“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)

‘Jason…chosen leader because his superior declines the honour, subordinate to his comrades, except once, in every trial of strength, skill, or courage, a great warrior only with the help of magical charms, jealous of honour but incapable of asserting it, passive in the face of crisis, timid and confused before trouble, tearful at insult, easily despondent, gracefully treacherous in his dealings with the love-sick Medea but cowering before her later threats and curses, coldly efficient in the time-serving murder of an (...) unsuspecting child , reluctant even in marriage.’ So Carspecken put the case against Jason's heroism. In the face of such an indictment, Lawall's plea in mitigation, ‘it must be admitted that [Jason] often reveals the qualities of a true gentleman’, seems somehow inadequate. Criticism since Carspecken has found various overlapping categories for Jason which both take account of the earlier negative judgements and preserve the centrality of his ‘personality’ and character in the poem: Jason is the quiet diplomat who works through consensus rather than force, his is a heroism of sex-appeal, he is an anti-hero, the embodiment of Sceptic ‘suspension of judgement’, or, alternatively, he is ‘one of us’, credible and lifelike. Carspecken himself tried a different tack: the poem is concerned not with individual heroism but with the heroism of the group. (shrink)

Late Antique Epistemology explores the techniques used by late antique philosophers to discuss truth. Non-rational ways to discover truth, or to reform the soul, have usually been thought inferior to the philosophically approved techniques of rational argument, suitable for the less philosophically inclined, for children, savages or the uneducated. Religious rituals, oracles, erotic passion, madness may all have served to waken courage or remind us of realities obscured by everyday concerns. What is unusual in the late antique classical philosophers is (...) that these techniques were reckoned as reliable as reasoned argument, or better still. Late twentieth century commentators have offered psychological explanations of this turn, but only recently had it been accepted that there might also have been philosophical explanations, and that the later antique philosophers were not necessarily deluded. (shrink)

‘Jason…chosen leader because his superior declines the honour, subordinate to his comrades, except once, in every trial of strength, skill, or courage, a great warrior only with the help of magical charms, jealous of honour but incapable of asserting it, passive in the face of crisis, timid and confused before trouble, tearful at insult, easily despondent, gracefully treacherous in his dealings with the love-sick Medea but cowering before her later threats and curses, coldly efficient in the time-serving murder of an (...) unsuspecting child (sic), reluctant even in marriage.’ So Carspecken put the case against Jason's heroism. In the face of such an indictment, Lawall's plea in mitigation, ‘it must be admitted that [Jason] often reveals the qualities of a true gentleman’, seems somehow inadequate. Criticism since Carspecken has found various overlapping categories for Jason which both take account of the earlier negative judgements and preserve the centrality of his ‘personality’ and character in the poem: Jason is the quiet diplomat who works through consensus rather than force, his is a heroism of sex-appeal, he is an anti-hero, the embodiment of Sceptic ‘suspension of judgement’, or, alternatively, he is ‘one of us’, credible and lifelike. Carspecken himself tried a different tack: the poem is concerned not with individual heroism but with the heroism of the group (cf. 1.1, 4.1773–81). (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)

The most famous and successful teacher of rhetoric at Athens in the fourth century was Isocrates, and he claimed for rhetoric an educational importance which Plato considered to be unmerited and misleading. He made rhetoric the basis of his whole educational system and claimed to teach his pupils to become not only good rhetoricians but good citizens. Plato attacked both aspects of this theory of education. In the Gorgias he exposed the claim of rhetoric to be considered valuable as an (...) instrument of education by showing that rhetorical excellence had no necessary connection with moral excellence. In the Protagoras he exposed the inconsistency of those who claimed to teach men to be good citizens—to teach πολιτιк τ?νη—without an absolute standard of moral values. Even if we believe that in the Gorgias and the Protagoras, as in other dialogues, Plato is representing faithfully the constructive views of the historic Socrates, we can hardly believe that he was unaware of the contemporary relevance of those views, and it is significant that he thought fit to publish them in the form of an attack on a teacher of rhetoric and an attack on a teacher of <ολιτικ τνη At any rate it is reasonable to suppose that the Athenian reading public would expect to find such a contemporary relevance and that they would interpret these attacks as being, in some measure at least, directed against Isocrates. (shrink)

It is argued that the characterization of the states of an infinite system of indistinguishable particles satisfying Bose-Einstein statistics which follows from the quantum-mechanical analog of de Finetti's theorem (2) can be used to interpret the nonuniqueness of the resolution into a convex combination of pure states of a quantum-mechanical mixed state.

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)