David Ellerman

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.

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.

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.

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.

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."

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.

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 ..

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 ›

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.

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.

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.