Chapter 1: Introduction. S = <{We}c<w; C,U,n,0,w> is the substructure formed by restricting the lattice <^P(w); C , U, n,0,w> to the re subsets We of the ...

We explore the interaction between Lebesgue measure and dominating functions. We show, via both a priority construction and a forcing construction, that there is a function of incomplete degree that dominates almost all degrees. This answers a question of Dobrinen and Simpson, who showed that such functions are related to the proof-theoretic strength of the regularity of Lebesgue measure for Gδ sets. Our constructions essentially settle the reverse mathematical classification of this principle.

We study the proof-theoretic strength and effective content of the infinite form of Ramsey's theorem for pairs. Let RT n k denote Ramsey's theorem for k-colorings of n-element sets, and let RT $^n_{ denote (∀ k)RT n k . Our main result on computability is: For any n ≥ 2 and any computable (recursive) k-coloring of the n-element sets of natural numbers, there is an infinite homogeneous set X with X'' ≤ T 0 (n) . Let IΣ n and BΣ (...) n denote the Σ n induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low 2 ) to models of arithmetic enables us to show that RCA 0 + IΣ 2 + RT 2 2 is conservative over RCA 0 + IΣ 2 for Π 1 1 statements and that $RCA_0 + I\Sigma_3 + RT^2_{ , is Π 1 1 -conservative over RCA 0 + IΣ 3 . It follows that RCA 0 + RT 2 2 does not imply BΣ 3 . In contrast, J. Hirst showed that $RCA_0 + RT^2_{ does imply BΣ 3 , and we include a proof of a slightly strengthened version of this result. It follows that $RT^2_{ is strictly stronger than RT 2 2 over RCA 0. (shrink)

The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, ε, such that the question of membership in this orbit is ${\Sigma _1^1 }$ -complete. This result and proof have a number of nice corollaries: the Scott rank of ε is $\omega _1^{{\rm{CK}}}$ + 1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of ε; for all finite α ≥ 9, there is a properly $\Delta (...) _\alpha ^0 $ orbit (from the proof). (shrink)

We show there is a non-recursive r.e. set A such that if W is any low r.e. set, then the join W $\oplus$ A is also low. That is, A is "almost deep". This answers a question of Jockusch. The almost deep degrees form an definable ideal in the r.e. degrees (with jump.).

We show that the double jump is definable in the computably enumerable sets. Our main result is as follows: let [Formula: see text] is the Turing degree of a [Formula: see text] set J ≥T0″}. Let [Formula: see text] such that [Formula: see text] is upward closed in [Formula: see text]. Then there is an ℒ property [Formula: see text] such that [Formula: see text] if and only if there is an A where A ≡T F and [Formula: see text]. (...) A corollary of this is that, for all n ≥ 2, the high n computably enumerable degrees are invariant in the computably enumerable sets. Our work resolves Martin's Invariance Conjecture. (shrink)

The main result of this paper is to show that for every recursive ordinal α ≠ 0 and for every nonrecursive r.e. degree d there is a r.e. set of rank α and degree d.

We state and prove the Translation Theorem. Then we apply the Translation Theorem to Soare's Extension Theorem, weakening slightly the hypothesis to yield a theorem we call the Modified Extension Theorem. We use this theorem to reprove several of the known results about orbits in the lattice of recursively enumerable sets. It is hoped that these proofs are easier to understand than the old proofs.

In this article we establish the existence of a number of new orbits in the automorphism group of the computably enumerable sets. The degree theoretical aspects of these orbits also are examined.

The purpose of this communication is to announce some recent results on the computably enumerable sets. There are two disjoint sets of results; the first involves invariant classes and the second involves automorphisms of the computably enumerable sets. What these results have in common is that the guts of the proofs of these theorems uses a new form of definable coding for the computably enumerable sets.We will work in the structure of the computably enumerable sets. The language is just inclusion, (...) ⊆. This structure is called ε.All sets will be computably enumerable non-computable sets and all degrees will be computably enumerable and non-computable, unless otherwise noted. Our notation and definitions are standard and follow Soare [1987]; however we will warm up with some definitions and notation issues so the reader need not consult Soare [1987]. Some historical remarks follow in Section 2.1 and throughout Section 3.We will also consider the quotient structure ε modulo the ideal of finite sets, ε*. ε* is a definable quotient structure of ε since “Χ is finite” is definable in ε; “Χ is finite” iff all subsets of Χ are computable. We use A* to denote the equivalent class of A under the ideal of finite sets. (shrink)

Let b and c be r.e. Turing degrees such that b>c. We show that there is an r.e. degree a such that b>a>c and all lattices containing a critical triple, including the lattice M5, cannot be embedded into the interval [c, a].

We analyze three applications of Ramsey’s Theorem for 4-tuples to infinite traceable graphs and finitely generated infinite lattices using the tools of reverse mathematics. The applications in graph theory are shown to be equivalent to Ramsey’s Theorem while the application in lattice theory is shown to be provable in the weaker system RCA0.

A result of Soare and Stob asserts that for any non-recursive r.e. setC, there exists a r.e.[C] setA such thatA⊕C is not of r.e. degree. A setY is called [of]m-REA (m-REA[C] [degree] iff it is [Turing equivalent to] the result of applyingm-many iterated ‘hops’ to the empty set (toC), where a hop is any function of the formX→X ⊕W e X . The cited result is the special casem=0,n=1 of our Theorem. Form=0,1, and any (m+1)-REA setC, ifC is not ofm-REA (...) degree, then for alln there exists an-r.e.[C] setA such thatA ⊕C is not of (m+n)-REA degree. We conjecture that this holds also form≥2. (shrink)

Soare [20] proved that the maximal sets form an orbit in${\cal E}$. We consider here${\cal D}$-maximal sets, generalizations of maximal sets introduced by Herrmann and Kummer [12]. Some orbits of${\cal D}$-maximal sets are well understood, e.g., hemimaximal sets [8], but many are not. The goal of this paper is to define new invariants on computably enumerable sets and to use them to give a complete nontrivial classification of the${\cal D}$-maximal sets. Although these invariants help us to better understand the${\cal D}$-maximal (...) sets, we use them to show that several classes of${\cal D}$-maximal sets break into infinitely many orbits. (shrink)

We show that if A and $\widehat{A}$ are automorphic via Φ then the structures $S_{R}(A)$ and $S_{R}(\widehat{A})$ are $\Delta_{3}^{0}-isomorphic$ via an isomorphism Ψ induced by Φ. Then we use this result to classify completely the orbits of hhsimple sets.

A computably enumerable (c.e.) degree is a maximal contiguous degree if it is contiguous and no c.e. degree strictly above it is contiguous. We show that there are infinitely many maximal contiguous degrees. Since the contiguous degrees are definable, the class of maximal contiguous degrees provides the first example of a definable infinite anti-chain in the c.e. degrees. In addition, we show that the class of maximal contiguous degrees forms an automorphism base for the c.e. degrees and therefore for the (...) Turing degrees in general. Finally we note that the construction of a maximal contiguous degree can be modified to answer a question of Walk about the array computable degrees and a question of Li about isolated formulas. (shrink)

Soare [20] proved that the maximal sets form an orbit in${\cal E}$. We consider here${\cal D}$-maximal sets, generalizations of maximal sets introduced by Herrmann and Kummer [12]. Some orbits of${\cal D}$-maximal sets are well understood, e.g., hemimaximal sets [8], but many are not. The goal of this paper is to define new invariants on computably enumerable sets and to use them to give a complete nontrivial classification of the${\cal D}$-maximal sets. Although these invariants help us to better understand the${\cal D}$-maximal (...) sets, we use them to show that several classes of${\cal D}$-maximal sets break into infinitely many orbits. (shrink)

We prove conjectures of Herrmann and Stob by showing that the dense simple sets are orbit complete w.r.t. the simple sets.

Now in a third edition, Introduction to Philosophy: Classical and Contemporary Readings is a highly acclaimed, topically organized collection that covers five major areas of philosophy--theory of knowledge, philosophy of religion, philosophy of mind, freedom and determinism, and moral philosophy. Editor Louis P. Pojman enhances the text's topical organization by arranging the selections into a pro/con format to help students better understand opposing arguments. He also includes accessible introductions to each chapter, subsection, and individual reading, a unique feature for an (...) anthology of this depth. While the book focuses on a compelling sampling of classical material--including selections from Plato, Aristotle, Descartes, Locke, Berkeley, Hume, and Kant--it also incorporates some of philosophy's best twentieth-century and contemporary work, featuring articles by Bertrand Russell, Richard Taylor, John Searle, Thomas Nagel, and others. This third edition contains an expanded glossary, more extensive section introductions, and twelve new selections: Karl Popper: "Epistemology without a Knowing Subject" Richard Rorty: "Dismantling Truth: Solidarity Versus Objectivity" Daniel Dennett: "Postmodernism and Truth" Bruce Russell: "The Problem of Evil: Too Much Suffering" David Chalmers: "Against Materialism: Can Consciousness Be Reductively Explained?" Baron Paul Henri d'Holbach: "A Defense of Determinism" Michael Levin: "A Compatibilist Defense of Moral Responsibility" Plato: Socratic Morality: "Crito" Herodotus: "Custom Is King" J. L. Mackie: "The Subjectivity of Values" Louis P. Pojman: "A Critique of Mackie's Theory of Moral Subjectivism" Thomas Nagel: "Moral Luck". (shrink)

In ”Skepticism,” Peter Klein distinguishes between the “Academic Skeptic” who proposes that we cannot have knowledge of a certain set of propositions and the “Pyrrhonian Skeptic” who refrains from opining about whether we can have knowledge. Klein argues that Academic Skepticism is plausibly supported by a “Closure Principle‐style” argument based on the claim that if x entails y and S has justification for x, then S has justification for y. He turns to contextualism to see if it can contribute to (...) the discussion between one who claims that we can have knowledge about some epistemically interesting class of propositions and the Academic Skeptic. He outlines the background of Pyrrhonian Skepticism, pointing out that the Pyrrhonist withholds assent concerning our knowledge‐bearing status because reason cannot provide an adequate basis for assent. He assesses three possible patterns of reasoning, and concludes that the Pyrrhonist view, that reason cannot resolve matters concerning the nonevident, is vindicated. (shrink)

First published in 1987. Routledge is an imprint of Taylor & Francis, an informa company.

First published in 1987. Routledge is an imprint of Taylor & Francis, an informa company.

Engineering is 'the people-serving profession'. The work of engineers involves interaction with clients, other engineers, and the public at large. More than any other profession, their work also directly involves and affects the environment. This book makes the case that engineers have special professional obligations to protect and enhance the environment, and the authors - one, an engineer and the other, a philosopher - seek to provide an ethical basis for these obligations. In exploring these ethical issues, the authors aim (...) to show that engineers make a difference. The text opens with a series of case studies in which engineers face complex and challenging decisions about the environment. Succeeding chapters examine different ideas about environmental ethics for engineers, including professional codes and both modern and historical discussions of environmental responsibility. The book concludes with a collection of readings that complement the text. Students, as well as practising engineers, will find much of interest in this well-argued and thought-provoking book. (shrink)

Gordon Baker and I had been colleagues at St John’s for almost ten years when we resolved, in 1976, to undertake the task of writing a commentary on Wittgenstein’s Philosophical Investigations. We had been talking about Wittgenstein since 1969, and when we cooperated in writing a long critical notice on the Philosophical Grammar in 1975, we found that working together was mutually instructive, intellectually stimulating and great fun. We thought that we still had much to say about Wittgenstein’s philosophy, and (...) it seemed to us that misinterpretations of passages in the Investigations were so extensive that it would be worth trying to write a detailed analytical commentary. It is difficult to recapture the excitement of those early days in being among the first to work on the microfilms and, subsequently, on the photocopies of Wittgenstein’s Nachlass. We spent many hundreds of hours poring over the typescripts and the often only semi-legible manuscripts, fascinated and privileged to be able to try to follow the development of the thoughts of a great philosophical genius. We talked endlessly about what we had found in Wittgenstein’s manuscripts and typescripts, and debated how it should be understood. The first fruit of our labours was Wittgenstein – Understanding and Meaning. Its guiding idea was to draw attention to the manner in which Wittgenstein linked the concepts of meaning, understanding and explanation, and so to bypass the connections between meaning, truth and truth-conditions that so fascinated philosophers of the 1970s, and to abandon the red-herring of assertion-conditions and anti-realism. After a hiatus of four years, during which time we wrote a controversial book entitled Frege – Logical Excavations and a polemical book on contemporary philosophy of language – Language, Sense. (shrink)

This paper advances the research on personalization of political communication by investigating whether this process of focusing on politicians instead of political issues plays a role on Twitter. Results of a content analysis of 5,530 tweets posted in the run-up to the German federal election provide evidence that Twitter communication refers more often to politicians than to issues. However, tweets containing personal characteristics about political leaders play only a marginal role. When distinguishing among different groups of actors on Twitter (journalists, (...) politicians, citizens), we find that citizens focus more on candidates than do journalists or politicians. Investigating the impact of a televised debate on Twitter communication, we observe that this person-centered event puts the focus on individual politicians instead of issues. (shrink)

... 243-) INTRODUCTION §§243- constitute the eighth 'chapter' of the book. Its point of departure is a natural query with respect to the conclusion of the ...

It has been over two decades since Miranda Fricker labeled epistemic injustice, in which an agent is wronged in their capacity as a knower. The philosophical literature has proliferated with variants and related concepts. By considering cases in popular music, we argue that it is worth distinguishing a parallel phenomenon of art-interpretive injustice, in which an agent is wronged in their creative capacity as a possible artist. In section 1, we consider the prosecutorial use of rap lyrics in court as (...) a central case of this injustice. In section 2, we distinguish art-interpretive injustice from other categories already discussed in recent literature. In section 3, we discuss the relationship between genre discourse and identity prejudice. The case for recognizing the category of art-interpretive injustice is that it allows one to recognize a class of harms as being importantly related in ways that one would otherwise overlook. (shrink)

Throughout its long history metaphysics has been variously conceived. At its most sublime, it has been taken to be the study of the super‐sensible, in particular of the existence of a god, the nature of the soul, and the possibility of an afterlife. When the young Ludwig Wittgenstein entered the lists, it was entirely reasonable to conceive of metaphysics in this manner. Its subject matter was held to be the language‐independent and thought‐independent de re necessities of the world. The Tractatus (...) is dominated by three problems: first, the nature of logic and logical necessity; secondly, the essence of the proposition and hence the characterization of the general propositional form; and thirdly, the intentionality of the proposition. Logical propositions are combinations of elementary propositions by means of truth‐functional logical connectives. The metaphysics adumbrated in the Tractatus, Wittgenstein thought, was required for the possibility of representation and for the possibility of logic. (shrink)

The conflict in Gaza and Israel that ignited on October 7, 2023 signals a catastrophic breakdown in the possibility of ethical dialogue in the region. The actions on both sides have revealed a dissolution of ethical restraints, with unimaginably cruel attacks on civilians, murder of children, destruction of health facilities, and denial of basic needs such as water, food, and shelter. There is a need both to understand the nature of the ethical singularity represented by this conflict and what, if (...) any, options are available to allow the reconstruction of communication between the warring parties. This article seeks to address these tasks by analysing the conflict as inherently an ethical one, in the sense that it exposes a rupture in the fabric of communicative relationships that has evolved systematically out of the deep cultural structures from which all protagonists have emerged. Drawing on the work of Levinas, Habermas, Arendt, and others, and referring to the specific circumstances in the region, it examines the ethical sources of the crisis and tries to identify conditions for its resolution. The possibility of reconciliation—that is, of refiguring relationships to open up a space for dialogue to create pathways to heal the ruptures—is examined. The dark legacy of the Holocaust is identified as an abiding cultural vulnerability for both societies. It is concluded, however, that the rich history of partnerships and collaborations between Jews and Palestinians provides a robust infrastructure on the basis of which a sustainable peace might be built, providing a much-needed source of hope. (shrink)

A succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. Suitable for all introductory mathematics undergraduates, Notes on Logic and Set Theory covers the basic concepts of logic: first-order logic, consistency, and the completeness theorem, before introducing the reader to the fundamentals of axiomatic set theory. Successive chapters examine the recursive functions, the axiom of choice, ordinal and cardinal arithmetic, and the incompleteness theorems. Dr. Johnstone has included numerous exercises designed (...) to illustrate the key elements of the theory and to provide applications of basic logical concepts to other areas of mathematics. (shrink)

By the time of his death in 2006, Sir Peter Strawson was regarded as one of the world's most distinguished philosophers. Unavailable for many years,_ Scepticism and Naturalism_ is a profound reflection on two classic philosophical problems by a philosopher at the pinnacle of his career. Based on his acclaimed Woodbridge lectures delivered at Columbia University in 1983, Strawson begins with a discussion of scepticism, which he defines as questioning the adequacy of our grounds for holding various beliefs. He then (...) draws deftly on Hume and Wittgenstein to argue that we must distinguish between 'hard', scientific naturalism; or 'soft', humanistic naturalism. In the remaining chapters the author takes up several issues in which sceptical doubts play an important role, in particular the nature of transcendental arguments and including the objectivity of moral philosophy, the mental and the physical, and the existence of abstract entities. _Scepticism and Naturalism_ is essential reading for those seeking an introduction to the work of one of the twentieth century’s most important and original philosophers. This reissue includes a substantial new foreword by Quassim Cassam and a fascinating intellectual autobiography by Strawson, which together form an excellent introduction to his life and work. (shrink)

Pekakak adalah salah satu tokoh yang memiliki pengetahuan mendalam dan perencanaan yang cermat. Saat fajar menyingsing, ia terlihat bertengger di atas pohon. Dengan kemampuan sempurna untuk mengontrol kecepatan penerbangan miliknya, seperti kapan harus memperlambat atau mempercepat, dia dapat dengan mudah menghitung cara optimal untuk menangkap ikan.

This article aims to read feminist new materialisms (Barad), together with ‘postulated’ linguistic or cultural primacy of Queer Theory (Butler), to show how both are engaged in similar critical-ethical endeavours. The central argument is that the criticism of Barad and new materialisms misses Butler’s materialistic insights due to a narrow interpretation of Butler's alleged social-constructivist position. There is, therefore, a specific focus on where they both make similar ethical appeals. Moreover, the article relies on Adorno's negative dialectic to highlight an (...) interpretation of Barad and Butler as being part of the same dialectical movement, in which materialism and idealism fluctuate in their mutual criticisms, thus continuing the procession towards 'new knowledge' and emancipation, or freedom, through their motions back and forth. (shrink)