This book describes a program of research in computable structure theory. The goal is to find definability conditions corresponding to bounds on complexity which persist under isomorphism. The results apply to familiar kinds of structures (groups, fields, vector spaces, linear orderings Boolean algebras, Abelian p-groups, models of arithmetic). There are many interesting results already, but there are also many natural questions still to be answered. The book is self-contained in that it includes necessary background material from recursion theory (ordinal notations, (...) the hyperarithmetical hierarchy) and model theory (infinitary formulas, consistency properties). (shrink)

We re-express a previous general result in a way which seems easier to remember, using the terminology of infinite games. We show how this can be applied to construct recursive linear orderings, showing, for example, that if there is a ▵ 0 2β + 1 linear ordering of type τ, then there is a recursive ordering of type ω β · τ.

We extend results of Harizanov and Barker. For a relation R on a recursive structure /oA, we give conditions guaranteeing that the image of R in a recursive copy of /oA can be made to have arbitrary ∑α0 degree over Δα0. We give stronger conditions under which the image of R can be made ∑α0 degree as well. The degrees over Δα0 can be replaced by certain more general classes. We also generalize the Friedberg-Muchnik Theorem, giving conditions on a pair (...) of relations R and S under which the images of R and S can be made ∑α0 and independent over Δα0 in a recursive copy of /oA. (shrink)

A subspace V of an infinite dimensional fully effective vector space V ∞ is called decidable if V is r.e. and there exists an r.e. W such that $V \oplus W = V_\infty$ . These subspaces of V ∞ are natural analogues of recursive subsets of ω. The set of r.e. subspaces forms a lattice L(V ∞ ) and the set of decidable subspaces forms a lower semilattice S(V ∞ ). We analyse S(V ∞ ) and its relationship with L(V (...) ∞ ). We show: Proposition. Let U, V, W ∈ L(V ∞ ) where U is infinite dimensional and $U \oplus V = W$ . Then there exists a decidable subspace D such that U |oplus D = W. Corollary. Any r.e. subspace can be expressed as the direct sum of two decidable subspaces. These results allow us to show: Proposition. The first order theory of the lower semilattice of decidable subspaces, Th(S(V ∞ )), is undecidable. This contrasts sharply with the result for recursive sets. Finally we examine various generalizations of our results. In particular we analyse S * (V ∞ ), that is, S(V ∞ ) modulo finite dimensional subspaces. We show S * (V ∞ ) is not a lattice. (shrink)

Let be a recursive structure, and let R be a recursive relation on . Harizanov isolated a syntactical condition which is necessary and sufficient for to have recursive copies in which the image of R is r.e. of arbitrary r.e. degree. We had conjectured that a certain extension of Harizanov's syntactical condition would be necessary and sufficient for to have recursive copies in which the image of R is ∑α0 of arbitrary ∑α0 degree, but this is not the case. Here (...) we give examples illustrating some restrictions on the possible ∑α0 degrees. In these examples, the image of R cannot be ∑α0 of degree d unless d possesses an “α-table”. (shrink)

Ash, C.J., Generalizations of enumeration reducibility using recursive infinitary propositional sentences, Annals of Pure and Applied Logic 58 173–184. We consider the relation between sets A and B that for every set S if A is Σ0α in S then B is Σ0β in S. We show that this is equivalent to the condition that B is definable from A in a particular way involving recursive infinitary propositional sentences. When α = β = 1, this condition is that B is (...) enumeration reducible to A. We establish further generalizations involving infinitely many sets and ordinals. (shrink)

In his celebrated masterpiece, Symposium, Plato imagines a high-society dinner-party in Athens in 416 BC at which the guests - including the comic poet Aristophanes and, of course, Plato's mentor Socrates - each deliver a short speech in praise of love. The sequence of dazzling speeches culminates in Socrates' famous account of the views of Diotima, a prophetess who taught him that love is our means of trying to attain goodness. And then into the party bursts the drunken Alcibiades, the (...) most popular and notorious Athenian of the time, who insists on praising Socrates himself rather than love, and gives us a brilliant sketch of this enigmatic character. The power, humour, and pathos of Plato's creation engages the reader on every page. This new translation is complemented by full explanatory notes and an illuminating introduction. (shrink)

C.J. Bartley places Ramunuja in his intellectual context. This study is particularly concerned with Ramanuja's engagement with opposing schools of thought and practice, rendering it a valuable contribution to the history of Indian thought.

In this paper, I defend teleological theories of belief against the exclusivity objection. I argue that despite the exclusive influence of truth in doxastic deliberation, multiple epistemic aims interact when we consider what to believe. This is apparent when we focus on the processes involved in specific instances (or concrete cases) of doxastic deliberation, such that the propositions under consideration are specified. First, I out- line a general schema for weighing aims. Second, I discuss recent attempts to defend the teleological (...) position in relation to this schema. And third, I develop and defend my proposal that multiple epistemic aims interact in doxastic deliberation—a possibility which, as of yet, has received no serious attention in the literature. (shrink)

The social, educational and political writings of Jean-Jacques Rousseau have become enormously influential in the 200 years since his death. But the breadth as well as the depth of Rousseau's achievement - he was amongst other things a creative writer and musical composer as well as a philosopher - is not always appreciated. In around 100 articles, alphabetically arranged and fully cross-referenced, N. J. H. Dent explores all facets of Rousseau's work and thoughts, while his subject's remarkable life is summarized (...) in a biographical introduction. Details of works by and about Rousseau are listed in an extensive bibliography. For students or general readers seeking an introduction to Rousseau's work, and for those already familiar with the material who require a convenient reference source, this dictionary is essential reading. (shrink)

In this paper, I discuss whether different interpretations of the ‘aim’ of belief—both the teleological and normative interpretations—have the resources to explain certain descriptive and normative features of suspended belief (suspension). I argue that, despite the recent efforts of theorists to extend these theories to account for suspension, they ultimately fail. The implication is that we must either develop alternative theories of belief that can account for suspension, or we must abandon the assumption that these theories ought to be able (...) to account for suspension. To close, I briefly consider some of the reasons we have in favour of pursing each of these options, and I suggest that it is worth exploring the possibility that suspension is best understood as its own attitude, independently of theories of belief’s ‘aim’. (shrink)

Kengo Miyazono, in his work Delusions and Beliefs, defends a teleo-functional account of delusions. In my contribution to this symposium, I question one of Miyazono’s motivations for appealing to teleo-functionalism over its main rival, dry-functionalism. Miyazono suggests that teleo-functionalism, unlike dry-functionalism, can account for the compatibility of the theses (i) that delusions are genuine doxastic states (doxasticism about delusions) and (ii) that delusions do not perform the typical causal roles of beliefs (the causal difference thesis). I argue, however, that there (...) are also ways for dry-functionalism to account for this compatibility. If what I have to say is correct, then Miyazono is perhaps too hasty—at least in one important respect—to opt for teleo-functionalism over dry-functionalism. Dry-functionalism has an equally good chance of satisfying some of Miyazono’s explanatory goals. (shrink)

One of the most extensive yet least conclusive methodological debates within religious studies revolves around the question of what, precisely, the phenomenology of religion is and what contribution it can make to the study of religion. I do not intend to answer this important question here. To do so satisfactorily would require a range of historical, philosophical and methodological inquiry which would go quite beyond the bounds of a single article. My intention in this paper is, by comparison, unambitious. It (...) is to take one view of what phenomenology of religion is and to consider an area outside that usually explored by students of religion which can, nonetheless, shed some light on how religions might be studied in a way which is in accordance with the phenomenology of religion so understood. What follows will offer an answer to the question of what contribution one particular understanding of phenomenology might make to the study of religion, but no attempt will be made to establish whether or not this particular understanding ought to be regarded as normative. (shrink)

New natural law advocates are somewhat notorious for their loose action theory, having a track record of counterintuitive claims. In response to criticisms, advocates have entrenched, further defending their questionable action theory. This paper first rehearses the basic criticism against the new natural law action theory. It then examines four recent attempts to revive this action theory and finds these attempts wanting. Within these attempts, certain patterns arise. Given a certain means A to a goal C, a search is made (...) to determine whether any middle means B is implied by A. The standards of this search vary wildly, however. By some standards, a middle means can be found; by others, every middle means can be easily swept aside. The same author will sometimes use both kinds of standards, depending upon the situation. One great weakness of the action theory, then, is a lack of consistency in applying a universal standard. (shrink)

The Statesman is Plato's neglected political work, but it is crucial for an understanding of the development of his political thinking. In some respects it continues themes from the Republic, particularly the importance of knowledge as entitlement to rule. But there are also changes: Plato has dropped the ambitious metaphysical synthesis of the Republic, changed his view of the moral psychology of the citizen, and revised his position on the role of law and institutions. In its presentation of the statesman's (...) expertise, the Statesman modifies, as well as defending in original ways, this central theme of the Republic. (shrink)

"This edition, which replaces the original Loeb edition..., offers text, translation, and annotation that are fully current with modern scholarship"--Front flap of dust jacket, volume 1.

Banking, in common with other areas of finance, is often considered an amoral field focused purely on risk and return. However, ethics does have an important role to play, both traditionally and as business and banking evolve. Based on a speech to a European Union conference on financing small and medium–sized enterprises (SMEs), this paper seeks to provide an overview of ethics in banking using three terms. Integrity is important to generate the trust necessary for any banking system to flourish, (...) responsibility highlights contemporary banks’ need to take into account the consequences of their lending policies, and affinity refers to a set of relatively novel ways in which depositors and borrowers can be brought closer together than they are in conventional western banking. (shrink)

Banking, in common with other areas of finance, is often considered an amoral field focused purely on risk and return. However, ethics does have an important role to play, both traditionally and as business and banking evolve. Based on a speech to a European Union conference on financing small and medium–sized enterprises , this paper seeks to provide an overview of ethics in banking using three terms. Integrity is important to generate the trust necessary for any banking system to flourish, (...) responsibility highlights contemporary banks’ need to take into account the consequences of their lending policies, and affinity refers to a set of relatively novel ways in which depositors and borrowers can be brought closer together than they are in conventional western banking. (shrink)