Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

*22. CALCULUS OF CLASSES.

Summary of *22.
In this number we reach what was historically the starting-point of
symbolic logic. The Greek letters used (except f, r, X, 0) are always to
stand for expressions of the form 2 (! x), or, where the Greek letters are
not apparent variables,.5 (Px). 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:
*2201. aC,/.=:xEa. )x.xeI Df
This defines " the class a is contained in the class A," or " all a's are i's."
*22'02. anr3=a (xea.xe/3) Df
This defines the logical product or common part of two classes a and,.
*22-03. avfu3=(xea.v.xeB) Df
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. -a = S (x~ e a) Df
This defines the negation of a class. It is read "not-a." It does not
contain every object x concerning which "x a" is not true, but only those
objects concerning which "x a" is false; i.e. it excludes those objects for
which " x a " is meaningless. Thus it consists of all objects, of the type next
below a, which are not members of a; but it does not contain objects of any
other type but this.
*22'05. a-/3=an-,/ Df
This definition gives an abbreviation which is often convenient.
The postulates required for the algebra of logic have been enumerated by
Huntington*. In our notation, they are as follows.
We assume a class K, with two rules of combination, namely v and n;
and we then require the following ten postulates:
* Trals. Amer. Math. Soc. Vol. 5, July 1904, p. 292.