Abstracts from papers

connected with the number two. It contains two unique elements; it has a law of double negation; and the number of its elements, in finite cases, is a power of 2. This presents a generalization of these characteristics, yielding a calculus containing n unique elements, a law of n-tuple negation, and containing, in finite cases, a number of elements which is a power of n. When n = 2, Boolean algebra results; but for each n^2, there is a new algebra. All these algebras may be developed in terms of two binary opera tors, " + " and "X". Any law of Boolean algebra composed only of the symbols " + ", "X", " = ", and real variables holds for all of them. They are capable of a spatial interpretation, their elements being represented as sectors or com binations of sectors of concentric circles, the number of concentric circles being determined by n. In every algebra, a+b may be represented by the area included either in a or b and aXb may be represented by their common area. Each algebra contains a unary function, definable in terms of "-J-" and "X", corresponding to the negation of Boolean algebra, and it is chiefly in respect to laws involving this function that the algebras differ.