   Syllogistic reasoning is just an exercise in elimination, namely the middle term is eliminated from the premises to give the conclusion. Elimination is a standard topic in the theory of equations, and Boole borrowed a simple elimination result regarding two equations to use in his algebra of logic—if the premises of a syllogism involved the classes \(X, Y\), and \(Z\), and one wanted to eliminate the middle term \(Y\), then Boole put the equations for the two premises in the form