*50. IDENTITY AND DIVERSITY AS RELATIONS.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
Summary of *50.
The purpose of the present number is primarily notational. For notational reasons, we must be able to express identity and diversity as relations, and not merely as propositional functions, i.e. we require a notation for and . We therefore put
In spite of the fact that diversity is merely the negation of identity, the kinds of propositions which employ diversity are quite different from the kinds that employ identity. Identity as a relation is required, to begin with, in the theory of unit classes, which is our reason for treating of it at this stage. It is next required, constantly, in the theory of mathematical induction (Part II, Section E). It is required also in showing that cardinal and ordinal similarity are reflexive. These are its principal uses.
Diversity, on the other hand, is required almost exclusively in the theory of series (Part V), and the first number in that theory will be devoted to diversity. Until that stage, diversity will seldom be referred to, with one important exception, namely in proving the associative law of multiplication in relation-arithmetic (*174).
The most important propositions on identity in the present number are the following:
50·16.
50·4.
50·5.
50·51.
50·52.
50·62.
50·63.
The most important propositions on diversity in the present number are the following:
*50·23.
*50·24.
*50·43.
*50·45.
*50·47.
It will be observed that all these propositions are concerned with or , both of which are satisfied if is a serial relation. The hypothesis or characterizes an asymmetrical relation, i.e. one which, if it holds between and , cannot hold between and .
*50·01.
*50·02.
Most of the propositions of this number are obvious, and call for no comment.
*50·1.
*50·11
*50·12.
*50·13.
*50·14.
*50·15.
*50·16.
Dem.
*50·17.
Dem.
*50·2.
Dem.
*50·21.
Dem.
*50·22.
*50·23.
*50·24.
Dem.