Volume 1 / Part 2 — *50. IDENTITY AND DIVERSITY AS RELATIONS.
Reading edition
*50. IDENTITY AND DIVERSITY AS RELATIONS.
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.
*50·3.
*50·31.
Dem.
*50·32.
*50·33.
Dem.
In the above proposition (*50·33), the hypothesis is equivalent to the hypothesis that more than one object exists of the type in question. This can be proved for all except the lowest type. For the lowest type, we can only prove the existence of at least one object: this is proved in *24·52. For the next type, we can prove the existence of at least two objects, namely and ; these are distinct, by*24·1. For a class of order , we can prove the existence of objects. But for the class of individuals we cannot prove, from our primitive propositions, that there is more than one object in the universe, and therefore we cannot prove . We might, of course, have included among our primitive propositions the assumption that more than one individual exists, or some assumption from which this would follow, such as But very few of the propositions which we might wish to prove depend upon this assumption, and we have therefore excluded it. It should be observed that most philosophers, being monists, deny this assumption.
*50·34.
Dem.
*50·35.
*50·4.
Dem.
*50·41.
Dem.
*50·42.
Dem.
*50·43.
This proposition is useful in the theory of series. "" is the characteristic of an asymmetrical relation.
*50·44.
Dem.
*50·45.
*50·46.
*50·47.
Dem.
This proposition is used in the theory of series. If is a serial relation, we shall have and .
*50·5.
Dem.
*50·51.
*50·52.
Dem.
*50·53.
Dem.
*50·54.
Dem.
*50·55.
Dem.
*50·56.
Dem.
*50·57.
Dem.
*50·58.
Dem.
*50·59.
Dem.
*50·6.
Dem.
*50·61.
Dem.
*50·62.
*50·63.
*50·64.
*50·65.
*50·7.
*50·71.
*50·72.
*50·73.
*50·74.
Dem.
*50·75.
*50·76.
Dem.
*50·761.