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New Foundations for Information Theory

Logical Entropy and Shannon Entropy

  • Book
  • © 2021

Overview

Authors:
  1. David Ellerman

    1. Faculty of Social Sciences, University of Ljubljana, Ljubljana, Slovenia

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  • Introduces a new foundation for information theory, based on logical entropy & its transform into Shannon entropy
  • Offers a new maximizing logical entropy approach to the MaxEntropy method
  • Presents a new logical entropy approach to quantum information theory

Part of the book series: SpringerBriefs in Philosophy (BRIEFSPHILOSOPH)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Logical Entropy

    • David Ellerman
    Pages 1-13

  3. The Relationship Between Logical Entropy and Shannon Entropy

    • David Ellerman
    Pages 15-22

  4. The Compound Notions for Logical and Shannon Entropies

    • David Ellerman
    Pages 23-37

  5. Further Developments of Logical Entropy

    • David Ellerman
    Pages 39-72

  6. Quantum Logical Information Theory

    • David Ellerman
    Pages 73-93

  7. Conclusion

    • David Ellerman
    Pages 95-100

  8. Back Matter

    Pages 101-113
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Keywords

About this book

This monograph offers a new foundation for information theory that is based on the notion of information-as-distinctions, being directly measured by logical entropy, and on the re-quantification as Shannon entropy, which is the fundamental concept for the theory of coding and communications.
Information is based on distinctions, differences, distinguishability, and diversity. Information sets are defined that express the distinctions made by a partition, e.g., the inverse-image of a random variable so they represent the pre-probability notion of information. Then logical entropy is a probability measure on the information sets, the probability that on two independent trials, a distinction or “dit” of the partition will be obtained. 


The formula for logical entropy is a new derivation of an old formula that goes back to the early twentieth century and has been re-derived many times in different contexts. As a probability measure, all the compound notions of joint, conditional, and mutual logical entropy are immediate. The Shannon entropy (which is not defined as a measure in the sense of measure theory)  and its compound notions are then derived from a non-linear dit-to-bit transform that re-quantifies the distinctions of a random variable in terms of bits—so the Shannon entropy is the average number of binary distinctions or bits necessary to make all the distinctions of the random variable. And, using a linearization method, all the set concepts in this logical information theory naturally extend to vector spaces in general—and to Hilbert spaces in particular—for quantum logical information theory which provides the natural measure of the distinctions made in quantum measurement.


Relatively short but dense in content, this work can be a reference to researchers and graduate students doing investigations in information theory,  maximum entropy methods in physics, engineering, and statistics, and to all those with a special interest in a new approach to  quantum information theory.

Authors and Affiliations

  • Faculty of Social Sciences, University of Ljubljana, Ljubljana, Slovenia

    David Ellerman

About the author

David Ellerman is an Associate Researcher at the Faculty of Social Sciences, University of Ljubljana, Slovenia. In 2003 he retired to academia after 10 years at the World Bank where he was the economic advisor and speech-writer for the Chief Economist Joseph Stiglitz.  In his prior university teaching, Ellerman taught over a twenty-year period in the Boston area in five disciplines: economics, mathematics, computer science, operations research, and accounting. He was educated at Massachusetts Institute of Technology (USA), and at Boston University where he has two Master’s degrees, one in Philosophy and one in Economics, and a doctorate in Mathematics. Ellerman has published eight books and many articles in scholarly journals in economics, logic, mathematics, physics, philosophy, and law.



Bibliographic Information

  • Book Title: New Foundations for Information Theory

  • Book Subtitle: Logical Entropy and Shannon Entropy

  • Authors: David Ellerman

  • Series Title: SpringerBriefs in Philosophy

  • DOI: https://doi.org/10.1007/978-3-030-86552-8

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Author(s), under exclusive license to Springer Nature Switzerland AG 2021

  • Softcover ISBN: 978-3-030-86551-1Published: 31 October 2021

  • eBook ISBN: 978-3-030-86552-8Published: 30 October 2021

  • Series ISSN: 2211-4548

  • Series E-ISSN: 2211-4556

  • Edition Number: 1

  • Number of Pages: XIII, 113

  • Number of Illustrations: 24 b/w illustrations

  • Topics: Philosophy, general, Coding and Information Theory

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