Contemporary evolutionary biology comprises a plural landscape of multiple co-existent conceptual frameworks and strenuous voices that disagree on the nature and scope of evolutionary theory. Since the mid-eighties, some of these conceptual frameworks have denounced the ontologies of the Modern Synthesis and of the updated Standard Theory of Evolution as unfinished or even flawed. In this paper, we analyze and compare two of those conceptual frameworks, namely Niles Eldredge’s Hierarchy Theory of Evolution (with its extended ontology (...) of evolutionary entities) and the Extended Evolutionary Synthesis (with its proposal of an extended ontology of evolutionary processes), in an attempt to map some epistemic bridges (e.g. compatible views of causation; niche construction) and some conceptual rifts (e.g. extra-genetic inheritance; different perspectives on macroevolution; contrasting standpoints held in the “externalism–internalism” debate) that exist between them. This paper seeks to encourage theoretical, philosophical and historiographical discussions about pluralism or the possible unification of contemporary evolutionary biology. (shrink)

Sugar, pork, beer, corn, cider, scrapple, and hoppin' John all became staples in the diet of colonial America. The ways Americans cultivated and prepared food and the values they attributed to it played an important role in shaping the identity of the newborn nation. In A Revolution in Eating, James E. McWilliams presents a colorful and spirited tour of culinary attitudes, tastes, and techniques throughout colonial America. Confronted by strange new animals, plants, and landscapes, settlers in the colonies and West (...) Indies found new ways to produce food. Integrating their British and European tastes with the demands and bounty of the rugged American environment, early Americans developed a range of regional cuisines. From the kitchen tables of typical Puritan families to Iroquois longhouses in the backcountry and slave kitchens on southern plantations, McWilliams portrays the grand variety and inventiveness that characterized colonial cuisine. As colonial America grew, so did its palate, as interactions among European settlers, Native Americans, and African slaves created new dishes and attitudes about food. McWilliams considers how Indian corn, once thought by the colonists as "fit for swine," became a fixture in the colonial diet. He also examines the ways in which African slaves influenced West Indian and American southern cuisine. While a mania for all things British was a unifying feature of eighteenth-century cuisine, the colonies discovered a national beverage in domestically brewed beer, which came to symbolize solidarity and loyalty to the patriotic cause in the Revolutionary era. The beer and alcohol industry also instigated unprecedented trade among the colonies and further integrated colonial habits and tastes. Victory in the American Revolution initiated a "culinary declaration of independence," prompting the antimonarchical habits of simplicity, frugality, and frontier ruggedness to define American cuisine. McWilliams demonstrates that this was a shift not so much in new ingredients or cooking methods, as in the way Americans imbued food and cuisine with values that continue to shape American attitudes to this day. (shrink)

Perceptual learning can be defined as practice-induced improvement in the ability to perform specific perceptual tasks. We previously proposed the Reverse Hierarchy Theory as a unifying concept that links behavioral findings of visual learning with physiological and anatomical data. Essentially, it asserts that learning is a top-down guided process, which begins at high-level areas of the visual system, and when these do not suffice, progresses backwards to the input levels, which have a better signal-to-noise ratio. This simple concept (...) has proved powerful in explaining a broad range of findings, including seemingly contradicting data. We now extend this concept to describe the dynamics of skill acquisition and interpret recent behavioral and electrophysiological findings. (shrink)

Human ecology, it is argued, even when embracing recent developments in the natural sciences and granting a place to culture, tends to justify excessively pessimistic conclusions about the prospects for creating a sustainable world order. This is illustrated through a study of the work and assumptions of Richard Newbold Adams and Stephen Bunker. It is argued that embracing hierarchy theory as this has been proposed and elaborated by Herbert Simon, Howard Pattee, T.F.H. Allen and others enables human ecology (...) to conceive humans both as part of nature and as cultural beings in a way that gives due regard to the ethical development of humanity. That is, ethical constraints need no longer be conceived of as 'unnatural'. Characterizing the nature of such constraints, conceived of as emerging from the Hegelian 'struggle for recognition', this argument is shown to justify some optimism about the future, and to give some idea of how society should be organized if ethical constraints, able to constrain humanity’s relationship to the rest of nature, are to prevail. (shrink)

Aquinas’ natural law theory contains a set of basic goods, such as survival, reproduction and the pursuit of truth. However, whether and how there is a hierarchical relationship among these goods remains disputed. Given the importance of Aquinas’ natural law theory for Christianity and the philosophy of law, this issue merits a closer investigation. By carefully examining various modern scholars’ theories and Aquinas’ texts, it is demonstrated that according to Aquinas, firstly, there are hierarchies of basic goods and (...) sins; secondly, these hierarchies are horizontal and vertical according to the order of participation and the dignity of substances, respectively.Contribution: This research reconsiders the modern debate over Aquinas’ hierarchical theory of basic goods and provides a more authentic understanding of Aquinas’ own view, which can be applied to his theory of sin. Aquinas’ natural law theory can hence be clarified in a more profound way. (shrink)

This article argues that the dynamics behind the generation of social pathologies in modern society also undermine the social-relational framework for recognition. It therefore claims that the theory of recognition is impotent in face of the kinds of normative power exerted by social hierarchies. The article begins by discussing the particular forms of social pathology and their relation to hierarchical forms of social structure that are based on domination, control and subordination and then shows how the internalization of the (...) norms that shape and hold together hierarchical social formations causes pathologies within the self. As a result of these processes, the recognitive aspects of social action that the theory of recognition posits are unable to overcome and in fact reproduce and in many instances reinforce the pathologies themselves. (shrink)

Both mindreading and stereotyping are forms of social cognition that play a pervasive role in our everyday lives, yet too little attention has been paid to the question of how these two processes are related. This paper offers a theory of the influence of stereotyping on mental-state attribution that draws on hierarchical predictive coding accounts of action prediction. It is argued that the key to understanding the relation between stereotyping and mindreading lies in the fact that stereotypes centrally involve (...) character-trait attributions, which play a systematic role in the action–prediction hierarchy. On this view, when we apply a stereotype to an individual, we rapidly attribute to her a cluster of generic character traits on the basis of her perceived social group membership. These traits are then used to make inferences about that individual’s likely beliefs and desires, which in turn inform inferences about her behavior. (shrink)

In this paper we introduce applicative theories which characterize the polynomial hierarchy of time and its levels. These theories are based on a characterization of the functions in the polynomial hierarchy using monotonicity constraints, introduced by Ben-Amram, Loff, and Oitavem.

In lieu of an abstract, here is a brief excerpt of the content:A THEORY OF BASIC GOODS: STRUCTURE AND HIERARCHY* I. FTEN, PERHAPS ALWAYS, moral theory emerges from particular problems. Just how is obscure. The logic of discovery is elusive; and it is harder to explain how we have come to see matters rightly than to recognize that we do, in fact, see them rightly. What counts as a theory, moreover, calls for explication as much as (...) does a theory's emergence. When we have in mind a theory that shapes our social vision, there is a progression from a central set of propositions, keyed to a range of paradigm cases and seen as a logical structure, to a " way of thinking " widely enough embraced to have political significance, at least for a given oommunity. In this essay I want to explore some fundamental questions raised by Germain Grisez's theory of natural law ethics.1 These questions are chiefly conceptual. But neither they nor Grisez's theory should be seen in a vacuum. Rather, I would begin by suggesting that the importance of his work, its emergence in some Catholic circles, is linked with two specific moral and political problems. The first is abortion; the second is the superpowers' policy of nuclear deterrence. *I thank David Blake, Robert Gordh, Carroll Kearley, Gary Mar. Martin Woods, and Linda Zagzebski for their comments on an earlier version of this essay. 1 For the theory's fullest presentation, see Germain Grisez, The Way of the Lord Jesus, vol. one, Ohristian Moral Principles, with the help of Joseph M. Boyle, Jr., Basil Cole, O.P., John Finnis, John A. Geinzer, Robert G. Kennedy, Patrick Lee, William E. May, and Russell Shaw (Chicago: Franciscan Herald Press, 1983). Especially critical are chapters two through twelve. 221 JAMES G. HANINK As a point of chronology, Grisez's early study of rubortion was the context for one of the first formulations of his natural law ethics.2 But why should this have been so? And why should abortion so engage Catholic thinkers, some of whom had felt a distance from explicitly Catholic positions, that Grisez's work should become pivotal? Answers to such questions must be tentative, but they include the following considerations. In a brief period, no more than a decade, legal abortion in the West became entrenched social policy.8 It was a policy, too, supported by dominant intellectual forces. But the new policy, defended in terms of individual freedom and a right to privacy, struck many Catholicsand non-Catholics-as vicious and tragic. It was vicious in that it made expendable the weakest and most vulnerable human beings. Human rights somehow became restricted to the strong and self-sufficient. The policy was tragic in its denial of human community. We are not, however much radical individualism supposes, merely a collection of separate egos. l/Ve are, rather, a community tied to a past and pledged to a future, so pledged, in part, by the children we bear and nurture.4 For in hearing and nurturing children we carry on the love of parents who have done as much for us; and we carry ourselves into a future where we hope our own love will be thus extended. But abortion, and surely unrestricted abortion, betrays this community of trust. Still, a policy so vicious and tragic can make us look for a fresh moral vision. And, of course, to the extent that we have contributed to,a climate in which abortion could become entrenched -either by our failures to support women in need or 2 Germain Grisez, Abortion: The Myths, the Realities, and the Arguments (New York and Cleveland: Corpus Books, 1970). a A milepost, of course, was the United States Supreme Court decision, Roe v. Wade, in 1973. Recent U.S. Government figures indicate that there are now well over a million legal abortions annually in this country. 4 For a provocative discussion of this theme, see Stanley Hauerwas's " The Moral Value of the Family " in his A Commimity of Character (Notre Dame: University of Notre Dame Press, 1981). A THEORY OF BASIC GOODS: STRUCTURE & HIERARCHY 223 by an acquiescence in our culture's individualism-the call for theoretical reconstruction is all the more urgent... (shrink)

Potentialists think that the concept of set is importantly modal. Using tensed language as an heuristic, the following bar-bones story introduces the idea of a potential hierarchy of sets: 'Always: for any sets that existed, there is a set whose members are exactly those sets; there are no other sets.' Surprisingly, this story already guarantees well-foundedness and persistence. Moreover, if we assume that time is linear, the ensuing modal set theory is almost definitionally equivalent with non-modal set theories; (...) specifically, with Level Theory, as developed in Part 1. (shrink)

In spite of feminist recognition that hierarchical organizations are an important location of male dominance, most feminists writing about organizations assume that organizational structure is gender neutral. This article argues that organizational structure is not gender neutral; on the contrary, assumptions about gender underlie the documents and contracts used to construct organizations and to provide the commonsense ground for theorizing about them. Their gendered nature is partly masked through obscuring the embodied nature of work.jobs and hierarchies, common concepts in organizational (...) thinking, assume a disembodies and universal worker. This worker is actually a man; men's bodies, sexuality, and relationships to procreation and paid work are subsumed in the image of the worker. Images of men's bodies and masculinity pervade organizational processes, marginalizing women and contributing to the maintenance of gender segregation in organizations. The positing of gender-neutral and disembodied organizational structures and work relations is part of the larger strategy of control in industrial capitalist societies, which, at least partly, are built upon a deeply embedded substructure of gender difference. (shrink)

The following bare-bones story introduces the idea of a cumulative hierarchy of pure sets: 'Sets are arranged in stages. Every set is found at some stage. At any stage S: for any sets found before S, we find a set whose members are exactly those sets. We find nothing else at S.' Surprisingly, this story already guarantees that the sets are arranged in well-ordered levels, and suffices for quasi-categoricity. I show this by presenting Level Theory, a simplification of (...) set theories due to Scott, Montague, Derrick, and Potter. (shrink)

We investigate the interactions of formula complexity in weak set theories with the axioms available there. In particular, we show that swapping bounded and unbounded quantification preserves formula complexity in presence of the axiom of foundation weakened to an arbitrary set base, while it does not if the axiom of foundation is further weakened to a proper class base. More attention is being paid to the necessary axioms employed in the positive results, than to the combinatorial strength of the positive (...) results themselves. (shrink)

This paper seeks to resolve a fairly simple question in ethics: Why do seemingly reasonable people disagree about ethical problems? My paper seeks both to analyze this question and attempts to find a solution. My premise is that disagreement happens because of differences in hierarchical value ranking, or quite simply because some problems are more important to some people than others. Theories of choice, however, influenced by concepts such as "freedom of choice," conceal the hierarchical nature of our choices, leading (...) to confusion over the nature and degree of our ethical disagreements. I propose then that ethical disagreements can be reduced when we understand hierarchy as an essential element in moral choice theory. (shrink)

We formulate epsilon substitution method for theories (H)α0 of absolute jump hierarchies, and give two termination proofs of the H-process: The first proof is an adaption of Mints M, Mints-Tupailo-Buchholz MTB, i.e., based on a cut-elimination of a specially devised infinitary calculus. The second one is an adaption of Ackermann Ack. Each termination proof is based on transfinite induction up to an ordinal θ(α0+ ω)0, which is best possible.

This article proposes two interpretations of the concept of belief hierarchies in Bayesian game theory: the behaviorist interpretation and the mentalist interpretation. On the former, belief hierarchies are derived from the players’ preferences over acts. On the latter, they are causal mechanisms that are responsible for the players’ choices and preferences over acts. The claim is that the epistemic program in game theory is potentially confronted with a dilemma regarding which interpretation should be adopted. If the behaviorist interpretation (...) of belief hierarchies is retained, then Bayesian game theory cannot account for the players’ reasoning that leads to the generation of specific belief hierarchies and thus to the implementation of a specific strategy profile in a game. If the mentalist interpretation is retained, then belief hierarchies appear to be unnecessarily and unrealistically complex objects and that as a result they fail to provide reasonable epistemic foundations for various solution concepts. In other words, neither interpretation seems to allow the epistemic program to fulfill all its aims. I suggest however that the mentalist interpretation can be arranged to escape this dilemma. (shrink)

This is part two of a two-part paper in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. In this part of the paper, we extend the base theory of the first part of the paper with hierarchically typed truth-predicates and principles about the interaction of partial ground and truth. We show that our (...) theory is a proof-theoretically conservative extension of the ramified theory of positive truth up to. (shrink)

The aims of this paper are: (1) to present tense-logical versions of such classical notions as saturated and special models; (2) to establish several fundamental existence theorems about these notions; (3) to apply these powerful techniques to tense complexity.In this paper we are concerned exclusively with quantifiedK 1 (for linear time) with constant domain. Our present research owes much to Bowen [2], Fine [5] and Gabbay [6].

The von Neumann hierarchy of sets is heavily used as a basic tool in classical set theory, being an underlying ingredient in many proofs and concepts. In constructive set theories like without the powerset axiom however, it loses much of its potency by ceasing to be a hierarchy of sets as its single stages become only classes. This article proposes an alternative cumulative hierarchy which does not have this drawback and provides examples of how it can (...) be used to prove new theorems in. (shrink)

The concepts of hierarchical organization, genetic determinism and biological specificity have played a crucial role in biology as a modern experimental science since its beginnings in the nineteenth century. The idea of genetic information and genetic determination was at the basis of molecular biology that developed in the 1940s with macromolecules, viruses and prokaryotes as major objects of research often labelled “reductionist”. However, the concepts have been marginalized or rejected in some of the research that in the late 1960s began (...) to focus additionally on the molecularization of complex biological structures and functions using systems approaches. This paper challenges the view that ‘molecular reductionism’ has been successfully replaced by holism and a focus on the collective behaviour of cellular entities. It argues instead that there are more fertile replacements for molecular ‘reductionism’, in which genomics, embryology, biochemistry, and computer science intertwine and result in research that is as exact and causally predictive as earlier molecular biology. (shrink)

Applied Evolutionary Epistemology is a scientific-philosophical theory that defines evolution as the set of phenomena whereby units evolve at levels of ontological hierarchies by mechanisms and processes. This theory also provides a methodology to study evolution, namely, studying evolution involves identifying the units that evolve, the levels at which they evolve, and the mechanisms and processes whereby they evolve. Identifying units and levels of evolution in turn requires the development of ontological hierarchy theories, and examining mechanisms and (...) processes necessitates theorizing about causality. Together, hierarchy and causality theories explain how biorealities form and diversify with time. This paper analyzes how Applied EE redefines both hierarchy and causality theories in the light of the recent explosion of network approaches to causal reasoning associated with studies on reticulate and macroevolution. Causality theories have often been framed from within a rigid, ladder-like hierarchy theory where the rungs of the ladder represent the different levels, and the elements on the rungs represent the evolving units. Causality then is either defined reductionistically as an upward movement along the strands of a singular hierarchy, or holistically as a downward movement along that same hierarchy. Upward causation theories thereby analyze causal processes in time, i.e. over the course of natural history or phylogenetically, as Darwin and the founders of the Modern Synthesis intended. Downward causation theories analyze causal processes in space, ontogenetically or ecologically, as the current eco-evo-devo schools are evidencing. This work demonstrates how macroevolution and reticulate evolution theories add to the complexity by examining reticulate causal processes in space–time, and the interactional hierarchies that such studies bring forth introduce a new form of causation that is here called reticulate causation. Reticulate causation occurs between units and levels belonging to different as well as to the same ontological hierarchies. This article concludes that beyond recognizing the existence of multiple units, levels, and mechanisms or processes of evolution, also the existence of multiple kinds of evolutionary causation as well as the existence of multiple evolutionary hierarchies needs to be acknowledged. This furthermore implies that evolution is a pluralistic process divisible into different kinds. (shrink)

Purpose : The purpose of this paper is to bring the concept of a 'hierarchy of action', as it is currently being used in other fields, into library and information science . Design/methodology/approach Hierarchy theory is adopted to describe three hierarchies of action, which include the human processes of semantic and social innovation, as well as a system of biological interpretence, from which human processes are thought to have evolved as a development of biosemiosis in nature. By (...) way of example, it is argued that a text is a complex achievement, and hierarchy theory shows how to account for this complexity; the everyday definition of text is augmented with accounts from different levels of observation. Findings: The concept of a hierarchy of action enables a person to account for a text as a meaning/symbolic product; include in that account the processes whereby texts are produced and used; and say why these processes are important to the health of the biosphere that is called home. Originality/value Hierarchy of action has been developed as a concept in biology and ecology; it belongs to a way of thinking whereby human reality, like nature, is construed as dynamical processes operating in symbiotic relationship with each other; it has not yet been adopted in LIS with reference to hierarchy theory and its potential is yet to be explored. Keywords Hierarchy theory, Complexity theory, Process philosophy, Natural philosophy, Hierarchies of action, Complex systems, Libraries, Information science Paper type Conceptual paper. (shrink)

A venerable tradition in the metaphysics of science commends ontological reduction: the practice of analysis of theoretical entities into further and further proper parts, with the understanding that the original entity is nothing but the sum of these. This tradition implicitly subscribes to the principle that all the real action of the universe (also referred to as its "causation") happens at the smallest scales-at the scale of microphysics. A vast majority of metaphysicians and philosophers of science, covering a wide swath (...) of the spectrum from reductionists to emergentists, defend this principle. It provides one pillar of the most prominent theory of science, to the effect that the sciences are organized in a hierarchy, according to the scales of measurement occupied by the phenomena they study. On this view, the fundamentality of a science is reckoned inversely to its position on that scale. This venerable tradition has been justly and vigorously countered-in physics, most notably: it is countered in quantum theory, in theories of radiation and superconduction, and most spectacularly in renormalization theories of the structure of matter. But these counters-and the profound revisions they prompt-lie just below the philosophical radar. This book illuminates these counters to the tradition principle, in order to assemble them in support of a vaster (and at its core Aristotelian) philosophical vision of sciences that are not organized within a hierarchy. In so doing, the book articulates the principle that the universe is active at absolutely all scales of measurement. This vision, as the book shows, is warranted by philosophical treatment of cardinal issues in the philosophy of science: fundamentality, causation, scientific innovation, dependence and independence, and the proprieties of explanation. (shrink)

The concepts of hierarchical organization, genetic determinism and biological specificity have played a crucial role in biology as a modern experimental science since its beginnings in the nineteenth century. The idea of genetic information and genetic determination was at the basis of molecular biology that developed in the 1940s with macromolecules, viruses and prokaryotes as major objects of research often labelled “reductionist”. However, the concepts have been marginalized or rejected in some of the research that in the late 1960s began (...) to focus additionally on the molecularization of complex biological structures and functions using systems approaches. This paper challenges the view that ‘molecular reductionism’ has been successfully replaced by holism and a focus on the collective behaviour of cellular entities. It argues instead that there are more fertile replacements for molecular ‘reductionism’, in which genomics, embryology, biochemistry, and computer science intertwine and result in research that is as exact and causally predictive as earlier molecular biology. (shrink)

Scientific progress in the 20th century has shown that the structure of the world is hierarchical. A philosophical analysis of the hierarchy will bear obvious significance for metaphysics and philosophy in general. Jonathan Schaffer’s paper, “Is There a Fundamental Level?”, provides a systematic review of the works in the field, the difficulties for various versions of fundamentalism, and the prospect for the third option, i.e., to treat each level as ontologically equal. The purpose of this paper is to provide (...) an argument for the third option. The author will apply Aristotle’s theory of matter and form to the discussion of the hierarchy and develop a theory of form realism, which will grant every level with “full citizenship in the republic of being.” It constitutes an argument against ontological and epistemological reductionism. A non-reductive theory of causation is also developed against the fundamental theory of causation. (shrink)

In both The Right and the Good and Foundations of Ethics, W. D. Ross maintains that any amount of the non-instrumental value of virtue outweighs any amount of the non-instrumental value of pleasure or avoidance of pain. The chapter raises two challenges to the status that Ross accords the value of virtue relative to the value of pleasure (pain). First, it argues that Ross fails to provide a good argument for thinking that virtue is always better than pleasure and that (...) it is in any case implausible to think that any amount of virtue (or avoidance of vice) is better than the avoidance of any amount of pain or suffering. Second, it argues that the inflexibility of Ross’s value theory exhibited in his claim about the relative value of virtue produces tension with and mars the attractive heterarchical structure of his theory of rightness or prima facie duties. (shrink)

Some philosophers of fiction – most famously Jerold Levinson1 - have tried to argue that fictional narrators can never be identified with real authors. This argument relies on the claim that narration involves genuine assertion (not just the pretense of assertion that lacks truthfulness) and that real authors are not in a position to assert anything about beings on the fictional plain - given that they don’t rationally believe in their existence. This debate on the status of narrators depends on (...) the idea that fictional beings and beings in the real worlds reside, as it were, at different levels. The assumption that there is a gap separating the levels of fiction and reality serves as a rationale for the claim that real authors could not possibly be en rapport with the fictional characters that they create (e.g. entertain beliefs about them, etc.). (shrink)

International human rights concepts and documents figure prominently within theories of global justice. Appeals to human rights often rely on theories and interpretations that rank human rights in relation to one another designating some as more important or more crucial than others such that they may or must be given priority. In this paper I argue that hierarchical ranking of human rights should be rejected by theorists of global justice because such ranking: (a) undermines the effectiveness with which human rights (...) operate as principled constraints on state actors; and (b) is inconsistent with the international legal framework and practice from which the human rights concepts and documents draw their rhetorical force. Instead theorists of global justice should accept the indivisibility principle, which states that all human rights are interrelated and interdependent and must be treated on the same footing and given the same emphasis. The attraction of hierarchical ordering rests on misguided concerns about the practicality of indivisibility that fail to properly distinguish between rights and claims and ignore important dimensions of human rights implementation and assessment. I argue that indivisibility is both pragmatically defensible, provides a better theoretical grounding for understanding complaints about violations, and offers a better framework for understanding the relationship between human rights and legitimacy than hierarchical ordering. (shrink)

Category-specific impairments of object recognition and naming are among the most intriguing disorders in neuropsychology, affecting the retrieval of knowledge about either living or nonliving things. They can give us insight into the nature of our representations of objects: Have we evolved different neural systems for recognizing different categories of object? What kinds of knowledge are important for recognizing particular objects? How does visual similarity within a category influence object recognition and representation? What is the nature of our semantic knowledge (...) about different objects? We review the evidence on category-specific impairments, arguing that deficits even for one class of object cannot be accounted for in terms of a single information processing disorder across all patients; problems arise at contrasting loci in different patients. The same apparent pattern of impairment can be produced by damage to different loci. According to a new processing framework for object recognition and naming, the hierarchical interactive theory, we have a hierarchy of highly interactive stored representations. HIT explains the variety of patients in terms of lesions at different levels of processing and different forms of stored knowledge used both for particular tasks and for particular categories of object. Key Words: category-specific deficits; functional imaging; hierarchical models; interactive activation models; neuropsychology; object recognition; perceptual and functional knowledge. (shrink)

We introduce a new hierarchy of computably enumerable degrees. This hierarchy is based on computable ordinal notations measuring complexity of approximation of${\rm{\Delta }}_2^0$functions. The hierarchy unifies and classifies the combinatorics of a number of diverse constructions in computability theory. It does so along the lines of the high degrees (Martin) and the array noncomputable degrees (Downey, Jockusch, and Stob). The hierarchy also gives a number of natural definability results in the c.e. degrees, including a definable (...) antichain. (shrink)

A modified version of Normann's hierarchy of domains with totality [9] is presented and is shown to be suitable for interpretation of Martin-Löf's intuitionistic type theory. This gives an interpretation within classical set theory, which is natural in the sense that $\Sigma$ -types are interpreted as sets of pairs and $\Pi$ -types as sets of choice functions. The hierarchy admits a natural definition of the total objects in the domains, and following an idea of Berger [3] (...) this makes possible an interpretation where a type is defined to be true if its interpretation contains a total object. In particular, the empty type contains no total objects and will therefore be false (in any non-empty context). In addition, there is a natural equivalence relation on the total objects, so we derive a hierarchy of topological spaces (quotient spaces wrt. the Scott topology), and give a second interpretation using this hierarchy. (shrink)

It is a well-known fact of mathematical logic, by now developed in considerable detail, that formalized mathematical theories can be ordered by relative interpretability, and the "strength" of a theory is indicated by where it stands in this ordering. Mutual interpretability is an equivalence relation, and what I call an ordering is a partial ordering modulo this equivalence. Of the theories that have been studied, the natural theories belong to a linearly ordered subset of this ordering.