“Small” large cardinal notions in the language of ZFC are those large cardinal notions that are consistent with V = L. Besides their original formulation in classical set theory, we have a variety of analogue notions in systems of admissible set theory, admissible recursion theory, constructive set theory, constructive type theory, explicit mathematics and recursive ordinal notations (as used in proof theory). On the face of it, it is surprising that such distinctively set-theoretical notions have analogues in such disaparate and (...) relatively constructive contexts. There must be an underlying reason why that is possible (and, incidentally, why “large” large cardinal notions have not led to comparable analogues). My long term aim is to develop a common language in which such notions can be expressed and can be interpreted both in their original classical form and in their analogue form in each of these special constructive and semi-constructive cases. This is a program in progress. What is done here, to begin with, is to show how that can be done to a considerable extent in the settings of classical and admissible set theory (and thence, admissible recursion theory). The approach taken here is to expand the language of set theory to allow us to talk about (possibly partial) operations applicable both to sets and to operations and to formulate the large cardinal notions in question in terms of operational closure conditions; at the same time only minimal existence axioms are posited for sets. The resulting system, called Operational Set Theory, is a partial adaptation to the set-theoretical framework of the explicit mathematics framework Feferman (1975). The.. (shrink)

In the sole extended break from his life and varing in this way we can associate a sysied career in England, Alan Turing spent the tem of logic with any constructive ordinal. It may be asked whether such a years 1936–1938 doing graduate work at..

Philosophers and others have often debated whether we have freewill: i.e. whether (in a sense I shall try to elucidate) our power to choose between X and Y is radically undetermined, so that if we choose X we yet might have chosen Y, and vice versa. My concern is not with that question but with a hypothetical one which arises from it: if we had such freewill, what implications, if any, would, that fact have for the sciences. My argument concentrates (...) on the social sciences, since the phenomena with which they deal inevitably involves human choice. (shrink)

Famous for the number-theoretic first-order statement known as Goodstein's theorem, author R. L. Goodstein was also well known as a distinguished educator. With this text, he offers an elementary treatment that employs Boolean algebra as a simple medium for introducing important concepts of modern algebra. The text begins with an informal introduction to the algebra of classes, exploring union, intersection, and complementation; the commutative, associative, and distributive laws; difference and symmetric difference; and Venn diagrams. Professor Goodstein proceeds to a detailed (...) examination of three different axiomatizations, and an outline of a fourth system of axioms appears in the examples. The final chapter, on lattices, examines Boolean algebra in the setting of the theory of partial order. Numerous examples appear at the end of each chapter, with full solutions at the end. (shrink)

This paper attempts to explore a number of conceptual issues surrounding genetic testing. It looks at the meaning of the terms, genetic information and genetic testing in relation to the definition set out by the Advisory Committee on Genetic Testing in the UK, and by the Task Force on Genetic Testing in the USA. It argues that the special arrangements that may be required for the regulation of genetic tests should not be determined by reference to the nature or technology (...) of the test, but by considering those morally relevant features that justify regulation. Failure to do so will lead to the regulation of genetic tests that need not be regulated, and would fail to cover other tests which should be regulated. The paper also argues that there is little in the nature of the properties of gene tests, using DNA or chromosomes, that in itself justifies a special approach. (shrink)

Consistent materialists are almost bound to suggest that ‘conscious experience’, if it exists at all, is no more than epiphenomenal. A correct understanding of the real requires that everything we do and say is no more than a product of whatever processes are best described by physics, without any privileged place, person, time or scale of action. Consciousness is a myth, or at least a figment. Plotinus was no materialist: for him, it is Soul and Intellect that are more real (...) than the phenomena we misdescribe as material. Nor does he suppose that consciousness depends on language (as Stoics and modern materialists have sometimes said): wordless experience is actually superior. And much of what counts towards our present consciousness is to be discarded. It is betternotto remember most of what now seems more significant to us; better to discard images; better that the intellect be ‘drunk’ than ‘sober’, losing any sense of separation between subject and object. The goal of the Plotinian intellectual is to join ‘the dance of immortal love’, but it is a mark of the good dancer that she is not conscious of what she does. There is therefore a strange confluence between Plotinus and modern materialists:ourexperience at least is transitory, deceitful, epiphenomenal, and ‘reality’ is to be encountered when we have shed our illusions. (shrink)

The notion of a ‘cosmopolites’ has diverged quite far from its philosophical origins, but may eventually serve a similar function. The hope of a global peace or any sort of global managemen...

The notion of autonomy commonly employed in medical ethics literature and practices is inadequate on three fronts: it fails to properly identify nonautonomous actions and choices, it gives a false account of which features of actions and choices makes them autonomous or nonautonomous, and it provides no grounds for the moral requirement to respect autonomy. In this paper I offer a more adequate framework for how to think about autonomy, but this framework does not lend itself to the kinds of (...) practical application assumed in medical ethics. A general problem then arises: the notion of autonomy used in medical ethics is conceptually inadequate, but conceptually adequate notions of autonomy do not have the practical applications that are the central concern of medical ethics. Thus, a revision both of the view of autonomy and the practice of “respect for autonomy” are in order. (shrink)

In composing this study of 'Ancient Mediterranean Philosophy', I have chosen to draw attention to other philosophical traditions than the Classical Greek and Latin , although we know much less about them. My working assumption is that ...