From Rota to the Logic of Partitions and the Notion of Logical Entropy
Abstract
This paper traces an intellectual journey that starts with some unfinished work of Gian-Carlo Rota on making a logic of equivalence relations or partitions. Rota understood the category-theoretic duality between subsets and partitions which implied there should be a logic of partitions dual to the usual Boolean logic of subsets. Rota also conjectured that subsets are to probability as partitions are to information. And just as probability starts quantitatively with the size of a subset, so he saw that information should start with some notion of size of a partition. In the duality between subsets and partitions, the elements of a subset are dual to the distinctions (or dits) of a partition, i.e., the ordered pairs of elements in different blocks of the partition. Since
logical probability starts with the normalized number of elements in a subset, the corresponding logical notion of information is the normalized number of distinctions in a partition so that is the definition of logical entropy. Finally, logical entropy is contrasted with the usual Shannon notion of entropy.
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DOI: http://dx.doi.org/10.23756/sp.v14i1.1738
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