 The development of the algebra of logic in the Laws of Thought proceeded much as in his 1847 book, with minor changes to his translation scheme, and with the selection operators replaced by classes. There is a new and very important theorem (correcting the one he had used in 1847), the Elimination Theorem, which says the following: given an equation \(F(x,y, z, \ldots) = 0\) in the class symbols \(x, y, z\), etc., the most general conclusion that follows from eliminating certain of the class symbols is obtained by (1) substituting 0s and 1s into \(F(x, y, z, \ldots)\) for the symbols to be eliminated, in all possible ways, then (2) multiplying these various substitution instances together and setting the product equal to 0. Thus eliminating \(y\) and \(z\) from \(F(x, y, z) = 0\) gives \(F(x, 0, 0)F(x, 0, 1)F(x, 1, 0)F(x, 1, 1) = 0\). This theorem also played an important role in Boole’s interpretation of Aristotle’s syllogistic.