 Burris and Sankappanavar (2013) use the fact that  Boole’s model, a partial-algebra, is isomorphic to the  restriction of the operations of addition, multiplication and subtraction in the ring \(Z^U\) to the idempotent elements of the ring. Here \(Z\) is the ring of integers, and \(U\) is the universe of discourse.  From this one can deduce that any Horn sentence which holds in \(Z\) when the variables are restricted to 0 and 1 will hold in \(Z^U\) when the variables are restricted to idempotent elements, and thus will hold in Boole’s partial-algebra.  This gives an expanded version of Boole’s Rule of  0 and 1, and since his main results (Expansion, Reduction, Elimination and Solution) can be expressed by such Horn sentences, one has a quick proof that they are indeed valid.  