Volume 1 / Part 2 — *51. UNIT CLASSES.
Reading edition
*51. UNIT CLASSES.
Summary of *51.
In this number we introduce a new descriptive function , meaning "the class of terms which are identical with ," which is the same thing as "the class whose only member is ." We are thus to have But . Hence we secure what we require by the following definition:
*51·01.
As a matter of notation, it might be thought that would do as well as , and that this definition is superfluous. But we need also the converse of this relation, and "" is not a sufficiently convenient symbol.
The propositions of this number are constantly used in what follows. It should be observed that the class whose members are and is , the class whose members are , , is , the class formed by adding to is , and the class formed by taking away from is . (If is not a member of , this is equal to .)
The distinction between and is one of the merits of Peano's symbolic logic, as well as of Frege's. On the basis of our theory of classes, the necessity for the distinction is of course obvious. But apart from this, the following consideration makes the necessity apparent. Let be a class; then the class whose only member is has only one member, namely , while may have many members. Hence the class whose only member is cannot be identical with [59].
The propositions of the present number which are most used are the following:
*51·15.
*51·16.
*51·2.
This proposition is useful because it enables us to replace membership of a class () by inclusion in the class ().
*51·211.
*51·221.
*51·222.
*51·23.
*51·4.
I.e. an existent class contained in a unit class must be identical with the unit class. From this proposition it will follow that 0 is the only cardinal which is less than 1.
*51·51.
For classes, has the same uses that ( has for functions; "" means "the only member of ." We have
*51·59.
*51·01.
*51·1.
Dem.
*51·11.
*51·12.
*51·13.
*51·131.
*51·14.
*51·141.
*51·15.
*51·16.
*51.161.
*51·17.
Dem.
The above proposition is used in the theory of selections (*83·71).
*51·2.
Dem.
The above proposition shows how to replace membership of a class by inclusion in a class; thus for example it gives:
Before Peano and Frege, the relation of membership () was regarded as merely a particular case of the relation of inclusion (). For this reason, the traditional formal logic treated such propositions as "Socrates is a man" as instances of the universal affirmative , "All is ," which is what we express by "." This involved a confusion of fundamentally different kinds of propositions, which greatly hindered the development and usefulness of symbolic logic. But by means of the above proposition (*51·2), we can always obtain a proposition stating an inclusion (namely "") which is equivalent to a given proposition stating membership of a class (namely "").
*51·21.
Dem.
*51·211.
Dem.
*51·22.
Dem.
*51·221.
Dem.
*51·222.
*51·23.
Dem.
*51·231.
Dem.
*51·232.
This proposition states that a member of must be either or , and vice versa, i.e. that is the class whose only members are and .
*51·233.
*51·234.
Dem.
*51·235.
Dem.
*51·236.
*51·237.
*51·238.
Dem.
*51·239.
Dem.
*51·24.
Dem.
*51·25.
*51·3.
*51·31.
Dem.
*51·34.
*51·35.
*51·36.
*51·36 is frequently used.
*51·37.
*51·4.
Dem.
*51·401.
Dem.
This proposition shows that unit classes are the smallest existent classes.
*51·41.
Dem.
The two following propositions are lemmas for *51·43.
*51·42.
Dem.
*51·421.
*51·43.
The following propositions are concerned with , i.e. with the relation of the only member of a unit class to that class. If is a unit class, is its only member. ( and are equal whenever either exists, and any proposition about the one is equivalent to the same proposition about the other.
*51·51.
Dem.
*51·511.
*51·52.
*51·53.
*51·54.
*51·55.
Dem.
*51·56.
Dem.
*51·57.
Dem.
*51·58.
*51·59.