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David Ellerman [79]David P. Ellerman [11]
  1. A New Logic, a New Information Measure, and a New Information-Based Approach to Interpreting Quantum Mechanics.David Ellerman - 2024 - Entropy Special Issue: Information-Theoretic Concepts in Physics 26 (2).
    The new logic of partitions is dual to the usual Boolean logic of subsets (usually presented only in the special case of the logic of propositions) in the sense that partitions and subsets are category-theoretic duals. The new information measure of logical entropy is the normalized quantitative version of partitions. The new approach to interpreting quantum mechanics (QM) is showing that the mathematics (not the physics) of QM is the linearized Hilbert space version of the mathematics of partitions. Or, putting (...)
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  2. An Introduction to Partition Logic.David Ellerman - 2014 - Logic Journal of the IGPL 22 (1):94-125.
    Classical logic is usually interpreted as the logic of propositions. But from Boole's original development up to modern categorical logic, there has always been the alternative interpretation of classical logic as the logic of subsets of any given (nonempty) universe set. Partitions on a universe set are dual to subsets of a universe set in the sense of the reverse-the-arrows category-theoretic duality--which is reflected in the duality between quotient objects and subobjects throughout algebra. Hence the idea arises of a dual (...)
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  3. Follow the Math!: The Mathematics of Quantum Mechanics as the Mathematics of Set Partitions Linearized to (Hilbert) Vector Spaces.David Ellerman - 2022 - Foundations of Physics 52 (5):1-40.
    The purpose of this paper is to show that the mathematics of quantum mechanics is the mathematics of set partitions linearized to vector spaces, particularly in Hilbert spaces. That is, the math of QM is the Hilbert space version of the math to describe objective indefiniteness that at the set level is the math of partitions. The key analytical concepts are definiteness versus indefiniteness, distinctions versus indistinctions, and distinguishability versus indistinguishability. The key machinery to go from indefinite to more definite (...)
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  4. Property and Contract in Economics: The Case for Economic Democracy.David P. Ellerman - 1992 - Blackwell.
    From a pre-publication review by the late Austrian economist, Don Lavoie, of George Mason University: -/- "The book's radical re-interpretation of property and contract is, I think, among the most powerful critiques of mainstream economics ever developed. It undermines the neoclassical way of thinking about property by articulating a theory of inalienable rights, and constructs out of this perspective a "labor theory of property" which is as different from Marx's labor theory of value as it is from neoclassicism. It traces (...)
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  5. Counting distinctions: on the conceptual foundations of Shannon’s information theory.David Ellerman - 2009 - Synthese 168 (1):119-149.
    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 (...)
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  6.  37
    New Foundations for Information Theory: Logical Entropy and Shannon Entropy.David Ellerman - 2021 - Springer Verlag.
    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 (...)
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  7. Inalienable rights: A litmus test for liberal theories of justice.David Ellerman - 2010 - Law and Philosophy 29 (5):571-599.
    Liberal-contractarian philosophies of justice see the unjust systems of slavery and autocracy in the past as being based on coercion—whereas the social order in modern democratic market societies is based on consent and contract. However, the ‘best’ case for slavery and autocracy in the past were consent-based contractarian arguments. Hence, our first task is to recover those ‘forgotten’ apologia for slavery and autocracy. To counter those consent-based arguments, the historical anti-slavery and democratic movements developed a theory of inalienable rights. Our (...)
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  8.  25
    Is “capitalism” a misnomer?: on Marx’s “capitalism” and Knight’s “civilization”.David Ellerman - 2024 - European Journal of the History of Economic Thought 1.
    The name “capitalism” derives from Marx’s false analogy between medieval land ownership and the “ownership of the means of production.” However, unlike medieval land, capital goods can be rented out, e.g., by Frank Knight’s entrepreneur, and then the capital owner does not hold those management or product rights. What then is the characteristic institution in our civilization? It is the voluntary renting of workers. What then is the relationship between Classical Liberalism, the dominant philosophy behind Economics, and a lifetime labour (...)
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  9. Quantum mechanics over sets: a pedagogical model with non-commutative finite probability theory as its quantum probability calculus.David Ellerman - 2017 - Synthese (12):4863-4896.
    This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of ℂ replaced by ℤ₂. Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability calculus. The previous attempts (...)
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  10. Neo-Abolitionism: Abolishing Human Rentals in favor of Workplace Democracy.David Ellerman - 2021 - Cham: SpringerNature.
    The abolition of slavery abolished not only the involuntary ownership of other people (workers) but also voluntary contractual forms of lifetime servitude. But that system of lifetime servitude was replaced by the current system of voluntary renting, hiring, employing, or leasing workers, i.e., the employment system. Hence the name “Neo-Abolitionism” for the idea of abolishing the employer–employee contract in favor of each firm being a workplace democracy. The three arguments against the human rental system are modern versions of old arguments (...)
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  11. Category theory and concrete universals.David P. Ellerman - 1988 - Erkenntnis 28 (3):409 - 429.
  12. The Labour Theory of Property and Marginal Productivity Theory.David Ellerman - 2016 - Economic Thought 5 (1):19.
    After Marx, dissenting economics almost always used 'the labour theory' as a theory of value. This paper develops a modern treatment of the alternative labour theory of property that is essentially the property theoretic application of the juridical principle of responsibility: impute legal responsibility in accordance with who was in fact responsible. To understand descriptively how assets and liabilities are appropriated in normal production, a 'fundamental myth' needs to be cleared away, and then the market mechanism of appropriation can be (...)
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  13. Logical Entropy: Introduction to Classical and Quantum Logical Information theory.David Ellerman - 2018 - Entropy 20 (9):679.
    Logical information theory is the quantitative version of the logic of partitions just as logical probability theory is the quantitative version of the dual Boolean logic of subsets. The resulting notion of information is about distinctions, differences and distinguishability and is formalized using the distinctions of a partition. All the definitions of simple, joint, conditional and mutual entropy of Shannon information theory are derived by a uniform transformation from the corresponding definitions at the logical level. The purpose of this paper (...)
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  14. The logic of partitions: Introduction to the dual of the logic of subsets: The logic of partitions.David Ellerman - 2010 - Review of Symbolic Logic 3 (2):287-350.
    Modern categorical logic as well as the Kripke and topological models of intuitionistic logic suggest that the interpretation of ordinary “propositional” logic should in general be the logic of subsets of a given universe set. Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms—which is reflected in the duality between quotient objects and subobjects throughout algebra. If “propositional” logic is thus seen as the logic of subsets of (...)
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  15. Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics.David P. Ellerman - 1995 - Rowman & Littlefield Publishers.
    Dramatic changes or revolutions in a field of science are often made by outsiders or 'trespassers,' who are not limited by the established, 'expert' approaches. Each essay in this diverse collection shows the fruits of intellectual trespassing and poaching among fields such as economics, Kantian ethics, Platonic philosophy, category theory, double-entry accounting, arbitrage, algebraic logic, series-parallel duality, and financial arithmetic.
  16.  60
    On Abstraction in Mathematics and Indefiniteness in Quantum Mechanics.David Ellerman - 2021 - Journal of Philosophical Logic 50 (4):813-835.
    ion turns equivalence into identity, but there are two ways to do it. Given the equivalence relation of parallelness on lines, the #1 way to turn equivalence into identity by abstraction is to consider equivalence classes of parallel lines. The #2 way is to consider the abstract notion of the direction of parallel lines. This paper developments simple mathematical models of both types of abstraction and shows, for instance, how finite probability theory can be interpreted using #2 abstracts as “superposition (...)
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  17.  3
    The Logic of Partitions: With Two Major Applications.David Ellerman - 2023 - London: College Publications.
    There are fundamentally two mathematical logics. One the Boolean logic of subsets, usually presented today in the special case of propositional logic, which has many sublogics and extensions, the most important being the intuitionistic logic usually modeled by the open subsets of a topological space. The other co-fundamental mathematical logic is the topic of this book, the logic of partitions. We are using ”logic” in a mathematical sense as being about basic mathematical objects, subsets of a universe set or partitions (...)
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  18. An introduction to logical entropy and its relation to Shannon entropy.David Ellerman - 2013 - International Journal of Semantic Computing 7 (2):121-145.
    The logical basis for information theory is the newly developed logic of partitions that is dual to the usual Boolean logic of subsets. The key concept is a "distinction" of a partition, an ordered pair of elements in distinct blocks of the partition. The logical concept of entropy based on partition logic is the normalized counting measure of the set of distinctions of a partition on a finite set--just as the usual logical notion of probability based on the Boolean logic (...)
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  19. On Property Theory.David Ellerman - 2014 - Journal of Economic Issues (3):601–624.
    A theory of property needs to give an account of the whole life-cycle of a property right: how it is initiated, transferred, and terminated. Economics has focused on the transfers in the market and has almost completely neglected the question of the initiation and termination of property in normal production and consumption (not in some original state or in the transition from common to private property). The institutional mechanism for the normal initiation and termination of property is an invisible-hand function (...)
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  20.  46
    Where do Adjunctions Come From? Chimera Morphisms and Adjoint Functors in Category Theory.David Ellerman - 2025 - Foundations 5 (10):1-22.
    Category theory has foundational importance because it provides conceptual lenses to characterize what is important and universal in mathematics—with adjunction seeming to be the primary lens. Our topic is a theory showing “where adjoints come from”. The theory is based on object-to-object “chimera morphisms”, “heteromorphisms”, or “hets” between the objects of different categories (e.g., the insertion of generators as a set-to-group map). After showing that heteromorphisms can be treated rigorously using the machinery of category theory (bifunctors), we show that all (...)
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  21. On Adjoint and Brain Functors.David Ellerman - 2016 - Axiomathes 26 (1):41-61.
    There is some consensus among orthodox category theorists that the concept of adjoint functors is the most important concept contributed to mathematics by category theory. We give a heterodox treatment of adjoints using heteromorphisms that parses an adjunction into two separate parts. Then these separate parts can be recombined in a new way to define a cognate concept, the brain functor, to abstractly model the functions of perception and action of a brain. The treatment uses relatively simple category theory and (...)
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  22.  85
    Logical information theory: new logical foundations for information theory.David Ellerman - 2017 - Logic Journal of the IGPL 25 (5):806-835.
    There is a new theory of information based on logic. The definition of Shannon entropy as well as the notions on joint, conditional, and mutual entropy as defined by Shannon can all be derived by a uniform transformation from the corresponding formulas of logical information theory. Information is first defined in terms of sets of distinctions without using any probability measure. When a probability measure is introduced, the logical entropies are simply the values of the probability measure on the sets (...)
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  23. Why Delayed Choice Experiments do NOT imply Retrocausality.David Ellerman - 2015 - Quantum Studies: Mathematics and Foundations 2 (2):183-199.
    There is a fallacy that is often involved in the interpretation of quantum experiments involving a certain type of separation such as the: double-slit experiments, which-way interferometer experiments, polarization analyzer experiments, Stern-Gerlach experiments, and quantum eraser experiments. The fallacy leads not only to flawed textbook accounts of these experiments but to flawed inferences about retrocausality in the context of delayed choice versions of separation experiments.
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  24. Does Classical Liberalism Imply Democracy?David Ellerman - 2015 - Ethics and Global Politics 8 (1):29310.
    There is a fault line running through classical liberalism as to whether or not democratic self-governance is a necessary part of a liberal social order. The democratic and non-democratic strains of classical liberalism are both present today—particularly in America. Many contemporary libertarians and neo-Austrian economists represent the non-democratic strain in their promotion of non-democratic sovereign city-states (startup cities or charter cities). We will take the late James M. Buchanan as a representative of the democratic strain of classical liberalism. Since the (...)
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  25.  47
    Sheaves of structures and generalized ultraproducts.David P. Ellerman - 1974 - Annals of Mathematical Logic 7 (2):163.
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  26.  30
    Born Again! The Born Rule as a feature of superposition.David Ellerman - 2025 - International Journal of Quantum Foundations 11:57-72.
    Where does the Born Rule come from? We ask: “What is the simplest extension of probability theory where the Born rule appears”? This is answered by introducing “superposition events” in addition to the usual discrete events. Two-dimensional matrices (e.g., incidence matrices and density matrices) are needed to mathematically represent the differences between the two types of events. Then it is shown that those incidence and density matrices for superposition events are the (outer) products of a vector and its transpose whose (...)
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  27. A Fundamental Duality in the Exact Sciences: The Application to Quantum Mechanics.David Ellerman - 2024 - Foundations 4 (2):175-204.
    There is a fundamental subsets–partitions duality that runs through the exact sciences. In more concrete terms, it is the duality between elements of a subset and the distinctions of a partition. In more abstract terms, it is the reverse-the-arrows of category theory that provides a major architectonic of mathematics. The paper first develops the duality between the Boolean logic of subsets and the logic of partitions. Then, probability theory and information theory (as based on logical entropy) are shown to start (...)
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  28.  75
    Marxism as a capitalist tool.David Ellerman - 2010 - Journal of Socio-Economics 39 (6):696-700.
    Just as the two sides in the Cold War agreed that Western Capitalism and Soviet Communism were "the" two alternatives, so the two sides in the intellectual Great Debate agreed on a common framing of questions with the defenders of capitalism taking one side and Marxists taking the other side of the questions. From the viewpoint of economic democracy (e.g., a labor-managed market economy), this late Great Debate between capitalism and socialism was as misframed as would be an antebellum 'Great (...)
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  29.  45
    (1 other version)The Quantum Logic of Direct-Sum Decompositions: The Dual to the Quantum Logic of Subspaces.David Ellerman - 2018 - Logic Journal of the IGPL 26 (1):1-13.
    ince the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the "quantum logic" of subspaces of a general vector space--which is then specialized to the closed subspaces of a Hilbert space. But there is a "dual" progression. The set notion of a partition is dual to the notion of a subset. Hence the Boolean logic of (...)
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  30.  45
    A critical analysis of different forms of employee ownership.David Ellerman & Tej Gonza - 2024 - International Review of Applied Economics 39 (2-3):261-276.
    From the 1970’s, there has been almost a half-century of development of employee-owned firms. There has been a wide variety of legal/capital structures that have been tried but too little analysis of which legal forms work or don’t work over the longer term, e.g. the transition from one generation of employee-owners to the next generation of employee owners. This paper provides a critical analysis of the major forms. These include the forms based on common ownership (Yugoslav self-managed firms and the (...)
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  31.  20
    On Tocqueville, Pauperism, and Employee Ownership.David Ellerman - 2025 - International Journal for Public Policy, Law and Development 2 (1):4-10.
    There seems to be two rather different philosophies of aid to development and poverty relief. (1) The progressive/social-democratic approach is for the government or aid agencies to do more and more good things for people. (2) The classical-liberal approach is to change the underlying conditions so that people are empowered to do good things for themselves. In this paper, we analyze Alexis de Tocqueville’s approach to these questions in his First Memoir and his (unfinished) Second Memoir on Pauperism.
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  32.  35
    Democratic Ownership: Scale through leveraged conversions.David Ellerman, Tej Gonza & Kosta Juri - 2024 - In Jerome Warren, Jamin Hübner, Lucio Biggiero & Kemi Ogunyemi, Routledge Handbook on Cooperative Economics and Management. London: Routledge. pp. 250-61.
  33.  43
    Intentionality and information theory.David P. Ellerman - 1986 - Behavioral and Brain Sciences 9 (1):143-144.
  34.  16
    A New Logical Measure for Quantum Information.David Ellerman - 2025 - Quantum Information and Computation 25:81-95.
    Starting at the logical level of the logic of partitions, dual to the usual Boolean logic of subsets, the notion of logical entropy, i.e., information as distinctions, is developed as the quantification of the distinctions of partitions—just as probability theory starts with the quantification of elements of subsets. Logical entropy is compared and contrasted with the usual notion of Shannon entropy. Then a semi-algorithmic procedure (from the mathematical folklore) is used to translate the notion of logical entropy at the set (...)
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  35.  14
    Opening the Gates to Plato's Heaven.David Ellerman - 2025 - International Journal of Education and Social Science Research 8 (1).
    Substance is rivalrous and form is non-rivalrous (in the sense that information can be shared without diminishing one’s own). The recipe to "open the gates to Plato's Heaven" is by minimizing the role of rivalrous substance and maximizing the role of non-rivalrous form. This creates a whole series of different processes, positive feedback processes, vicious or virtuous circles, cumulative circular causality, and increasing returns phenomena, which are analysed in this paper.
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  36. On the Labor Theory of Property.David P. Ellerman - 1985 - Philosophical Forum 16 (4):293.
     
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  37. Category Theory and Set Theory as Theories about Complementary Types of Universals.David Ellerman - 2017 - Logic and Logical Philosophy 26 (2):145-162.
    Instead of the half-century old foundational feud between set theory and category theory, this paper argues that they are theories about two different complementary types of universals. The set-theoretic antinomies forced naïve set theory to be reformulated using some iterative notion of a set so that a set would always have higher type or rank than its members. Then the universal u F = {x | F(x)} for a property F(.) could never be self-predicative in the sense of uF ∈ (...)
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  38.  18
    Worker Cooperatives and Other 'Cooperatives'.David Ellerman & Tej Gonza - 2024 - In Jerome Warren, Jamin Hübner, Lucio Biggiero & Kemi Ogunyemi, Routledge Handbook on Cooperative Economics and Management. London: Routledge. pp. 85-95.
    The short answer is whenever the actual activity of the “cooperative” is not carried out by the members but by employees. The problem is, of course, not in cooperation per se but in the hiring, employing, renting, or leasing of people to carry out the supposedly “cooperative” activities of the “cooperative” (Ellerman, 2021). Consider the case of a typical consumer cooperative. What is the cooperative activity carried out by the consumer‑members? They do not consume cooperatively; that would be a commune (...)
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  39. The Democratic Worker-Owned Firm : A New Model for the East and West.David Ellerman - 2015 - Routledge.
    When this book was first published in 1990, there were massive economic changes in the East and significant economic challenges to the West. This critical analysis of democratic theory discusses the principles and forces that push both socialist and capitalist economies toward a common ground of workplace democratization. This book is a comprehensive approach to the theory and practice of the "Democratic firm" – from philosophical first principles to legal theory and finally to some of the details of financial structure. (...)
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  40. Marxian Exploitation Theory: A Brief Exposition, Analysis, and Critique.David P. Ellerman - 1983 - Philosophical Forum 14 (3):315.
  41. Reclaiming Democratic Classical Liberalism.David Ellerman - 2020 - In D. Hardwick & L. Marsh, Reclaiming Liberalism. Palgrave Studies in Classical Liberalism. pp. 1-39.
    This essay shows that the principles of classical liberalism (e.g., James Buchanan) do not apply to the firm based on the employer-employee relationship. However, there is a deeper democratic classical liberalism tradition based on inalienable rights, but it rules out the employment or human rental relation.
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  42. The Case for Workplace Democracy.David Ellerman - 2018 - In James Muldoon, Council Democracy: Towards a Democratic Socialist Politics. Routledge. pp. 210-227.
    In this chapter I seek to provide a theoretical defense of workplace democracy that is independent from and outside the lineage of Marxist and communist theory. Common to the council movements, anarcho- syndicalism and many other forms of libertarian socialism was the idea “that workers’ self- management was central.” Yet the idea of workers’ control has not been subject to the same theoretical development as Marx’s theory, not to mention capitalist economic theory. This chapter aims to contribute at a theoretical (...)
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  43. How Category Theory Works.David Ellerman - manuscript
    The purpose of this paper is to show that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, and universal constructions in the Sets, the category of sets and functions. The analysis extends directly to other concrete categories (groups, rings, vector spaces, etc.) where the objects are sets with a certain type of structure and the morphisms are functions that preserve that structure. Then the elements & distinctions-based definitions can be (...)
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  44. Reclaiming Democratic Classical Liberalism.David Ellerman - 2020 - In Reclaiming Democratic Classical Liberalism. pp. 1-39.
    Classical liberalism is skeptical about governmental organizations "doing good" for people. Instead governments should create the conditions so that people individually (Adam Smith) and in associations (Tocqueville) are empowered to do good for themselves. The market implications of classical liberalism are well-known, but the implications for organizations are controversial. We will take James Buchanan as our guide (with assists from Mill and Dewey). Unpacking the implications of classical liberalism for the "science of associations" (Tocqueville) requires a tour through the intellectual (...)
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  45. Workplace Democracy and Human Development: The Example of the Postsocialist Transition Debate.David Ellerman - 2010 - Journal of Speculative Philosophy 24 (4):333-353.
    In the 1990s , a debate raged across the whole postsocialist world as well as in Western development agencies such as the World Bank about the best approach to the transition from various forms of socialism or communism to a market economy and political democracy. One of the most hotly contested topics was the question of the workplace being organized based on workplace democracy (e.g., various forms of worker ownership) or based on the conventional employer-employee relationship. Well before 1989, many (...)
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  46. Category theory and set theory as theories about complementary types of universals.David P. Ellerman - 2017 - Logic and Logical Philosophy 26 (2):1-18.
    Instead of the half-century old foundational feud between set theory and category theory, this paper argues that they are theories about two different complementary types of universals. The set-theoretic antinomies forced naïve set theory to be reformulated using some iterative notion of a set so that a set would always have higher type or rank than its members. Then the universal u_{F}={x|F(x)} for a property F() could never be self-predicative in the sense of u_{F}∈u_{F}. But the mathematical theory of categories, (...)
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  47. On the self-predicative universals of category theory.David Ellerman - manuscript
    This paper shows how the universals of category theory in mathematics provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in Hegel and similar ideas of paradigmatic exemplars in ordinary thought. The paper also shows how the always-self-predicative universals of category theory provide the "opposite bookend" to the never-self-predicative universals of iterative set theory and thus that the paradoxes arose from having (...)
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  48. On Concrete Universals: A Modern Treatment using Category Theory.David Ellerman - 2014 - AL-Mukhatabat.
    Today it would be considered "bad Platonic metaphysics" to think that among all the concrete instances of a property there could be a universal instance so that all instances had the property by virtue of participating in that concrete universal. Yet there is a mathematical theory, category theory, dating from the mid-20th century that shows how to precisely model concrete universals within the "Platonic Heaven" of mathematics. This paper, written for the philosophical logician, develops this category-theoretic treatment of concrete universals (...)
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  49. On Double-Entry Bookkeeping: The Mathematical Treatment.David Ellerman - 2014 - Accounting Education: An International Journal 23 (5):483-501.
    Double-entry bookkeeping (DEB) implicitly uses a specific mathematical construction, the group of differences using pairs of unsigned numbers ("T-accounts"). That construction was only formulated abstractly in mathematics in the 19th century—even though DEB had been used in the business world for over five centuries. Yet the connection between DEB and the group of differences (here called the "Pacioli group") is still largely unknown both in mathematics and accounting. The precise mathematical treatment of DEB allows clarity on certain conceptual questions and (...)
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  50. Probability Theory with Superposition Events.David Ellerman - manuscript
    In finite probability theory, events are subsets S⊆U of the outcome set. Subsets can be represented by 1-dimensional column vectors. By extending the representation of events to two dimensional matrices, we can introduce "superposition events." Probabilities are introduced for classical events, superposition events, and their mixtures by using density matrices. Then probabilities for experiments or `measurements' of all these events can be determined in a manner exactly like in quantum mechanics (QM) using density matrices. Moreover the transformation of the density (...)
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