   Applying weak elimination to the primary equational expressions was not  sufficient to derive all of the valid syllogisms. For example, in the cases  where the premises had primary expressions \(ay = 0\) and \(cy = 0\), this elimination gave \(0 = 0\), even when there was a non-trivial  conclusion.  Boole introduced the alternative equational expressions (see MAL, p. 32) of categorical propositions to be able to derive all of the valid syllogisms. 