In [Countable thin [Formula: see text] classes, Ann. Pure Appl. Logic 59 79–139], Cenzer, Downey, Jockusch and Shore proved the density of degrees containing members of countable thin [Formula: see text] classes. In the same paper, Cenzer et al. also proved the existence of degrees containing no members of thin [Formula: see text] classes. We will prove in this paper that the c.e. degrees containing no members of thin [Formula: see text] classes are dense in the c.e. degrees. We will (...) also prove that the c.e. degrees containing members of thin [Formula: see text] classes are dense in the c.e. degrees, improving the result of Cenzer et al. mentioned above. Thus, we obtain a new natural subclass of c.e. degrees which are both dense and co-dense in the c.e. degrees, while the other such class is the class of branching c.e. degrees 113–130] for nonbranching degrees and [T. A. Slaman, The density of infima in the recursively enumerable degrees, Ann. Pure Appl. Logic 52 155–179] for branching degrees). (shrink)

Argues that a set is the mereological whole of the singleton sets of its members (following Lewis's Parts of Classes), and that the singleton set of X is the state of affairs of X's having some unit-making property.

This Article attempts to clarify a proposition of certain Canadian authors that while class actions represent a significant part of our court activities, they may not truly be compensating our citizens. I argue that leading up to the present study, we did not know for certain whether a class action was an effective mechanism to compensate class members. Through empirical data collected up by the Class Actions Lab from the past twelve years from cases filed in (...) the province of Quebec, District of Montreal, analyzed through the lens of a collective approach to compensation, I demonstrate that Quebec citizens are in fact being compensated by class actions. (shrink)

Most human societies exhibit a distinct class structure, with an elite, middle classes, and a bottom class, whereas animals form simple dominance hierarchies in which individuals with higher fighting ability do not appear to form coalitions to “oppress” weaker individuals. Here, we extend our model of primate coalitions and find that a division into a bottom class and an upper class is inevitable whenever fitness-enhancing resources, such as food or real estate, are exploitable or tradable and (...) the members of the bottom class cannot easily leave the group. The model predicts that the bottom class has a near flat, low payoff and always comprises at least half the society. The upper class may subdivide into one or more middle class(es), resulting in improved payoff for the topmost members (elite). The model predicts that the bottom class on its own is incapable of mounting effective counter-coalitions against the upper class, except when receiving support from dissatisfied members of the middle class(es). Such counter-coalitions can be prevented by keeping the payoff to the lowest-ranked members of the middle classes (through concessions) well above that of the bottom class. This simple model explains why classes are also absent in nomadic hunter-gatherers and predominate in (though are not limited to) societies that produce and store food. Its results also agree well with various other known features of societies with classes. (shrink)

A set of infinite binary sequences ${\cal C} \subseteq 2$ℕ is negligible if there is no partial probabilistic algorithm that produces an element of this set with positive probability. The study of negligibility is of particular interest in the context of ${\rm{\Pi }}_1^0 $ classes. In this paper, we introduce the notion of depth for ${\rm{\Pi }}_1^0 $ classes, which is a stronger form of negligibility. Whereas a negligible ${\rm{\Pi }}_1^0 $ class ${\cal C}$ has the property that one (...) cannot probabilistically compute a member of ${\cal C}$ with positive probability, a deep ${\rm{\Pi }}_1^0 $ class ${\cal C}$ has the property that one cannot probabilistically compute an initial segment of a member of ${\cal C}$ with high probability. That is, the probability of computing a length n initial segment of a deep ${\rm{\Pi }}_1^0 $ class converges to 0 effectively in n.We prove a number of basic results about depth, negligibility, and a variant of negligibility that we call tt-negligibility. We provide a number of examples of deep ${\rm{\Pi }}_1^0 $ classes that occur naturally in computability theory and algorithmic randomness. We also study deep classes in the context of mass problems, examine the relationship between deep classes and certain lowness notions in algorithmic randomness, and establish a relationship between members of deep classes and the amount of mutual information with Chaitin’s Ω. (shrink)

I argue that causation is a contrastive relation: c-rather-than-C* causes e-rather-than-E*, where C* and E* are contrast classes associated respectively with actual events c and e. I explain why this is an improvement on the traditional binary view, and develop a detailed definition. It turns out that causation is only well defined in ‘uniform’ cases, where either all or none of the members of C* are related appropriately to members of E*.

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out several examples. One motivation is to see whether some classes whose set of countable members is very complex become classifiable when we consider only computable members. We follow recent work (...) by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from non-classifiable. For real closed fields we show that the isomorphism problem is Δ1 1 complete (the maximum possible), and for others we show that it is of relatively low complexity. We show that the isomorphism problem for algebraically closed fields, Archimedean real closed fields, or vector spaces is Π0 3 complete. (shrink)

We examine how degrees of computably enumerable equivalence relations under computable reduction break down into isomorphism classes. Two ceers are isomorphic if there is a computable permutation of ω which reduces one to the other. As a method of focusing on nontrivial differences in isomorphism classes, we give special attention to weakly precomplete ceers. For any degree, we consider the number of isomorphism types contained in the degree and the number of isomorphism types of weakly precomplete ceers contained in the (...) degree. We show that the number of isomorphism types must be 1 or ω, and it is 1 if and only if the ceer is self-full and has no computable classes. On the other hand, we show that the number of isomorphism types of weakly precomplete ceers contained in the degree can be any member of $[0,\omega ]$. In fact, for any $n \in [0,\omega ]$, there is a degree d and weakly precomplete ceers ${E_1}, \ldots,{E_n}$ in d so that any ceer R in d is isomorphic to ${E_i} \oplus D$ for some $i \le n$ and D a ceer with domain either finite or ω comprised of finitely many computable classes. Thus, up to a trivial equivalence, the degree d splits into exactly n classes.We conclude by answering some lingering open questions from the literature: Gao and Gerdes [11] define the collection of essentially FC ceers to be those which are reducible to a ceer all of whose classes are finite. They show that the index set of essentially FC ceers is ${\rm{\Pi }}_3^0$-hard, though the definition is ${\rm{\Sigma }}_4^0$. We close the gap by showing that the index set is ${\rm{\Sigma }}_4^0$-complete. They also use index sets to show that there is a ceer all of whose classes are computable, but which is not essentially FC, and they ask for an explicit construction, which we provide.Andrews and Sorbi [4] examined strong minimal covers of downwards-closed sets of degrees of ceers. We show that if $\left$ is a uniform c.e. sequence of non universal ceers, then $\left\{ {{ \oplus _{i \le j}}{E_i}|j \in \omega } \right\}$ has infinitely many incomparable strong minimal covers, which we use to answer some open questions from [4].Lastly, we show that there exists an infinite antichain of weakly precomplete ceers. (shrink)

This article explores an under-examined theme, which is who or what is the working class and what is wrong with the situation that members of this class share. It argues that class divisions impose a unique harm for a diverse and interdependent group within capitalist societies both in spite and because of differences among group members. Class matters not just because it creates economic groups in which some are rich and others are poor, but because competition (...) creates conditions that militate against solidarity, toward cleavage and conflict. Class is a constraint on collective self-determination and, therefore, a source of domination. (shrink)

Inspired by and responding to Avtar Brah's ‘The Scent of Memory’, this piece attempts to reinscribe race into an auto-ethnographic narrative where previously whiteness was unmarked. It explores the dynamics of gender, race and class through the author's personal history as a white English woman and class migrant, and through discussion of the broader political and historical context of that trajectory. The discussion includes analysis of the impact of British Conservative politician Enoch Powell's infamous ‘rivers of blood’ speech (...) in 1968 on the author's white English working-class culture of origin in Wolverhampton, where Powell was a Member of Parliament. The article considers the speech's continuing ramifications in the twenty-first century and in more middle-class contexts, as evidenced by the recent evocation of the speech by historian David Starkey in discussion of the ‘riots’ of August 2011 in British cities. The personal history is reconstructed through a series of memory scenes that trace and retrace the author's experience and understanding of race and its intersections with class and gender; this is attempted in full cognisance of the constructed nature of memory, and of the performance of identity that autobiography entails. The piece draws on the work of the class migrant white French writer Annie Ernaux, with whom the author has been in dialogue since 1997. (shrink)

In his paper "Why a Class Can't Change Its Members," Richard Sharvy appears to establish the impossibility of the existence of a variable class—that is, a class that at one time has a member that is not a member of it at another time. I first indicate the importance of Sharvy's argument for our understanding of the concept of identity in the contexts of time and modality, and I summarize his argument. Sharvy says that a (...) class C that has one (non-variable) group of members at t2 and another (nonvariable) group of members at t2 would be identical with both the class C1 that always has the first group as members and the class C2 that always has the second group as members. This is an impossibility, since in general, one thing cannot be identical with two.I then criticize Sharvy's argument by pointing out a weakness in the defense of the claim that C = C1 and C = C2. This weakness is due to an ambiguity in Sharvy's Principle of Extensionality. (shrink)

The notion of a "class as many" was central to Bertrand Russell''s early form of logicism in his 1903 Principles of Mathematics. There is no empty class in this sense, and the singleton of an urelement (or atom in our reconstruction) is identical with that urelement. Also, classes with more than one member are merely pluralities — or what are sometimes called "plural objects" — and cannot as such be themselves members of classes. Russell did not formally (...) develop this notion of a class but used it only informally. In what follows, we give a formal, logical reconstruction of the logic of classes as many as pluralities (or plural objects) within a fragment of the framework of conceptual realism. We also take groups to be classes as many with two or more members and show how groups provide a semantics for plural quantifier phrases. (shrink)

Classes are the objects of the second sort of second-order set theory. They have sets as their members and behave like sets, but paradoxes tell us that many classes cannot be sets. Then, what are classes? Predicativism about classes suggests that classes are predicates of sets, and this article investigates the question from the predicativist point of view in light of recent developments in the use of classes in set theory. Predicativism has been considered too restrictive and unable to accommodate (...) the use of classes in set theory. This diagnosis, however, is only true of a certain specific type of predicativism. In this article, we propose a new type of predicativism, which we call liberal predicativism, and argue that predicativism is still a highly viable option, and our liberal version provides a sufficiently versatile and workable nominalist concept of classes for set theory. (shrink)

This paper explores how gender can operate as a disguise for class in an examination of the self-sacrifice of the Maiden in Euripides' Children of Herakles. In Part I, I discuss the role of human sacrifice in terms of its radical potential to transform society and the role of class struggle in Athens. In Part II, I argue that the representation of women was intimately connected with the social and political life of the polis. In a discussion of (...) iconography, the theater industry and audience I argue that female characters became one of the means by which different groups promoted partisan interests based on class and social status. In Part III, I show how the Maiden solicits the competing interests of the theater audience. After discussing the centrality (as a heroine from an aristocratic family) and marginality (as a woman and associated with other marginal social groups) of the Maiden's character, I draw upon the funeral oration as a comparative model with which to understand the quite different role of self-sacrifice in tragedy. In addition to representing and mystifying the interests of elite, lower class and marginal groups, the play glorifies a subordinate character whose contradictory social status (both subordinate and elite) embodies the social position of other “marginal” members of Athenian society. The play stages a model for taking political action to transform the social system and for commemorating the tragic costs of such undertakings. (shrink)

This paper explores how gender can operate as a disguise for class in an examination of the self-sacrifice of the Maiden in Euripides' Children of Herakles. In Part I, I discuss the role of human sacrifice in terms of its radical potential to transform society and the role of class struggle in Athens. In Part II, I argue that the representation of women was intimately connected with the social and political life of the polis. In a discussion of (...) iconography, the theater industry and audience I argue that female characters became one of the means by which different groups promoted partisan interests based on class and social status. In Part III, I show how the Maiden solicits the competing interests of the theater audience. After discussing the centrality and marginality of the Maiden's character, I draw upon the funeral oration as a comparative model with which to understand the quite different role of self-sacrifice in tragedy. In addition to representing and mystifying the interests of elite, lower class and marginal groups, the play glorifies a subordinate character whose contradictory social status embodies the social position of other “marginal” members of Athenian society. The play stages a model for taking political action to transform the social system and for commemorating the tragic costs of such undertakings. (shrink)

It is well known that there may be many causal explanations that are consistent with a given set of data. Recent work has been done to represent the common aspects of these explanations into one representation. In this paper, we address what is less well known: how do the relationships common to every causal explanation among the observed variables of some DAG process change in the presence of latent variables? Ancestral graphs provide a class of graphs that can encode (...) conditional independence relations that arise in DAG models with latent and selection variables. In this paper we present a set of orientation rules that construct the Markov equivalence class representative for ancestral graphs, given a member of the equivalence class. These rules are sound and complete. We also show that when the equivalence class includes a DAG, the equivalence class representative is the essential graph for the said DAG. (shrink)

This paper investigates a quasi-variety of representable integral commutative residuated lattices axiomatized by the quasi-identity resulting from the well-known Wajsberg identity → q ≤ → p if it is written as a quasi-identity, i. e., → q ≈ 1 ⇒ → p ≈ 1. We prove that this quasi-identity is strictly weaker than the corresponding identity. On the other hand, we show that the resulting quasi-variety is in fact a variety and provide an axiomatization. The obtained results shed some light (...) on the structure of Archimedean integral commutative residuated chains. Further, they can be applied to various subvarieties of MTL-algebras, for instance we answer negatively Hájek's question asking whether the variety of ΠMTL-algebras is generated by its Archimedean members. (shrink)

'Natural class' trope nominalism makes a trope's being of a certain sort--its nature--a matter of its membership in a certain natural class of actual tropes. It has been objected that on this theory had even a single member of the class of red tropes not existed, for example, then the type 'being red' would not have been instantiated and nothing would have been red. I argue that natural class trope nominalism can avoid this implication by (...) way of counterpart theory as applied to properties. (shrink)

Necessary and sufficient conditions are presented for the (first-order) theory of a universal class of algebraic structures (algebras) to have a model completion, extending a characterization provided by Wheeler. For varieties of algebras that have equationally definable principal congruences and the compact intersection property, these conditions yield a more elegant characterization obtained (in a slightly more restricted setting) by Ghilardi and Zawadowski. Moreover, it is shown that under certain further assumptions on congruence lattices, the existence of a model completion (...) implies that the variety has equationally definable principal congruences. This result is then used to provide necessary and sufficient conditions for the existence of a model completion for theories of Hamiltonian varieties of pointed residuated lattices, a broad family of varieties that includes lattice-ordered abelian groups and MV-algebras. Notably, if the theory of a Hamiltonian variety of pointed residuated lattices has a model completion, it must have equationally definable principal congruences. In particular, the theories of lattice-ordered abelian groups and MV-algebras do not have a model completion, as first proved by Glass and Pierce, and Lacava, respectively. Finally, it is shown that certain varieties of pointed residuated lattices generated by their linearly ordered members, including lattice-ordered abelian groups and MV-algebras, can be extended with a binary operation to obtain theories that do have a model completion. (shrink)

In transfinite arithmetic 2n is defined as the cardinality of the family of all subsets of some set v with cardinality n. However, in the arithmetic of recursive equivalence types 2N is defined as the RET of the family of all finite subsets of some set v of nonnegative integers with RET N. Suppose v is a nonempty set. S is a class over v, if S consists of finite subsets of v and has v as its union. Such (...) a class is an intersecting class over v, if every two members of S have a nonempty intersection. An IC over v is called a maximal IC , if it is not properly included in any IC over v. It is known and readily proved that every MIC over a finite set v of cardinality n ≥ 1 has cardinality 2n-1. In order to generalize this result we introduce the notion of an ω-MIC over v. This is an effective analogue ot the notion of an MIC over v such that a class over a finite set v is an ω-MIC iff it is an MIC. We then prove that every ω-MIC over an isolated set v of RET N ≥ 1 has RET 2N-1. This is a generalization, for while there only are χ0 finite sets, there are ϰ isolated sets, where c denotes the cardinality of the continuum, namely all the finite sets and the c immune sets. MSC: 03D50. (shrink)

The author reviews the planned withdrawal of healthcare from the primary public hospital, Grady Memorial Hospital, of Atlanta, Georgia. Prior to passage of the Affordable Care Act of 2010, at least half of the patients had no public or private health insurance and their care was financially supported by State and County funding as well as supplementation from Emory University. New administration in the elected positions of the State and County and at the University reached agreement to decrease care. The (...) University President stated that the Hippocratic Oath was no longer valid for salaried physicians who must follow the direction of their business manager. The author, a tenured Professor, was forcibly transferred to the Veterans Hospital, which also was withholding standard medical care, including life-saving medications. Organizing multiple concerned organizations including members of the King Center, the Carter Center, the Open Door Community Center, Gay Rights, etc., (Grady Coalition) multiple public protests were successful in raising the necessary funds to continue adequate care at Grady and begin an investigation of the Veterans Affairs Hospital. (shrink)

"You have to come to my wedding," Kavita told me, turning to face me where I sat next to her on the couch. "You can come with the other people from the street. You will get everything you need for your *research* there." "I will come, I will come!" I replied enthusiastically. I had only met Kavita and her two younger sisters, Arthi and Deepti (see Figure 2.1), mere minutes before this invitation was extended. I had initially come to Pulan (...) that day in October 2012 to meet another woman, Heena, whose family rents a room on the third story of Kavita's family's home. Heena and I had been sitting in the furniture refurbishing store she operates with her husband on the main street of Pulan when Deepti, Kavita's youngest sister, passed by. Heena introduced us and told me to go with Deepti to meet her family. When we reached the family's three-story house-the largest in the gali-Deepti led me past the empty rooms on the ground floor, which I would eventually begin renting, to the second-story living room. There, we found Kavita and Arthi organizing clothing and jewelry they had purchased earlier in the day for the upcoming wedding festivities. Kavita made room for me to sit next to her on the couch and began asking me about myself. I immediately warmed to her because of her open, friendly smile and sharp, staccato Hindi, which I delighted in being able to understand. I explained that I had come to India to study how women's lives are different in rural and urban areas, and Kavita assured me that she and her family could help. She noted that her parents had come to Udaipur from Ram Nagar, a large village thirty-five kilometers north of the city, and that the family would be returning for her and her older brother Krishna's weddings the following month. Their weddings would be held five days apart to help reduce the difficulties of family members traveling from outside Udaipur. Prompted by the description of my research, Kavita commented on differences that she recognized between the village and the city. The biggest difference, she suggested, was the experience of caste, namely that in the village, people from different jatis live separately, whereas in the city, people are "mixed." As I would come to learn when visiting Ram Nagar for various functions, there is a fair amount of caste and religious diversity in the village. Although spatial and ritual segregation was rather strictly maintained during religious observances, it is likely more flexible in everyday life. The segregation during ritual functions-the occasions for which Kavita also traveled to the village-likely informed her sense of a lack of "mixing" in the village as. The majority of residents in the area of Ram Nagar where the family maintains a home were also from the Mali (lit: gardener) jati, although Mali was not a majority jati in Pulan. (shrink)

A Π01 class is an effectively closed set of reals. We study properties of these classes determined by cardinality, measure and category as well as by the complexity of the members of a class P. Given an effective enumeration {Pe:e < ω} of the Π01 classes, the index set I for a certain property is the set of indices e such that Pe has the property. For example, the index set of binary Π01 classes of positive measure is (...) Σ02 complete. Various notions of boundedness are discussed and classified. For example, the index set of the recursively bounded classes is Σ03 complete and the index set of the recursively bounded classes which have infinitely many recursive members is Π04 complete. Consideration of the Cantor-Bendixson derivative leads to index sets in the transfinite levels of the hyperarithmetic hierarchy. (shrink)

Cenzer, D., R. Downey, C. Jockusch and R.A. Shore, Countable thin Π01 classes, Annals of Pure and Applied Logic 59 79–139. A Π01 class P {0, 1}ω is thin if every Π01 subclass of P is the intersection of P with some clopen set. Countable thin Π01 classes are constructed having arbitrary recursive Cantor- Bendixson rank. A thin Π01 class P is constructed with a unique nonisolated point A and furthermore A is of degree 0’. It is shown (...) that no set of degree ≥0” can be a member of any thin Π01 class. An r.e. degree d is constructed such that no set of degree d can be a member of any thin Π01 class. It is also shown that between any two distinct comparable r.e. degrees, there is a degree that contains a set which is of rank one in some thin Π01 class. It is shown that no maximal set can have rank one in any Π01 class, while there exist maximal sets of rank 2. The connection between Π01 classes, propositional theories and recursive Boolean algebras is explored, producing several corollaries to the results on Π01 classes. For example, call a recursive Boolean algebra thin if it has no proper nonprincipal recursive ideals. Then no thin recursive Boolean algebra can have a maximal ideal of degree ≥0”. (shrink)

Sensory classification is a central theme of Mohan Matthen's wonderful book, Seeing, Doing, and Knowing. ( All page references are to Matthen 2005 unless otherwise indicated.) My plan for this commentary is simple: I shall list a series of claims that Matthen makes about the classes involved in sensory classification. Each member of the series is admirable, and seems credible on its own. The question at the end is whether we can hold them all, together.

In this paper the authors argue that if Tarski’s definition of truth for the calculus of classes is correct, then set theories which assert the existence of proper classes (classes which are not the member of anything) are incorrect.

The central result of this paper provides a simple equational basis for the join, IRLLG, of the variety LG of lattice-ordered groups (-groups) and the variety IRL of integral residuated lattices. It follows from known facts in universal algebra that IRLLG=IRL×LG. In the process of deriving our result, we will obtain simple axiomatic bases for other products of classes of residuated structures, including the class IRL×s LG, consisting of all semi-direct products of members of IRL by members of LG. (...) We conclude the paper by presenting a general method for constructing such semi-direct products, including wreath products. (shrink)

Over last half of the twentieth century, a silent revolution in post-material values made significant advances around the world. The formation of post-material values also resulted in expanded participation in post-material experiences such as joining voluntary groups, pursuing creativity and independence in the world of work, and engaging in political actions—experiences that go beyond a strict focus on accumulating economic wealth and material possessions. Because social class position matters for being a post-materialist, a class divide exists between middle- (...) class='Hi'>class post-materialists and working-class materialists who occupy the lower end of the social class spectrum. This article will show that such a divide occurs, first for participation in post-material experiences and second for the attainment of life satisfaction. Using data from the World Values Survey, evidence will be provided in this article showing that members of the working class participate less in post-material experiences than others and, as a consequence, enjoy lower life satisfaction than others, especially post-materialists. In light of working-class-supported right-wing populism’s recent emergence in the USA and Europe, this social class divide takes on a special significance in explaining shifting trends in politics and public policy. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Graduate Student Member Spotlights Blog for SBCS:Chera Jo WattsChera Jo WattsMy name is Chera Jo Watts, and I am a first-year doctoral student at the University of Georgia in the Department of Religion and Institute for African American Studies. I am a mother, writer, gardener, yoga practitioner, and artist striving toward what Darlene Clark Hines labels a "Black Studies Mindset." As a first-generation college graduate from a poverty- (...) class='Hi'>class background, my degrees include a Bachelor of Science in Psychology, a Master of Arts in Religion, and a Graduate Certificate in African American Studies from the University of Georgia. My broad research interests include African American women's religion and literature, focusing primarily on Womanism, and bridging the gap between the Academy and the everyday. I assert that we have much to learn from our ancestors and from each other while living and operating in what Black Buddhist bell hooks labels an imperialist white supremacist capitalist cispatriarchy. These teachings facilitate personal and communal healing as we continuously dismantle white supremacy in the tangible ways we can from the spaces that we occupy. Also, I currently serve as the Graduate Teaching Assistant in the Institute for African American Studies at the University of Georgia.Within SBCS, I am a graduate student member, and I strongly encourage other graduate students to consider membership in this organization. This community consists of imaginative and generous scholars, and you could make numerous international connections within the field of religious studies (along with other disciplines). As the current Executive Associate for Digital Services, I currently support the Society in a professional role through web updates, monitoring email, and various tasks as instructed by the governing board.How do your research interests relate to the SBCS? What are you working on at the moment?I am currently learning from published Womanists-Buddhists, and SBCS has published several journal articles focused on this thread of Womanist thought and scholarship (especially see Volumes 34, 2014, and 36, 2016). I am grateful to be in conversation with these folks through their work.In May 2022, I finished my Master's Thesis (published in ProQuest) under the direction of Dr. Carolyn Jones Medine. My thesis focuses on Womanist Buddhist thought in the works of Alice Walker while paying special attention to her first novel, The Third Life of Grange Copeland, and then tracing the importance of her Buddhist practice across other works, such as essays and poetry. My first peer-reviewed journal [End Page 273] article was also published in 2022. Reading between the Times: An Ongoing Womanist Buddhist Project may be accessed online through MDPI, and it was included as part of a Special Edition of the journal which focused on spirituality, identity, and resistance in African American literature. At the moment, I have a few journal articles in process which extend this work, and I expanded the analysis to include Black Buddhist bell hooks. For hooks, I am especially interested in her focus on living from a "love ethic," and I'm grateful for an upcoming panel presentation at the 2023 bell hooks Symposium hosted by Berea College this June. Finally, along with Dr. Carolyn Jones Medine, I am co-authoring an Oxford Bibliography entry for "Alice Walker," in which we acknowledge the importance of Walker's spiritual practices, including Buddhism, on her life and writings.Contact information: [email protected] | University of Georgia | Institute for African American Studies and Department of Religion. [End Page 274]Chera Jo WattsExecutive Associate for Digital ServicesCopyright © 2023 University of Hawai'i Press... (shrink)

A new reduction class is presented for the satisfiability problem for well-formed formulas of the first-order predicate calculus. The members of this class are closed prenex formulas of the form ∀ x∀ yC. The matrix C is in conjunctive normal form and has no disjuncts with more than three literals, in fact all but one conjunct is unary. Furthermore C contains but one predicate symbol, that being unary, and one function symbol which symbol is binary.

For a class K of structures, closed under isomorphism, the index set is the set I of all indices for computable members of K in a universal computable numbering of all computable structures for a fixed computable language. We study the complexity of the index set of class of structures with decidable theories. We first prove the result for the class of all structures in an arbitrary finite nontrivial language. After the complexity is found, we prove similar (...) results for some well-known classes of structures, such as directed graphs, undirected graphs, partial orders and lattices. (shrink)

In December 1984 Angela Weir and Elizabeth Wilson, two founding members of Feminist Review, published an article assessing contemporary British feminism and its relationship to the left and to class struggle. They suggested that the women's movement in general, and socialist-feminism in particular, had lost its former political sharpness. The academic focus of socialist-feminism has proved more interested in theorizing the ideological basis of sexual difference than the economic contradictions of capitalism. Meanwhile the conditions of working-class and black (...) women have been deteriorating. In this situation, they argue, feminists can only serve the general interests of women through alliance with working-class movements and class struggle. Weir and Wilson represent a minority position within the British Communist Party (the CP), which argues that ‘feminism’ is now being used by sections of the left, in particular the dominant ‘Eurocommunist’ left in the CP, to justify their moves to the right, with an accompanying attack on traditional forms of trade union militancy. Beatrix Campbell, who is aligned to the dominant position within the CP, has been one target of Weir and Wilson's criticisms. In several articles from 1978 onwards, and in her book Wigan Pier Revisited, Beatrix Campbell has presented a very different analysis of women and the labour movement. She has criticized the trade union movement as a ‘men's movement’, in the sense that it has always represented the interests of men at the expense of women. And she has described the current split within the CP as one extending throughout the left between the politics of the ‘old’ and the ‘new’: traditional labour movement politics as against the politics of those who have rethought their socialism to take into account the analysis and importance of popular social movements – in particular feminism, the peace and anti-racist movements. In reply to this debate, Anne Phillips has argued that while women's position today must be analysed in the context of the capitalist crisis, it is not reducible to the dichotomy ‘class politics’ versus ‘popular alliance’. Michèle Barrett, in another reply to Weir and Wilson, has argued that they have presented a reductionist and economistic approach to women's oppression, which caricatures rather than clarifies much of the work in which socialist-feminists have been engaged. To air these differences between socialist-feminists over the question of feminism and class politics, and to see their implications for the women's movement and the left, Feminist Review has decided to bring together the main protagonists of this debate for a fuller, more open discussion. For this discussion Feminist Review drew up a number of questions which were put to the participants by Clara Connolly and Lynne Segal. (Michèle Barrett was present in a personal capacity.) They cover the recent background to socialist-feminist politics, the relationship of feminism to Marxism, the role of feminists in le ft political parties and the labour movement, the issue of racism and the prospects for the immediate future. The discussion was lengthy and what follows is an edited version of the transcript. (shrink)

Annotation. There has never been a writer quite like Christopher Pearson. A BETTER CLASS OF SUNSET brings together Pearson's enduring columns and essays, chronicling the twists and turns of Australian cultural life over three intriguing decades. From climate change to homosexuality and the welfare of Indigenous Australians, Pearson was an unflagging champion of unfashionable causes, a conscientious objector to political correctness and a skilled dissector of muddle-headed arguments. They reveal a true man of letters with a rare literary voice; (...) passionate, moving and often very funny, Pearson's thinking seldom strays from his abiding principles. For all those who aspire to be members of what Pearson called Club Sensible, A BETTER CLASS OF SUNSET is an indispensible addition to the bookshelf. (shrink)

We present some characterizations of the members of Δta2, that class of the topological arithmetical hierarchy which is just large enough to include several fundamental types of sets of points in Euclidean spaces ℝk. The limit characterization serves as a basic tool in further investigations. The characterization by effective difference chains of effectively exhaustible sets yields only a hierarchy within a subfield of Δta2. Effective difference chains of transfinite (but constructive) order types, consisting of complements of effectively exhaustible sets, (...) as well as another closely related concept, yield a rich hierarchy within the whole class Δta2. The presentation always first reports analogies between Hausdorff's difference hierarchy within the Borel class ΔB2 and Ershov's hierarchy within the class Δ02 of the arithmetical hierarchy; after that the counterparts for Δta2 are developed. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

A $\Pi^0_1$ class can be defined as the set of infinite paths through a computable tree. For classes $P$ and $Q$, say that $P$ is Medvedev reducible to $Q$, $P \leq_M Q$, if there is a computably continuous functional mapping $Q$ into $P$. Let $\mathcal{L}_M$ be the lattice of degrees formed by $\Pi^0_1$ subclasses of $2^\omega$ under the Medvedev reducibility. In "Non-branching degrees in the Medvedev lattice of $\Pi \sp{0}\sb{1} classes," I provided a characterization of nonbranching/branching and a classification (...) of the nonbranching degrees. In this paper, I present a similar classification of the branching degrees. In particular, $P$ is separable if there is a clopen set $C$ such that $P \cap C \neq \emptyset \neq P \cap C^c$ and $P \cap C \perp_M P \cap C^c$. By the results in the first paper, separability is an invariant of a Medvedev degree and a degree is branching if and only if it contains a separable member. Further define $P$ to be hyperseparable if, for all such $C$, $P \cap C \perp_M P \cap C^c$ and totally separable if, for all $X,Y \in P$, $X \perp_T Y$. I will show that totally separable implies hyperseparable implies separable and that the reverse implications do not hold, that is, that these are three distinct types of branching degrees. Along the way I will show some related results and present a combinatorial framework for constructing $\Pi^0_1$ classes with priority arguments. (shrink)

Important examples of $\Pi^0_1$ classes of functions $f \in {}^\omega\omega$ are the classes of sets (elements of ω 2) which separate a given pair of disjoint r.e. sets: ${\mathsf S}_2(A_0, A_1) := \{f \in{}^\omega2 : (\forall i < 2)(\forall x \in A_i)f(x) \neq i\}$ . A wider class consists of the classes of functions f ∈ ω k which in a generalized sense separate a k-tuple of r.e. sets (not necessarily pairwise disjoint) for each k ∈ ω: ${\mathsf S}_k(A_0,\ldots,A_k-1) (...) := \{f \in {}^\omega k : (\forall i < k) (\forall x \in A_i) f(x) \neq i\}$ . We study the structure of the Medvedev degrees of such classes and show that the set of degrees realized depends strongly on both k and the extent to which the r.e. sets intersect. Let ${\mathcal S}^m_k$ denote the Medvedev degrees of those ${\mathsf S}_k(A_0,\ldots,A_{k-1})$ such that no m + 1 sets among A 0,...,A k-1 have a nonempty intersection. It is shown that each ${\mathcal S}^m_k$ is an upper semi-lattice but not a lattice. The degree of the set of k-ary diagonally nonrecursive functions $\mathsf{DNR}_k$ is the greatest element of ${\mathcal S}^1_k$ . If 2 ≤ l < k, then 0 M is the only degree in ${\mathcal S}^1_l$ which is below a member of ${\mathcal S}^1_k$ . Each ${\mathcal S}^m_k$ is densely ordered and has the splitting property and the same holds for the lattice ${\mathcal L}^m_k$ it generates. The elements of ${\mathcal S}^m_k$ are exactly the joins of elements of ${\mathcal S}^1_i$ for $\lceil{k \over m}\rceil \leq i \leq k$. (shrink)

An ingenious study in the interplay between mereology and set theory, this book is launched innocuously enough with the thesis that a class just is the mereological sum or "fusion" of its sub-classes. The sub-classes of a class are parts of a class in the literal sense of the word "part," as trigonometry is literally a part of mathematics. We are thus urged to resist the suggestion that the word "part" applies first and foremost to the spatial (...) parts of a material object. The atomic parts of a class being seen to be its singleton or "unit" sub-classes, a class is now to be understood on the most basic level as a fusion of singletons but for the distressing fact that how a class might have only one member was never explained to us when we first learned that a class was supposed to be "a many combined into one.". (shrink)

The naive theory of properties states that for every condition there is a property instantiated by exactly the things which satisfy that condition. The naive theory of properties is inconsistent in classical logic, but there are many ways to obtain consistent naive theories of properties in nonclassical logics. The naive theory of classes adds to the naive theory of properties an extensionality rule or axiom, which states roughly that if two classes have exactly the same members, they are identical. In (...) this paper we examine the prospects for obtaining a satisfactory naive theory of classes. We start from a result by Ross Brady, which demonstrates the consistency of something resembling a naive theory of classes. We generalize Brady’s result somewhat and extend it to a recent system developed by Andrew Bacon. All of the theories we prove consistent contain an extensionality rule or axiom. But we argue that given the background logics, the relevant extensionality principles are too weak. For example, in some of these theories, there are universal classes which are not declared coextensive. We elucidate some very modest demands on extensionality, designed to rule out this kind of pathology. But we close by proving that even these modest demands cannot be jointly satisfied. In light of this new impossibility result, the prospects for a naive theory of classes are bleak. (shrink)

The article demonstrates that post-Soviet academic debates about theoretical concepts and visions of truth can be usefully interpreted in terms of different “class positions” of knowledge producers. One academic faction is interested in academic freedom, autonomy, and corporate solidarity, as the social and cultural capitals of its members are involved with the global symbolic market. The capitals of the other group are invested into the slightly modified Soviet academic system and local symbolic fields. Intellectuals necessarily are aligned with more (...) powerful social actors and thus become involved in divisions and struggles that they cannot escape. (shrink)

We present some characterizations of the members of Δta2, that class of the topological arithmetical hierarchy which is just large enough to include several fundamental types of sets of points in Euclidean spaces ℝk. The limit characterization serves as a basic tool in further investigations. The characterization by effective difference chains of effectively exhaustible sets yields only a hierarchy within a subfield of Δta2. Effective difference chains of transfinite order types, consisting of complements of effectively exhaustible sets, as well (...) as another closely related concept, yield a rich hierarchy within the whole class Δta2. The presentation always first reports analogies between Hausdorff's difference hierarchy within the Borel class ΔB2 and Ershov's hierarchy within the class Δ02 of the arithmetical hierarchy; after that the counterparts for Δta2 are developed. (shrink)

We often, in effect, take it for granted that some word or phrase is what is called ‘the name of a class, be that class empty or non-empty’. We do so whenever in effect we either wonder about, or mean to be taking a view on, the number of members a certain suppositious class has, on the ultrasimple number scale: ‘None, more than none’.

Zermelo, Frege and Russell accepted a common theme regarding classes. Classes were determined by other entities—functions, concepts, properties or conditions—and a class was only acceptable in the theory if there was such a determining entity. Thus, the existence of a class was taken to be dependent on a concept, function, condition, etc. whose satisfaction or fulfillment by an element determined the element to be a member of the class. This feature was behind Russell’s “no class (...) theory,” where a class was not only taken to be “determined by” a function but was considered to be “replaceable” by its determining function, as well as Frege’s problematic claim that every concept determined a class. Thus, Russell could identify the class of φs with, and Frege could take there to be a class of φs whose elements were all things falling under φ. Treating classes in such a manner does not provide for an adequate ontological assay of classes. If classes are entities, then whether or not there exists a class cannot depend on there being a property or function or concept or condition specifying membership in the class. Such a concept or property merely allows us to provide a definite description of the class. It is not the basis for the existence of the class. Russell’s “no class” theory of the first edition of Principia takes class abstracts and the classes they purportedly denote as “incomplete symbols.” As applied to the class abstract signs, the notion of an “incomplete symbol” means that such signs are only meaningful in contexts, in fact, are contextually defined. Thus, their meaning is not supplied by an entity such a sign purportedly denotes. As applied to a purported entity, to say of such a “thing” that it is an incomplete symbol is to say that it doesn’t exist. Thus, Russell classified the King of France in 1905, the king, as well as the definite description, as an incomplete symbol and called propositions “incomplete symbols” when he denied that there were such entities. (shrink)

Helix–loop–helix (HLH) genes encode for transcription factors affecting a whole variety of developmental programs, including neurogenesis. At least seven functional classes (denoted I to VII) of HLH genes exist,1 with subclass members exhibiting homo‐ and heterodimerisation for proper DNA binding and transcriptional regulation of downstream target genes. In the developing nervous system, members of class II, V and VI have been most extensively studied concerning their roles in neural programming. In contrast, the function of class I proteins (such (...) as E12 and E47) is poorly defined and the orthodox view relegates them to general dimerisation duties that are necessary for the activity of the other classes. However, closer scrutiny of the spatiotemporal expression patterns of class I factors, combined with recent biochemical evidence, would suggest that class I proteins possess specific functions during early neural differentiation. This essay supports this possibility, in addition to putting forward the hypothesis that, outside their general dimerisation activity, class I genes have independent roles in regulating neurogenesis. BioEssays 25:709–716, 2003. © 2003 Wiley Periodicals, Inc. (shrink)