An incomplete degree is cuppable if it can be joined by an incomplete degree to a complete degree. For sets fulfilling some type of simplicity property one can now ask whether these sets are cuppable with respect to a certain type of reducibilities. Several such results are known. In this paper we settle all the remaining cases for the standard notions of simplicity and all the main strong reducibilities.

The task of computing a function F with the help of an oracle X can be viewed as a search problem where the cost measure is the number of queries to X. We ask for the minimal number that can be achieved by a suitable choice of X and call this quantity the query complexity of F. This concept is suggested by earlier work of Beigel, Gasarch, Gill, and Owings on “Bounded query classes”. We introduce a fault tolerant version and (...) relate it with Ulam's game. For many natural classes of functions F we obtain tight upper and lower bounds on the query complexity of F. Previous results like the Nonspeedup Theorem and the Cardinality Theorem appear in a wider perspective. (shrink)

We present a simple proof of a Theorem of Khutoretskij on the number of incomparable one-one numberings of an r.e. family of r.e. sets. The proof directly generalizes to effective domains. In the second part, applying a Theorem of Goncharov, we show that for anyk≧ there exist total recursive functions having exactlyk recursive isomorphism classes. Using a Theorem of Selivanov, it is shown that a certain notion of computability via gödelization is different from Lacombe's notion ofV-recursiveness. Finally, we discuss the (...) complexity (w.r.t.T-degrees) of translating a Gödelnumbering into a direct sum of Friedbergnumberings. (shrink)

Weakly semirecursive sets have been introduced by Jockusch and Owings . In the present paper their investigation is pushed forward by utilizing r.e. partial orderings, which turn out to be instrumental for the study of degrees of subclasses of weakly semirecursive sets.

For a fixed set A, the number of queries to A needed in order to decide a set S is a measure of S's complexity. We consider the complexity of certain sets defined in terms of A: $ODD^A_n = \{(x_1, \dots ,x_n): {\tt\#}^A_n(x_1, \dots, x_n) \text{is odd}\}$ and, for m ≥ 2, $\text{MOD}m^A_n = \{(x_1, \dots ,x_n):{\tt\#}^A_n(x_1, \dots ,x_n) \not\equiv 0 (\text{mod} m)\},$ where ${\tt\#}^A_n(x_1, \dots ,x_n) = A(x_1)+\cdots+A(x_n)$ . (We identify A(x) with χ A (x), where χ A is (...) the characteristic function of A.) If A is a nonrecursive semirecursive set or if A is a jump, we give tight bounds on the number of queries needed in order to decide ODD A n and $\text{MOD}m^A_n: \bullet\text{ODD}^A_n$ can be decided with n parallel queries to A, but not with n - 1. $\bullet \text{ODD}^A_n$ can be decided with $\lceil log(n + 1)\rceil$ sequential queries to A but not with $\lceil log(n + 1)\rceil - 1. \bullet\text{MOD}m^A_n$ can be decided with $\lceil n/m\rceil + \lfloor n/m\rfloor$ parallel queries to A but not with $\lceil n/m\rceil + \lfloor n/m\rfloor - 1. \bullet\text{MOD}m^A_n$ can be decided with $\lceil log(\lceil n/m\rceil + \lfloor n/m\rfloor + 1)\rceil$ sequential queries to A but not with $\lceil log(\lceil n/m\rceil + \lfloor n/m\rfloor + 1)\rceil - 1$ . The lower bounds above hold for nonrecursive recursively enumerable sets A as well. (Interestingly, the lower bounds for recursively enumerable sets follow by a general result from the lower bounds for semirecursive sets.) In particular, every nonzero truth-table degree contains a set A such that ODD A n cannot be decided with n - 1 parallel queries to A. Since every truth-table degree also contains a set B such that ODD B n can be decided with one query to B, a set's query complexity depends more on its structure than on its degree. For a fixed set A, $Q(n,A) = \{S: S \text{can be decided with n sequential queries to} A\},\\Q_\parallel(n, A) = \{S: S \text{can be decided with n parallel queries to} A\}.$ We show that if A is semirecursive or recursively enumerable, but is not recursive, then these classes form non-collapsing hierarchies: $\bullet Q(0,A) \subset Q(1,A) \subset Q(2,A) \subset\cdots\\ \bullet Q_\parallel(0, A) \subset Q_\parallel(1, A) \subset Q_\parallel(2,A) \subset\cdots$ The same is true if A is a jump. (shrink)

Most theories of learning consider inferring a function f from either observations about f or, questions about f. We consider a scenario whereby the learner observes f and asks queries to some set A. If I is a notion of learning then I[A] is the set of concept classes I-learnable by an inductive inference machine with oracle A. A and B are I-equivalent if I[A] = I[B]. The equivalence classes induced are the degrees of inferability. We prove several results about (...) when these degrees are trivial, and when the degrees are omniscient. (shrink)

Online supplement, "Mulsow: Additions to Notes drawn from the 2002 edition of Moderne aus dem Untergrund" full versions of nearly 300 notes that were truncated in the print edition. Hosted on H. C. Erik Midelfort's website. Martin Mulsow's seismic reinterpretation of the origins of the Enlightenment in Germany won awards and renown in its original German edition, and now H. C. Erik Midelfort's translation makes this sensational book available to English-speaking readers. In Enlightenment Underground, Mulsow shows that even in (...) the late seventeenth century some thinkers in Germany ventured to express extremely dangerous ideas, but did so as part of a secret underground. Scouring manuscript collections across northern Europe, Mulsow studied the writings of countless hitherto unknown radical jurists, theologians, historians, and dissident students who pushed for the secularization of legal, political, social, and religious knowledge. Often their works circulated in manuscript, anonymously, or as clandestinely published books. Working as a philosophical microhistorian, Mulsow has discovered the identities of several covert radicals and linked them to circles of young German scholars, many of whom were connected with the vibrant radical cultures of the Netherlands, England, and Denmark. The author reveals how radical ideas and contributions to intellectual doubt came from Socinians and Jews, church historians and biblical scholars, political theorists, and unemployed university students. He shows that misreadings of humorous or ironic works sometimes gave rise to unintended skeptical thoughts or corrosively political interpretations of Christianity. This landmark book overturns stereotypical views of the early Enlightenment in Germany as cautious, conservative, and moderate, and replaces them with a new portrait that reveals a movement far more radical, unintended, and puzzling than previously suspected. (shrink)

Since at least the advent of Jack Kevorkian’s “suicide machine” the major argument against adopting physician-assisted suicide laws has been that they will lead us down a slippery slope to state-sa...

This collection of texts (originally published in German under the title Holzwege) is Heidegger's first post-war book and contains some of the major expositions of his later philosophy. Of particular note are 'The Origin of the Work of Art', perhaps the most discussed of all of Heidegger's essays, and 'Nietzsche's Word 'God is Dead',' which sums up a decade of Nietzsche research. Although translations of the essays have appeared individually in a variety of places, this is the first English translation (...) to bring them all together as Heidegger intended. The text is taken from the last edition of the work, which contains the author's final corrections together with important marginal annotations that provide considerable insight into the development of his thought. This fresh and accurate new translation will be an invaluable resource for all students of Heidegger, whether they work in philosophy, literary theory, religious studies, or intellectual history. (shrink)

Explores the moral dimensions of the current global role of the U.S. military.

An axiomatic theory is formulated which describes a class of “yes-no” experiments, involving a fixed basic source, a fixed basic detector, and various filters. It is assumed that all filters considered can be constructed from a setP of primitive filters by composition and stochastic selection. Two physically plausible axioms are formulated which allow us to define the concept of asystem in the present context (cf. Definition2.4). To each system we can attach anorder unit module ( $^\circ \hat V, ^\circ \hat (...) V_ + , |1\rangle , \langle 1|$ ) (cf. Definition5.1) whereby ( $^\circ \hat V, ^\circ \hat V_ + , |1\rangle ,$ ) is acomplete, separable order unit space. Two additional axioms are proposed which have the effect that the space ( $^\circ \hat V, ^\circ \hat V_ + , |1\rangle ,$ ) becomes isomorphic to the order unit space underlying a JB-algebra, at least in the case where $^\circ \hat V$ isfinite dimensional (cf. Corollary7.9). (shrink)

Christian Georg Martin offers an argumentative reconstruction of the whole work, reading it as a critical ontology, namely as the attempt to abstract from all presuppositions and to immanently unfold conceptual determinations characterizing ...

Semi-hyperhypersimple c.e. sets, also known as diagonals, were introduced by Kummer. He showed that by considering an analogue of hyperhypersimplicity, one could characterize the sets which are the Halting problem relative to arbitrary computable numberings. One could also consider half of splittings of maximal or hyperhypersimple sets and get another variant of maximality and hyperhypersimplicity, which are closely related to the study of automorphisms of the c.e. sets. We investigate the Turing degrees of these classes of c.e. sets. In (...) particular, we show that the analogue of a theorem of Martin fails for these classes. (shrink)

The present paper is the first part of a work which follows up on H. Kummer: “A constructive approach to the foundations of quantum mechanics,”Found. Phys. 17, 1–63 (1987). In that paper we deduced the JB-algebra structure of the space of observables (=detector space) of quantum mechanics within an axiomatic theory which uses the concept of a filter as primitive under the restrictive assumption that the detector space is finite-dimensional. This additional hypothesis will be dropped in the present paper.It (...) turns out that the relevant mathematics for our approach to a quantum mechanical system with infinite-dimensional detector space is the noncommutative spectral theory of Alfsen and Shultz.We start off with the same situation as in the previous paper (cf. Sects. 1 and 2 of the present paper). By postulating four axioms (Axioms S, DP, R, and SP of Sec. 3), we arrive in a natural way at the mathematical setting of Alfsen and Shultz, which consists of a dual pair of real ordered linear spaces 〈Y, M〉: A base norm space, called the strong source space (which, however, in slight contrast to the setting of Alfsen and Shultz, is not 1-additive) and an order unit space, called the weak detector space, which is the norm and order dual space of Y. The last section of part I contains the guiding example suggested by orthodox quantum mechanics. We observe that our axioms are satisfied in this example. In the second part of this work (which will appear in the next issue of this journal) we shall postulate three further axioms and derive the JB-algebra structure of quantum mechanics. (shrink)

And, as The Promise of Salvation makes clear through abundant empirical evidence, religion will not disappear as long as these promises continue to help people ...

Wealthy individuals have many adventures to choose from in today’s world, including visiting the moon and spending time in the deepest canyons of the ocean. But the most coveted waiting list is spo...

In the present paper we give a precise definition of a hidden-variable theory for quantum mechanics, whereby we adopt the weakest possible definition of a hidden-variable theory, which is compatible with the assumption that the bounded observables of a quantum mechanical system are represented by the elements of the real part Ar of a W*-algebra A (of the most general type) and the states are represented by the “normal states” (in the mathematical sense) of A. We then go on to (...) show that an example put forward by Bell in 1966 satisfies our definition (Sec. 2). Finally we make use of Bell's famous theorem to show that for a sufficiently non-commutative W*-algebra A no hidden-variable theory in our sense exists (Theorem 3.3 and its corollaries). (shrink)

While I admire the reviewed discoveries on the social techniques by which monkeys and apes cope with the conflicts within their communities, I am worried about some of the high-level interpretations given by the authors. In my view the processes reviewed are not ‘building blocks of morality’ but objects of human moral judgements. Their interpretation as ‘shared solutions’ is not supported by a demonstrated ability of nonhuman primates to identify their own acts with those of the other group members. There (...) is no evidence of a ‘concerted effort by community members to find shared solutions’ rather than individual learning of these solutions. The postulate of a ‘community concern’ implies a concept of community which is not supported by evidence. Applying terms coined for humans to other species is useful only if one eventually justifies it by appropriate experimentation. Possible methods are suggested. (shrink)

Bd. 1. A-B -- Bd. 2. C-F -- Bd. 3. G-Inn -- Bd. 4. Ins-Loc.

The aim of this paper is to provide a proof-theoretic characterization of relevant logics including fusion and fission connectives, as well as Ackermann’s truth constant. We achieve this by employing the well-established methodology of labelled sequent calculi. After having introduced several systems, we will conduct a detailed proof-theoretic analysis, show a cut-admissibility theorem, and establish soundness and completeness. The paper ends with a discussion that contextualizes our current work within the broader landscape of the proof theory of relevant logics.

Classic text considersgeneral theory of computability, computable functions, operations on computable functions, Turing machines self-applied, unsolvable decision problems, applications of general theory, mathematical logic, Kleene hierarchy, computable functionals, classification of unsolvable decision problems and more.

Martin Heidegger's Contributions to Philosophy reflects his famous philosophical "turning." In this work, Heidegger returns to the question of being from its inception in Being and Time to a new questioning of being as event.

Background Neuroimaging and lesion studies suggested that the dorsolateral prefrontal and posterior parietal cortices mediate visual metacognitive awareness. The causal evidence provided by non-invasive brain stimulation, however, is inconsistent. -/- Objective/hypothesis Here we revisit a major figure discrimination experiment adding a new Kanizsa figure task trying to resolve whether bilateral continuous theta-burst transcranial magnetic stimulation (cTBS) over these regions affects perceptual metacognition. Specifically, we tested whether subjective visibility ratings and/or metacognitive efficiency are lower when cTBS is applied to these two (...) regions in comparison to an active control region. -/- Methods A within-subjects design including three sessions spaced by one-week intervals was implemented. In each session, every participant was administered bilateral cTBS to either prefrontal, control or parietal cortices. Two concurrent tasks were performed, a real and an illusory figure task, stabilising objective performance with use of an adaptive staircase procedure. -/- Results When performing the replicated task, cTBS was found insufficient to disrupt neither visibility ratings nor metacognitive efficiency. However, with use of Kanizsa style illusory figures, cTBS over the dorsolateral prefrontal, but not over the posterior parietal cortex, was observed to significantly diminish metacognitive efficiency. -/- Conclusion(s) Real and illusory figure tasks demonstrated different cTBS effects. A possible explanation is the involvement of the prefrontal cortex in the creation of expectations, which is necessary for efficient metacognition. Failure to replicate previous findings for the real figure task, however, cannot be said to support, conclusively, the notion that these brain regions have a causal role in metacognitive awareness. This inconsistent finding may result from certain limitations of our study, thereby suggesting the need for yet further investigation. (shrink)