This is a contribution to the study of the Muchnik and Medvedev lattices of non-empty Π01 subsets of 2ω. In both these lattices, any non-minimum element can be split, i. e. it is the non-trivial join of two other elements. In fact, in the Medvedev case, ifP > MQ, then P can be split above Q. Both of these facts are then generalised to the embedding of arbitrary finite distributive lattices. A consequence of this is that both lattices have decidible (...) ∃-theories. (shrink)

Let w and M be the countable distributive lattices of Muchnik and Medvedev degrees of non-empty Π1 0 subsets of 2ω, under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are lattice-embeddable below any non-zero element of w . We show that many countable distributive lattices are lattice-embeddable below any non-zero element of M.

We investigate the notion of hyperimmunity with respect to how it can be applied to Π{\sp 0}{\sb 1} classes and their Muchnik degrees. We show that hyperimmunity is a strong enough concept to prove the existence of Π{\sp 0}{\sb 1} classes with intermediate Muchnik degree—in contrast to Post's attempts to construct intermediate c.e. degrees.

The property of smallness for Π01 classes is introduced and is investigated with respect to Medvedev and Muchnik degree. It is shown that the property of containing a small Π01 class depends only on the Muchnik degree of a Π01 class. A comparison is made with the idea of thinness for Π01 classesmsthm.

The property of smallness for Π0 1 classes is introduced and is investigated with respect to Medvedev and Muchnik degree. It is shown that the property of containing a small Π0 1 class depends only on the Muchnik degree of a Π0 1 class. A comparison is made with the idea of thinness for Π0 1 classesmsthm.

An infinite binary sequence is complex if the Kolmogorov complexity of its initial segments is bounded below by a computable function. We prove that a Π₁⁰ class P contains a complex element if and only if it contains a wtt-cover for the Cantor set. That is, if and only if for every Y⊆ω there is an X in P such that X≥wtt Y. We show that this is also equivalent to the Π₁⁰ class's being large in some sense. We give (...) an example of how this result can be used in the study of scattered linear orders. (shrink)

An infinite binary sequence is complex if the Kolmogorov complexity of its initial segments is bounded below by a computable function. We prove that a $\Pi _{1}^{0}$ class P contains a complex element if and only if it contains a wtt-cover for the Cantor set. That is, if and only if for every Y ⊆ ω there is an X in P such that X ⩾wtt Y. We show that this is also equivalent to the $\Pi _{1}^{0}$ class's being large (...) in some sense. We give an example of how this result can be used in the study of scattered linear orders. (shrink)

We use the notions of effective dimension and Kolmogorov complexity to describe a geometry on the set of infinite binary sequences. Geometric concepts that we define and use include angle, projections and scalar multiplication. A question related to compressibility is addressed using these ideas.

We consider two axioms of second-order arithmetic. These axioms assert, in two different ways, that infinite but narrow binary trees always have infinite paths. We show that both axioms are strictly weaker than Weak König's Lemma, and incomparable in strength to the dual statement (WWKL) that wide binary trees have paths.

We divide the class of infinite computable trees into three types. For the first and second types, 0' computes a nontrivial self-embedding while for the third type 0'' computes a nontrivial self-embedding. These results are optimal and we obtain partial results concerning the complexity of nontrivial self-embeddings of infinite computable trees considered up to isomorphism. We show that every infinite computable tree must have either an infinite computable chain or an infinite Π01 antichain. This result is optimal and has connections (...) to the program of reverse mathematics. (shrink)

This paper continues the study of the metric topology on $2^{\mathbb {N}}$ that was introduced by S. Binns. This topology is induced by a directional metric where the distance from $Y\in2^{\mathbb {N}}$ to $X\in2^{\mathbb {N}}$ is given by \[\limsup_{n}\frac{C(X\upharpoonright n|Y\upharpoonright n)}{n}.\] This definition is closely related to the notions of effective Hausdorff and packing dimensions. Here we establish that this is a path-connected topology on $2^{\mathbb {N}}$ and that under it the functions $X\mapsto\operatorname{dim}_{\mathcal{H}}X$ and $X\mapsto\operatorname{dim}_{p}X$ are continuous. We also (...) investigate the scalar multiplication operation that was introduced by Binns. The multiplication of a real $X\in2^{\mathbb {N}}$ by an element $\alpha\in[0,1]$ represents a dilution of the information in $X$ by a factor of $\alpha$ . Our main result is to show that every regular real is the dilution of a real of Hausdorff dimension 1. That is, that the information in every regular real can be maximally compressed. (shrink)

Recall that [Formula: see text] is the lattice of Muchnik degrees of nonempty effectively compact sets in Euclidean space. We solve a long-standing open problem by proving that [Formula: see text] is dense, i.e. satisfies [Formula: see text]. Our proof combines an oracle construction with hyperarithmetical theory.

A Brief History of Time: From the Big Bang to Black Holes is a popular-science book on cosmology (the study of the origin and evolution of the universe) by British physicist Stephen Hawking. It was first published in 1988. Hawking wrote the book for readers who have no prior knowledge of the universe and people who are interested in learning.

A comprehensive reassessment of the concept of mimesis in the history of ancient Greek aesthetics and philosophy of art, with particular attention to Plato, Aristotle, Hellenistic philosophy, and neoplatonism. There is also a wide-ranging review of arguments pro and contra the idea of artistic mimesis from the Renaissance to modern literar theory. The book challenges standard accounts in numerous respects and builds a new dialectical model with which to make sense of the entire history of mimeticist thinking in aesthetics.

Aboutness has been studied from any number of angles. Brentano made it the defining feature of the mental. Phenomenologists try to pin down the aboutness-features of particular mental states. Materialists sometimes claim to have grounded aboutness in natural regularities. Attempts have even been made, in library science and information theory, to operationalize the notion. But it has played no real role in philosophical semantics. This is surprising; sentences have aboutness-properties if anything does. Aboutness is the first book to examine through (...) a philosophical lens the role of subject matter in meaning. A long-standing tradition sees meaning as truth-conditions, to be specified by listing the scenarios in which a sentence is true. Nothing is said about the principle of selection--about what in a scenario gets it onto the list. Subject matter is the missing link here. A sentence is true because of how matters stand where its subject matter is concerned. Stephen Yablo maintains that this is not just a feature of subject matter, but its essence. One indicates what a sentence is about by mapping out logical space according to its changing ways of being true or false. The notion of content that results--directed content--is brought to bear on a range of philosophical topics, including ontology, verisimilitude, knowledge, loose talk, assertive content, and philosophical methodology. Written by one of today's leading philosophers, Aboutness represents a major advance in semantics and the philosophy of language. (shrink)

In Return to Reason, Stephen Toulmin argues that the potential for reason to improve our lives has been hampered by a serious imbalance in our pursuit of ...

Stephen Hetherington's prominent career within epistemology has been a series of distinctive, bold, varied and provocative arguments and ideas. Bringing together Hetherington's unique body of writing for the first time, this collection features previously published as well as new material that link his approaches to key issues including knowledge, justification, fallibility, scepticism and the Gettier Problem. Advancing our understanding of the systemic nature of Hetherington's thinking, Stephen Hetherington on Epistemology presents his distinctive perspective on some of philosophy's central (...) questions about knowledge - an inviting blend of forensic detail and 'big picture' proposals. (shrink)

As government pressure on major technology companies builds, both firms and legislators are searching for technical solutions to difficult platform governance puzzles such as hate speech and misinformation. Automated hash-matching and predictive machine learning tools – what we define here as algorithmic moderation systems – are increasingly being deployed to conduct content moderation at scale by major platforms for user-generated content such as Facebook, YouTube and Twitter. This article provides an accessible technical primer on how algorithmic moderation works; examines some (...) of the existing automated tools used by major platforms to handle copyright infringement, terrorism and toxic speech; and identifies key political and ethical issues for these systems as the reliance on them grows. Recent events suggest that algorithmic moderation has become necessary to manage growing public expectations for increased platform responsibility, safety and security on the global stage; however, as we demonstrate, these systems remain opaque, unaccountable and poorly understood. Despite the potential promise of algorithms or ‘AI’, we show that even ‘well optimized’ moderation systems could exacerbate, rather than relieve, many existing problems with content policy as enacted by platforms for three main reasons: automated moderation threatens to further increase opacity, making a famously non-transparent set of practices even more difficult to understand or audit, further complicate outstanding issues of fairness and justice in large-scale sociotechnical systems and re-obscure the fundamentally political nature of speech decisions being executed at scale. (shrink)

Defenders of pragmatic encroachment in epistemology (or what I call practicalism) need to address two main problems. First, the view seems to imply, absurdly, that knowledge can come and go quite easily—in particular, that it might come and go along with our variable practical interests. We can call this the stability problem. Second, there seems to be no fully satisfying way of explaining whose practical interests matter. We can call this the “whose stakes?” problem. I argue that both problems can (...) be addressed in roughly the same terms. More exactly, I argue that by first clarifying the whose stakes? problem an answer to the stability problem naturally falls out. (shrink)

Elizabeth Anscombe famously argued that "modern moral philosophy" centrally involved unsupported notions of obligation and culpability. Modern Moral Philosophy: From Grotius to Kant exhibits, for the first time, resources that modern moral philosophers had to respond to Anscombe's challenge, also enhancing our own philosophical grasp of morality and its foundations.

The ever-increasing application of algorithms to decision-making in a range of social contexts has prompted demands for algorithmic accountability. Accountable decision-makers must provide their decision-subjects with justifications for their automated system’s outputs, but what kinds of broader principles should we expect such justifications to appeal to? Drawing from political philosophy, I present an account of algorithmic accountability in terms of the democratic ideal of ‘public reason’. I argue that situating demands for algorithmic accountability within this justificatory framework enables us to (...) better articulate their purpose and assess the adequacy of efforts toward them. (shrink)

In this paper I elucidate various ways in which understanding can be seen as an excellence of the mind or intellectual virtue. Along the way, I take up the neglected issue of what it might mean to be an “understanding person”—by which I mean not a person who understands a number of things about the natural world, but a person who steers clear of things like judgmentalism in her evaluation of other people, and thus is better able to take up (...) different perspectives and view them with a sympathetic eye. Being an understanding person in this sense seems to be a character-level virtue that interestingly combines moral and epistemic elements; it also seems to be a virtue particularly needed in our age of deep political division, where it is commonly said that failures of mutual understanding are partly to blame for this problem. (shrink)

Keith Augustine’s critical evaluation of the essay contest sponsored by the Bigelow Institute of Consciousness Studies (BICS) is an interesting but problematic review. It mixes reasonable and detailed criticisms of the contest and many of the winning essays with a disappointing reliance on some of the most trite and superficial criticisms of parapsychological research. Ironically, Augustine criticizes the winning essays for using straw-man arguments and cherry-picked evidence even though many of his own arguments commit these same errors.

Many experimental philosophers are philosophers by training and professional affiliation, but some best work in experimental philosophy has been done by people who do not have advanced degrees in philosophy and do not teach in philosophy departments. This chapter explains that the experimental philosophy is the empirical investigation of philosophical intuitions, the factors that affect them, and the psychological and neurological mechanisms that underlie them. It explores what are philosophical intuitions, and why do experimental philosophers want to study them using (...) the methods of empirical science. The positive program in experimental philosophy shares the goal of the substantial part of traditional philosophy that is concerned with the analysis of important philosophical concepts. The negative program has implications for philosophical projects whose goal is conceptual analysis. There have been a number of responses to the challenge posed by experimental philosophy's negative program. The chapter also focuses on the expertise defense. (shrink)

Why is there anything, rather than nothing at all? This question often serves as a debating tactic used by theists to attack naturalism. Many people apparently regard the question—couched in such stark, general terms—as too profound for natural science to answer. It is unanswerable by science, I argue, not because it’s profound or because science is superficial but because the question, as it stands, is ill-posed and hence has no answer in the first place. In any form in which it (...) is well-posed, it has an answer that naturalism can in principle provide. The question therefore gives the foes of naturalism none of the ammunition that many on both sides of the debate think it does. (shrink)

In this article, from the 2009 Oxford University Press collection Epistemic Value, I criticize existing accounts of epistemic normativity by Alston, Goldman, and Sosa, and then offer a new view.

Philosophical Perspectives on Art presents a series of essays devoted to two of the most fundamental topics in the philosophy of art: the distinctive character of artworks and what is involved in understanding them as art. In Part I, Stephen Davies considers a wide range of questions about the nature and definition of art. Can art be defined, and if so, which definitions are the most plausible? Do we make and consume art because there are evolutionary advantages to doing (...) so? Has art completed the mission that guided its earlier historical development, and if so, what is to become of it now? Should architecture be classified as an art form? -/- Part II turns to the interpretation and appreciation of art. What is the target and purpose of the critic's interpretation? Is interpretation primarily directed at uncovering artists' intended meanings? Can apparently contradictory interpretations of a given piece both be true? Are interpretative evaluations entailed by descriptions of a work's aesthetic and artistic characteristics? In addition to providing fresh answers to these and other central questions in aesthetics, Davies considers the nature and content of metaphor, and the relation between the expressive qualities of a work of art and the emotions of its creator. (shrink)

In November 2022, OpenAI released ChatGPT, an incredibly sophisticated chatbot. Its capability is astonishing: as well as conversing with human interlocutors, it can answer questions about history, explain almost anything you might think to ask it, and write poetry. This level of achievement has provoked interest in questions about whether a chatbot might have something similar to human intelligence or even consciousness. Given that the function of a chatbot is to process linguistic input and produce linguistic output, we consider the (...) question whether a sophisticated chatbot might have inner speech. That is: Might it talk to itself, internally? We explored this via a conversation with ‘Playground’, a chatbot which is very similar to ChatGPT but more flexible in certain respects. We asked it questions which, plausibly, can only be answered if one first produces some inner speech. Here, we present our findings and discuss their philosophical significance. (shrink)

The long human odyssey of self-discovery has reached a crucial stage: everything we do affects everyone and everything else - and we know it. The next hundred years will bring more change than we can easily imagine: more opportunities for more people to achieve the fulfilment of a good life, and more risks that could result in catastrophic harm to the entire planet.Viewed geopolitically, the main question is whether the world-views of the world's most important and influential powers - China (...) and America (the one fundamentally Confucian, the other essentially individualist) - can be made to work together constructively.At the same time, on a deeper level, the even greater question is how the irreversible fact of urbanisation may nurture healthy and mature human individuality, such that the accumulated wisdom of the world's great cultures becomes mutually transforming and enriching.This bold and wide-ranging book explores those questions, with all the risks and opportunities they hold for generations still to com"--Publisher information. (shrink)

The field of modern day bioethics is relatively young and continues to constantly evolve in parallel with the ever increasingly complex nature of contemporary medical practice. These advances present clinicians with an array of therapeutic options that would have not seemed possible only a generation ago. Given these medical advances and the expansion of the academic and medicolegal field of bioethics, one would have thought that clinical decision making would have become easier. However paradoxically this has not proved to be (...) the case. Each advance may provide an answer to a particular clinical question which is almost immediately replaced by a whole new set of ethical problems that require consideration. (shrink)

Traces the rise and fall of successive international systems from medieval times to the present, showing how international order is created, how it is maintained, and why it breaks down.