  In the final chapter on logic, Chapter XV, Boole presented his analysis of the conversions and syllogisms of Aristotelian logic.   He now considered this ancient logic to be a weak, fragmented attempt at a logical system.   This neglected chapter is quite interesting because it is the only chapter where he analyzed particular propositions, making essential use of additional letters like \(v\) to encode some.   This is also the chapter where he stated (incompletely) the rules for working with some.