The logical entropy formula Given a partition on a finite universe set U, the set of distinctions or dits is the set of ordered pairs of elements in distinct blocks of the partition. The logical entropy of the partition is the normalized cardinality of the dit set: . The logical entropy can be interpreted probabilistically [...]
Archives for Partition logic
From Partition Logic to Information Theory
A new logic of partitions has been developed that is dual to ordinary logic when the latter is interpreted as the logic of subsets rather than the logic of propositions. For a finite universe, the logic of subsets gave rise to finite probability theory by assigning to each subset its relative cardinality as a Laplacian probability. The analogous development for the dual logic of partitions gives rise to a notion of logical entropy that is related in a precise manner to Claude Shannon’s entropy.
The implication operation on partitions
In a 2001 commemorative volume for my mathematical mentor, Gian-Carlo Rota, three of his associates noted that “the only operations on the family of equivalence relations fully studied, understood and deployed are the binary join $latex \lor$ and meet $latex \land$ operations.” This note defines the apparently new operation of implication for partitions, an operation that was key to the development of the logic of partitions that is dual to the usual logic of subsets.
From propositional logic to subset logic to partition logic
The modern category-theoretic treatment of logic, the variables in “propositional” logic should be interpreted as subsets of some given nonempty universe set U, i.e., propositional logic is subset logic. Since partitions on a set are dual to subsets of the set, the idea arises of a dual logic of partitions.