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 p on U making a distinction, i.e., if u and
u' were in different blocks of p. 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 |UxU| 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." (forthcoming in
Synthese
