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

Authors
David Ellerman
University of Ljubljana
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 Philosophy   Metaphysics   Philosophy of Language   Logic   Epistemology   Computer Science, general   Philosophy of Science
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DOI 10.1007/s11229-008-9333-7
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References found in this work BETA

A Mathematical Theory of Communication.Claude E. Shannon - 1948 - Bell System Technical Journal 27:379–423.
Lattice Theory.Garrett Birkhoff - 1940 - Journal of Symbolic Logic 5 (4):155-157.
Lattice Theory.Garrett Birkhoff - 1950 - Journal of Symbolic Logic 15 (1):59-60.

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Citations of this work BETA

An Introduction to Partition Logic.David Ellerman - 2014 - Logic Journal of the IGPL 22 (1):94-125.
An Introduction to Logical Entropy and its Relation to Shannon Entropy.David Ellerman - 2013 - International Journal of Semantic Computing 7 (2):121-145.

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