The class of all recursive functions fails to possess a natural hierarchical structure, generated predicatively from "within". On the other hand, many (proof-theoretically significant) sub-recursive classes do. This paper attempts to measure the limit of predicative generation in this context, by classifying and characterizing those (predictably terminating) recursive functions which can be successively defined according to an autonomy condition of the form: allow recursions only over well-orderings which have already been "coded" at previous levels. The question is: how can a (...) recursion code a well-ordering? The answer lies in Girard's theory of dilators, but is reworked here in a quite different and simplified framework specific to our purpose. The "accessible" recursive functions thus generated turn out to be those provably recursive in $(\prod_{1}^{1}-CA)_{0}$. (shrink)

Girard’s maxim, that Peano Arithmetic is a theory of one inductive definition, is re-examined in the light of a weak theory EA formalising basic principles of Nelson’s predicative Arithmetic.

Wilfred Sieg and Stanley S. Wainer. Program Transformation and Proof Transformation.

There is a very simple way in which the safe/normal variable discipline of Bellantoni–Cook recursion [S. Bellantoni, S. Cook, A new recursion theoretic characterization of the polytime functions, Computational Complexity 2 97–110] can be imposed on arithmetical theories like PA: quantify over safes and induct on normals. This weakens the theory severely, so that the provably recursive functions become more realistically computable . Earlier results of D. Leivant [Intrinsic theories and computational complexity, in: D. Leivant , Logic and Computational Complexity, (...) Lecture Notes in Computer Science, vol. 960, Springer-Verlag, 1995, pp. 177–194] are re-worked and extended in this new context, giving proof-theoretic characterizations of complexity classes between Grzegorczyk’s E2 and E3. (shrink)

Following our [6], though with somewhat different methods here, further variants of Goodstein sequences are introduced in terms of parameterized Ackermann–Péter functions. Each of the sequences is shown to terminate, and the proof-theoretic strengths of these facts are calibrated by means of ordinal assignments, yielding independence results for a range of theories: PRA, PA,$\Sigma ^1_1$-DC$_0$, ATR$_0$, up to ID$_1$. The key is the so-called “Hardy hierarchy” of proof-theoretic bounding finctions, providing a uniform method for associating Goodstein-type sequences with parameterized normal (...) form representations of positive integers. (shrink)

This work is derived from the SERC "Logic for IT" Summer School Conference on Proof Theory held at Leeds University. The contributions come from acknowledged experts and comprise expository and research articles which form an invaluable introduction to proof theory aimed at both mathematicians and computer scientists.

This book demonstrates how an interest in political argument leads naturally to a philosophical way of thinking. It analyses both practising politicans and political theorists and is a much needed introudction to two vitally important, and independent, disciplines.

This book examines the underlying theoretical issues concerning the nature of political freedom. Arguing that most previous discussions of such freedom have been too narrowly focused, it explores both conservativism from Edmund Burke to its present resurgence, the radical tradition of Karl Marx, as well as the orthodox liberal model of freedom of John Locke, John Stuart Mill and Isaiah Berlin. Political Freedom argues that these three accounts of political freedom - conservative, liberal and radical - all have internal weaknesses (...) which render them unsatisfactory. In the second part of the book George Brenkert develops an alternative theory of political freedom. Using the guiding concept of empowerment, his model explores individual rights, democratic participation in government and workplace, and the need to provide the material and educational resources to allow individuals to effectively exercise their rights to self-determination. It is a clear and bold attack on the view that there is no link between freedom and power. (shrink)

Clearly and succinctly describes the standards of God-like living as taught by the Orthodox Church. Eleven chapters deal with our relationships with God, our selves and our neighbors from both the personal and churchly perspectives. Readers will find it a veritable source book of biblical and patristic material on the practical aspects of Orthodox life. Among the topics covered are issues of personal religious life, family life, sex ethics, bioethics, the Christian and culture, the state, peace and war, economic responsibilities, (...) inter-church relations and hundreds of other ethics-related issues. Thoroughly indexed, the material in the book is easy to get to for reference. 2nd printing. (shrink)

This article seeks to identify some of the major perspectives in Eastern Orthodox Christianity which provide direction for bioethical-decision making. The article first identifies some historical, theological, and liturgical sources in the Eastern Orthodox tradition which have implications for bioethics. The manuscript also seeks to address the question of the place of religious bioethics within public discussion of issues in bioethics and health care policy. Keywords: bioethics, Eastern Orthodox, faith, liturgy, secular, tradition CiteULike Connotea Del.icio.us What's this?

Eastern Orthodox Christianity carries forward a moral tradition from the earliest Christian period, in the belief that scriptural and patristic teaching remains applicable to the contemporary economic sphere of life. The Church Fathers focused on the ownership of property and the ethical acquisition of wealth and its use; they stressed special concern for the poor and disadvantaged. Carried forward through the Byzantine and modern eras, these early Christian understandings now can be applied through a basic and elementary natural law morality (...) to business activities. The Orthodox approach embodies traditional virtue and character ethics as well. The essay concludes by applying these Orthodox approaches to two current issues: the charging of interest and internet ethics. (shrink)

ABSTRACT In the course of her defence of German protesters against Peter Singer's lectures, Jenny Teichman claims that the right to make use of a public platform is not covered by the principle of freedom of expression. I argue that this view is mistaken, and that she is also wrong to focus on whether Singer deserved a public platform. Instead I suggest that what matters is whether there was an attempt to prevent communication between a speaker and willing hearers. But (...) I agree with Teichman that there are some relevant differences between speaking from a public platform and speaking privately. In particular I argue that protesters have a right, though not one based on freedom of expression, to interrupt a public speaker. Such a right is on a par with the speaker's freedom to draw public attention to the fact that she has opinions which she thinks worthy of a hearing, and in exercising it a protester must not prevent those who wish to hear the speech from doing so. Finally I offer an argument against the view that disrupting Singer's lectures could be seen as a justified interference with freedom of expression. (shrink)

Book reviewed in this article: Rights and Advocacy for Retarded People. By Stanley S. Herr. Lexington.

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)