David Ellerman

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).

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 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 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.

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.

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.

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.

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 matrices induced by the experiments or `measurements' is the Lüders mixture operation as in QM. And finally by moving the machinery into the n-dimensional vector space over ℤ₂, different basis sets become different outcome sets. That `non-commutative' extension of finite probability theory yields the pedagogical model of quantum mechanics over ℤ₂ that can model many characteristic non-classical results of QM.

In her recent book Private Government, Elizabeth Anderson makes a powerful but pragmatic case against the abuses experienced by employees in conventional corporations. The purpose of this review-essay is to contrast Anderson’s pragmatic critique of many abuses in the employment relation with a principled critique of the employment relationship itself. This principled critique is based on the theory of inalienable rights that descends from the Reformation doctrine of the inalienability of conscience down through the Enlightenment in the abolitionist, democratic, and feminist movements. That theory was the basis for the abolition of the voluntary slavery or self-sale contract, the voluntary non-democratic constitution (pactum subjectionis), and the voluntary coverture marriage contract in today’s democratic countries. When understood in modern terms, that same theory applies as well against the voluntary self-rental or employment contract that is the basis for our current economic system.

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.”

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 history of the voluntary slavery contract and the voluntary non-democratic constitution. The argument concludes that the classical liberal endorsement of sovereign individuals acting in the marketplace generalizes to the joint action of individuals as the principals in their own organizations and associations.

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 level by providing a justification and defense of self- managed workplaces that is independent of the particular historical tradition of the council movements. There is a clear and definitive case for workplace democracy based on first principles that descends to modern times through the Reformation and Enlightenment in the abolitionist, democratic and feminist movements. By the twentieth century, the arguments had been scattered and lost  – like the bones of some ancient beast scattered in a desert  – partly due to misconceptions, mental blocks and misinterpretations embodied in Marxism, liberalism and economic theory. When one has worked through some of these intellectual roadblocks, then one may be better able to reassemble the case for workplace democracy from well- known first principles developed in the abolitionist, democratic and feminist movements.

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.

Much of the recent discussion in progressive circles [e.g., Stiglitz; Galbraith; Piketty] has focused the obscene mal-distribution of wealth and income as if that was "the" problem in our economic system. And the proposed redistributive reforms have all stuck to that framing of the question. To put the question in historical perspective, one might note that there was a similar, if not more extreme, mal-distribution of wealth, income, and political power in the Antebellum system of slavery. Yet, it should be obvious to modern eyes that redistributions in favor of the slaves, while leaving the institution of owning workers intact, would not address the root of the problem. The system of slavery was eventually abolished in favor of the system we have today which differs in two important respects: the workers are only rented, hired, or employed ; and the rental relationship between employer and employee is voluntary. Today, the root of the problem is the whole institution for the voluntary renting of human beings, the employment system itself, not the terms of the contract or the accumulated consequences in the form of the mal-distribution of income and wealth.

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.

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.

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.

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.

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).