Let p be a set. A function φ is uniformly σ 1 (p) in every admissible set if there is a σ 1 formula φ in the parameter p so that φ defines φ in every σ 1 -admissible set which includes p. A theorem of Van de Wiele states that if φ is a total function from sets to sets then φ is uniformly σ 1R in every admissible set if anly only if it is E-recursive. A function is (...) ES p -recursive if it can be generated from the schemes for E-recursion together with a selection scheme over the transitive closure of p. The selection scheme is exactly what is needed to insure that the ES p - recursively enumerable predicates are closed under existential quantification over the transitive closure of p. Two theorems are established: a) If the transitive closure of p is countable than a total function on sets is ES p -recursive if and only if it is uniformly σ 1 (p) in every admissible set. b) For any p, if φ is a function on the ordinal numbers then φ is ES p -recursive if and only if it is uniformly ∑ 1 (p) in every admissible set. (shrink)

The fundamental ideas concerning computation and recursion naturally find their place at the interface between logic and theoretical computer science. The contributions in this book, by leaders in the field, provide a picture of current ideas and methods in the ongoing investigations into the pure mathematical foundations of computability theory. The topics range over computable functions, enumerable sets, degree structures, complexity, subrecursiveness, domains and inductive inference. A number of the articles contain introductory and background material which it is hoped will (...) make this volume an invaluable resource for specialists and non-specialists alike. (shrink)

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)

We show that there is a low T-upper bound for the class of K-trivial sets, namely those which are weak from the point of view of algorithmic randomness. This result is a special case of a more general characterization of ideals in $\Delta _2^0 $ T-degrees for which there is a low T-upper bound.

Sanskrit text with Hindi verse translation of Aṣṭāvakragītā, work on Vedanta philosophy.

We study the global properties of [Formula: see text], the Turing degrees of the n-r.e. sets. In Theorem 1.5, we show that the first order of [Formula: see text] is not decidable. In Theorem 1.6, we show that for any two n and m with n < m, [Formula: see text] is not a Σ1-substructure of [Formula: see text].

Related Works: Part II: C. T. Chong, Yue Yang. $\Sigma_2$ Induction and Infinite Injury Priority Argument, Part II: Tame $\Sigma_2$ Coding and the Jump Operator. Ann. Pure Appl. Logic, vol. 87, no. 2, 103--116. Mathematical Reviews : MR1490049 Part III: C. T. Chong, Lei Qian, Theodore A. Slaman, Yue Yang. $\Sigma_2$ Induction and Infinite Injury Priority Argument, Part III: Prompt Sets, Minimal Paries and Shoenfield's Conjecture. Mathematical Reviews : MR1818378.

T. A. Cavanaugh defends double-effect reasoning (DER), also known as the principle of double effect. DER plays a role in anti-consequentialist ethics (such as deontology), in hard cases in which one cannot realize a good without also causing a foreseen, but not intended, bad effect (for example, killing non-combatants when bombing a military target). This study is the first book-length account of the history and issues surrounding this controversial approach to hard cases. It will be indispensable in theoretical ethics, applied (...) ethics (especially medical and military), and moral theology. It will also interest legal and public policy scholars. (shrink)

Classical text on Nyaya and Vaiśeṣika philosophy; with Āloka Sanskrit commentary.

§1. Introduction. Natural sets that can be enumerated by a computable function always seem to be either actually computable or of the same complexity as the Halting Problem, the complete r.e. set K. The obvious question, first posed in Post [1944] and since then called Post's Problem is then just whether there are r.e. sets which are neither computable nor complete, i.e., neither recursive nor of the same Turing degree as K?Let be the r.e. degrees, i.e., the r.e. sets modulo (...) the equivalence relation of equicomputable with the partial order induced by Turing computability. This structure is a partial order with least element 0, the degree of the computable sets, and greatest element 1 or 0′, the degree of K. Post's problem then asks if there are any other elements of.The solution of Post's problem by Friedberg [1957] and Muchnik [1956] was followed by various algebraic or order theoretic results that were interpreted as saying that the structure was in some way well behaved:Theorem 1.1. Every countable partial ordering or even uppersemilattice can be embedded into.Theorem 1.2. For every nonrecursive r.e. degreeathere are r.e. degreesb, c < asuch thatb ∨ c = a.Theorem 1.3. For every pair of nonrecursive r.e. degreesa < bthere is an r.e. degreecsuch thata < c < b. (shrink)

Work on Nyaya philosophy; includes Siddhāntamuktāvalī autocommentary and Dinakarī of Dinakarabhaṭṭa, 18th cent.

Jaina interpretation of Tarkasaṅgraha of Annambhaṭṭa, work on Hindu logic ; includes Sanskrit commentary.

Critical edition with eight commentaries, on Nyaya and Vaiśeṣika philosophy.

Classical work on the basic concepts and terminology of Nyaya and Vaiśeṣika school in Hindu philosophy; includes Phakkikā commentary by Kṣamākalyāṇagaṇi, 18th cent.

Aphoristic work on Nyaya and Vaiśeṣika school of Hindu philosophy; includes two Sanskrit commentaries.

Ghazzālī, 1058-1111; views on behaviorism ; religious aspects; Islam.

Compendium of Indian epistemology and logic.

Supercommentary on portion of Tattvacintāmaṇi of Gaṅgeśa, dealing with an argument liable to a valid objection (satpratipakṣa), one of the five forms of fallacious middle term hetvābhāsa.

Aphoristic work on Nyāya and Vaiśeṣika school of Hindu philosophy; includes Sanskrit commentary with Hindi translation.

Contributed research papers on Indic philosophy.

Classical commentary on Nyāyasūtra of Gautama, aphoristic work on Nyaya philosophy.

Sacks [23] asks if the metarecursively enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as [Formula: see text] or, equivalently, that of the truth set of [Formula: see text].

T. A. Cavanaugh's Hippocrates' Oath and Asclepius' Snake: The Birth of the Medical Profession articulates the Oath as establishing the medical profession's unique internal medical ethic - in its most basic and least controvertible form, this ethic mandates that physicians help and not harm the sick. Relying on Greek myth, drama, and medical experience (e.g., homeopathy), the book shows how this medical ethic arose from reflection on the most vexing medical-ethical problem -- injury caused by a physician -- and argues (...) that deliberate iatrogenic harm, especially the harm of a doctor choosing to kill (physician assisted suicide, euthanasia, abortion, and involvement in capital punishment), amounts to an abandonment of medicine as an exclusively therapeutic profession. The book argues that medicine as a profession necessarily involves stating before others what one stands for: the good one seeks and the bad one seeks to avoid on behalf of the sick, and rejects the view that medicine is purely a technique lacking its own unique internal ethic. It concludes noting that medical promising (as found in the White Coat Ceremony through which U. S. medical students matriculate) implicates medical autonomy which in turn merits respect, including honoring professional conscientious objections. (shrink)

Exgesis on Mīmāṃsābhāṣya, Śabarasvāmi's commentary on Jaiminī's Mīmāṃsāsūtra, basic aphoristic work of the Mīmāṃsā school in Hindu philosophy; includes supercommentaries.