Differences, a sequel to the former work. Analysis In 1857, Boole published the treatise "On the Comparison of Transcendent, with Certain Applications to the Theory of Definite Integrals", in which he studied the sum of residues of a rational function. Among other results, he proved what is now called Boole's identity: for any real numbers ak > 0, bk, and t > 0. Generalisations of this identity play an important role in the theory of the Hilbert transform. Binary logic In 1847, Boole published the pamphlet Mathematical Analysis of Logic. He later regarded it as a flawed exposition of his logical system and wanted An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities to be seen as the mature statement of his views. Contrary to widespread belief, Boole never intended to criticise or disagree with the main principles of Aristotle's logic. Rather he intended to systematise it, to provide it with a foundation, and to extend its range of applicability. Boole's initial involvement in logic was prompted by a current debate on quantification, between Sir William Hamilton who supported the theory of "quantification of the predicate", and Boole's supporter Augustus De Morgan who advanced a version of De Morgan duality, as it is now called. Boole's approach was ultimately much further reaching than either sides' in the controversy. It founded what was first known as the "algebra of logic" tradition. Among his many innovations is his principle of wholistic reference, which was later, and probably independently, adopted by Gottlob Frege and by logicians who subscribe to standard first-order logic. A 2003 article provides a systematic comparison and critical evaluation of Aristotelian logic and Boolean logic; it also reveals the centrality of holistic reference in Boole's philosophy of logic. 1854 definition of the universe of discourse In every discourse, whether of the mind conversing with its own thoughts, or of the individual in his intercourse with others, there is an assumed or expressed limit within which the subjects of its operation are confined. The most unfettered discourse is that in which the words we use are understood in the widest possible application, and for them, the limits of discourse are co-extensive with those of the universe itself. But more usually we confine ourselves to a less spacious field. Sometimes, in discoursing of men we imply (without expressing the limitation) that it is