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

Ellerman, David (2009) Counting Distinctions: On the Conceptual Foundations of Shannon's Information Theory. [Published Article]

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    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."

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    Item Type: Published Article
    Keywords: logical entropy, distinctions, Shannon entropy, logical information theory
    Subjects: Specific Sciences > Computation/Information
    General Issues > History of Philosophy of Science
    Specific Sciences > Mathematics
    Specific Sciences > Probability/Statistics
    Depositing User: David Ellerman
    Date Deposited: 22 Dec 2011 19:46
    Last Modified: 22 Dec 2011 19:46
    Item ID: 8967
    Journal or Publication Title: Synthese
    Publisher: Springer (Springer Science+Business Media B.V.)

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