David Ellerman

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 abstracted in purely arrow-theoretic way for abstract category theory. In short, the language of elements & distinctions is the conceptual language in which the category of sets is written, and abstract category theory gives the abstract arrows version of those definitions.

Classical physics and quantum physics suggest two meta-physical types of reality: the classical notion of a objectively definite reality with properties "all the way down," and the quantum notion of an objectively indefinite type of reality. The problem of interpreting quantum mechanics (QM) is essentially the problem of making sense out of an objectively indefinite reality. These two types of reality can be respectively associated with the two mathematical concepts of subsets and quotient sets (or partitions) which are category-theoretically dual to one another and which are developed in two mathematical logics, the usual Boolean logic of subsets and the more recent logic of partitions. Our sense-making strategy is "follow the math" by showing how the logic and mathematics of set partitions can be transported in a natural way to Hilbert spaces where it yields the mathematical machinery of QM--which shows that the mathematical framework of QM is a type of logical system over ℂ. And then we show how the machinery of QM can be transported the other way down to the set-like vector spaces over ℤ₂ showing how the classical logical finite probability calculus (in a "non-commutative" version) is a type of "quantum mechanics" over ℤ₂, i.e., over sets. In this way, we try to make sense out of objective indefiniteness and thus to interpret quantum mechanics.

Nancy MacLean’s book, Democracy in Chains, raised questions about James M. Buchanan’s commitment to democracy. This paper investigates the relationship of classical liberalism in general and of Buchanan in particular to democratic theory. Contrary to the simplistic classical liberal juxtaposition of “coercion vs. consent,” there have been from Antiquity onwards voluntary contractarian defenses of non-democratic government and even slavery—all little noticed by classical liberal scholars who prefer to think of democracy as just “government by the consent of the governed” and slavery as being inherently coercive. Historically, democratic theory had to go beyond that simplistic notion of democracy to develop a critique of consent-based non-democratic government, e.g., the Hobbesian pactum subjectionis. That critique was based firstly on the distinction between contracts or constitutions of alienation (translatio) versus delegation (concessio). Then the contracts of alienation were ruled out based on the theory of inalienable rights that descends from the Reformation doctrine of inalienability of conscience down through the Enlightenment to modern times in the abolitionist and democratic movements. While he developed no theory of inalienability, the mature Buchanan explicitly allowed only a constitution of delegation, contrary to many modern classical liberals or libertarians who consider the choice between consent-based democratic or non-democratic governments (e.g., private cities or shareholder states) to be a pragmatic one. But Buchanan seems to not even realize that his at-most-delegation dictum would also rule out the employer-employee or human rental contract which is a contract of alienation “within the scope of the employment.”

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 is to give the direct generalization to quantum logical information theory that similarly focuses on the pairs of eigenstates distinguished by an observable, i.e., qudits of an observable. The fundamental theorem for quantum logical entropy and measurement establishes a direct quantitative connection between the increase in quantum logical entropy due to a projective measurement and the eigenstates that are distinguished by the measurement. Both the classical and quantum versions of logical entropy have simple interpretations as “two-draw” probabilities for distinctions. The conclusion is that quantum logical entropy is the simple and natural notion of information for quantum information theory focusing on the distinguishing of quantum states.

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 states is the partition join operation at the set level that prefigures at the quantum level projective measurement as well as the formation of maximally-definite state descriptions by Dirac’s Complete Sets of Commuting Operators. This development is measured quantitatively by logical entropy at the set level and by quantum logical entropy at the quantum level. This follow-the-math approach supports the Literal Interpretation of QM—as advocated by Abner Shimony among others which sees a reality of objective indefiniteness that is quite different from the common sense and classical view of reality as being “definite all the way down”.

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 all required the brackets to take values in ℤ₂. But the usual QM brackets <ψ|ϕ> give the "overlap" between states ψ and ϕ, so for subsets S,T⊆U, the natural definition is <S|T>=|S∩T| (taking values in the natural numbers). This allows QM/sets to be developed with a full probability calculus that turns out to be a non-commutative extension of classical Laplace-Boole finite probability theory. The pedagogical model is illustrated by giving simple treatments of the indeterminacy principle, the double-slit experiment, Bell's Theorem, and identical particles in QM/Sets. A more technical appendix explains the mathematics behind carrying some vector space structures between QM over ℂ and QM/Sets over ℤ₂.

Category theory has foundational importance because it provides conceptual lenses to characterize what is important and universal in mathematics—with adjunctions being the primary lens. If adjunctions are so important in mathematics, then perhaps they will isolate concepts of some importance in the empirical sciences. But the applications of adjunctions have been hampered by an overly restrictive formulation that avoids heteromorphisms or hets. By reformulating an adjunction using hets, it is split into two parts, a left and a right semiadjunction. Semiadjunctions (essentially a formulation of universal mapping properties using hets) can then be combined in a new way to define the notion of a brain functor that provides an abstract model of the intentionality of perception and action (as opposed to the passive reception of sense-data or the reflex generation of behavior).

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 roots of such ideas in some fascinating and largely forgotten strands of the history of economics. It draws attention to the question of "responsibility" which neoclassicism has utterly lost sight of. It is startlingly fresh in its overall approach, and unusually well written in its presentation. ... It constitutes a better case for its economic democracy viewpoint than anything else in the literature."

Liberalism is based on the juxtaposition of consent to coercion. Autocracy and slavery were based on coercion whereas today’s political democracy and economic “employment system” are based on consent to voluntary contracts. This article retrieves an almost forgotten dark side of contractarian thought that based autocracy and slavery on explicit or implicit voluntary contracts. The democratic and antislavery movements forged arguments not simply in favor of consent but arguments that voluntary contracts to alienate aspects of personhood were invalid—which made the underlying rights inalienable. Once understood, those arguments apply as well to today’s self-rental contract, the employer-employee contract.

Classical physics and quantum physics suggest two meta-physical types of reality: the classical notion of a objectively definite reality with properties "all the way down," and the quantum notion of an objectively indefinite type of reality. The problem of interpreting quantum mechanics is essentially the problem of making sense out of an objectively indefinite reality. These two types of reality can be respectively associated with the two mathematical concepts of subsets and quotient sets which are category-theoretically dual to one another and which are developed in two dual mathematical logics, the usual Boolean logic of subsets and the more recent logic of partitions. Our sense-making strategy is "follow the math" by showing how the mathematics of set partitions can be transported in a natural way to complex vector spaces where it yields the mathematical machinery of QM. And then we show how the machinery of QM can be transported the other way down to set-like vector spaces over ℤ₂ yielding a rather fulsome "toy" or pedagogical model of "quantum mechanics over sets." In this way, we try to make sense out of objective indefiniteness and thus to interpret quantum mechanics.

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 it the other way around, the math of partitions is a skeletal version of the math of QM. The key concepts throughout this progression from logic, to logical information, to quantum theory are distinctions versus indistinctions, definiteness versus indefiniteness, or distinguishability versus indistinguishability. The distinctions of a partition are the ordered pairs of elements from the underlying set that are in different blocks of the partition and logical entropy is defined (initially) as the normalized number of distinctions. The cognate notions of definiteness and distinguishability run throughout the math of QM, e.g., in the key non-classical notion of superposition (= ontic indefiniteness) and in the Feynman rules for adding amplitudes (indistinguishable alternatives) versus adding probabilities (distinguishable alternatives).

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 all required the brackets to take values in ℤ₂. But the usual QM brackets <ψ|ϕ> give the "overlap" between states ψ and ϕ, so for subsets S,T⊆U, the natural definition is <S|T>=|S∩T| (taking values in the natural numbers). This allows QM/sets to be developed with a full probability calculus that turns out to be a non-commutative extension of classical Laplace-Boole finite probability theory. The pedagogical model is illustrated by giving simple treatments of the indeterminacy principle, the double-slit experiment, Bell's Theorem, and identical particles in QM/Sets. A more technical appendix explains the mathematics behind carrying some vector space structures between QM over ℂ and QM/Sets over ℤ₂.

Early democratic theorists such as Kant considered the effects of being a servant or, in modern terms, an employee to be so negative that such dependent people should be denied the vote. John Stuart Mill and John Dewey also noted the negative effects of the employment relation on the development of democratic habits and civic virtues but rather than deny the franchise to employees, they pushed for workplace democracy where workers would be a member of their company rather than an employee. In spite of the continuing prevalence of the employment relation and the lack of workplace democracy, this topic now seems to be something of a "third rail" in deliberative democratic theory

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 of the socialist countries had started experimenting with various forms of "self-management" operating in more of a market setting, Yugoslavia being the most developed example. Thus one "path to the market" would have been to push those experiments all the way to some Western form of employee ownership or worker cooperatives operating in a full market environment. Alternatively, all these decentralizing experiments could be condemned as "vestiges of communism" to be eradicated by renationalizing all the decentralized firms and then privatizing by some alternative means.

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 Debate' between the private or public ownership of slaves. Even though the Great Debate between capitalism and socialism is now in the dustbin of intellectual history, Marxism still plays an important role in sustaining the misframing of the questions so that the defenders of the present employment system do not have to face the real questions that separate that system from a system of economic democracy. In that sense, Marxism has become the ultimate capitalist tool.

This paper presents an argument for the democratic (or 'labor-managed') firm based on ordinary jurisprudence. The standard principle of responsibility in jurisprudence ('Assign legal responsibility in accordance with de facto responsibility') implies that the people working in a firm should legally appropriate the assets and liabilities produced in the firm (the positive and negative fruits of their labor). This appropriation is normally violated due to the employment or self-rental contract. However, we present an inalienable rights argument that descends from the Reformation and Enlightenment which argues that the self-rental contract, like the self-sale or voluntary slavery contract, is inherently invalid. The key intuition of the inalienable rights theory is that one cannot in fact voluntarily transfer de facto responsibility for one's actions to another person. One can only voluntarily co-operate with another person, but then one is de facto jointly responsible for the results. Just as the legal authorities legally reconstruct the criminous employer and employee as a partnership with shared responsibility, so justice demands that every firm be legally reconstructed as a partnership of all who work (working employers and employees) in the enterprise, i.e., as a democratic firm.

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 events” in addition to the ordinary events. The goal is to use the second notion of abstraction to shed some light on the notion of an indefinite superposition in quantum mechanics.

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 of distinctions. The compound notions of joint, conditional, and mutual entropies are obtained as the values of the measure, respectively, on the union, difference, and intersection of the sets of distinctions. These compound notions of logical entropy satisfy the usual Venn diagram relationships since they are values of a measure. The uniform transformation into the formulas for Shannon entropy is linear so it explains the long-noted fact that the Shannon formulas satisfy the Venn diagram relations--as an analogy or mnemonic--since Shannon entropy is not a measure on a given set. What is the logic that gives rise to logical information theory? Partitions are dual to subsets, and the logic of partitions was recently developed in a dual/parallel relationship to the Boolean logic of subsets. Boole developed logical probability theory as the normalized counting measure on subsets. Similarly the normalized counting measure on partitions is logical entropy--when the partitions are represented as the set of distinctions that is the complement to the equivalence relation for the partition. In this manner, logical information theory provides the set-theoretic and measure-theoretic foundations for information theory. The Shannon theory is then derived by the transformation that replaces the counting of distinctions with the counting of the number of binary partitions it takes, on average, to make the same distinctions by uniquely encoding the distinct elements--which is why the Shannon theory perfectly dovetails into coding and communications 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 of a fixed universe 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 size as a probability. The analogous construction for the dual logic of partitions gives rise to a notion of logical entropy that is precisely related to Claude Shannon's entropy. In this manner, the new logic of partitions provides a logico-conceptual foundation for information-theoretic entropy or information content.

Jamie Morgan's commentary (Morgan, 2016) on my paper 'The Labour Theory of Property and Marginal Productivity Theory' (Ellerman, 2016) and Ted Burczak's later comments (Burczak, 2016) raise a number of issues that surely will occur to other readers and that need to be addressed. I take the occasion to expand upon the arguments and to explore some related issues. In the narrative that unfolds, Frank H. Knight plays the role of the sophisticated defender of the system of renting, hiring and employing human beings. He was quite clear that the social role of economics is to develop an idealised model, the competitive free enterprise model, and then to frame the normative discussion in terms of that model. Knight would agree with the whole thread of heterodox 'criticism' that the actual economy falls far short of the ideal – which is why I largely eschewed the descriptive shortcomings of ideal model as a purported model of the actual economy. Instead my paper focused on developing a critique of the key part of the idealised model, the marginal productivity theory of distribution under competitive conditions. That critique is based on the usual juridical principle of imputing legal responsibility in accordance with factual responsibility – the principle whose property-theoretic application is the modern treatment of the labour theory of property. Historically, heterodox economics faced a fork in the road in the 19th century: whether to criticise 'the system' by developing the inchoate 'labour theory' as a theory of value or a theory of property. Marx and much of left-wing economics took the labour-theory-of-value road, whereas my paper is part of a modern attempt – Thomas Hodgskin (1832) being an earlier attempt – to take the labour-theory-of-property road. As we will see, much of the debate still revolves around these two roads. Read David Ellerman's original paper "The Labour Theory of Property and Marginal Productivity Theory" › Read Jamie Morgan's commentary on David Ellerman's paper ›

Following the development of the selectionist theory of the immune system, there was an attempt to characterise many biological mechanisms as being ‘selectionist’ as juxtaposed with ‘instructionist’. However, this broad definition would group Darwinian evolution, the immune system, embryonic development, and Chomsky’s principles-and-parameters language-acquisition mechanism together under the ‘selectionist’ umbrella, even though Chomsky’s mechanism and embryonic development are significantly different from the selectionist mechanisms of biological evolution and the immune system. Surprisingly, there is an abstract way using two dual mathematical logics to make the distinction between genuinely selectionist mechanisms and what are better called ‘generative’ or symmetry-breaking mechanisms. This distinction is outlined in this note.

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 subsets has a dual logic of partitions. Then the dual progression is from that logic of set partitions to the quantum logic of direct-sum decompositions of a general vector space--which can then be specialized to the direct-sum decompositions of a Hilbert space. This allows the quantum logic of direct-sum decompositions to express measurement by any self-adjoint operators. The quantum logic of direct-sum decompositions is dual to the usual quantum logic of subspaces in the same sense that the logic of partitions is dual to the usual Boolean logic of subsets.

Two aspects of the fine Flecha-Cruz paper can be usefully elaborated. The Mondragon cooperatives differ not only from capitalist firms but also from most other cooperatives in the doctrine of the 'priority of labor over capital' which means that the people working in any sort of cooperative will be members and will not be rented as employees. Also the Mondragon system of internal capital accounts solves the equity-structure problem that has plagued many modern cooperatives structured as non-profits or traditional worker cooperatives with 'membership shares'.