Categorical logic has shown that modern logic is essentially the logic of subsets (or "subobjects"). Partitions are dual to subsets so there is a dual logic of partitions where a "distinction" [an ordered pair of distinct elements (u,u′) from the universe U ] is dual to an "element". An element being in a subset is analogous to a partition π on U making a distinction, i.e., if u and u′ were in different blocks of π. Subset logic leads to finite (...) probability theory by taking the (Laplacian) probability as the normalized size of each subset-event of a finite universe. The analogous step in the logic of partitions is to assign to a partition the number of distinctions made by a partition normalized by the total number of ordered pairs |U|² from the finite universe. That yields a notion of "logical entropy" for partitions and a "logical information theory." The logical theory directly counts the (normalized) number of distinctions in a partition while Shannon's theory gives the average number of binary partitions needed to make those same distinctions. Thus the logical theory is seen as providing a conceptual underpinning for Shannon's theory based on the logical notion of "distinctions.". (shrink)

Self-taught mathematician and father of Boolean algebra, George Boole (1815-1864) published An Investigation of the Laws of Thought in 1854. In this highly original investigation of the fundamental laws of human reasoning, a sequel to ideas he had explored in earlier writings, Boole uses the symbolic language of mathematics to establish a method to examine the nature of the human mind using logic and the theory of probabilities. Boole considers language not just as a mode of expression, but as a (...) system one can use to understand the human mind. In the first 12 chapters, he sets down the rules necessary to represent logic in this unique way. Then he analyses a variety of arguments and propositions of various writers from Aristotle to Spinoza. One of history's most insightful mathematicians, Boole is compelling reading for today's student of intellectual history and the science of the mind. (shrink)

Scientific knowledge grows at a phenomenal pace--but few books have had as lasting an impact or played as important a role in our modern world as The Mathematical Theory of Communication, published originally as a paper on communication theory more than fifty years ago. Republished in book form shortly thereafter, it has since gone through four hardcover and sixteen paperback printings. It is a revolutionary work, astounding in its foresight and contemporaneity. The University of Illinois Press is pleased and honored (...) to issue this commemorative reprinting of a classic. (shrink)