# Counting distinctions: on the conceptual foundations of Shannon’s information theory

*Synthese*168 (1):119-149 (2009)

**Abstract**

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."

**Keywords**

**Categories**

(categorize this paper)

**PhilPapers/Archive ID**

ELLCDO-2

**Upload history**

**Added to PP index**

2009-01-28

**Total views**

400 ( #17,260 of 65,558 )

**Recent downloads (6 months)**

10 ( #53,807 of 65,558 )

How can I increase my downloads?

**Downloads since first upload**

*This graph includes both downloads from PhilArchive and clicks on external links on PhilPapers.*