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”. (shrink)

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. (shrink)

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 (...) probability theory by taking the (Laplacian) probability as the normalized size of each subset-event of a finite universe. The analogous step in the logic of partitions is to assign to a partition the number of distinctions made by a partition normalized by the total number of ordered pairs |U|² from the finite universe. That yields a notion of "logical entropy" for partitions and a "logical information theory." The logical theory directly counts the (normalized) number of distinctions in a partition while Shannon's theory gives the average number of binary partitions needed to make those same distinctions. Thus the logical theory is seen as providing a conceptual underpinning for Shannon's theory based on the logical notion of "distinctions.". (shrink)

This study focused on the adaptability and attitude of Grade 6 students in the Modular Learning Approach. This study aims to create a comprehensive review of the adaptability and attitude of elementary students in the Modular Learning Approach Especially. It seeks to describe and analyze the following: the profile of the respondents in terms of sex and education of the parents; the adaptability of the respondents toward the relationship between the student's profile in terms of their sex and educational attainment (...) of parents; the attitude of the respondent's significant relationship between sex and educational attainment. The researcher used the quantitative survey questionnaires to gather data from 128 Grade 6 students in five Elementary Schools in Bongabon, Nueva Ecija who enrolled during the School Year 2020-2021. The study revealed that the respondent's demographic profile in sex has no significant relationship to their adaptability and attitude in the modular learning approach. Yet the parents' educational attainment of the respondents has a significant relationship to adaptability in terms of learning assistance and support in the modular learning approach. The results further signified that parents' educational attainment is a basis for adaptability in modular learning. -/- . (shrink)

Saunders Mac Lane famously remarked that "Bourbaki just missed" formulating adjoints in a 1948 appendix (written no doubt by Pierre Samuel) to an early draft of Algebre--which then had to wait until Daniel Kan's 1958 paper on adjoint functors. But Mac Lane was using the orthodox treatment of adjoints that only contemplates the object-to-object morphisms within a category, i.e., homomorphisms. When Samuel's treatment is reconsidered in view of the treatment of adjoints using heteromorphisms or hets (object-to-object morphisms between objects in (...) different categories), then he, in effect, isolated the concept of a left representation solving a universal mapping problem. When dualized to obtain the concept of a right representation, the two halves only need to be united to obtain an adjunction. Thus Samuel was only a now-simple dualization away for formulating adjoints in 1948. Apparently, Bodo Pareigis' 1970 text was the first and perhaps only text to give the heterodox "new characterization" (i.e., heteromorphic treatment) of adjoints. Orthodox category theory uses various relatively artificial devices to avoid formally recognizing hets--even though hets are routinely used by the working mathematician. Finally we consider a "philosophical" question as to whether the most important concept in category theory is the notion of an adjunction or the notion of a representation giving a universal mapping property (where adjunctions arise as the special case of a bi-representation of dual universal mapping problems). (shrink)

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 (...) logic of partitions. That dual logic is described here. Partition logic is at the same mathematical level as subset logic since models for both are constructed from (partitions on or subsets of) arbitrary unstructured sets with no ordering relations, compatibility or accessibility relations, or topologies on the sets. Just as Boole developed logical finite probability theory as a quantitative treatment of subset logic, applying the analogous mathematical steps to partition logic yields a logical notion of entropy so that information theory can be refounded on partition logic. But the biggest application is that when partition logic and the accompanying logical information theory are "lifted" to complex vector spaces, then the mathematical framework of quantum mechanics is obtained. Partition logic models indefiniteness (i.e., numerical attributes on a set become more definite as the inverse-image partition becomes more refined) while subset logic models the definiteness of classical physics (an entity either definitely has a property or definitely does not). Hence partition logic provides the backstory so the old idea of "objective indefiniteness" in QM can be fleshed out to a full interpretation of quantum mechanics. (shrink)

Following the development of the selectionist theory of the immune system, there was an attempt to characterize many biological mechanisms as being "selectionist" as juxtaposed to "instructionist." But this broad definition would group Darwinian evolution, the immune system, embryonic development, and Chomsky's language-acquisition mechanism as all being "selectionist." Yet Chomsky's mechanism (and embryonic development) are significantly different from the selectionist mechanisms of biological evolution or the immune system. Surprisingly, there is a very abstract way using two dual mathematical logics to (...) make the distinction between genuinely selectionist mechanisms and what are better called "generative" mechanisms. This note outlines that distinction. (shrink)

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. (shrink)

This paper is part of a larger project to better understand the limitations of the economic theory of agency and incentives.1 The economic approach focuses on extrinsic incentives whereas a better understanding of human organization requires an understanding of intrinsic motivation and the complementary or substitutive relationships with extrinsic motivation. I will focus on different treatments of informational or “panoptic” questions regarding transparency or non-transparency in the management of firms or other human organizations. This context will give the questions definiteness (...) but the ideas might also be applied in a broader social context. As a conceptual framework, I will use the seminal work of Douglas McGregor. The economic approach to agency and incentives theory is based essentially on what McGregor called the “Theory X” worldview, and the alternative explored here is essentially what he called the “Theory Y” view of individuals in organizations. (shrink)

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 (...) of the market, the market mechanism of appropriation. Does this mechanism satisfy an appropriate normative principle? The standard normative juridical principle is to assign or impute legal responsibility according to de facto responsibility. It is given a historical tag of being "Lockean" but the basis is contemporary jurisprudence, not historical exegesis. Then the fundamental theorem of the property mechanism is proven which shows that if "Hume's conditions" (no transfers without consent and all contracts fulfilled) are satisfied, then the market automatically satisfies the Lockean responsibility principle, i.e., "Hume implies Locke." As a major application, the results in their contrapositive form, "Not Locke implies Not Hume," are applied to a market economy based on the employment contract. It is shown the production based on the employment contract violates the Lockean principle (all who work in an employment enterprise are de facto responsible for the positive and negative results) and thus Hume's conditions must also be violated in the marketplace (de facto responsible human action cannot be transferred from one person to another—as is readily recognized when and employer and employee together commit a crime). (shrink)

Liberal thought (in the sense of classical liberalism) is based on the juxtaposition of consent to coercion. Autocracy and slavery were seen as based on coercion whereas today's political democracy and economic 'employment system' are based on consent to voluntary contracts. This paper retrieves an almost forgotten dark side of contractarian thought that based autocracy and slavery on explicit or implicit voluntary contracts. To answer these 'best case' arguments for slavery and autocracy, the democratic and abolitionist movements forged arguments not (...) simply in favour of consent, but arguments that voluntary contracts to legally alienate aspects of personhood were invalid 'even with consent' – which made the underlying rights inherently inalienable. Once understood, those arguments have the perhaps 'unintended consequence' of making the neo-abolitionist case for ruling out today's self-rental contract, the employer-employee contract. The paper has to also retrieve these inalienable rights arguments since they have been largely lost on the Left, not to mention in liberal thought. (shrink)

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. (shrink)

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. (shrink)

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.”. (shrink)

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. (shrink)

It seems undeniable that some cultures encourage individuals to act in ways that harm others, and/or to believe that there is nothing wrong when another acts in a way that harms them. And when this is the case it also seems undeniable that it would be better if the scope for such cultures to guide individuals' decision-making were minimized or even eliminated. From these observations a number of people have inferred that groups which exhibit bad cultures ought not to be (...) permitted to hold or exercise group rights. Susan Okin's liberal feminist critique of multiculturalism is one of the most interesting and persuasive examples of this type of argument. In this paper I take a closer look at Okin's critique and ask whether it actually does give one reason to be skeptical about group rights. I conclude that it does not. For although the worries which animate Okin's critique are good ones to have, it is a mistake to think that they are about group rights. Moreover, insofar as Okin's concerns are well motivated they are not actually worries about morally problematic cultures but rather worries about groups' internal political and social structures. These worries should not be equated. Ultimately (I argue) it is not only misleading but counterproductive to focus on the viciousness of a group's culture as a potential reason for disqualifying it from holding or exercising rights. (shrink)

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, (...) dating from the mid-twentieth century, includes a theory of always-self-predicative universals--which can be seen as forming the "other bookend" to the never-self-predicative universals of set theory. The self-predicative universals of category theory show that the problem in the antinomies was not self-predication per se, but negated self-predication. They also 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. (shrink)

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 (...) one theory (e.g., Frege's Paradise) where universals could be either self-predicative or non-self-predicative (instead of being always one or always the other). (shrink)

Evolutionary economics often focuses on the comparison between economic competition and the process of natural selection to select the fitter members of a given population. But that neglects the other "half" of an evolutionary process, the mechanism for the generation of new possibilities that is key to dynamic efficiency. My topic is the process of parallel experimentation which I take to be a process of multiple experiments running concurrently with some form of common goal, with some semi-isolation between the experiments, (...) with benchmarking comparisons made between the experiments, and with the "migration" of discoveries between experiments wherever possible to ratchet up the performance of the group. The thesis is that parallel experimentation is a fundamental dynamic efficiency scheme to enhance and accelerate variation, innovation, and learning in contexts of genuine uncertainty or known ignorance. Within evolutionary biology, this type of parallel experimentation scheme was developed in Sewall Wright's shifting balance theory of evolution. It addressed the rather neglected topic of how a population on a low fitness peak might eventually be able to go "downhill" against selective pressures, traverse a valley of low fitness, and then ascend a higher fitness peak. The theme of parallel experimentation is used to recast and pull together dynamic and pluralistic theories in economics, political theory, philosophy of science, and social learning. (shrink)

Recent developments in pure mathematics and in mathematical logic have uncovered a fundamental duality between "existence" and "information." In logic, the duality is between the Boolean logic of subsets and the logic of quotient sets, equivalence relations, or partitions. The analogue to an element of a subset is the notion of a distinction of a partition, and that leads to a whole stream of dualities or analogies--including the development of new logical foundations for information theory parallel to Boole's development of (...) logical finite probability theory. After outlining these dual concepts in mathematical terms, we turn to a more metaphysical speculation about two dual notions of reality, a fully definite notion using Boolean logic and appropriate for classical physics, and the other objectively indefinite notion using partition logic which turns out to be appropriate for quantum mechanics. The existence-information duality is used to intuitively illustrate these two dual notions of reality. The elucidation of the objectively indefinite notion of reality leads to the "killer application" of the existence-information duality, namely the interpretation of quantum mechanics. (shrink)

The notion of a partition on a set is mathematically dual to the notion of a subset of a set, so there is a logic of partitions dual to Boole's logic of subsets (Boolean logic is usually mis-specified as "propositional" logic). The notion of an element of a subset has as its dual the notion of a distinction of a partition (a pair of elements in different blocks). Boole developed finite logical probability as the normalized counting measure on elements of (...) subsets so there is a dual concept of logical entropy which is the normalized counting measure on distinctions of partitions. Thus the logical notion of information is a measure of distinctions. Classical logical entropy naturally extends to the notion of quantum logical entropy which provides a more natural and informative alternative to the usual Von Neumann entropy in quantum information theory. The quantum logical entropy of a post-measurement density matrix has the simple interpretation as the probability that two independent measurements of the same state using the same observable will have different results. The main result of the paper is that the increase in quantum logical entropy due to a projective measurement of a pure state is the sum of the absolute squares of the off-diagonal entries ("coherences") of the pure state density matrix that are zeroed ("decohered") by the measurement, i.e., the measure of the distinctions ("decoherences") created by the measurement. (shrink)

John Tomasi's new book, Free Market Fairness, has been well-received as "one of the very best philosophical treatments of libertarian thought, ever" and as a "long and friendly conversation between Friedrich Hayek and John Rawls—a conversation which, astonishingly, reaches agreement". The book does present an authoritative state-of-the-debate across the spectrum from right-libertarianism on the one side to high liberalism on the other side. My point is not to question where Tomasi comes down with his own version of "market democracy" as (...) a remix of Hayek and Rawls. My point is to use his sympathetic restatements of views across the liberal spectrum to show the basic misframings and common misunderstandings that cut across the liberal-libertarian viewpoints surveyed in the book. As usual, the heart of the debate is not in the answers to carefully framed questions, but in the framing itself. (shrink)

Advancements in artificial intelligence and (social) robotics raise pertinent questions as to how these technologies may help shape the society of the future. The main aim of the chapter is to consider the social and conceptual disruptions that might be associated with social robots, and humanoid social robots in particular. This chapter starts by comparing the concepts of robots and artificial intelligence and briefly explores the origins of these expressions. It then explains the definition of a social robot, as well (...) as the definition of humanoid robots. A key notion in this context is the idea of anthropomorphism: the human tendency to attribute human qualities, not only to our fellow human beings, but also to parts of nature and to technologies. This tendency to anthropomorphize technologies by responding to and interacting with them as if they have human qualities is one of the reasons why social robots (in particular social robots designed to look and behave like human beings) can be socially disruptive. As is explained in the chapter, while some ethics researchers believe that anthropomorphization is a mistake that can lead to various forms of deception, others — including both ethics researchers and social roboticists — believe it can be useful or fitting to treat robots in anthropomorphizing ways. The chapter explores that disagreement by, among other things, considering recent philosophical debates about whether social robots can be moral patients, that is, whether it can make sense to treat them with moral consideration. Where one stands on this issue will depend either on one’s views about whether social robots can have, imitate, or represent morally relevant properties, or on how people relate to social robots in their interactions with them. Lastly, the chapter urges that the ethics of social robots should explore intercultural perspectives, and highlights some recent research on Ubuntu ethics and social robots. (shrink)

The region of South East Asia has developed over the past several years as one of the most rapidly growing aviation markets in the world. Despite the latest economic recession around the world, South East Asia is still reaching substantial growth. With the amount of flexible income increasing for the most inhabitants of the region, the aviation trade can expect to continue this movement. Singapore has been the dominant hub in the region for many years, however, Vietnam is contemplating the (...) practicality of investing in a new airport that they hope would eventually challenge Singapore’s current superiority. (shrink)

This article is a review of Erik Olin Wright’s 2010 book Envisioning Real Utopias. The review focuses on certain topics such as his understanding of ‘capitalism,’ his conception of worker cooperatives, and the general issues surrounding markets, the Left, and Marxism.

This article criticizes what I call "Raunchy" feminist art by employing discussions of pornography and objectification from Eaton and Nussbaum. Artists considered include Carolee Schneeman, Cindy Sherman, Lisa Yuskavage, and Jenny Saville. The article includes by citing examples of feminist art dealing with erotic material in a more productive manner: Eija-Liisa Ahtila, Kiki Smith, and Marlene Dumas.

Arthur C. Danto, philosopher and art critic for The Nation from 1984-2009, offered interpretations of artworks by a wide array of artists, including Eva Hesse, Judy Chicago, and Cindy Sherman, whose "disturbatory" works were either ignored or denounced by mainstream critics at the time. Danto's championing of feminist art was deliberate and delightful; he openly endorsed the Guerilla Girls! His feminist art critical writings ultimately shaped the early development of what has come to be known as "feminist aesthetics" particularly (...) his focus on defiant artworks by female artists who sought to reclaim agency and expression of the female body. He revived interest in the neglected topic of beauty while simultaneously advancing radical political goals within the artworld. His call to arms urged art viewers to experience the new--what he often called "art on the edge and over"--wherein, "The experience of art becomes a moral adventure rather than merely an aesthetic interlude" (in Art on the Edge and Over: Searching for Art's Meaning in Contemporary Society 1970s-1990s, 1996). (shrink)

This study examines playgrounds as lenses on urban transitions to explain the link between urban transformations and changes in the discourse of play and childhood. Specifically, it compares Soviet public playgrounds and post-Soviet privatized playscapes in the city of Yekaterinburg, Russia, through primary observation and secondary data analysis. Using the framework of social reproduction developed by Cindy Katz and Saskia Sassen to explain how the local forces affect cities, my analysis shows that the shift in the discourse of play (...) and childhood in the post-Soviet period is hinged on global influences combined with local transformations, from the abandonment of Soviet ideals of communal play spaces to the embracement of today’s consumerist play places. Whereas the old Soviet playgrounds have uncertain purposes, in contemporary Yekaterinburg private playgrounds offer a narrative of play in terms of leisure, love, and convenience for parents. Children turn into consumers of private play, leaving most of the Soviet playgrounds as idle spaces in the city. This article argues that Yekaterinburg’s shift toward participating in the globalized economy combined with its transition from the Soviet ideals maintains social relations and reproduces social inequalities in childhood, as this condition favors consumerist narratives of play. I conclude that the playgrounds in Yekaterinburg are bystanders of new global ecologies whereby social, political, and economic transformations become an impetus to reproduce new ways of seeing the social importance and meaning of play and playgrounds. (shrink)

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). (shrink)

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 (...) with the quantitative versions of subsets and partitions. Some basic universal mapping properties in the category of Sets are developed that precede the abstract duality of category theory. But by far the main application is to the clarification and interpretation of quantum mechanics. Since classical mechanics illustrates the Boolean worldview of full distinctness, it is natural that quantum mechanics would be based on the indefiniteness of its characteristic superposition states, which is modeled at the set level by partitions (or equivalence relations). This approach to interpreting quantum mechanics is not a jury-rigged or ad hoc attempt at the interpretation of quantum mechanics but is a natural application of the fundamental duality running throughout the exact sciences. (shrink)

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 (...) second task is to recover that theory and to consider several other applications of the theory. Finally, the liberal theories of justice expounded by John Rawls and by Robert Nozick are briefly examined from this perspective. (shrink)

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.

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. (...) The argument for economic democracy supports private property, free markets and entrepreneurship for instance, but fundamentally it replaces the employer/employee relationship with democratic membership in the firm. For students, teachers, policy makers and others interested in the application of democracy to the workplace, this book will serve as a manifesto and a standard reference on the topic. (shrink)

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. (shrink)

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. (shrink)

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 (...) a universe set, then the question naturally arises of a dual logic of partitions on a universe set. This paper is an introduction to that logic of partitions dual to classical subset logic. The paper goes from basic concepts up through the correctness and completeness theorems for a tableau system of partition logic. (shrink)

This paper considers ethical concerns with regard to replacing human relations with humanoid robots. Many have written about the impact that certain types of relations with robots may have on us, and why we should be concerned about robots replacing human relations. There has, however, been no consideration of this issue from an African philosophical perspective. Ubuntu philosophy provides a novel perspective on how relations with robots may impact our own moral character and moral development. This paper first discusses what (...) humanoid robots are, why and how humans tend to anthropomorphise them, and what the literature says about robots crowding out human relations. It then explains the ideal of becoming “fully human”, which pertains to being particularly moral in character. In ubuntu philosophy, we are not only biologically human, but must strive to become better, more moral versions of ourselves, to become fully human. We can become fully human by having other regarding traits or characteristics within the context of interdependent, or humane, relationships (such as by exhibiting human equality, reciprocity, or solidarity). This concept of becoming fully human is important in ubuntu philosophy. Having explained that idea, the main argument of the paper is then put forward: that treating humanoid robots as if they are human is morally concerning if they crowd out human relations, because such relations prevent us from becoming fully human. This is because we cannot experience human equality, solidarity, and reciprocity with robots, which can be seen to characterise interdependent, or humane, relations with human beings. (shrink)

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 (...) of subsets is the normalized counting measure of the subsets (events). Thus logical entropy is a measure on the set of ordered pairs, and all the compound notions of entropy (join entropy, conditional entropy, and mutual information) arise in the usual way from the measure (e.g., the inclusion-exclusion principle)--just like the corresponding notions of probability. The usual Shannon entropy of a partition is developed by replacing the normalized count of distinctions (dits) by the average number of binary partitions (bits) necessary to make all the distinctions of the partition. (shrink)

It has come to be widely accepted that jus post bellum includes responsibilities to rebuild. Consequently, duties to establish a sustainable peace are increasingly defined in terms of duties to protect and promote international human rights, including duties to effectively investigate human rights violations, to ensure access to effective remedy, and to transform institutional and legal contexts that have facilitated or sustained human abuse. But what are investigations by transitional bodies seeking when they take on these tasks? Often, investigators present (...) themselves as seeking the truth and claim value for their findings based on having produced a better description or explanation than was antecedently available. But is truth worth pursuing for its own sake in transitions from conflict or must it contribute to some other goal to be valuable? In what follows I argue that the value of truth in transitions from conflict lies in the role of truth in ascribing knowledge. The connection between truth and knowledge makes it important to preserve distinctively epistemological grounds for accepting and rejecting claims and narratives. These distinctively epistemological grounds explain how, if it is conceived of as adequate responsiveness to experience, truth is a legitimate and important goal in transitions from violence that is worth pursuing for its own sake. In particular, I argue that there is value in insisting that adequate responsiveness to experience serve as an arbiter in the reception of claims and narratives because when responsiveness to experience plays an arbitral role it becomes possible for bodies such as truth commissions to serve as vehicles by which groups may attain or ascribe knowledge. (shrink)

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 (...) fundamental norm of classical liberalism is consent, we must start with the intellectual history of the voluntary slavery contract, the coverture marriage contract, and the voluntary non-democratic constitution (or pactum subjectionis). Next we recover the theory of inalienable rights that descends from the Reformation doctrine of the inalienability of conscience through the Enlightenment (e.g., Spinoza and Hutcheson) in the abolitionist and democratic movements. Consent-based governments divide into those based on the subjects' alienation of power to a sovereign and those based on the citizens' delegation of power to representatives. Inalienable rights theory rules out that alienation in favor of delegation, so the citizens remain the ultimate principals and the form of government is democratic. Thus the argument concludes in agreement with Buchanan 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. (shrink)

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 (...) understood. On the normative side, neoclassical theory represents marginal productivity theory as showing that (a metaphorical version of) the imputation principle is satisfied ('people get what they produce') in competitive enterprises. Since that shows the moral commitment of neoclassical economics to the imputation principle, the labour theory of property is presented here as the actual non-metaphorical application of the imputation principle to property appropriation. The property-theoretic analysis at the firm level shows how the neoclassical (and much heterodox) analysis in terms of 'distributive shares' wholly misframed the basic questions. Finally, the paper shows how the imputation principle (modernised labour theory of property) is systematically violated in the present wage labour system of renting persons. The paper can be seen as taking up the recent challenge posed by Donald Katzner for a dialogue between neoclassical and heterodox microeconomics. (shrink)

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.

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. (shrink)

The lattice operations of join and meet were defined for set partitions in the nineteenth century, but no new logical operations on partitions were defined and studied during the twentieth century. Yet there is a simple and natural graph-theoretic method presented here to define any n-ary Boolean operation on partitions. An equivalent closure-theoretic method is also defined. In closing, the question is addressed of why it took so long for all Boolean operations to be defined for partitions.

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 (...) is focused on the interpretation and application of the mathematical concepts. (shrink)

Historically, culture has been treated as an object in international documents. One consequence of this is that cultural rights in international law have been understood as rights of access and consumption. Recently, an alternative conception of culture, and of what cultural rights protect, has emerged from international documents treating indigenous peoples. Within these documents culture is treated as an activity rather than a good. This activity is ascribed to peoples as well as persons, and protecting the capacity of both peoples (...) and persons to engage in culture is taken to be as basic a component of human dignity as are freedom of movement, freedom of speech, and freedom from torture. It is not an accident that this treatment of culture has emerged from international documents treating indigenous peoples. However, the value of this treatment of culture extends beyond the human rights of indigenous peoples. Treating culture as an activity establishes an understanding of what cultural rights protect that clarifies the relationship between cultural rights and other mechanisms for protecting minorities and frames the role of cultural communities in the realization of human dignity as an important physical and political issue, not just a psychological one. For these reasons, the norms regarding cultural rights that are emerging from international documents treating indigenous peoples are a much-needed step forward for peoples' rights more generally. (shrink)

The controversy that accompanied the publication and reprinting of cartoons depicting the Prophet Muhammed as part of a 2005 editorial in the Danish newspaper Jyllands-Posten has been widely interpreted as yet another illustration of an ineliminable tension between multiculturalism and liberalism. Such an interpretation would have us believe that what is at issue in defending the cartoons is our commitment to civil liberties as a mainstay of liberal democracy. But is this really what is at issue? A closer examination suggests (...) that what is actually being defended in this case is not civil liberty but civil privilege. In particular, what is at issue is the privilege to exclude and define Muslims. (shrink)

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. (shrink)

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 (...) it immediately yields the generalization of the double-entry method to multi-dimensional vectors typically representing the different types of property involved in an enterprise or household. (shrink)

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 (...) along with a new concept to abstractly model the functions of a brain. (shrink)