*22. CALCULUS OF CLASSES.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
Summary of *22.
In this number we reach what was historically the starting-point of symbolic logic. The Greek letters used (except , , , ) are always to stand for expressions of the form , or, where the Greek letters are not apparent variables, . The small Latin letters may either be such as have a meaning in isolation, or may represent classes or relations; this is possible in virtue of the notes at the ends of *20 and *21. We put:
*22·01.
This defines "the class is contained in the class ," or "all 's are 's."
*22·02.
This defines the logical product or common part of two classes and .
*22·03.
This defines the logical sum of two classes; it is the class consisting of all the members of one together with all the members of the other.
*22·04.
This defines the negation of a class. It is read "not-." It does not contain every object concerning which "" is not true, but only those objects concerning which "" is false; i.e. it excludes those objects for which "" is meaningless. Thus it consists of all objects, of the type next below , which are not members of ; but it does not contain objects of any other type but this.
*22·05.
This definition gives an abbreviation which is often convenient.
The postulates required for the algebra of logic have been enumerated by Huntington[54]. In our notation, they are as follows.
We assume a class , with two rules of combination, namely and ; and we then require the following ten postulates:
I a. is in the class whenever and are in the class.
I b. is in the class whenever and are in the class.
II a. There is an element such that for every element .
II b. There is an element such that for every element .
III a. whenever , , and are in the class.
III b. whenever , , and are in the class.
IV a. whenever , , , , , , , and are in the class.
IV b. whenever , , , , , , , and ( are in the class.
V. If the elements and in postulates II a and II b exist and are unique, then for every element there is an element such that and .
VI. There are at least two elements, and , in the class, such that .
The form of the above postulates is such that they are mutually independent, i.e. any nine of them are satisfied by interpretations of the symbols which do not satisfy the remaining one.
For our purposes, "" must be replaced by "." and will be the null-class and the universal class, which are defined in *24. Then the above ten postulates are proved below, as follows:
I a. in *22·37, namely ""
I b. in *22·36, namely ""
II a. in *24·24, namely ""
II b. in *24·26, namely ""
III a. in *22·57, namely ""
III b. in *22·51, namely ""
IV a. in *22·69, namely ""
IV b. in *22·68, namely ""
V. in *24·21·22, namely "" and ""
VI. in *24·1, namely ""
Hence, assuming Huntington's analysis of the postulates for the formal algebra of logic, the propositions proved in what follows suffice to establish that this algebra holds for classes. The corresponding propositions of *23 and *25 prove that it holds for relations, substituting , , , , for , , , , .
The principal propositions of the present number are the following:
(1) Those embodying the formal rules:
*22·51.