 The next step in Boole’s system was to translate the four kinds of categorical propositions into equations, for example “All \(X\) is \(Y\)” becomes \(X = XY\), and “Some \(X\) is \(Y\)” becomes \(V = XY\), where \(V\) is a new symbol. To eliminate the middle term in a syllogism Boole borrowed an elimination theorem from ordinary algebra, but it was too weak for his algebra of logic. This would be remedied in his 1854 book. Boole found that he could not always derive the desired conclusions with the above translation of particular propositions (i.e., those with existential import), so he added the variants \(X = VY\), \(Y = VX\), and \(VX = VY\) (see the entry on  Boole).