent of . For, since , we have ; and since , we have , i.e. . Calling the selected referent of , it is evident that we may replace by any other member of , and still have a member of . (This is proved in *80·4.) Thus if has any members at all, we can get as many members as there are members of by merely altering the selected referent of , leaving the other selected referents unchanged.
In the present section, we first prove various simple properties of . Most of these are almost immediate consequences of
*80·14.
The most useful of them are
*80·2.
*80·291.
*80·3.
*80·33.
We then have various propositions (*80·4—·46) concerned with when . Of these the most important are the following:
*80·41.
I.e. given a selective relation , the selected referent of (where ) may be replaced by any other term having the relation to , and we shall still have a selective relation.
*80·45.
We then have a set of propositions (*80·5—·54) connecting ( with and . These are chiefly useful as leading to the next set
(*80·6—·69), connecting with and . The most useful of these are the following:
*80·6.
*80·65.
*80·66.
We have next a set of propositions (*80·7—·78) dealing with the relations of and when (e.g.) and . These propositions are seldom used, but they would be useful in considering division.
We next have a set of propositions (*80·8—·84) dealing with the relations of and . The most useful are
*80·81.
*80·82.
Finally, we have four propositions (*80·9—·93) on and one on . The most useful of these is
*80·9.
*80·01.
*80·1.
*80·11.
*80·12.
*80·13.
*80·14.
*80·15.
*80·16.
Dem.
*80·17.
Dem.
This proposition is used in the theory of ordinal multiplication (*172·162).
*80·2.
Dem.
*80·21.