 Since symbolical algebra was just the syntactic side of ordinary algebra, Boole needed ways to interpret the usual operations and constants of algebra to create his algebra of logic for classes. Multiplication was interpreted as intersection, leading to his one new law, the idempotent law \(XX = X\) for multiplication, rediscovering a logical law already formulated by Leibniz. Addition was defined as union, provided one was dealing with disjoint classes; and subtraction as class difference, provided one was subtracting a subclass from a class. In other cases, the addition and subtraction operations were simply undefined, or as Boole wrote, uninterpretable. The usual laws of arithmetic told Boole that 1 must be the universe and \(1 - X\) must be the complement of \(X\).