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)

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 ...

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)

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)

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.

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].

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 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.

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

As I write this, in November 1971, people are dying in East Bengal from lack of food, shelter, and medical caxc. The suffering and death that are occurring there now axe not inevitable, 1101; unavoidable in any fatalistic sense of the term. Constant poverty, a cyclone, and a civil war have turned at least nine million people into destitute refugees; nevertheless, it is not beyond Lhe capacity of the richer nations to give enough assistance to reduce any further suffering to (...) very small proportions. The decisions arid actions of human beings can prevent this kind of suffering. Unfortunately, human beings have not made the necessary decisions. At the individual level, people have, with very few exceptions, not responded to the situation in any signihczmt way. Generally speaking, people have not given large sums to relief funds; they have not written t0 their parliamentaxy representatives demanding increased government assistance; they have not demonstrated in the streets, held symbolic fasts, or done anything else directed toward providing thc refugees with the means to satisfy their essential needs. At thc government level, no govcrmmcnt has given the sort of massive aid that would enable the refugees to survive fm: more than a few days. Britain, for instance, has given rather more than most countries. It has, to date, given £14,750,000. For comparative purposes, B1~itain’s share of thc nonreccverabla development costs of the Anglo-French Concorde project is already in excess of £275,000,000, and on present estimates will reach £440,000,¤00. The implication is that the British government values a supersonic transport more than thirty times as.. (shrink)

This paper argues that a set of questioning attitudes are among the foundations of human and animal minds. While both verbal questioning and states of curiosity are generally explained in terms of metacognitive desires for knowledge or true belief, I argue that each is better explained by a prelinguistic sui generis type of mental attitude of questioning. I review a range of considerations in support of such a proposal and improve on previous characterizations of the nature of these attitudes. I (...) then broaden their explanatory scope to include a number of forms of exploratory search. The paper has three main goals: to characterize the nature of the questioning attitudes, outlining their causal role and type of content; to argue that they are fundamental components of the mind, being widespread among animals and not constructed during ontogeny out of other attitudes; and to suggest that they explain a great deal more behavior than one might think. (shrink)

Since its original publication in 1975, this groundbreaking work has awakened millions of people to the existence of "speciesism"—our systematic disregard of nonhuman animals—inspiring a worldwide movement to transform our attitudes to animals and eliminate the cruelty we inflict on them. In Animal Liberation, author Peter Singer exposes the chilling realities of today’s "factory farms" and product-testing procedures—destroying the spurious justifications behind them, and offering alternatives to what has become a profound environmental and social as well as moral issue. (...) An important and persuasive appeal to conscience, fairness, decency, and justice, it is essential reading for the supporter and the skeptic alike. (shrink)

Single-premise epistemic closure is the principle that: if one is in an evidential position to know that P where P entails Q, then one is in an evidential position to know that Q. In this paper, I defend the viability of opposition to closure. A key task for such an opponent is to precisely formulate a restricted closure principle that remains true to the motivations for abandoning unrestricted closure but does not endorse particularly egregious instances of closure violation. I focus (...) on two brands of epistemic theory that naturally incorporate closure restrictions. The first type holds that the truth value of a knowledge ascription is relative to a relevant question. The second holds that the truth value of a knowledge ascription is relative to a relevant topic. For each approach, I offer a formalization of a leading theory from the literature and use this formalization to evaluate the theory’s adequacy in terms of a precise set of desiderata. I conclude that neither theory succeeds in meeting these desiderata, casting doubt on the viability of the underlying approaches. Finally, I offer a novel variant of the topic-sensitive approach that fares better. (shrink)

I lay out the framework for my theory of sensory imagination in “Imagining as a guide to possibility.” Sensory imagining involves mental imagery , and crucially, in describing the content of imagining, I distinguish between qualitative content and assigned content. Qualitative content derives from the mental image itself; for visual imaginings, it is what is “pictured.” For example, visually imagine the Philadelphia Eagles defeating the Pittsburgh Steelers to win their first Super Bowl. You picture the greenness of the field and (...) the football’s brown oblong shape. Some of what you imagine isn’t explicitly pictured, however. That it is Sunday, that it is the Super Bowl: these facts are assigned. (shrink)

Political decisions have the potential to greatly impact our lives. Think of decisions in relation to abortion or climate change, for example. This makes political legitimacy an important normative concern. But what makes political decisions legitimate? Are they legitimate in virtue of having support from the citizens? Democratic conceptions of political legitimacy answer in the affirmative. Such conceptions righly highlight that legitimate political decision-making must be sensitive to disagreements among the citizens. But what if democratic decisions fail to track what (...) there is most reason to do? What if a democratically elected government fails to take measures necessary to protect its population from threats related to climate change? Peter argues that the legitimacy of political decisions doesn't just depend on respect for the citizens' will; and defends a novel hybrid conception of political legitimacy, called the Epistemic Accountability conception. According to this conception, political legitimacy also depends on how political decision-making responds to evidence for what there is most reason to do. Grounds for Political Legitmacy starts with an overview of the main ways in which philosophers have thought about political legitimacy, and identifies the epistemic accountability conception as an overlooked alternative. It then develops the epistemic accountability conception of political legitimacy and discusses its implications for legitimate political decision-making. Considering the norms that should govern political debate, it examines the role of experts in politics, and probes the responsibilities of democratically elected political leaders and as well as of citizens. (shrink)

In at least some cases of justified perceptual belief, our perceptual experience itself, as opposed to beliefs about it, evidences and thereby justifies our belief. While the phenomenon is common, it is also mysterious. There are good reasons to think that perceptions cannot justify beliefs directly, and there is a significant challenge in explaining how they do. After explaining just how direct perceptual justification is mysterious, I considerMichael Huemers (Skepticism and the Veil of Perception, 2001) and Bill Brewers (Perception and (...) Reason, 1999) recent, but radically different, attempts to eliminate it. I argue that both are unsuccessful, though a consideration of their mistakes deepens our appreciation of the mystery. (shrink)

This book offers a reassessment of the political philosophy of the British Idealists, a group of once influential and now neglected nineteenth-century Hegelian philosophers, whose work has been much misunderstood. Peter Nicholson focuses on F. H. Bradley's idea of morality and moral philosophy; T. H. Green's theory of the Common Good, of the social nature of rights, of freedom, and of state interference; and Bernard Bosanquet's notorious theory of the General Will. By examining the arguments offered by the Idealists (...) and by their critics the author is able to penetrate the deep layers of hostile comment laid down by several generations of later writers and to show that these ideas, once properly understood, are not only defensible but interesting and important. (shrink)

In the Western world it is usually taken as given that we all want happiness, and our educational arrangements tacitly acknowledge this. Happiness, Hope, and Despair argues, however, that education has an important role to play in deepening our understanding of suffering and despair as well as happiness and joy. Education can be uncomfortable, unpredictable, and unsettling; it can lead to greater uncertainty and unhappiness. Drawing on the work of Søren Kierkegaard, Miguel de Unamuno, Fyodor Dostoevsky, Simone Weil, Paulo Freire, (...) and others, Peter Roberts shows why these features of educational life need not be feared; to the contrary, they can be seen as a source of hope and human fulfilment. After years of negotiating an education system dominated by the language of competition, performance, and economic advancement, students and teachers often long for something different; they seek not just measurable success but also opportunities to ask searching questions of themselves and the world they encounter. Happiness, Hope, and Despair makes an important contribution toward meeting this need. It fosters a rethinking of the nature, purpose, and value of education, and opens up possibilities for further scholarly and professional inquiry. (shrink)

This book presents and evaluates the late philosophy (Spätphilosophie) of F. W. J. Schelling (1775-1854) across a wide range of issues, ranging from relation between pure thinking and being, to the philosophy of mythology and religion, to the philosophy of history, to questions concerning the philosophy of nature and freedom. Simultaneously, it discusses Hegel's treatment of similar issues, and systematically compares the two thinkers. This is the first time, in an English-language publication, that these two major German Idealists have been (...) compared in such detail along such a broad front. The book begins with three chapters exploring the development of Schelling's thinking concerning transcendental philosophy, nature and teleology, human freedom, and the theory of history, from his earliest publications up to his middle years. Against this background, the book then presents Schelling's distinction between "positive" and "negative" philosophy, the defining mark of his late philosophy. It explores his theory of pure a priori thinking (negative philosophy), and his account of the transition from negative to positive philosophy. The major components of Schelling's positive philosophy, including his conception of "un-pre-thinkable being", and his theories of mythology and revelation, are then discussed. Throughout, a comparative assessment of Hegel's approach similar issues is sustained. Schelling emerges as a philosopher who traced his own highly distinctive path through the thicket of problems bequeathed by Kant, and whose systematic responses to these problems still merit serious consideration as alternatives to those of Hegel. (shrink)

Our topic is the theory of topics. My goal is to clarify and evaluate three competing traditions: what I call the way-based approach, the atom-based approach, and the subject-predicate approach. I develop criteria for adequacy using robust linguistic intuitions that feature prominently in the literature. Then I evaluate the extent to which various existing theories satisfy these constraints. I conclude that recent theories due to Parry, Perry, Lewis, and Yablo do not meet the constraints in total. I then introduce the (...) issue-based theory—a novel and natural entry in the atom-based tradition that meets our constraints. In a coda, I categorize a recent theory from Fine as atom-based, and contrast it to the issue-based theory, concluding that they are evenly matched, relative to our main criteria of adequacy. I offer tentative reasons to nevertheless favour the issue-based theory. (shrink)

This is the first volume of a projected three-volume set on the subject of innateness. The extent to which the mind is innate is one of the central questions in the human sciences, with important implications for many surrounding debates. By bringing together the top nativist scholars in philosophy, psychology, and allied disciplines these volumes provide a comprehensive assessment of nativist thought and a definitive reference point for future nativist inquiry. The Innate Mind: Structure and Content, concerns the fundamental architecture (...) of the mind, addressing such question as: What capacities, processes, representations, biases, and connections are innate? How do these innate elements feed into a story about the development of our mature cognitive capacities, and which of them are shared with other members of the animal kingdom? The editors have provided an introduction giving some of the background to debates about innateness and introducing each of the subsequent essays, as well as a consolidated bibliography that will be a valuable reference resource for all those interested in this area. The volume will be of great importance to all researchers and students interested in the fundamental nature and powers of the human mind. Together, the three volumes in the series will provide the most intensive and richly cross-disciplinary investigation of nativism ever undertaken. They point the way toward a synthesis of nativist work that promises to provide a new understanding of our minds and their place in the natural order. (shrink)