 Volume II augments the algebra of logic for classes developed in Volume I so that it can handle existential statements. First, using modern semantics, Schröder proved that one cannot use equations to express “Some \(X\) is \(Y\)”. However, he noted that one can easily express it with a negated equation, namely \(XY \ne 0\). Volume II, a study of the calculus of classes using both equations and negated equations, attempted to cover the same topics covered in Vol. I, in particular there was considerable effort devoted to finding an Elimination Theorem. After dealing with several special cases, Schröder recommended this topic as an important research area—the quest for an Elimination Theorem would be known as the Elimination Problem.