Algebra of logic and algebra of classes. The so-called “algebra of logic” had its origin in the fundamental work of the English mathematician and logician, George Boole (1815–1864), and is therefore usually called Boolean algebra. Although there is a very extensive literature on this subject, we shall present just enough to furnish a partial setting for the material of the next section.

Let us consider a set of “statements” which may be either true or false. For concreteness, we shall consider the class C of all books in the Library of Congress, and statements about any book x of C. As examples, we may use the following statements:

(α): x has exactly 300 pages,

(β): x was published in 1929,

(γ): x is printed in the French language,

(δ): The date of accession of x was 50 B.C.,

(ε): x has at least one page.

It will be noted that (δ) is true for no book, and (ε) for all books of C.

From a given set of statements, such as those listed above, it is possible to construct others by use of the familiar connectives “and,” “or,” and “not.” (The word ”or” is used in the ordinary sense of either … or … or both.) Thus, we may get compound statements of the following types: —(α) and (β): x has exactly 300 pages and was published in 1929; (α) or (β): x has exactly 300 pages or was published in 1929; not (α): x does not have exactly 300 pages.