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 February 2010
Arbitrage and Graphical Gridlock
The arbitrage-free law (or Kirchhoff’s voltage law) Recently I emailed a friend to complain when his organization used this 3 gear image as their logo. What was my complaint? Read on. The basic idea of arbitrage is to “get something for nothing” by trading commodities or currencies around some circle ending up with more than [...]
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.
Series-Parallel Duality: Part I: Combating series chauvinism
This post describes the duality between the usual (series) addition and the dual parallel addition. This duality is normally considered in electrical circuit theory and combinatorics, but it has a much wider applications. In Part I of this post, the focus is on developing series-parallel dual formulas—in contrast to the usual focus on formulas using only the series sum.
Series-Parallel Duality: Part II: Finanical arithmetic
In financial arithmetic and in the appraisal literature, it has been noticed that the basic formulas occur in pairs, one being the reciprocal of the other. This Part II of the series-parallel duality post shows that these reciprocal formulas are an example of the SP duality normally associated with electrical circuit theory.
The Math of Double-Entry Bookkeeping: Part I (scalars)
Double-entry bookkeeping illustrates one of the most astonishing examples of intellectual insulation between disciplines—the very opposite of intellectual trespassing.
The Math of Double-Entry Bookkeeping: Part II (vectors)
Although double-entry bookkeeping (DEB) has been used in the business world for 5 centuries, the mathematical formulation of the double entry method is almost completely unknown.
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.