Using new techniques for controlling the categoricity spectrum of a structure, we construct a structure with degree of categoricity but infinite spectral dimension, answering a question of Bazhenov, Kalimullin and Yamaleev. Using the same techniques, we construct a computably categorical structure of non-computable Scott rank.

We review the current knowledge concerning strong jump-traceability. We cover the known results relating strong jump-traceability to randomness, and those relating it to degree theory. We also discuss the techniques used in working with strongly jump-traceable sets. We end with a section of open questions.

We define the Scott complexity of a countable structure to be the least complexity of a Scott sentence for that structure. This is a finer notion of complexity than Scott rank: it distinguishes between whether the simplest Scott sentence is $\Sigma _{\alpha }$, $\Pi _{\alpha }$, or $\mathrm {d-}\Sigma _{\alpha }$. We give a complete classification of the possible Scott complexities, including an example of a structure whose simplest Scott sentence is $\Sigma _{\lambda + 1}$ for $\lambda $ a limit (...) ordinal. This answers a question left open by A. Miller.We also construct examples of computable structures of high Scott rank with Scott complexities $\Sigma _{\omega _1^{CK}+1}$ and $\mathrm {d-}\Sigma _{\omega _1^{CK}+1}$. There are three other possible Scott complexities for a computable structure of high Scott rank: $\Pi _{\omega _1^{CK}}$, $\Pi _{\omega _1^{CK}+1}$, $\Sigma _{\omega _1^{CK}+1}$. Examples of these were already known. Our examples are computable structures of Scott rank $\omega _1^{CK}+1$ which, after naming finitely many constants, have Scott rank $\omega _1^{CK}$. The existence of such structures was an open question. (shrink)

EveryK-trivial set is computable from an incomplete Martin-Löf random set, i.e., a Martin-Löf random set that does not compute the halting problem.

We show the existence of noncomputable oracles which are low for Demuth randomness, answering a question in [15]. We fully characterize lowness for Demuth randomness using an appropriate notion of traceability. Central to this characterization is a partial relativization of Demuth randomness, which may be more natural than the fully relativized version. We also show that an oracle is low for weak Demuth randomness if and only if it is computable.

A problem is a multivalued function from a set of instances to a set of solutions. We consider only instances and solutions coded by sets of integers. A problem admits preservation of some computability-theoretic weakness property if every computable instance of the problem admits a solution relative to which the property holds. For example, cone avoidance is the ability, given a noncomputable set A and a computable instance of a problem ${\mathsf {P}}$, to find a solution relative to which A (...) is still noncomputable.In this article, we compare relativized versions of computability-theoretic notions of preservation which have been studied in reverse mathematics, and prove that the ones which were not already separated by natural statements in the literature actually coincide. In particular, we prove that it is equivalent to admit avoidance of one cone, of $\omega $ cones, of one hyperimmunity or of one non- $\Sigma ^{0}_1$ definition. We also prove that the hierarchies of preservation of hyperimmunity and non- $\Sigma ^{0}_1$ definitions coincide. On the other hand, none of these notions coincide in a nonrelativized setting. (shrink)

We introduce several highness notions on degrees related to the problem of computing isomorphisms between structures, provided that isomorphisms exist. We consider variants along axes of uniformity, inclusion of negative information, and several other problems related to computing isomorphisms. These other problems include Scott analysis (in the form of back-and-forth relations), jump hierarchies, and computing descending sequences in linear orders.

We extend the notion of limitwise monotonic functions to include arbitrary computable domains. We then study which sets and degrees are support increasing limitwise monotonic on various computable domains. As applications, we provide a characterization of the sets S with computable increasing η-representations using support increasing limitwise monotonic sets on ℚ and note relationships between the class of order-computable sets and the class of support increasing limitwise monotonic sets on certain domains.

It is commonly believed that Edmund Husserl (1859-1938), well known as the founder of phenomenology and as the teacher of Heidegger, was unable to free himself from the framework of a classical metaphysics of subjectivity. Supposedly, he never abandoned the view that the world and the Other are constituted by a pure transcendental subject, and his thinking in consequence remains Cartesian, idealistic, and solipsistic. The continuing publication of Husserl’s manuscripts has made it necessary to revise such an interpretation. Drawing upon (...) both Husserl’s published works and posthumous material, Husserl’s Phenomenology incorporates the results of the most recent Husserl research. It is divided into three parts, roughly following the chronological development of Husserl’s thought, from his early analyses of logic and intentionality, through his mature transcendental-philosophical analyses of reduction and constitution, to his late analyses of intersubjectivity and lifeworld. It can consequently serve as a concise and updated introduction to his thinking. (shrink)

... Let me start my investigation by taking a brief look at the way in which self-awareness is expressed linguistically, as in the sentences "I am tired" or ...

We show that the fact that the first player wins every instance of Galvin’s “racing pawns” game is equivalent to arithmetic transfinite recursion. Along the way we analyze the satisfaction relation for infinitary formulas, of “internal” hyperarithmetic comprehension, and of the law of excluded middle for such formulas.

In this entry, I tackle the phenomenon known as "faultless disagreement", considered by many authors to pose a challenge to the main views on the semantics of subjective expressions. I first present the phenomenon and the challenge, then review the main answers given by contextualist, absolutist and relativist approaches to the expressions in question. I end with signaling two issues that might shape future discussions about the role played by faultless disagreement in semantics.

ABSTRACT In this paper, I present data involving the use of the Romanian slur ‘țigan’, consideration of which leads to the postulation of a sui-generis, irreducible type of use of slurs. This type of use is potentially problematic for extant theories of slurs. In addition, together with other well-established uses, it shows that there is more variation in the use of slurs than previously acknowledged. I explain this variation by construing slurs as polysemous. To implement this idea, I appeal to (...) a rich-lexicon account of polysemy. I show how such a theory can be applied to slurs and discuss several important issues that arise. (shrink)

In this chapter, we focus on the debate over publicly maintained racist monuments as it manifests in the mid-2010s Anglosphere, primarily in the United States and South Africa. After pointing to some representative examples of racist monuments, we discuss ways a monument can be thought racist and neutrally categorize removalist and preservationist arguments heard in the monument debate. We suggest that both extremist and moderate removalist goals are likely to be self-defeating and that when concerns of civic sustainability are put (...) on moral par with those of fairness and justice, something like a Mandela-era preservationist policy is best: one which removes the most offensive of the minor racist monuments, but which focuses on closing the monumentary gap between peoples and reframing existing racist monuments. (shrink)

Using the concept of social denial, this article puts the American Psychological Association's (APA’s) pattern of willful blindness, identified by independent reviewer David Hoffman, in historical context by examining the contributions of Cold War social scientists to the CIA's KUBARK torture manual, and discusses the implications of this history for the reform of the APA's ethics policies. David Hoffman found that the leadership of the APA colluded with Department of Defense (DoD) to ensure that the APA's ethical policies were no (...) stronger than those issued by the DoD. While the independent reviewer did not find evidence of collaboration between the CIA and the APA, this was not due to a lack of effort on the part of the APA, which was anxious to establish good relations and so promote the use of psychology in the national security arena. While Hoffman did not find that the APA knew that its collaborations would facilitate the development of abusive interrogation techniques, it showed a marked, motivated lack of interest in whether or not the DoD or CIA was abusing prisoners. The APA maintained its strategic ignorance even while engaging in a public relations campaign designed to give the impression that it was deeply concerned about multiple reports of psychologist involvement in a system of torture. This willful ignorance was not unprecedented and follows a predictable pattern of knowing and not-knowing to which all psychologists should attend. (shrink)

Premodern Buddhists are sometimes characterized as veritable "mind scientists" whose insights anticipate modern research on the brain and mind. Aiming to complicate this story, Dan Arnold confronts a significant obstacle to popular attempts at harmonizing classical Buddhist and modern scientific thought: since most Indian Buddhists held that the mental continuum is uninterrupted by death, they would have no truck with the idea that everything about the mental can be explained in terms of brain events. Nevertheless, a predominant stream of Indian (...) Buddhist thought, associated with the seventh-century thinker Dharmakirti, turns out to be vulnerable to arguments modern philosophers have leveled against physicalism. By characterizing the philosophical problems commonly faced by Dharmakirti and contemporary philosophers such as Jerry Fodor and Daniel Dennett, Arnold seeks to advance an understanding of both first-millennium Indian arguments and contemporary debates on the philosophy of mind. The issues center on what modern philosophers have called _intentionality_--the fact that the mind can be about other things. Tracing an account of intentionality through Kant, Wilfrid Sellars, and John McDowell, Arnold argues that intentionality cannot, in principle, be explained in causal terms. Elaborating some of Dharmakirti's central commitments, Arnold shows that despite his concern to refute physicalism, Dharmakirti's causal explanations of the mental mean that modern arguments from intentionality cut as much against his project as they do against physicalist philosophies of mind. This is evident in the arguments of some of Dharmakirti's contemporaneous Indian critics, whose critiques exemplify the same logic as modern arguments from intentionality. Elaborating these various strands of thought, Arnold shows that seemingly arcane arguments among first-millennium Indian thinkers can illuminate matters still very much at the heart of contemporary philosophy. (shrink)

In _Buddhists, Brahmins, and Belief_, Dan Arnold examines how the Brahmanical tradition of Purva Mimamsa and the writings of the seventh-century Buddhist Madhyamika philosopher Candrakirti challenged dominant Indian Buddhist views of epistemology. Arnold retrieves these two very different but equally important voices of philosophical dissent, showing them to have developed highly sophisticated and cogent critiques of influential Buddhist epistemologists such as Dignaga and Dharmakirti. His analysis--developed in conversation with modern Western philosophers like William Alston and J. L. Austin--offers an innovative (...) reinterpretation of the Indian philosophical tradition, while suggesting that pre-modern Indian thinkers have much to contribute to contemporary philosophical debates. In logically distinct ways, Purva Mimamsa and Candrakirti's Madhyamaka opposed the influential Buddhist school of thought that emphasized the foundational character of perception. Arnold argues that Mimamsaka arguments concerning the "intrinsic validity" of the earliest Vedic scriptures are best understood as a critique of the tradition of Buddhist philosophy stemming from Dignaga. Though often dismissed as antithetical to "real philosophy," Mimamsaka thought has affinities with the reformed epistemology that has recently influenced contemporary philosophy of religion. Candrakirti's arguments, in contrast, amount to a principled refusal of epistemology. Arnold contends that Candrakirti marshals against Buddhist foundationalism an approach that resembles twentieth-century ordinary language philosophy--and does so by employing what are finally best understood as transcendental arguments. The conclusion that Candrakirti's arguments thus support a metaphysical claim represents a bold new understanding of Madhyamaka. (shrink)

[Requested essay for George Washington Leadership Institute curriculum, Fred W. Smith National Library for the Study of George Washington, Mt. Vernon.] Honor is often equated with integrity, dignity, courage, and unimpeachable reputation. But what is the underlying essence of honor that explains those associations? This essay provides a framework for thinking about honor, and explores a theory of honor that understands it in terms of agonism---that is, as an ethic regulating our pursuit of prestige according to principles of fair and (...) respectful contest. We begin by distinguishing between honor-as-prestige and honor as an ethical system. We then canvass honor’s commitment to fair competition, noblesse oblige, integrity, independence, and martial valor. After forwarding the agonistic theory of honor, we conclude with some suggestions for leaders interested in fostering honor-mindedness in their organizations. (shrink)

[Updated 2/23/21: complete chapter scan] In this chapter I sketch a rightist approach to monumentary policy in a diverse polity beleaguered by old ethnic grievances. I begin by noting the importance of tribalism, memorialization, and social trust. I then suggest a policy which 1) gradually narrows the gap between peoples in the heritage landscape, 2) conserves all but the most offensive of the least beloved racist monuments, 3) avoids recrimination (i.e., “keeps it positive”) and eschews ideological commentary in new monuments (...) or revisions to old ones, 4) as much as politically feasible, recognizes only the offense of willing tribemates, and 5) responds to aesthetic and other “irrational” offenses more than to “objective” historical or philosophical critiques. (shrink)

If one comes to Phénoménologie de la perception after having read Sein und Zeit (or Prolegomena zur Geschichte des Zeitbegriffs) one will be in for a surprise. Both works contain a number of both implicit and explicit references to Husserl, but the presentation they give is so utterly different, that one might occasionally wonder whether they are referring to the same author. Thus nobody can overlook that Merleau-Ponty’s interpretation of Husserl differs significantly from Heidegger’s. It is far more charitable. In (...) fact, when evaluating the merits of respectively Husserl and Heidegger, Merleau-Ponty often goes very much against the standard view. This is not only the case in his notorious remark on the very first page of Phénoménologie de la Perception where he declares that the whole of Sein und Zeit is nothing but an explication of Husserl’s notion of Lifeworld, but also - to give just one further example - in one of his Sorbonne-lectures, where Merleau-Ponty writes that Husserl took the issue of historicity far more seriously than Heidegger. (shrink)

How should modern medicine's dramatic new powers to sustain life be employed? How should limited resources be used to extend and improve the quality of life? In this collection, Dan Brock, a distinguished philosopher and bioethicist and co-author of Deciding for Others (Cambridge, 1989), explores the moral issues raised by new ideals of shared decision making between physicians and patients. The book develops an ethical framework for decisions about life-sustaining treatment and euthanasia, and examines how these life and death decisions (...) are transformed in health policy when the focus shifts from what is best for a patient to what is just for all patients. Professor Brock combines acute philosophical analysis with a deep understanding of the realities of clinical health policy. This is a volume for philosophers concerned with medical ethics, health policy professionals, physicians interested in bioethics, and undergraduate courses in biomedical ethics. (shrink)

Dan Zahavi engages with classical phenomenology, philosophy of mind, and a range of empirical disciplines to explore the nature of selfhood. He argues that the most fundamental level of selfhood is not socially constructed or dependent upon others, but accepts that certain dimensions of the self and types of self-experience are other-mediated.

The Oxford Handbook of the History of Phenomenology contains thirty-seven new essays by leading scholars in the field. The essays all highlight historical influences, connections, and developments and provide an in-depth coverage of the development of phenomenology; one that allows for a better comprehension and assessment of the continuity as well as diversity of the phenomenological tradition. The handbook is divided into three distinct parts. The first part contains chapters that address the way phenomenology has been influenced by earlier periods (...) or figures in the history of philosophy. The second part contains chapters targeting prominent phenomenologists: How was their work affected by earlier figures, how did their own views change over time, and what kind of influence did they exert on subsequent thinkers? The contributions in the third part trace various core topics such as subjectivity, intersubjectivity, embodiment, spatiality, and imagination in the work of different phenomenologists, in order to explore how the notions were transformed, enriched, and expanded up through the century. The handbook will be a source of insight for philosophers, students of philosophy, and for people working in other disciplines of the humanities, social sciences, and sciences, who are interested in the phenomenological tradition. It is an authoritative guide to how phenomenology started, how it developed, and where it is heading. (shrink)

Many times, what we say proves to be wrong. It might turn out that what we took to be a comforting remark was, in fact, making things worse. Or that a joke was inappropriate. Or that yelling out loud was rude. More importantly for this paper, there are plenty of cases in which what we said turns out to be false: we spoke without paying attention, we were misinformed or tricked, or we made a reasoning mistake. -/- A particular instance (...) of this latter phenomenon is when someone changes their perspective and doesn't find their previous assertions true anymore. For example, imagine that you used to like licorice as a child, and that you went around saying things like "Licorice is tasty." But, growing up, you find licorice too strong and quite boring; your tastes have changed. While the taste of licorice itself has not changed, you are not going around anymore uttering "Licorice is tasty." In fact, in certain cases, you might even think that what you said in your childhood was wrong - at least in light of your present preferences. -/- Sometimes we "take back" the assertions we made when we realize they are no longer true. Such speech acts of taking back can be performed in various ways: more formally, as effected in a court of law or when an official speaks to the press, or more informally in our day-by-day interactions. These acts of taking back are known as retraction. Retraction is usually achieved by employing certain linguistic markers ('I was wrong', 'I was mistaken', 'I retract', "Scratch that' etc.) that signal that the retractor takes their previous speech act to be faulty in a certain way. -/- This phenomenon has formed the basis for a popular argument used by relativists about a variety of natural language expressions (predicates of taste, epistemic modals, moral and aesthetic claims etc.) in support of their view. Recently, several considerations (mostly from contextualists about the same expressions), both from the armchair and based on empirical studies, have been offered to undercut the support retraction was thought to provide for relativism. In this paper, I consider and re-evaluate that support in light of both types of considerations, showing that neither of them decisively undermines it. However, to survive the contextualist counterattack, the relativist needs to pay a price: that is, she will be forced to make some concessions or rethink certain claims. I show what I think those concessions should be and what needs to be rethought (and how). The upshot is that, while the support retraction offers relativism is not as strong as initially believed, appeal to retraction remains a legitimate move in the debate. In section 1, I introduce the views in the debate and clarify the phenomenon of retraction and its dialectical role. Armchair data are investigated in section 2, proper experimental studies in section 3, while section 4 summarizes and concludes. (shrink)

The Oxford Handbook of Contemporary Phenomenology presents twenty-eight essays by some of the leading figures in the field, and gives an authoritative overview of the type of work and range of topics found and discussed in contemporary phenomenology. It is the definitive guide to what is currently going on in phenomenology, and offers a rich source of insight and stimulation for philosophers, students of philosophy, and for people working in other disciplines of the humanities, social sciences, and sciences, who are (...) interested in the state of phenomenology today. The essays explore a diverse range of topics, including intentionality, embodiment, perception, naturalism, temporality, self-consciousness, language, knowledge, ethics, politics, art and religion, and will make it clear that phenomenology, far from being a tradition of the past, is alive and in a position to make valuable contributions to contemporary thought. (shrink)

This revised edition includes a new Preface outlining developments in Relevance Theory since 1986, discussing the more serious criticisms of the theory, and ...

Human, cognition, interaction, and culture are thoroughly intertwined. Without cognition and interaction, there would be no culture. Without culture, cognition and interaction would be very different affairs, as they are among other social species. The effect of culture on mental life has always been a main concern of the social sciences and, after a long period of almost total neglect, it is more and more taken into consideration in cognitive psychology. The effect of cognition, and in particular of the ability (...) to attribute mental states to others, on interaction has become a important topic of investigation in developmental and social psychology. Little attention, on the other hand, is paid to the effect of cognition on culture. All would agree, of course that the human mind is what makes human culture possible. Quite generally, however, the mind is seen as a mere enabler of culture, a pure opportunity with no constraints attached, nothing that might contribute to shaping, or at least to biasing cultural contents. Against this neglect of cognition, I will argue that understanding the mind is doubly important to the study of culture. Psychological considerations are crucial both to a proper characterization of what is cultural and to a proper explanation of cultural phenomena. Most anthropologists may be too savvy to still talk of the mind as a “blank slate”, but they too often meet any discussion of psychological factors in culture with a blank stare. Most evolutionary theorists who see cultural evolution as an analogue of biological evolution by natural selection postulate rather than investigate the few particulars of the human mind.. (shrink)

This chapter determines a major empirical hurdle for any future discipline of memetics. It mainly shows that one can find very similar copies of some cultural item, link these copies through a causal chain of events which faithfully reproduced those items, and nevertheless not have an example of memetic inheritance. In addition, the stability of cultural patterns is proof that fidelity in copying is high despite individual variations. It is also believed that what is offered as an explanation is precisely (...) what needs to be explained; what is offered as a solution is in fact the very problem to be solved. Moreover, the issue is whether the relative stability observed in cultural transmission is proof of replication. The example of the acquisition of language is briefly addressed. The Darwinian model of selection is informative, and in various ways, for thinking about culture. Imitation is of course well worth investigating. On the other hand, the grand project of memetics is misled. (shrink)

Non-skeptical robust realists about normativity, mathematics, or any other domain of non- causal truths are committed to a correlation between their beliefs and non- causal, mind-independent facts. Hartry Field and others have argued that if realists cannot explain this striking correlation, that is a strong reason to reject their theory. Some consider this argument, known as the Benacerraf–Field argument, as the strongest challenge to robust realism about mathematics, normativity, and even logic. In this article I offer two closely related accounts (...) for the type of explanation needed in order to address Field's challenge. I then argue that both accounts imply that the striking correlation to which robust realists are committed is explainable, thereby discharging Field's challenge. Finally, I respond to some objections and end with a few unresolved worries. (shrink)

The relationship of self, and self-awareness, and experience: exploring classical phenomenological analyses and their relevance to contemporary discussions in ...

According to the Monist Interpretation of Descartes, there is really only one corporeal substance—the entire extended plenum. Evidence for this interpretation seems to be provided by Descartes in the Synopsis of the Meditations, where he claims that all substances are incorruptible. Finite bodies, being corruptible, would then fail to be substances. On the other hand, ‘body, taken in the general sense,’ being incorruptible, would be a corporeal substance. In this paper, I defend a Pluralist Interpretation of Descartes, according to which (...) there are many corporeal substances. In particular, I show that none of the claims in the Synopsis about incorruptibility and substance entail either that finite bodies are not substances, or that the only corporeal substance is the entire plenum. (shrink)

Reason, we are told, is what makes us human, the source of our knowledge and wisdom. If reason is so useful, why didn't it also evolve in other animals? If reason is that reliable, why do we produce so much thoroughly reasoned nonsense? In their groundbreaking account of the evolution and workings of reason, Hugo Mercier and Dan Sperber set out to solve this double enigma. Reason, they argue with a compelling mix of real-life and experimental evidence, is not geared (...) to solitary use, to arriving at better beliefs and decisions on our own. What reason does, rather, is help us justify our beliefs and actions to others, convince them through argumentation, and evaluate the justifications and arguments that others address to us. In other words, reason helps humans better exploit their uniquely rich social environment. This interactionist interpretation explains why reason may have evolved and how it fits with other cognitive mechanisms. It makes sense of strengths and weaknesses that have long puzzled philosophers and psychologists--why reason is biased in favor of what we already believe, why it may lead to terrible ideas and yet is indispensable to spreading good ones.--. (shrink)

Philosophers of physics have long debated whether the Past State of low entropy of our universe calls for explanation. What is meant by “calls for explanation”? In this article we analyze this notion, distinguishing between several possible meanings that may be attached to it. Taking the debate around the Past State as a case study, we show how our analysis of what “calling for explanation” might mean can contribute to clarifying the debate and perhaps to settling it, thus demonstrating the (...) fruitfulness of this analysis. Applying our analysis, we show that two main opponents in this debate, Huw Price and Craig Callender, are, for the most part, talking past each other rather than disagreeing, as they employ different notions of “calling for explanation”. We then proceed to show how answering the different questions that arise out of the different meanings of “calling for explanation” can result in clarifying the problems at hand and thus, hopefully, to solving them. (shrink)

Discussions of “privilege” have become increasingly common, but it’s often unclear what exactly people mean by “privilege.” Even well-known writings about privilege rarely take the time to define the word and explain what it means. The confusion this creates is one reason why debates about privilege are often contentious and unproductive. This essay aims to demystify privilege, presupposing no prior knowledge of philosophy. With a clear definition, it is easier to discuss some of the main debates about privilege: Is there (...) such a thing as “white privilege”? What about “male privilege”? And what’s the point of talking about privilege, anyway? (shrink)

In a well-known paper, Reginald Jackson expresses a sentiment not uncommon among readers of Locke: “Among the merits of Locke’s Essay…not even the friendliest critic would number consistency.”2 This unflattering opinion of Locke is reiterated by Maurice Mandelbaum: “Under no circumstances can [Locke] be counted among the clearest and most consistent of philosophers.”3 The now familiar story is that there are innumerable inconsistencies and internal problems contained in Locke’s Essay. In fact, it is probably safe to say that there is (...) not another canonical, well-respected, and seminal philosopher whose work is so widely thought to be swarming with inconsistencies. I, however, do not think that the common, unflattering view of Locke is accurate as a general view of the Essay. But despite my wishes to the contrary, I do believe that Locke’s chapter ‘Of Identity and Diversity’ (2.27)4 leads to (at least) one intractable problem, a problem that is the subject of this paper. (shrink)

Several authors believe that there are certain facts that are striking and cry out for explanation—for instance, a coin that is tossed many times and lands in the alternating sequence HTHTHTHTHTHT…. According to this view, we have prima facie reason to believe that such facts are not the result of chance. I call this view the striking principle. Based on this principle, some have argued for far-reaching conclusions, such as that our universe was created by intelligent design, that there are (...) many universes other than the one we inhabit, and that there are no mathematical or normative facts. Appealing as the view may initially seem, I argue that we lack sufficient reason to accept it. (shrink)

Dan Zahavi presents a rich new study of the philosophy of Edmund Husserl, the founder of phenomenology. What kind of philosophical project was Husserl engaged in? What is ultimately at stake in so-called phenomenological analyses? In this volume Zahavi makes it clear why Husserl had such a decisive influence on 20th-century philosophy.

We examine the sequences A that are low for dimension, i.e. those for which the effective dimension relative to A is the same as the unrelativized effective dimension. Lowness for dimension is a weakening of lowness for randomness, a central notion in effective randomness. By considering analogues of characterizations of lowness for randomness, we show that lowness for dimension can be characterized in several ways. It is equivalent to lowishness for randomness, namely, that every Martin-Löf random sequence has effective dimension (...) 1 relative to A, and lowishness for K, namely, that the limit of KA/K is 1. We show that there is a perfect [Formula: see text]-class of low for dimension sequences. Since there are only countably many low for random sequences, many more sequences are low for dimension. Finally, we prove that every low for dimension is jump-traceable in order nε, for any ε > 0. (shrink)

The idea that there are some facts that call for explanation serves as an unexamined premise in influential arguments for the inexistence of moral or mathematical facts and for the existence of a god and of other universes. This book is the first to offer a comprehensive and critical treatment of this idea. It argues that calling for explanation is a sometimes-misleading figure of speech rather than a fundamental property of facts.

LetEbe a computably enumerable equivalence relation on the setωof natural numbers. We say that the quotient set$\omega /E$realizesa linearly ordered set${\cal L}$if there exists a c.e. relation ⊴ respectingEsuch that the induced structure is isomorphic to${\cal L}$. Thus, one can consider the class of all linearly ordered sets that are realized by$\omega /E$; formally,${\cal K}\left = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}$. In this paper we study the relationship between computability-theoretic properties ofEand algebraic properties of linearly ordered sets realized (...) byE. One can also define the following pre-order$ \le _{lo} $on the class of all c.e. equivalence relations:$E_1 \le _{lo} E_2 $if every linear order realized byE1is also realized byE2. Following the tradition of computability theory, thelo-degrees are the classes of equivalence relations induced by the pre-order$ \le _{lo} $. We study the partially ordered set oflo-degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among thelo-degrees. (shrink)

The De hebdomadibus (DH) of Boethius presents a problem with the idea that ordinary finite substances are good and then proposes a solution to the problem. Careful reconstruction of Boethius’ arguments reveals that his solution relies on an account of finite goodness that he does not make explicit. Moreover, accounts of finite goodness that commentators have supplied to the DH should be rejected. Instead, the account of finite goodness given in book III of the Consolatio successfully resolves the problem raised (...) in the DH. (shrink)