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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. Synthese, 168 (1). pp. 119-149. ISSN 1573-0964

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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."


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Item Type: Published Article or Volume
Creators:
CreatorsEmailORCID
Ellerman, Daviddavid@ellerman.org
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: 23 Dec 2011 00:46
Last Modified: 23 Dec 2011 00:46
Item ID: 8967
Journal or Publication Title: Synthese
Publisher: Springer (Springer Science+Business Media B.V.)
Subjects: Specific Sciences > Computation/Information
General Issues > History of Philosophy of Science
Specific Sciences > Mathematics
Specific Sciences > Probability/Statistics
Date: May 2009
Page Range: pp. 119-149
Volume: 168
Number: 1
ISSN: 1573-0964
URI: https://philsci-archive-dev.library.pitt.edu/id/eprint/8967

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