Peter Abelard (1079–1142 ce) was the most wide‐ranging philosopher of the twelfth century. He quickly established himself as a leading teacher of logic in and near Paris shortly after 1100. After his affair with Heloise, and his subsequent castration, Abelard became a monk, but he returned to teaching in the Paris schools until 1140, when his work was condemned by a Church Council at Sens. His logical writings were based around discussion of the “Old Logic”: Porphyry's Isagoge, aristotle'S Categories and (...) On Interpretation and boethius'S textbook on topical inference. They comprise a freestanding Dialectica (“Logic”; probably c.1116), a set of commentaries (known as the Logica [Ingredientibus], c. 1119) and a later (c. 1125) commentary on the Isagoge (Logica Nostrorum Petititoni Sociorum or Glossulae). In a work Abelard called his Theologia, issued in three main versions (between 1120 and c.1134), he attempted a logical analysis of trinitarian relations and explored the philosophical problems surrounding God's claims to omnipotence and omniscience. The Collationes (“Debates,” also known as “Dialogue between a Christian, a Philosopher and a Jew”; probably c.1130) present a rational investigation into the nature of the highest good, in which the Christian and the Philosopher (who seems to be modeled on a philosopher of pagan antiquity) are remarkably in agreement. The unfinished Scito teipsum (“Know thyself,” also known as the “Ethics”; c.1138) analyses moral action. (shrink)

The London suicide bombings of July 7, 2005 were partly the revolt of moral earnestness against a liberal society that, enchanted by the fantasy of rationalist anthropology, surrenders its passionate members to a degrading consumerism. The “humane” liberalism variously espoused by Jürgen Habermas, John Rawls, and Jeffrey Stout offers a dignifying alternative; but it is fragile, and each of its proponents looks for allies among certain kinds of religious believer. Stanley Hauerwas, however, counsels Christians against cooperation. On the one hand, (...) he is right to resist, insofar as liberalism illiberally excludes theology from public discourse. On the other hand, not all humane liberalism does this: Stout's, for example, is genuinely polyglot, requiring not a common secularist language but a common ethic of communicating. Such a liberal ethic and its attendant anthropology merit the support of Christians: there may be more to be said about the Kingdom of God than respect, tolerance, and fairness, but there will not be less. The Christian has good theological reasons to expect some concord with other inhabitants of secular space. Ethical distinctiveness is no measure of theological integrity; and neither theology (pace Barth) nor biblical narrative (pace Richard Hays) should be expected to do all of the ethical running. If Christians are to be thorough in their moral theology and intelligible in their public statements, then they must borrow non‐theological material, formulate abstract concepts, and engage in casuistical analysis. Nevertheless, if an anxious insistence on distinctiveness is a mistake, concern for theological integrity is not. When the moral theologian borrows ethical material from elsewhere, he should integrate it into a theological vision structured by the Christian salvation‐historical narrative, which will sometimes modify the meaning of what is incorporated. So in affirming humane, polyglot liberalism, the moral theologian will at the same time make salutary qualifications. One of these is the assertion of the need of liberal institutions to own and promote their moral and anthropological commitments. In such a confessionally liberal society, universities in general, and the Arts and Humanities in particular, would recover their vocation to form citizens in communicative virtues and to offer them a dignifying, morally serious vision of human being that could save future generations from a degrading consumerism on the one hand and violent over‐reaction on the other. (shrink)

Commentary on "On Specification and the Senses," by Thomas A. Stoffregen and Benoît G. Bardy: Behavioral and Brain Sciences, 24, 195-261 (2001).
The target article's value lies not in its defence of specification, or the "global array" concept, but in its challenge to the paradigm of 5+ senses, and its examples of multiple receptor types cooperatively participating in specific information pick-up tasks. Rather than analysing our perceptual endowment into 5+ senses, it is more revealing to type perceptual systems according to task.

This paper argues that two areas vital to the teaching profession's own development and to the development of its standing in society have been neglected in inservice education and training. The first, an understanding and development of the 'public' dimension of teaching, suggests that teachers have duties and concerns which transcend those of professionals in the private sector because the public domain is a necessary focus for the promotion of collective life as opposed to individual interests. The second, an appreciation (...) of the 'ecological' context of teaching, locates its practice within wider political and social issues and deepens the teaching profession's understanding of itself. The evidence of neglect of these areas is derived from questionnaire data drawn by the authors from primary and secondary schools on their inservice priorities. (shrink)

As the 20th century began, few effective therapies existed. This soon changed with major therapeutic discoveries turning the century into what has been called the golden age of therapeutics.1 The emphasis of most of these developments was on medicines for common disorders as they presented the greatest need. However, it also allowed pharmaceutical manufacturers to produce blockbuster drugs that provided a large return on investment. Rare disorders were overlooked because most are genetic in origin and scientific knowledge was lacking, making (...) it difficult to find treatments. The first genetic disorder with a therapy was phenylketonuria. The biochemist who proposed the first successful treatment in the 1950s was actively discouraged as phenylketonuria was genetic and ‘incurable’.2 Another reason was an absence of incentives to invest in developing medicines for which the return was limited, given the small numbers of potential patients. This changed from 1983 as several countries began introducing orphan drug policies, with Canada being a notable exception. As the millennium changed, some drugs for rare disorders were available, but the sequencing of the human genome in the early 2000s provided scientific knowledge to develop many more. The new therapies are often costly because they require millions, even billions, of dollars and years to develop, test and satisfy regulatory standards. …. (shrink)

Abstractα/β barrels have an ill‐defined origin. Evidence exists which favours their divergent evolution from a common ancestral barrel and convergent evolution to a stable fold. However, recent sequence and structural information for the flavin oxidase/dehydrogenase family of barrel enzymes indicate that sub‐families of α/β barrels have evolved divergently. The modular fusion of barrel domains with core structures from other gene families has also contributed to the evolution of related but catalytically distinct enzyme molecules within each sub‐family of the flavin oxidases/dehydrogenases. (...) An analysis of the structures and sequences of the flavin oxidases/dehydrogenases has now enabled studies focusing on the evolutionary origins and modular assembly of this important family of proteins to be initiated. (shrink)

ABSTRACT The individual-community relationship has always been one of the most fundamental topics of social sciences. In sociology, this is known as the micro-macro relationship while in economics it refers to the processes, through which, individual actions lead to macroeconomic phenomena. Based on philosophical discourse and systems theory, many sociologists even use the term "emergence" in their understanding of micro-macro relationship, which refers to collective phenomena that are created by the cooperation of individuals, but cannot be reduced to individual actions. (...) "Emergence" theories attempt to explain the nature of society as a complex system by examining how individuals and their relationships lead to the creation of integrated and macro-social phenomena such as markets, educational systems, cultural beliefs, and shared social practices. As a prelude to activity, every researcher has to answer the question from the methodological point of view, how is it possible to study the behavior of social groups and how can we gain knowledge about the laws related to social groups? Anyone who deals with humanities and social sciences or any reality and phenomenon that affects human beings, inevitably deals with the reality that is emerging. In fact, emergence occurs when one level of reality emerges radically from another level. Examples of emergent levels of reality include how the mind emerges from the body; or the way society emerges from human beings. Therefore, when there is an emerging factor, different scientific disciplines should be used, because it is inevitable to talk about social affairs, psychology and neurobiology, as well as physical and even chemical. -/- INTRODUCTION Since its beginning in the 19th century, sociology has faced the fundamental question of what a social phenomena stands for? At a glance, sociologists deal with the study of social groups, collective behavior, institutions, social structures, social networks, and social dynamics, and ultimately all such phenomena are composed of individuals. Therefore, social phenomena seem to have no independent ontological status, and in this case, sociology is ultimately about reducible individuals. Different approaches about the micro-macro relationship, using the concept of "emergence", argue that although collective phenomena are created by individuals, they cannot be reduced to individual action. The term "emergence" is used to distinguish "emergent results" from " Procedural results". Results are called Procedural that can only be calculated by adding or subtracting the causes that act together. On the contrary, they are emergent outcomes that are qualitatively novel compared to the causes from which they originate. An example of such emergent results is mental characteristics that emerge from neural processes, but are not considered among the characteristics of neural processes of components - which are their origin. Philosophical debates about emergentism and reductionism focus on the mind-brain relationship, but as many philosophers have noted, they can be generalized to apply to any hierarchical set of features. The cognitive revolution reactivated a 19th century controversy between theorists of Identity and dualism. The theorists of Identity believe in reductionist materialism and elimination of metaphysics, according to which the mind is nothing more than the biological, while the dualists believe that the mind and the brain are distinct from each other. Emergence has been considered as a third way between the theories of Identity and dualism. -/- METHODOLOGY Interdisciplinarity Is fundamentally indicated by the dominance of open systems as well as social and emerging systems at different levels. Therefore, as long as we are facing a new emergentism, there will necessarily be different fields and an interdisciplinary method. The commonality of social sciences and cognitive sciences on the issue of emergence provides the possibility of interaction and exchange of concepts between these two for deeper research. Therefore, in this research, apart from the concepts of Subvenience and downward causation, we have used multiple realization and wildly disjunctive, which are used to explain the emerging concept in cognitive sciences, including neurobiology, psychology, and philosophy. For analytical and interdisciplinary approaches, after examining the concepts of subvenience, and wildly disjunctive, Individualistic and collectivist emergentism have been reviewed. And then their adequacy is criticized and evaluated in meeting strong emergence conditions. -/- FINDINGS Philosophers of the mind have been able to define the conditions of irreducibility and downward causation, which are essential for strong emergence, by using the principle of subvenience with the help of multiple realization and wildly disjunctive principles. Therefore, by definition, emergentist individualism suffers from the problem of how to explain downward causality and irreducibility, and emergentist collectivism faces the challenge of reification. The analysis of the challenges facing the two methodological approaches of individualism and collectivism leads us to three unresolved issues about the emerging phenomenon, which are realism, downward causality, and mechanism. Despite the differences in the stances of these two approaches regarding the aforementioned triple problems, individualists and collectivists both agree that some social characteristics are reducible; and some others are irreducible due to their complexity, the determination of which, depends on the case study of the mechanism of emergence of each social characteristic. -/- CONCLUSION An emergence that is consistent with reductionism is called a weak emergence. And strong emergence includes the principle of irreducibility and downward causality. According to strong emergence, the characteristics of social phenomena have downward causal power, which cannot be reduced to the causal forces. The principle of subvenience, together with multiple realization and wildly disjunctive, can provide the conditions of irreducibility and downward causality, which are essential conditions of strong emergence. Based on the philosophical analysis of two methodological approaches, individualist and collectivist, it can be concluded that emergentist individualism suffers from the problem of how to explain downward causality and irreducibility, and reification collectivism faces the challenge of reification. The analysis of the problems faced by these two approaches leads us to three unresolved issues, which are realism, downward causality, and mechanism. individualist emergentists acknowledge the existence of emergent social characteristics, but despite this, they believe that these characteristics are not real but merely analytical constructs that require explanation based on individuals and their mutual actions. But collectivists argue that emergentism yields a categorical ontology and supports social realism. They accept the dependence of social characteristics on individuals, but claim that social entities and structures are ontologically independent. Individualists and collectivists both agree that some social characteristics are reducible; and others are irreducible due to their complexity; and the only way to recognize this is to engage in empirical studies of the mechanisms and temporal processes of "emergence" that create social characteristics. -/- NOVELTY The newness of this compact article is as follows: a) Presenting an interdisciplinary approach using the analysis of philosophy of cognitive sciences, including the philosophy of mind, neurobiology and psychology to formulate the issue of social emergence; b) Criticizing individualist and collectivist methodological approaches to the issue of social emergence and explaining the sufficiency and shortcomings of both in providing weak and strong emerging conditions; c) Explaining and categorizing the challenges facing methodological approaches in explaining the phenomenon of social emergence, in three categories: realism, downward causality, and mechanism. BIBLIOGRAPHY Beckermann, A., Flohr, H., & Kim, J. (eds.) (1992). Emergence or reduction? Essays on the prospects of nonreductive physicalism. Berlin: Walter de Gruyter. Bhaskar, R. (1979). The possibility of Naturalism. New York: Routledge. Bhaskar, R., Danermark, B. & Price, L. (2018). Interdisciplinarity and wellbeing: A critical realist general theory of interdisciplinarity. London and New York: Routledge. Blau, P.M. (1981). Introduction: diverse views of social structure and their common denominator. In Blau, P.M. & Merton, R.K. (eds.), Continuity in Structural Inquiry (pp. 1–23). Beverly Hills: Sage. Blau, P.M. (1987). Microprocess and macrostructure. In K.S. Cook (ed.), Social Exchange Theory (pp. 83–100). Newbury Park, CA: Sage. Brooks, R.A., & Maes, P. (eds). (1994). Artificial life IV; proceedings of the Fourth International Workshop on the Synthesis and Simulation of Living Systems. Cambridge, Mass: MIT Press. Clark, A. (1997). Being there: Putting brain, body, and world together again. Cambridge, Mass.: MIT Press. Coleman, J. S. (1987). Microfoundations and macrosocial behavior. In: J. C. Alexander, B. Giesen, R. Munch, & Neil J. Smelser (Eds.). The Micro-Macro Link (pp. 153–73). Berkeley: University of California. Coleman, J. S. (1990). Foundations of social theory. Cambridge, MA: Harvard University Press. Elster, J. (1985). Making sense of Marx. New York: Cambridge University Press. Fodor, J.A. (1974). Special sciences (Or: The disunity of science as a working hypothesis). Synthese, 28(2), 97–115. Fodor, J.A. (1989). Making mind matter more. Philosophical Topics, 17, 59-79. Giddens, A. (1984). The constitution of society: Outline of the theory of structuration. Berkeley: University of California Press. Hempel, C.G. (1965). Aspects of scientific explanation and other essays in the philosophy of science. New York: The Free Press. Horgan, T. (1993). From supervenience to superdupervenience: meeting the demands of a material world. Mind, 102, 555-586. Humphreys, P. (1997). How properties emerge”. Philosophy of Science, 64, 1-17. Kim, J. (1999). Making sense of emergence. Philosophical Studies, 95, 3–36. Kincaid, H. (1997). Individualism and the unity of science. New York: Rowman & Littlefield. Lewis, D.K. (1986). The plurality of worlds. Oxford: Oxford University Press. Nigel, G., & Conte, R. (eds.) (1995). Artificial societies; The computer simulation of social life. London: UCL Press. O’Connor, T. (1994). Emergent properties. American Philosophical Quarterly, 31, 91-104. Popper, K.R. (1968). Conjectures and refutations: The Growth of Scientific Knowledge. New York: Harper & Row. Sawyer, R. K. (2001). Emergence in sociology: contemporary philosophy of mind and some implications for sociological theory. American Journal of Sociology, 107, 551-585. Sawyer, R. K. (2004). The mechanisms of emergence. Philosophy of the Social Sciences, 34, 260–282. Sawyer, R. K. (2005). Social emergence: societies as complex systems. Cambridge: Cambridge University Press. Sawyer, R.K. (2003). Nonreductive individualism: Part 2, Social causation. Philosophy of the Social Sciences, 33, 203–224. Winch, P. (1993). The idea of a social science and its relation to philosophy‭ (SAMT, Trans.). Tehran, Iran: SAMT. (shrink)

A set of amino acid side chains that confer specificity for the coenzyme NADPH and the substrate glutathione in the flavoprotein disulphide oxidoreductase, glutathione reductase, has been identified. Systematic replacement of these amino acid residues in the coenzyme‐binding site switches the specificity of the enzyme from its natural strong preference for NADPH to a marked preference For NADH. The amino acids replaced all lie in a structural motif within the dinucleotide‐binding domain of the protein. Since this domain is a feature (...) common to most dehydrogenases (reductases) that use nicotinamide coenzymes, it may be that the coenzyme specificities of all such enzymes can be manipulated in this way. Similarly, amino acid residues involved in the selective recognition of trypanothione by trypanothione reductase, an enzyme related to glutathione reductase and exclusive to trypanosomatids, were identified. Suitable mutation of the corresponding residues in E. coli glutathione reductase switched its substrate specificity towards trypanothione. A better understanding of the substrate specificity of these enzymes could open up a route to the chemotherapy of trypanosomal infections. (shrink)

Wana Manitpisitkul1, Sabrina Lee2, Matthew Cooper31Department of Pharmacy, University of Maryland Medical Center, Baltimore, MD, USA; 2Solid Organ Transplant Program, University of Utah Health Care, Salt Lake City, UT, USA; 3Department of Surgery, University of Maryland School of Medicine, Baltimore, MD, USA: Addition of mycophenolate mofetil to calcineurin-based immunosuppressive therapy has led to a significant improvement in graft survival and reduction of acute rejection in renal transplant recipients. However, in clinical practice, MMF dose reduction, interruption, or discontinuation due to hematological (...) and gastrointestinal side-effects occurred in up to 50% of the patients. Large retrospective analyses have demonstrated that patients requiring MMF dose manipulation due to adverse events experienced a higher rate of rejection and graft loss. Enteric-coated mycophenolate sodium was developed with the goal of improving upper GI side-effects. Here, we review the efficacy and safety of EC-MPS in de novo kidney transplant recipient, and in stable renal transplant patients who were converted from MMF. The changes in GI-related adverse events using patient-reported outcome instruments are also reviewed.Keywords: enteric-coated mycophenolate sodium, mycophenolate mofetil, kidney transplant, efficacy, gastrointestinal tolerability. (shrink)

Inborn errors of vitamin B12 (cobalamin) metabolism are associated with homocystinuria and methylmalonic aciduria, either alone or in combination. A number of these disorders have provided the first evidence for the existence of important steps in the transport or metabolism of cobalamin in eukaryotic cells. Eight complementation classes have been defined on the basis of somatic cell hybridization studies. Although the majority of patients present in infancy or early childhood, some are not diagnosed until adolescence or later. For some of (...) these disorders, prenatal diagnosis and therapy with cobalamin during pregnancy has been attempted. Although only males have been described with cblE disease, all of these disorders are presumed to be autosomal recessive in inheritance. The clinical and laboratory aspects of the different complementation classes (cblA–cblG) are reviewed here. (shrink)

There is growing political pressure for farmers to use more sustainable agricultural practices to protect people and the planet. The farming press could encourage farmers to adopt sustainable practices through its ability to manipulate discourse and spread awareness by changing the salience of issues or framing topics in specific ways. We sought to understand how the UK farming press framed sustainable agricultural practices and how the salience of these practices changed over time. We combined a media content analysis of the (...) farming press alongside 60 qualitative interviews with farmers and agricultural advisors to understand whether the farming press influenced farmers to try more sustainable practices. Salience of sustainable agricultural practices grew between 2009 and 2020. Many of the practices studied were framed by the press around economic and agronomic aspects, and farmer respondents said the most common reasons for trying sustainable agricultural practices were for economic and agronomic reasons. The farming press tended to use more positive rather than negative tones when covering sustainable agricultural practices. Respondents used the farming press as a source of information, though many did not fully trust these outlets as they believed the farming press were mouthpieces for agribusinesses. Whilst a minority of farmers stated they were motivated to try a new sustainable agricultural practice after learning about it in the farming press, this was rare. Instead, the farming press was used by respondents to raise their awareness about wider agricultural topics. We reflect on the role and power given to agribusinesses by the farming press and what this means for agricultural sustainability. (shrink)

The fundamental ideas concerning computation and recursion naturally find their place at the interface between logic and theoretical computer science. The contributions in this book, by leaders in the field, provide a picture of current ideas and methods in the ongoing investigations into the pure mathematical foundations of computability theory. The topics range over computable functions, enumerable sets, degree structures, complexity, subrecursiveness, domains and inductive inference. A number of the articles contain introductory and background material which it is hoped will (...) make this volume an invaluable resource for specialists and non-specialists alike. (shrink)

Logicians and theoretical physicists will also benefit from this book.

Recent events, such as the widespread acceptance of the book, Final Exit, show that the very foundation upon which all humane medical practice is ba sed is in danger of being replaced with ethical relativism. Cameron's book could very well be a rallying point for those in the profession who know something is wrong. Endorsed by C. Everett Koop.

By constructing a maximal incomplete d.r.e. degree, the nondensity of the partial order of the d.r.e. degrees is established. An easy modification yields the nondensity of the n-r.e. degrees and of the ω-r.e. degrees.

A proof is given that 0′ is definable in the structure of the degrees of unsolvability. This answers a long-standing question of Kleene and Post, and has a number of corollaries including the definability of the jump operator.

We show that there exists a properly Σ2 minimal degree b, and moreover that b can be chosen to join with 0′ to 0″ – so that b is a 0″ complement for every degree a such that 0′ ≤ a < 0″.

We prove that there exists a nonzero recursively enumerable Turing degree possessing a strong minimal cover.

The logical properties of the 'if-then' connective of ordinary English differ markedly from the logical properties of the material conditional of classical, two-valued logic. This becomes apparent upon examination of arguments in conversational English which involve (noncounterfactual) usages of if-then'. A nonclassical system of propositional logic is presented, whose conditional connective has logical properties approximating those of 'if-then'. This proposed system reduces, in a sense, to the classical logic. Moreover, because it is equivalent to a certain nonstandard three-valued logic, its (...) decision procedure is almost as efficient as that of the classical logic. It therefore provides a rational and convenient system in which to formalize English arguments. (shrink)

Computability theory concerns information with a causal–typically algorithmic–structure. As such, it provides a schematic analysis of many naturally occurring situations. Emil Post was the first to focus on the close relationship between information, coded as real numbers, and its algorithmic infrastructure. Having characterised the close connection between the quantifier type of a real and the Turing jump operation, he looked for more subtle ways in which information entails a particular causal context. Specifically, he wanted to find simple relations on reals (...) which produced richness of local computability-theoretic structure. To this extent, he was not just interested in causal structure as an abstraction, but in the way in which this structure emerges in natural contexts. Post’s programme was the genesis of a more far reaching research project. In this article we will firstly review the history of Post’s programme, and look at two interesting developments of Post’s approach. The first of these developments concerns the extension of the core programme, initially restricted to the Turing structure of the computably enumerable sets of natural numbers, to the Ershov hierarchy of sets. The second looks at how new types of information coming from the recent growth of research into randomness, and the revealing of unexpected new computability-theoretic infrastructure. We will conclude by viewing Post’s programme from a more general perspective. We will look at how algorithmic structure does not just emerge mathematically from information, but how that emergent structure can model the emergence of very basic aspects of the real world. (shrink)

The formal systems of logic have ordinarily been regarded as independent of biology, but recent developments in evolutionary theory suggest that biology and logic may be intimately interrelated. In this book, William Cooper outlines a theory of rationality in which logical law emerges as an intrinsic aspect of evolutionary biology. This biological perspective on logic, though at present unorthodox, could change traditional ideas about the reasoning process. Cooper examines the connections between logic and evolutionary biology and illustrates how (...) logical rules are derived directly from evolutionary principles, and therefore have no independent status of their own. Laws of decision theory, utility theory, induction, and deduction are reinterpreted as natural consequences of evolutionary processes. Cooper's connection of logical law to evolutionary theory ultimately results in a unified foundation for an evolutionary science of reason. It will be of interest to professionals and students of philosophy of science, logic, evolutionary theory, and cognitive science. (shrink)

Computability theory originated with the seminal work of Gödel, Church, Turing, Kleene and Post in the 1930s. This theory includes a wide spectrum of topics, such as the theory of reducibilities and their degree structures, computably enumerable sets and their automorphisms, and subrecursive hierarchy classifications. Recent work in computability theory has focused on Turing definability and promises to have far-reaching mathematical, scientific, and philosophical consequences. Written by a leading researcher, Computability Theory provides a concise, comprehensive, and authoritative introduction to contemporary (...) computability theory, techniques, and results. The basic concepts and techniques of computability theory are placed in their historical, philosophical and logical context. This presentation is characterized by an unusual breadth of coverage and the inclusion of advanced topics not to be found elsewhere in the literature at this level. The book includes both the standard material for a first course in computability and more advanced looks at degree structures, forcing, priority methods, and determinacy. The final chapter explores a variety of computability applications to mathematics and science. Computability Theory is an invaluable text, reference, and guide to the direction of current research in the field. Nowhere else will you find the techniques and results of this beautiful and basic subject brought alive in such an approachable and lively way. (shrink)

What can computers do in principle? What are their inherent theoretical limitations? These are questions to which computer scientists must address themselves. The theoretical framework which enables such questions to be answered has been developed over the last fifty years from the idea of a computable function: intuitively a function whose values can be calculated in an effective or automatic way. This book is an introduction to computability theory (or recursion theory as it is traditionally known to mathematicians). Dr Cutland (...) begins with a mathematical characterisation of computable functions using a simple idealised computer (a register machine); after some comparison with other characterisations, he develops the mathematical theory, including a full discussion of non-computability and undecidability, and the theory of recursive and recursively enumerable sets. The later chapters provide an introduction to more advanced topics such as Gildel's incompleteness theorem, degrees of unsolvability, the Recursion theorems and the theory of complexity of computation. Computability is thus a branch of mathematics which is of relevance also to computer scientists and philosophers. Mathematics students with no prior knowledge of the subject and computer science students who wish to supplement their practical expertise with some theoretical background will find this book of use and interest. (shrink)

We prove the following three theorems on the enumeration degrees of ∑20 sets. Theorem A: There exists a nonzero noncuppable ∑20 enumeration degree. Theorem B: Every nonzero Δ20enumeration degree is cuppable to 0′e by an incomplete total enumeration degree. Theorem C: There exists a nonzero low Δ20 enumeration degree with the anticupping property.

By constructing a maximal incomplete d.r.e. degree, the nondensity of the partial order of the d.r.e. degrees is established. An easy modification yields the nondensity of the n-r.e. degrees and of the ω-r.e. degrees.