Computability theory is used to evaluate the complexity of classifying various kinds of Lebesgue spaces and associated isometric isomorphism problems.

We prove uniformly computable versions of the Implicit Function Theorem in its differentiable and non-differentiable forms. We show that the resulting operators are not computable if information about some of the partial derivatives of the implicitly defining function is omitted. Finally, as a corollary, we obtain a uniformly computable Inverse Function Theorem, first proven by M. Ziegler.

This paper reports the results of a questionnaire administered to 1178 undergraduate students and discusses how they responded to ten situations which asked them to assess their personal evaluation of the ethical acceptability, how society would similarly assess the situation and how business persons would respond. Multiple versions of the instrument were developed to investigate if the sex of the person involved in the situation would influence the respondents' perception of the ethical action involved. No differences were identified. Further, the (...) image of business persons as less ethical than society in general seems to have evaporated. (shrink)

Let equation image. We show that for many reducibilities, the requirement that a relation be intrinsically reducible to the α-th jump of a countable mode A has a syntactic equivalent. Furthermore, we show that many reducibilities coincide in such a situation.

We show that any partial order with a Σ3 enumeration can be effectively embedded into any partial order obtained by imposing a strong reducibility such as ≤tt on the c. e. sets. As a consequence, we obtain that the partial orders that result from imposing a strong reducibility on the sets in a level of the Ershov hiearchy below ω + 1 are co-embeddable.

We consider the complexity (in terms of the arithmetical hierarchy) of the various quantifier levels of the diagram of a computably presented metric structure. As the truth value of a sentence of continuous logic may be any real in [0, 1], we introduce two kinds of diagrams at each level: the closed diagram, which encapsulates weak inequalities of the form $$\phi ^\mathcal {M}\le r$$, and the open diagram, which encapsulates strict inequalities of the form $$\phi ^\mathcal {M}< r$$. We show (...) that the closed and open $$\Sigma _N$$ diagrams are $$\Pi ^0_{N+1}$$ and $$\Sigma ^0_N$$ respectively, and that the closed and open $$\Pi _N$$ diagrams are $$\Pi ^0_N$$ and $$\Sigma ^0_{N + 1}$$ respectively. We then introduce effective infinitary formulas of continuous logic and extend our results to the hyperarithmetical hierarchy. Finally, we demonstrate that our results are optimal. (shrink)

Sufficient conditions are given for the computation of an arc that accesses a point on the boundary of an open subset of the plane from a point within the set. The existence of a not-computably-accessible but computable point on a computably compact arc is also demonstrated.

We use the compactness theorem of continuous logic to give a new proof that $$L^r([0,1]; {\mathbb {R}})$$ isometrically embeds into $$L^p([0,1]; {\mathbb {R}})$$ whenever $$1 \le p \le r \le 2$$. We will also give a proof for the complex case. This will involve a new characterization of complex $$L^p$$ spaces based on Banach lattices.

We study the following classes: Q* (r 1 A 1 ,..., r kA k ) which is defined to be the collection of all sets that can be computed by a Turing machine that on any input makes a total of r i queries to A i for all i ∈ {1,..., k}. Q(r 1A 1 ,...,r kA k ) which is defined like Q* (r 1A 1 ,..., r kA k ) except that queries to A i must be (...) made before queries to A i+1 for all i ∈ {1,..., k - 1}. QC(r 1A 1 ,..., r kA k ) which is defined like Q(r 1A 1 ,..., r kA k ) except that the Turing machine must halt even if given incorrect answers to some of its queries. We show that if A 1 ,..., A k are jumps that are not too close together, then all three of these classes are identical and are not changed if we permute (r 1A 1 ,..., r kA k ). This extends a result of Beigel's [1]. Since the second class is not affected by permutations, we say that these sets commute with each other. We also show that jumps that are too close together may not commute. We also characterize the commutative sequences of sets obtained by iterating the jump operation through an ordinal notation. (shrink)

We classify the asymptotic densities of the sets according to their level in the Ershov hierarchy. In particular, it is shown that for, a real is the density of an n‐c.e. set if and only if it is a difference of left‐ reals. Further, we show that the densities of the ω‐c.e. sets coincide with the densities of the sets, and there are ω‐c.e. sets whose density is not the density of an n‐c.e. set for any.

Suppose \ is a computable real. We extend previous work of Clanin, Stull, and McNicholl by determining the degrees of categoricity of the separable \ spaces whose underlying measure spaces are atomic but not purely atomic. In addition, we ascertain the complexity of associated projection maps.

Call a set A n-correctable if every set Turing reducible to A via a Turing machine that on any input makes at most n queries is Turing reducible to A via a Turing machine that on any input makes at most n-queries and on any input halts no matter what answers are given to its queries. We show that if a c.e. set A is n-correctable for some n ≥ 2, then it is n-correctable for all n. We show that (...) this is the optimal such result by constructing a c.e. set that is 1-correctable but not 2-correctable. The former result is obtained by examining the logical closure properties of c.e. sets that are 2-correctable. (shrink)

The theory of inner functions plays an important role in the study of bounded analytic functions. Inner functions are also useful in applied mathematics. Two foundational results in this theory are Frostman's Theorem and the Factorization Theorem. We prove a uniformly computable version of Frostman's Theorem. We then show that the Factorization Theorem is not uniformly computably true. We then show that for an inner function u with infinitely many zeros, the Blaschke sum of u provides the exact amount of (...) information necessary to compute the factorization of u. Along the way, we prove some uniform computability results for Blaschke products; these results play a key role in the analysis of factorization. We also give some computability results concerning zeros and singularities of analytic functions. We use Type-Two Effectivity as our foundation. (shrink)

We study the weak truth-table and truth-table degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truth-table reducible to any initial segment of any scattered computable linear ordering. Countable $\Pi _{1}^{0}$ subsets of 2ω and Kolmogorov complexity play a major role in the proof.

We examine the reverse mathematics and computability theory of a form of Ramsey's theorem in which the linear n-tuples of a binary tree are colored.

This book contains programmatic essays that focus on broad-ranging proposals for re-envisioning a discipline of comparative philosophy of religions. It also contains a number of case studies focussing on the interpretation of particular religio-historical data from comparatively oriented philosophical perspectives.

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)

A partial ordering $\mathbb{P}$ is $n$-Ramsey if, for every coloring of $n$-element chains from $\mathbb{P}$ in finitely many colors, $\mathbb{P}$ has a homogeneous subordering isomorphic to $\mathbb{P}$. In their paper on Ramsey properties of the complete binary tree, Chubb, Hirst, and McNicholl ask about Ramsey properties of other partial orderings. They also ask whether there is some Ramsey property for pairs equivalent to $\mathit{ACA}_{0}$ over $\mathit{RCA}_{0}$. A characterization theorem for finite-level partial orderings with Ramsey properties has been proven by (...) the second author. We show, over $\mathit{RCA}_{0}$, that one direction of the equivalence given by this theorem is equivalent to $\mathit{ACA}_{0}$, and the other is provable in $\mathit{ATR}_{0}$. We answer Chubb, Hirst, and McNicholl’s second question by showing that there is a primitive recursive partial ordering $\mathbb{P}$ such that, over $\mathit{RCA}_{0}$, “$\mathbb{P}$ is 2-Ramsey” is equivalent to $\mathit{ACA}_{0}$. (shrink)

We show, relative to the base theory RCA₀: A nontrivial tree satisfies Ramsey's Theorem only if it is biembeddable with the complete binary tree. There is a class of partial orderings for which Ramsey's Theorem for pairs is equivalent to ACA₀. Ramsey's Theorem for singletons for the complete binary tree is stronger than $B\sum_{2}^{0}$ , hence stronger than Ramsey's Theorem for singletons for ω. These results lead to extensions of results, or answers to questions, of Chubb, Hirst, and McNicholl (...) [3]. (shrink)

In this benchmark five-volume study, originally published between 1922 and 1955, Surendranath Dasgupta examines the principal schools of thought that define Indian philosophy. A unifying force greater than art, literature, religion, or science, Professor Dasgupta describes philosophy as the most important achievement of Indian thought, arguing that an understanding of its history is necessary to appreciate the significance and potentialities of India's complex culture. Volume I offers an examination of the Vedas and the Brahmanas, the earlier Upanisads, and the six (...) systems of Indian philosophy. (shrink)

The paper builds on the postulate of “myths we live by,” which shape our imaginative life (and hence our social expectations), but which are also open to reflective study and reinvention. It applies this principle, in particular, to the concepts of love and vulnerability. We are accustomed to think of the condition of vulnerability in an objectifying and distancing way, as something that affects the bearers of specific (disadvantaged) social identities. Against this picture, which can serve as a pretext for (...) paternalist and controlling attitudes to the groups in question – notably to women – Anderson urges us to reimagine our vulnerability as a condition not merely of exposure to violence but of openness to mutual affection, love, and friendship. Hegel’s celebrated image of the owl of Minerva, which takes wing only with the coming of dusk, suggests an association of (philosophical) wisdom with negativity – with the experience of death or loss. Anderson, by contrast, proposes an alternative and more hopeful image of the dawn of enlightenment, in the guise of new ethical dispositions shaped by an emancipatory (and so, no longer an oppressive or sexist) conception of our capacity for love. Her main interlocutors or influences in this piece are Judith Butler (who contributes the idea of a “relational ontology”), Michèle Le Doeuff, and Mary Midgley. (shrink)

The point of formalisation is to model various aspects of natural language. Perhaps the main use to which formalisation is put is to model and explain inferential relations between different sentences. Judged solely by this objective, a formalisation is successful in modelling the inferential network of natural language sentences to the extent that it mirrors this network. There is surprisingly little literature on the criteria of good formalisation, and even less on the question of what it is for a formalisation (...) to mirror the inferential network of a natural language or some fragment of it. This paper takes some exploratory steps towards a quantitative account of the main ingredient in the goodness of a formalisation. We introduce and critically examine a mathematical model of how well a formalisation mirrors natural-language inferential relations. (shrink)

Open Compositionality: Towards a Methodology of Language offers a fresh view into human languages as supermodular, highly interactive cognitive capacities allowing for human unique thought and communication. It is a novel account of semantics as decision-making in tandem with the cognition first methodology.

Objectives: Evidence based patient choice seems based on a strong liberal individualist interpretation of patient autonomy; however, not all patients are in favour of such an interpretation. The authors wished to assess whether ideals of autonomy in clinical practice are more in accordance with alternative concepts of autonomy from the ethics literature. This paper describes the development of a questionnaire to assess such concepts of autonomy.Methods: A questionnaire, based on six moral concepts from the ethics literature, was sent to aneurysm (...) patients and their surgeons. The structure of the questionnaire was assessed by factor analysis, and item reduction was based on reliability.Results: Ninety six patients and 58 surgeons participated. The questionnaire consisted of four scales. Two of the scales reflected the paternalistic and consumerist poles of the liberal individualist model, one scale reflected concepts of Socratic autonomy and of procedural independence, and the fourth scale reflected ideals of risk disclosure.Discussion: The Ideal Patient Autonomy Scale is a 14 item normative instrument. It is clearly distinct from the generally used psychological preference questionnaires that assess preferences for physician-patient roles. (shrink)

Imagination in Inquiry investigates the nature, kinds, component elements, functions, scope, and uses of the imagination that are at work in inquiry. It develops a homeostatic model and discusses its applications in various branches of philosophy, from the philosophy of science and the philosophy of technology to ethics and aesthetics.