We say that A≤LRB if every B-random set is A-random with respect to Martin–Löf randomness. We study this relation and its interactions with Turing reducibility, classes, hyperimmunity and other recursion theoretic notions.

We show that the identity bounded Turing degrees of computably enumerable sets are not dense.

We prove a number of results in effective randomness, using methods in which Π⁰₁ classes play an essential role. The results proved include the fact that every PA Turing degree is the join of two random Turing degrees, and the existence of a minimal pair of LR degrees below the LR degree of the halting problem.

A set $B\subseteq\mathbb{N}$ is called low for Martin-Löf random if every Martin-Löf random set is also Martin-Löf random relative to B . We show that a $\Delta^0_2$ set B is low for Martin-Löf random if and only if the class of oracles which compress less efficiently than B , namely, the class $\mathcal{C}^B=\{A\ |\ \forall n\ K^B(n)\leq^+ K^A(n)\}$ is countable (where K denotes the prefix-free complexity and $\leq^+$ denotes inequality modulo a constant. It follows that $\Delta^0_2$ is the largest arithmetical (...) class with this property and if $\mathcal{C}^B$ is uncountable, it contains a perfect $\Pi^0_1$ set of reals. The proof introduces a new method for constructing nontrivial reals below a $\Delta^0_2$ set which is not low for Martin-Löf random. (shrink)

We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress strings, when they are used as oracles. The results are put into context and several connections are made with various central issues in modern algorithmic randomness and computability.

Given two infinite binary sequences A,B we say that B can compress at least as well as A if the prefix-free Kolmogorov complexity relative to B of any binary string is at most as much as the prefix-free Kolmogorov complexity relative to A, modulo a constant. This relation, introduced in Nies [14] and denoted by A≤LKB, is a measure of relative compressing power of oracles, in the same way that Turing reducibility is a measure of relative information. The equivalence classes (...) induced by ≤LK are called LK degrees and there is a least degree containing the oracles which can only compress as much as a computable oracle, also called the ‘low for K’ sets. A well-known result from Nies [14] states that these coincide with the K-trivial sets, which are the ones whose initial segments have minimal prefix-free Kolmogorov complexity. We show that with respect to ≤LK, given any non-trivial sets X,Y there is a computably enumerable set A which is not K-trivial and it is below X,Y. This shows that the local structures of and Turing degrees are not elementarily equivalent to the corresponding local structures in the LK degrees. It also shows that there is no pair of sets computable from the halting problem which forms a minimal pair in the LK degrees; this is sharp in terms of the jump, as it is known that there are sets computable from which form a minimal pair in the LK degrees. We also show that the structure of LK degrees below the LK degree of the halting problem is not elementarily equivalent to the or structures of LK degrees. The proofs introduce a new technique of permitting below a set that is not K-trivial, which is likely to have wider applications. (shrink)

We study ideals in the computably enumerable Turing degrees, and their upper bounds. Every proper ideal in the c.e. Turing degrees has an incomplete upper bound. It follows that there is no prime ideal in the c.e. Turing degrees. This answers a question of Calhoun [2]. Every proper ideal in the c.e. Turing degrees has a low2 upper bound. Furthermore, the partial order of ideals under inclusion is dense.

A new approach for a uniform classification of the computably approximable real numbers is introduced. This is an important class of reals, consisting of the limits of computable sequences of rationals, and it coincides with the 0'-computable reals. Unlike some of the existing approaches, this applies uniformly to all reals in this class: to each computably approximable real x we assign a degree structure, the structure of all possible ways available to approximate x. So the main criterion for such classification (...) is the variety of the effective ways we have to approximate a real number. We exhibit extreme cases of such approximation structures and prove a number of related results. (shrink)

We say that A ≤LR B if every B-random number is A-random. Intuitively this means that if oracle A can identify some patterns on some real γ. In other words. B is at least as good as A for this purpose. We study the structure of the LR degrees globally and locally (i.e., restricted to the computably enumberable degrees) and their relationship with the Turing degrees. Among other results we show that whenever α in not GL₂ the LR degree of (...) α bounds $2^{\aleph _{0}}$ degrees (so that, in particular, there exist LR degrees with uncountably many predecessors) and we give sample results which demonstrate how various techniques from the theory of the c.e. degrees can be used to prove results about the c.e. LR degrees. (shrink)

TheK-trivial sets form an ideal in the Turing degrees, which is generated by its computably enumerable members and has an exact pair below the degree of the halting problem. The question of whether it has an exact pair in the c.e. degrees was first raised in [22, Question 4.2] and later in [25, Problem 5.5.8].We give a negative answer to this question. In fact, we show the following stronger statement in the c.e. degrees. There exists aK-trivial degreedsuch that for all (...) degreesa, bwhich are notK-trivial anda > d, b > dthere exists a degreevwhich is notK-trivial anda > v, b > v. This work sheds light to the question of the definability of theK-trivial degrees in the c.e. degrees. (shrink)

We show that there is a strong minimal pair in the computably enumerable Turing degrees, i.e. a pair of nonzero c.e. degrees a and b such that a∩b = 0 and for any nonzero c.e. degree x ≤ a, b ∪ x ≥ a.

We study the classes of hypersimple and semicomputable sets as well as their intersection in the weak truth table degrees. We construct degrees that are not bounded by hypersimple degrees outside any non-trivial upper cone of Turing degrees and show that the hypersimple-free c.e. wtt degrees are downwards dense in the c.e. wtt degrees. We also show that there is no maximal (w.r.t. ≤wtt) hypersimple wtt degree. Moreover, we consider the sets that are both hypersimple and semicomputable, characterize them as (...) the (bi-infinite) c.e. cuts of computable orderings of ℕ of order type ω+ω* and study their wtt degrees. We show that there are hypersimple degrees that are not bounded by any hypersimple semicomputable degree, investigate relationships with the join and look for maximal and minimal elements of related classes. (shrink)

The strong weak truth table (sw) reducibility was suggested by Downey, Hirschfeldt, and LaForte as a measure of relative randomness, alternative to the Solovay reducibility. It also occurs naturally in proofs in classical computability theory as well as in the recent work of Soare, Nabutovsky, and Weinberger on applications of computability to differential geometry. We study the sw-degrees of c.e. reals and construct a c.e. real which has no random c.e. real (i.e., Ω number) sw-above it.

We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on reducibilities that measure the initial segment complexity of reals and the power of reals to compress strings, when they are used as oracles. The results are put into context and several connections are made with various central issues in modern algorithmic randomness and computability.

We study Δ2 reals x in terms of how they can be approximated symmetrically by a computable sequence of rationals. We deal with a natural notion of ‘approximation representation’ and study how these are related computationally for a fixed x. This is a continuation of earlier work; it aims at a classification of Δ2 reals based on approximation and it turns out to be quite different than the existing ones (based on information content etc.).

We study the approximation properties of computably enumerable reals. We deal with a natural notion of approximation representation and study their wtt-degrees. Also, we show that a single representation may correspond to a quite diverse variety of reals.

We investigate notions of randomness in the space ${{\mathcal C}(2^{\mathbb N})}$ of continuous functions on ${2^{\mathbb N}}$ . A probability measure is given and a version of the Martin-Löf test for randomness is defined. Random ${\Delta^0_2}$ continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. For any ${y \in 2^{\mathbb N}}$ , (...) there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set. The set of zeroes of a random continuous function is always a random closed set. (shrink)

We extend the hierarchy defined in [5] to cover all hyperarithmetical reals. An intuitive idea is used or the definition, but a characterization of the related classes is obtained. A hierarchy theorem and two fixed point theorems are presented.

We study inversions of the jump operator on ${\mathrm{\Pi }}_{1}^{0}$ classes, combined with certain basis theorems. These jump inversions have implications for the study of the jump operator on the random degrees—for various notions of randomness. For example, we characterize the jumps of the weakly 2-random sets which are not 2-random, and the jumps of the weakly 1-random relative to 0′ sets which are not 2-random. Both of the classes coincide with the degrees above 0′ which are not 0′-dominated. A (...) further application is the complete solution of [24, Problem 3.6.9]: one direction of van Lambalgen's theorem holds for weak 2-randomness, while the other fails. Finally we discuss various techniques for coding information into incomplete randoms. Using these techniques we give a negative answer to [24, Problem 8.2.14]: not all weakly 2-random sets are array computable. In fact, given any oracle X, there is a weakly 2-random which is not array computable relative to X. This contrasts with the fact that all 2-random sets are array computable. (shrink)

We show that in the c.e. weak truth table degrees if b < c then there is an a which contains no hypersimple set and b < a < c. We also show that for every w < c in the c.e. wtt degrees such that w is hypersimple, there is a hypersimple a such that w < a < c. On the other hand, we know that there are intervals which contain no hypersimple set.

We show that there exists a real α such that, for all reals β, if α is linear reducible to β then β≤Tα. In fact, every random real satisfies this quasi-maximality property. As a corollary we may conclude that there exists no ℓ-complete Δ2 real. Upon realizing that quasi-maximality does not characterize the random reals–there exist reals which are not random but which are of quasi-maximal ℓ-degree–it is then natural to ask whether maximality could provide such a characterization. Such hopes, (...) however, are in vain since no real is of maximal ℓ-degree. (shrink)

Let h : ℕ → ℚ be a computable function. A real number x is called h-monotonically computable if there is a computable sequence of rational numbers which converges to x h-monotonically in the sense that h|x – xn| ≥ |x – xm| for all n andm > n. In this paper we investigate classes h-MC of h-mc real numbers for different computable functions h. Especially, for computable functions h : ℕ → ℚ, we show that the class h-MC coincides (...) with the classes of computable and semi-computable real numbers if and only if Σi∈ℕ) = ∞and the sum Σi∈ℕ) is a computable real number, respectively. On the other hand, if h ≥ 1 and h converges to 1, then h-MC = SC no matter how fast h converges to 1. Furthermore, for any constant c > 1, if h is increasing and converges to c, then h-MC = c-MC. Finally, if h is monotone and unbounded, then h-MC contains all ω-mc real numbers which are g-mc for some computable function g. (shrink)

Review by: Johanna N. Y. Franklin The Bulletin of Symbolic Logic, Volume 19, Issue 1, Page 115-118, March 2013.

Global Economy, Global Justice explores a vital question that is suppressed in most economics texts: "what makes for a good economic outcome?" Neoclassical theory embraces the normative perspective of "welfarism" to assess economic outcomes. This volume demonstrates the fatal flaws of this perspective--flaws that stem from objectionable assumptions about human nature, society and science. Exposing these failures, the book obliterates the ethical foundations of global neoliberalism. George DeMartino probes heterodox economic traditions and philosophy in search of an ethically viable (...) alternative to welfarism. Drawing on the work of Amartya Sen, DeMartino proposes the egalitarian principle of the "global harmonization of capabilities" to guide economics. This principle provides a basis for resisting oppression the world over while nevertheless demanding respect for cultural diversity. DeMartino puts this principle to work adjudicating contemporary debates over global policy regimes, and completes thebook with a set of deeply egalitarian global policies for the year 2025. Global Economy, Global Justice 's engaging prose will appeal to those seeking to understand the intersection between economics and political philosophy. Its focus on the normative foundations of contemporary policy disputes makes it unique in the literature on globalization. (shrink)

Written in the 1960s, TRUTH AND METHOD is Gadamer's magnum opus.

George Berkeley (1685-1753) is the superstar of Irish Philosophy. He entered Trinity College, Dublin, in 1700 and became a fellow in 1707. In 1724 he resigned his Fellowship to become Dean of Derry, and in 1734 he was made Bishop of Cloyne. He settled in Oxford in 1752 and died the following year. The work of George Berkeley is marked by its diversity and range. His writings take in such topics as mathematics, psychology, politics, health, economics, deism and (...) education, as well as that with which he is most associated - philosophy. Whatever topic he dealt with, his grasp of the subject matter was always impressive and his criticisms of his contemporaries often acute. Among the most noteable of the British Empiricists, he took his starting point from Locke's new 'way of ideas', but he rejected abstract ideas and the possibility of real existence outside perception. It is for this doctrine that he is most known - that there is no such thing as matter, and that things are collections of ideas which can exist only in our minds and only for so long as they are perceived. This philosophy, summed up in his phrase 'to be is to be perceived', he termed 'immaterialism'. A.C. Fraser's collection includes a series of what were previously unpublished notes by Berkeley on all the main topics of his philosophy. This, the 1901 edition, was the first complete edition of his works. (shrink)

A brilliant and comprehensive history of the creation of the modern Western mind. Soul Machine takes us back to the origins of modernity, a time when a crisis in religious authority and the scientific revolution led to searching questions about the nature of human inner life. This is the story of how a new concept—the mind—emerged as a potential solution, one that was part soul and part machine, but fully neither. In this groundbreaking work, award-winning historian George Makari shows (...) how writers, philosophers, physicians, and anatomists worked to construct notions of the mind as not an ethereal thing, but a natural one. From the ascent of Oliver Cromwell to the fall of Napoleon, seminal thinkers like Hobbes, Locke, Diderot, and Kant worked alongside often-forgotten brain specialists, physiologists, and alienists in the hopes of mapping the inner world. Conducted in a cauldron of political turmoil, these frequently shocking, always embattled efforts would give rise to psychiatry, mind sciences such as phrenology, and radically new visions of the self. Further, they would be crucial to the establishment of secular ethics and political liberalism. Boldly original, wide-ranging, and brilliantly synthetic, Soul Machine gives us a masterful, new account of the making of the modern Western mind. (shrink)

Throughout The Blessed and Boundless God, he proves his doctrine by demonstrating God's incomparableness in His being, attributes, works, and words. Swinnock is a pastor-theologian who views theology as the means by which we grow in acquaintance with God and, consequently, in godliness.

Berkeley's idealism started a revolution in philosophy. As one of the great empiricist thinkers he not only influenced British philosophers from Hume to Russell and the logical positivists in the twentieth century, he also set the scene for the continental idealism of Hegel and even the philosophy of Marx. There has never been such a radical critique of common sense and perception as that given in Berkeley's Principles of Human Knowledge (1710). His views were met with disfavour, and his response (...) to his critics was the Three Dialogues between Hylas and Philonous. This edition of Berkeley's two key works has an introduction which examines and in part defends his arguments for idealism, as well as offering a detailed analytical contents list, extensive philosophical notes, and an index. (shrink)

George Herbert Mead (1863-1931) had a powerful influence on the development of American pragmatism in the twentieth century. He also had a strong impact on the social sciences. This classic book represents Mead's philosophy of experience, so central to his outlook. The present as unique experience is the focus of this deep analysis of the basic structure of temporality and consciousness. Mead emphasizes the novel character of both the present and the past. Though science is predicated on the assumption (...) that the present is predictable based on a thorough knowledge of the past, the experience of the present, says Mead, is an utterly unique moment comparable to no other, and when it is past the novel character of that unique experience is irrevocable. The emergence of novelty within the perceived rational order of reality is the crux of the problem that Mead explores. The present, in his words, is "the emergent event . . . something which is more than the processes that have led up to it and which by its change, continuance, or disappearance, adds to later passages a content they would not otherwise have possessed." The present as "the seat of reality" heavily conditions our retrospective view of the past as much as it helps to shape the future. The novelty of every present experience causes us to reconstruct our preceding experiences to make sense of the past, which is naturally assumed to be the main cause of what we presently experience. Our perspective on reality is thus relative to the conditioning of each new event and it changes continuously as the effects of the present shift our view of the past and future. This emphasis on the integrative, holistic nature of reality, in which everything past, present, and future is a condition of everything else, makes Mead's philosophy highly relevant to today's scientific picture of a quantum universe, where chance and probability play a role in the emergence of reality. Also of great interest is the way in which he extends his basic analysis of temporal-spatial reality to the emergence of mind and consciousness as a natural development of the evolutionary process. This stimulating and provocative work attests to John Dewey's praise of Mead as "the most original mind in philosophy in America" of his generation. (shrink)

touch 27 Thirrdly, the straining of the eye 28 The occasions which suggest distance have in their own nature no relation to it 29 A difficult case proposed by Dr. Barrow as repugnant to all the known theories 30 This case contradicts a ...

Why does scholarship on transnational labor regulation (TLR) consistently fails to search for improvements in working conditions, and instead devotes itself to relentless efforts for identifying administrative processes, semantics, and amalgamations of stakeholders? This article critiques TLR from a pro-worker perspective, through the philosophical work of Georg Lukács, and the concepts of reification and commodification. A set of theoretically grounded criteria is developed and these are applied against selected contemporary cases of TLR. In the totality that is capitalism, reification of (...) social relations of production conceals completely the experiences of workers. In TLR, managerialist and process-oriented scholarship is dominant, verifiable outcomes and positive improvements in conditions of employment are not sought, and worse, meaningless procedures are celebrated as positive achievements. (shrink)

The Oxford Philosophical Texts series consists of authoritative teaching editions of canonical texts in the history of philosophy from the ancient world down to modern times. Each volume provides a clear, well laid out text together with a comprehensive introduction by a leading specialist,giving the student detailed critical guidance on the intellectual context of the work and the structure and philosophical importance of the main arguments. Endnotes are supplied which provide further commentary on the arguments and explain unfamiliar references and (...) terminology, and a fullbibliography and index are also included.The series aims to build up a definitive corpus of key texts in the Western philosophical tradition, which will form a reliable and enduring resource for students and teachers alike.In his Principles of Human Knowledge Berkeley makes the striking claim that physical things consist of nothing but ideas, and so do not exist outside the mind. This establishes Berkeley as the founder of the idealist tradition in philosophy. Berkeley argues vigorously that once we correct ourunderstanding of the physical, we can find a new proof of the existence of God, refute sceptical attacks on human knowledge, and resolve many difficulties and paradoxes raised by the advance of science.The text printed in this volume is the 1734 edition of the Principles which is generally agreed to represent Berkeley's mature thought. Also included are the four important letters between George Berkeley and Samuel Johnson, written in 1729-30. The text is supplemented by a comprehensiveintroduction which looks at the structure and main arguments of the text, as well as discussing Berkeley's life, influences, and general philosophy. In addition the volume includes an analysis of the text, a glossary, detailed notes, and a full bibliography with guidance on further reading.This new edition of Berkeley's most famous work, published alongside his other masterpiece, the Three Dialogues (also edited by Jonathan Dancy) provides the student with a thorough introduction to the central ideas of one of the world's greatest philosophers. (shrink)