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 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 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)

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)

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.

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)

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 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 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.).

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 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 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)

V.1 Re-establishing an initial union among philosophy, science, and wisdom by recovering our understanding of philosophy, science : how philosophy, science, is, and always has been, chiefly a study of the one and the many -- v.2. An introduction to ragamuffin Thomism.

This book challenges the presupposition among professional philosophers that René Descartes is the Father of Modern Philosophy. It demonstrates by intensive textual analysis of Descartes's Discourse and Meditations that he inaugurated a new type of sophistry rather than a new way of conducting philosophy. Transcendental Sophistry is a synthesis of Renaissance humanism and Christian theology, especially the theology of creation. This striking re-evaluation of the achievement of Descartes opens the history of Western philosophy to radical reinterpretation.

Peter Kwasniewski shows that St. Thomas contemplates the nature of ecstasy at key stages in the development of his thought and that it plays a crucial role in his doctrine of love. St. Thomas recognized that all love involves ecstatic transcendence, whether it be the creature's self-oblation to the Creator, the reverence of an inferior for a superior, a superior's generosity toward an inferior, or the mutual affection and help of equals joined in friendship. Love of persons for their (...) own sake generates an ecstatic love in which the self is borne as a gift to another subject by sharing a common life aspiring to common goods. Kwasniewski also examines Aquinas on the question of whether or not God experiences ecstasy, and if so, in what ways. (shrink)

Through a radical reinterpretation of classical philosophy as an organizational psychology, The Moral Psychology of St. Thomas: An Introduction to Ragamuffin Ethics just as radically reinterprets St. Thomas Aquinas's moral teaching to be a behavioristic psychology chiefly designed to synthesize right reason and right pleasure to help a person excel at living life as a whole. In the process of so doing, this work demonstrates how the skill of prudential living is a necessary condition for becoming a grand master of (...) leadership in any and every profession. (shrink)

Emerging neuroscientific insights are changing our understanding of what it means to be human. The resulting reconceptualization continues to impact law and the fit between law and morality. This book takes account of those developments and suggests that normative theory, particularly in its non-instrumental iterations, will be challenged, most profoundly. If we are, as the science suggests, nothing more than the coincidence of mechanical forces, then law and normative theory that depend on the immaterial and that would draw distinctions between (...) the "mental" or "emotional" and the more manifestly physical are misguided, so misguided that they would actually undermine human thriving. Indeed, they already do. The ramifications of that conclusion are profound. Trialectic posits and investigates the impact of those ramifications on law. (shrink)

THINK Currency. THINK Relevancy. THINK Critically. THINK Critically is a cutting-edge, self-reflective guide for improving critical thinking skills through careful analysis, reasoned inference, and thoughtful evaluation of contemporary culture and ideas. An engaging visual design developed with extensive student feedback and 15-page chapters makesTHINK Critically the textbook your students will actually read. It delivers the core concepts of critical thinking in a way they can easily understand. Additionally, engaging examples and masterful exercises help students learn to clarify ideas, analyze arguments, (...) and evaluate reasoning. (shrink)

Scientists engaged in global health research are increasingly faced with barriers to access and use of human tissues from the developing world communities where much of their research is targeted. In part, the problem can be traced to distrust of researchers from affluent countries, given the history of 'scientific-imperialism' and 'biocolonialism' reflected in past well publicized cases of exploitation of research participants from low to middle income countries. To a considerable extent, the failure to adequately engage host communities, the opacity (...) of informed consent, and the lack of fair benefit-sharing have played a significant role in eroding trust. These ethical considerations are central to biomedical research in low to middle income countries and failure to attend to them can inadvertently contribute to exploitation and erode trust. A 'tissue trust' may be a plausible means for enabling access to human tissues for research in a manner that is responsive to the ethical challenges considered. Preventing exploitation and restoring trust while simultaneously promoting global health research calls for innovative approaches to human tissues research. A tissue trust can reduce the risk of exploitation and promote host capacity as a key benefit. (shrink)

In this paper I sketch an account of moral blame and blameworthiness. I begin by clarifying what I take blame to be and explaining how blameworthiness is to be analyzed in terms of it. I then consider different accounts of the conditions of blameworthiness and, in the end, settle on one according to which a person is blameworthy for φ-ing just in case, in φ-ing, she violates one of a particular class of moral requirements governing the attitudes we bear, and (...) our mental orientation, toward people and other objects of significant moral worth. These requirements embody the moral stricture that we accord to these others a sufficient level of respect, one that their moral worth demands. This is a familiar theme which has its roots in P. F. Strawson’s pioneering views on moral responsibility. My development of it leads me to the conclusion that acting wrongly is not a condition of blameworthiness: violating a moral requirement to perform, or refrain from performing, an action is neither necessary nor sufficient for being blameworthy. All we are ever blameworthy for, I will argue, are certain aspects of our mental bearing toward others. We can be said to be blameworthy for our actions only derivatively, in the sense that those actions are the natural manifestations of the things for which we are strictly speaking blameworthy. (shrink)

Many of us are inclined to accept something like the following principle: We can only be properly morally assessed for what is in our control. And yet our ordinary practices seem to frequently violate this principle. The resulting tension, and the attempt to resolve it, is the problem of moral luck. For example, we tend to punish and think worse of the negligent driver who kills a child than we do the equally negligent driver who was lucky there was no (...) child in his path. Thus, the lucky outcomes of our actions do seem to affect the extent to which we hold and are held responsible, but these are not things over which we exercise control. And, as Thomas Nagel famously illustrated in his response to Bernard Williams (the two of which papers form the founding documents of the moral luck debate), the influence of luck is not limited to outcomes. For the circumstances in which we find ourselves and, indeed, our very constitution are also shaped by luck. Since the publication of Williams's and Nagel's papers, the existence and breadth of moral luck has been hotly debated. This debate is not a mere intellectual trifle but, as the essays in this volume illustrate, a debate which lies at the heart of free will, responsibility, identity, causation, and self-creation. (shrink)

There is a debate in normative ethics about whether or not our moral obligations depend solely on either our evidence concerning, or our beliefs about, the world. Subjectivists maintain that they do and objectivists maintain that they do not. I shall offer some arguments in support of objectivism and respond to the strongest argument for subjectivism. I shall also briefly consider the significance of my discussion to the debate over whether one’s future voluntary actions are relevant to one’s current moral (...) obligations. (shrink)

Philosophy and Poetry is the 33rd volume in the Midwest Studies in Philosophy series. It begins with contributions in verse from two world class poets, JohnAshbery and Stephen Dunn, and an article by Dunn on the creative processthat issued in his poem. The volume features new work from an internationalcollection of philosophers exploring central philosophical issues pertinent topoetry as well as the connections between the two domains.

Waar de een zijn leven geeft voor een goed geweten, haalt de ander zijn schouders erover op. Het geweten: een achterhaald begrip of hard toe aan herwaardering?

This volume, an expanded edition of the philosophy of language issue of the journal Midwest Studies in Philosophy (1977), includes essays by some of the ...

David Lewis has offered a reply to the standard argument for the claim that the truth of determinism is incompatible with anyone’s being able to do otherwise than she in fact does. Helen Beebee has argued that Lewis’s compatibilist strategy is untenable. In this paper I show that one recent attempt to defend Lewis’s view against this argument fails and then go on to offer my own defense of Lewis’s view.

In this lucid and penetrating book, Peter A. Schouls considers Locke's major writings in terms of the closely related ideas of freedom, progress, mastery, reason, and education.