Counting Distinctions: On the Conceptual Foundations of Shannon's Information Theory
 Citation data:

Synthese, ISSN: 15730964, Vol: 168, Issue: 1, Page: 119149
 Publication Year:
 2009
article description
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 subsetevent 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 U² 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."