This volume presents the philosophical and heuristic framework Cantor developed and explores its lasting effect on modern mathematics. "Establishes a new plateau for historical comprehension of Cantor's monumental contribution to mathematics." --The American Mathematical Monthly.

This is a book review of Cantorian set theory and limitations of size by Michael Hallett.

According to Cantor (Mathematische Annalen 21:545–586, 1883 ; Cantor’s letter to Dedekind, 1899 ) a set is any multitude which can be thought of as one (“jedes Viele, welches sich als Eines denken läßt”) without contradiction—a consistent multitude. Other multitudes are inconsistent or paradoxical. Set theoretical paradoxes have common root—lack of understanding why some multitudes are not sets. Why some multitudes of objects of thought cannot themselves be objects of thought? Moreover, it is a logical truth that such multitudes do (...) exist. However we do not understand this logical truth so well as we understand, for example, the logical truth $${\forall x \, x = x}$$ . In this paper we formulate a logical truth which we call the productivity principle. Rusell (Proc Lond Math Soc 4(2):29–53, 1906 ) was the first one to formulate this principle, but in a restricted form and with a different purpose. The principle explicates a logical mechanism that lies behind paradoxical multitudes, and is understandable as well as any simple logical truth. However, it does not explain the concept of set. It only sets logical bounds of the concept within the framework of the classical two valued $${\in}$$ -language. The principle behaves as a logical regulator of any theory we formulate to explain and describe sets. It provides tools to identify paradoxical classes inside the theory. We show how the known paradoxical classes follow from the productivity principle and how the principle gives us a uniform way to generate new paradoxical classes. In the case of ZFC set theory the productivity principle shows that the limitation of size principles are of a restrictive nature and that they do not explain which classes are sets. The productivity principle, as a logical regulator, can have a definite heuristic role in the development of a consistent set theory. We sketch such a theory—the cumulative cardinal theory of sets. The theory is based on the idea of cardinality of collecting objects into sets. Its development is guided by means of the productivity principle in such a way that its consistency seems plausible. Moreover, the theory inherits good properties from cardinal conception and from cumulative conception of sets. Because of the cardinality principle it can easily justify the replacement axiom, and because of the cumulative property it can easily justify the power set axiom and the union axiom. It would be possible to prove that the cumulative cardinal theory of sets is equivalent to the Morse–Kelley set theory. In this way we provide a natural and plausibly consistent axiomatization for the Morse–Kelley set theory. (shrink)

The usual objections to infinite numbers, and classes, and series, and the notion that the infinite as such is self-contradictory, may... be dismissed as groundless. There remains, however, a very grave difficulty, connected with the contradiction [of the class of all classes not members of themselves]. This difficulty does not concern the infinite as such, but only certain very large infinite classes.

The principle of “equivalence of care” in prison medicine is a principle by which prison health services are obliged to provide prisoners with care of a quality equivalent to that provided for the general public in the same country. It is cited in numerous national and international directives and recommendations.The principle of equivalence is extremely relevant from the point of view of normative ethics but requires adaptation from the point of view of applied ethics. From a clinical point of view, (...) the principle of equivalence is often insufficient to take account of the adaptations necessary for the organization of care in a correctional setting. The principle of equivalence is cost-effective in general, but has to be overstepped to ensure the humane management of certain special cases. (shrink)

It is well known that the validity of Choice Principles is problematic in non-standard Set Theories which do not abide by the Limitation of Size Principle. In this paper we discuss the consistency of various Choice Principles with respect to the Generalized Positive Comprehension Principle . The Principle GPC allows to take as sets those classes which can be specified by Generalized Positive Formulae, e.g. the universe. In particular we give a complete characterization of which choice (...) principles hold in Hyperuniverses. Hyperuniverses are structures which arose independently in Non-well-founded Set Theory and in Mathematical Semantics of Concurrent Programming Languages and are hitherto the only existing models of GPC. Hyperuniverses are naturally endowed with a κ-compact uniform κ-topology and are uniformly isomorphic to their exponential space, i.e. the space of their closed subsets endowed with the Exponential Uniformity. (shrink)

The principle of subsidiarity can be an effective tool for adjusting economic systems and establishing the social rule-of-law concept in public organisation. In considering the usability of the principle of subsidiarity, the following aspects thereof are discussed: the role in the development of the European social model; employment in determining limits for the powers of the public and private sectors; the application in the market and planned economy systems; and the social justice function in the development of a social welfare (...) state. The principle of subsidiarity as an instrument for individuals’ interests, balancing and adjusting in the regulation of economic systems, helps to establish the relative sizes of public policy and the market, as subsidiarity can help ensure a more equitable distribution of public goods while guaranteeing the maintenance of the creative potency of operators and minimal satisfaction of vital needs of individuals unable to contribute to the production of goods. It amounts to a constitutional measure determining the balance of productive and protective state, thus ensuring the stability of the state organisation. In the concept of a social rule-of-law state, subsidiarity draws clear limits to the state’s powers, institutionally bringing together efforts of all members of the community to achieve welfare in the state. (shrink)

We study reflection principles in Kelley-Morse set theory with urelements (KMU). We first show that First-Order Reflection Principle is not provable in KMU with Global Choice. We then show that KMU + Limitation of Size + Second-Order Reflection Principle is mutually interpretable with KM + Second-Order Reflection Principle. Furthermore, these two theories are also shown to be bi-interpretable with parameters. Finally, assuming the existence of a κ+-supercompact cardinal κ in KMU, we construct a model of KMU + (...) Second-Order Reflection Principle where Limitation of Size fails. (shrink)

BackgroundThe Western-European concept of libertarian rights-based autonomy, which advocates respect for individual rights, may conflict with African cultural values and norms. African communitarian ethics focuses on the interests of the collective whole or community, rather than rugged individualism. Hence collective decision-making processes take precedence over individual autonomy or consent. This apparent conflict may impact informed consent practice during biomedical research in African communities and may hinder ethical principlism in African bioethics. This study explored African biomedical researchers' perspectives regarding informed consent (...) and potential limitations to the principle of respect for autonomy in African communities.MethodsWe conducted a qualitative study based on in-depth interviews with 12 biomedical researchers, five females and seven males aged 34 to 74 years, currently working at an African university. Interviews lasted 35–40 min each and involved semi-structured open-ended interviews, which allowed participants to offer information about their perceptions and feelings regarding respect for autonomy and informed consent as practised in Africa. Empirical data from the interviews were recorded, transcribed, and analysed using thematic content analysis, together with an interrogation of relevant scientific literature about African communitarian ethics, making evaluations and drawing inferences consistent with the empirical bioethics approach.ResultsBased on these interviews and analysis of relevant literature, we found that informed consent is difficult to apply in an African context because it derives from a Western conception of libertarian rights-based autonomy. Most respondents pointed out that it was challenging to implement informed consent in the African setting. Furthermore, communalism, customary beliefs, spirituality, and relational autonomy are predominant in most African communities, as exemplified by the African moral philosophies of Ubuntu/botho and Ukama, which emphasize communitarianism over individual rights. We also found that language, education, poverty, and cultural beliefs are barriers to obtaining proper informed consent in African communities.ConclusionsWe conclude that there are limitations to applying the principle of respect for autonomy and informed consent in African communities, especially in the context of human biomedical research. We recommend using a more relational approach, such as Ross’s prima facie duties, to implement informed consent in African communities. (shrink)

The harm principle, understood as the normative requirement that conduct should be criminalized only if it is harmful, has difficulty in dealing with those core cases of criminal wrongdoing that can occur without causing any direct harm. Advocates of the harm principle typically find it implausible to hold that these core cases should not be crimes and so usually seek out some indirect harm that can justify criminalizing the seemingly harmless conduct. But this strategy justifies criminalization of a wide range (...) of conduct on the basis of the fear, worry, and anxiety it generates among those who are not the direct victims of the conduct, and thereby undermines the limiting role of the harm principle by permitting the very move it was meant to prevent: the criminalization of harmless conduct on the ground of others’ feelings about it. The best way to avoid this dilemma is to recognize that people have rights, operating independently of the harm principle, to be treated in certain ways just because they are persons. The existence of such rights provides a ground for both criminalizing conduct and limiting the scope of criminalization because these rights point both to conduct that people must be permitted to engage in and conduct that might well be criminalized. A complete account of criminal law will therefore require the harm principle to work together with an independent account of rights. (shrink)

Stakeholder engagement and dialogue have a central role in defining the relations between organisations and their internal and external interlocutors. Drawing upon the analysis of dialogic motifs, power–conflict dynamics and sociopolitical perspectives, and based on a set of interviews with the stakeholders of a consumer-owned cooperative, the research explores the dialogic potential of stakeholder engagement. The analysis revealed a fragmented picture where the co-design and co-implementation aspects were mainly related to the non-business areas of cooperative life, while business logic dominated (...) the most central aspects. Stakeholder engagement was mainly related to consensus building, while dialogic engagement based on a pluralistic understanding was only partially considered and then neglected. The social capital in the local area, the growing size of the organisation and the related power structure embrace stakeholder engagement, influencing the orientation of the dialogic dynamic. The analysis indicates that a dialogic exchange is a relative concept which depends on the interests involved and the topics discussed. It also reveals that the key factors in the democratisation of stakeholder engagement are a mutual understanding and long-term opportunities. Common sociopolitical aspects are also important, but they do not necessarily guarantee the creation of dialogism paths. The research contributes to the critical dialogic literature in revealing whether and how stakeholder engagement has been implemented in a specific setting. It also shows the limitations of voluntarist stakeholder engagement initiatives. (shrink)

Se realiza un estudio de los principios básicos o tradicionales de la bioética y su influencia en la práctica de la limitación del esfuerzo terapéutico en las Unidades de Cuidados Intensivos, como condicionante que favorece su aplicación en los pacientes en estado terminal. Se aborda la necesidad de una bioética no importada de otros países, que se corresponda a las características de la realidad latinoamericana y a cada contexto sociocultural. El trabajo tiene como objetivo fundamental resaltar la importancia de estos (...) principios básicos en la discusión ético-clínica para la toma de decisiones, a partir de una bioética que se corresponda a la realidad de los sistemas imperantes. A study of basic or traditional principles of bioethics and its influence in the practice of the limits of the therapeutic effort in the intensive care units, as a requirement that favors its application in terminal ill patients was carried out. The need of having bioethics principles that match Latin-American reality and each sociocultural context, not imported ones from other countries, was stated. This paper main objective is to highlight the importance of these basic principles in ethic-clinic discussion for decision-making, taking as the starting point a bioethics that corresponds to the reality of the dominant system. (shrink)

It is often argued that higher-level special-science properties cannot be causally efficacious since the lower-level physical properties on which they supervene are doing all the causal work. This claim is usually derived from an exclusion principle stating that if a higher-level property F supervenes on a physical property F* that is causally sufficient for a property G, then F cannot cause G. We employ an account of causation as difference-making to show that the truth or falsity of this principle is (...) a contingent matter and derive necessary and sufficient conditions under which a version of it holds. We argue that one important instance of the principle, far from undermining non-reductive physicalism, actually supports the causal autonomy of certain higher-level properties. (shrink)

History casts a spell on our minds more powerful than science or religion. It does not root us in the past at all. It rather flatters us with the belief in our ability to recreate the world in our image. It is a form of self-assertion that brooks no opposition or dissent and shelters us from the experience of time. So argues Constantin Fasolt in The Limits of History , an ambitious and pathbreaking study that conquers history's power by carrying (...) the fight into the center of its domain. Fasolt considers the work of Hermann Conring (1606-81) and Bartolus of Sassoferrato (1313/14-57), two antipodes in early modern battles over the principles of European thought and action that ended with the triumph of historical consciousness. Proceeding according to the rules of normal historical analysis--gathering evidence, putting it in context, and analyzing its meaning--Fasolt uncovers limits that no kind of history can cross. He concludes that history is a ritual designed to maintain the modern faith in the autonomy of states and individuals. God wants it, the old crusaders would have said. The truth, Fasolt insists, only begins where that illusion ends. With its probing look at the ideological underpinnings of historical practice, The Limits of History demonstrates that history presupposes highly political assumptions about free will, responsibility, and the relationship between the past and the present. A work of both intellectual history and historiography, it will prove invaluable to students of historical method, philosophy, political theory, and early modern European culture. (shrink)

Offers possible solutions to such medical dilemmas as who should receive organ transplants.

The neo-Fregean program in the philosophy of mathematics seeks a foundation for a substantial part of mathematics in abstraction principles—for example, Hume’s Principle: The number of Fs D the number of Gs iff the Fs and Gs correspond one-one—which can be regarded as implicitly definitional of fundamental mathematical concepts—for example, cardinal number. This paper considers what kind of abstraction principle might serve as the basis for a neo- Fregean set theory. Following a brief review of the main difficulties confronting (...) the most widely discussed proposal to date—replacing Frege’s inconsistent Basic Law V by Boolos’s New V which restricts concepts whose extensions obey the principle of extensionality to those which are small in the sense of being smaller than the universe—the paper canvasses an alternative way of implementing the limitation of size idea and explores the kind of restrictions which would be required for it to avoid collapse. (shrink)

The Precautionary Principle (PP) is an influential principle of risk management. It has been widely introduced into environmental legislation, and it plays an important role in most international environmental agreements. Yet, there is little consensus on precisely how to understand and formulate the principle. In this paper I prove some impossibility results for two plausible formulations of the PP as a decision-rule. These results illustrate the difficulty in making the PP consistent with the acceptance of any trade-offs between catastrophic risks (...) and more ordinary goods. (shrink)

I propose a new theory of mereology based on a mereological reflection principle. Reflective mereology has natural fusion principles but also refutes certain principles of classical mereology such as Universal Fusion and Fusion Uniqueness. Moreover, reflective mereology avoids Uzquiano’s cardinality problem–the problem that classical mereology tends to clash with set theory when they both quantify over everything. In particular, assuming large cardinals, I construct a model of reflective mereology and second-order ZFCU with Limitation of Size. In (...) the model, classical mereology holds when the quantifiers are restricted to the urelements. (shrink)

Drawing on arguments in Doris (2002, 2022) [Lack of Character: Personality and Moral Behavior. Cambridge: Cambridge University Press; Character Trouble: Undisciplined Essays on Moral Agency and Personality. Oxford: Oxford University Press], this essay argues that good character is typically an insufficient “bulwark” against misconduct in military organizations, for two reasons: (1) the situational sensitivity of behavior and (2) the relatively small effect sizes associated with personality variables. Additionally, what is known about moral development and education gives limited reason to think (...) organizations are likely to have much success in inculcating reliably good character in their members. Instead, military organizations concerned to reduce misconduct should focus on developing organizational cultures structured by carefully crafted and assiduously enforced moral rules. (shrink)

This essay argues against the commonly held view that "ought" implies "can" in the domain of morality. More specifically, I contest the notion that nobody should ever be held morally responsible for failing to avoid the infliction of any harm that he or she has not been able to avoid through all reasonably feasible precautions in the carrying out of some worthwhile activity. The article explicates the concept of a moral right in order to show why violations of moral rights (...) can occur even when no one has acted wrongfully in any fashion. In so doing, it will effectively be maintaining that strict liability (i.e., liability irrespective of the presence or absence of culpability) exists in morality as well as in law. When we take account of the distinction between exoneration and extenuation, we can see that scrupulously thorough precautions are never sufficient to constitute an excuse in morality. Having made that point with some extended examples, the article goes on to consider a number of possible objections - objections that lead into discussions of some basic distinctions within moral philosophy and some central principles within deontic logic. (shrink)

This article considers what features ought to be looked for in an effective billing and charging system. It then looks at different approaches to billing that are adopted and tests them against the features of an ideal system. The author acknowledges that a billing framework which is effective, economically defensible and ethical is in fact impossible and the issue becomes what trade-offs are appropriate in any given situation. The author suggests that there is no one-size-fits-all panacea and what is (...) worse, there are some situations in which no billing framework, however designed, will meet all of the challenges in fair and effective billing. It is suggested that the best approach is to make a principled choice between approaches that are each unsatisfactory to varying degrees. Discussions about billing frequently are either business strategies which develop techniques to maximise income. Others perhaps take a "ethics" approach and focus on what is fair to the client. This article seeks to avoid this conflict by recognising the conflicting tensions and to look for solutions that serve the interests of both lawyers and clients where possible. (shrink)

The maximum entropy principle is widely used to determine non-committal probabilities on a finite domain, subject to a set of constraints, but its application to continuous domains is notoriously problematic. This paper concerns an intermediate case, where the domain is a first-order predicate language. Two strategies have been put forward for applying the maximum entropy principle on such a domain: applying it to finite sublanguages and taking the pointwise limit of the resulting probabilities as the size n of the (...) sublanguage increases; selecting a probability function on the language as a whole whose entropy on finite sublanguages of size n is not dominated by that of any other probability function for sufficiently large n. The entropy-limit conjecture says that, where these two approaches yield determinate probabilities, the two methods yield the same probabilities. If this conjecture is found to be true, it would provide a boost to the project of seeking a single canonical inductive logic—a project which faltered when Carnap's attempts in this direction succeeded only in determining a continuum of inductive methods. The truth of the conjecture would also boost the project of providing a canonical characterisation of normal or default models of first-order theories. Hitherto, the entropy-limit conjecture has been verified for languages which contain only unary predicate symbols and also for the case in which the constraints can be captured by a categorical statement of quantifier complexity. This paper shows that the entropy-limit conjecture also holds for categorical statements of complexity, for various non-categorical constraints, and in certain other general situations. (shrink)

This review article offers a European perspective on the pluralistic, principles-based model of intellectual property (IP) advanced by Robert Merges in his book Justifying Intellectual Property. After introducing Merges’s model and theory of IP with reference to IP theories generally, other pluralistic legal models, and patterns of judicial reasoning in the patent and copyright fields, the article argues that European jurisprudence offers broad support for Merges’s operational model of IP, while also challenging certain aspects of his wider analysis. They (...) include his ‘one size fits all’ foundational theory of IP, his account of key IP rules and practices, and his choice and conception of IP’s midlevel principles. Through this critique the article draws attention to the utilitarian bias of Merges’s model; a bias which undermines its pluralistic claims, in part by undermining Merges’s own foundational theory of IP. The result is to underline the limits of a regime unconcerned with its own normative basis, and the need for more rather than less discussion of IP theory, including more work of the type that Merges’s book undertakes. (shrink)

Drawing on arguments in Doris (2002, 2022) [Lack of Character: Personality and Moral Behavior. Cambridge: Cambridge University Press; Character Trouble: Undisciplined Essays on Moral Agency and Personality. Oxford: Oxford University Press], this essay argues that good character is typically an insufficient “bulwark” against misconduct in military organizations, for two reasons: (1) the situational sensitivity of behavior and (2) the relatively small effect sizes associated with personality variables. Additionally, what is known about moral development and education gives limited reason to think (...) organizations are likely to have much success in inculcating reliably good character in their members. Instead, military organizations concerned to reduce misconduct should focus on developing organizational cultures structured by carefully crafted and assiduously enforced moral rules. (shrink)

This book makes an important contribution to ongoing efforts in the fields of medical law and bioethics to answer the challenges posed by the limitations of the principle of respect for autonomy, especially as these pertain to human research ethics. The principle of respect for autonomy seems to have become firmly embedded in human research ethics since its inclusion in the 1947 Nuremberg Code, which was a response to atrocities committed by Nazi doctors. Nonetheless, there is an increasing awareness of (...) the limitations of the principle of respect for autonomy and the underlying conceptualisations of human autonomy. (shrink)

Since its inception as an international requirement to protect patients and healthy volunteers taking part in medical research, informed consent has become the primary consideration in research ethics. Despite the ubiquity of consent, however, scholars have begun to question its adequacy for contemporary biomedical research. This book explores this issue, reviewing the application of consent to genetic research, clinical trials, and research involving vulnerable populations. For example, in genetic research, information obtained from an autonomous research participant may have significant bearing (...) on the interests of family members who have not consented to the study. This casts doubt on the adequacy of consent for such studies. This book also questions the assumptions that informed consent is essential and that it satisfactorily protects the principle of individual autonomy. It reviews recent empirical studies that challenge the possibility of truly informed consent and highlights the extent to which consent is governed by social norms and expectations. It also investigates how consent might be of secondary importance in some circumstances, for example when a research project appears to protect a public or community interest. (shrink)

This book makes an important contribution to ongoing efforts in the fields of medical law and bioethics to answer the challenges posed by the limitations of the principle of respect for autonomy, especially as these pertain to human research ethics. The principle of respect for autonomy seems to have become firmly embedded in human research ethics since its inclusion in the 1947 Nuremberg Code, which was a response to atrocities committed by Nazi doctors. Nonetheless, there is an increasing awareness of (...) the limitations of the principle of respect for autonomy and the underlying conceptualisations of human autonomy. (shrink)

Our aim is to draw a set of distinctions among types of goods which has significant implications for theories of distributive justice. We begin by providing a general account of two sets of properties--fungibility and nonfungibility, divisibility and indivisibility--and argue that goods can be distinguished according to these criteria. Further, we contend that these distinctions entail complications for structural principles of distributive justice (i.e., principles such as maximin that distribute payoffs to positions). As an example we consider James (...) Fishkin’s discussion of structural principles, arguing that Fishkin’s view that (i) value, structure, and assignment are independent holds only to the degree that the goods considered are fungible and divisible, (ii) structural principles face difficulties beyond those which Fishkin identifies and addresses with his principle of nontyranny, since structural principles cannot accommodate highly non¬fungible, indivisible goods, and (iii) these difficulties can be managed through the application of a value-sensitivity proviso. We then show that two important goods, medical care and advanced education, are highly nonfungible and indivisible and thus support the distinctions drawn earlier. Finally, we specify the nature of complementary contributions as well as coordination problems between structural principles and the value-sensitivity proviso in their application to distributive justice issues. -/- . (shrink)

The Forrest-Armstrong argument, as reconfigured by David Lewis, is a reductio against an unrestricted principle of recombination. There is a gap in the argument which Lewis thought could be bridged by an appeal to recombination. After presenting the argument, I show that no plausible principle of recombination can bridge the gap. But other plausible principles of plenitude can bridge the gap, both principles of plenitude for world contents and principles of plenitude for world structures. I conclude that (...) the Forrest-Armstrong argument, when fortified in one of these ways, demands that unrestricted recombination be rejected. The appropriate restriction comes from a consideration of what world structures are possible. I argue that, although there are too many worlds to form a set, for any world, the individuals at that world do form a set. To defend it I invoke a principle of Limitation of Size together with an iterative conception of structure. (shrink)

The idea that healthcare should become more person-centred is extremely influential. By using recent English policy developments as a case study, this article aims to critically analyse an important element of person-centred care, namely, the belief that to treat patients as persons is to think that care should be ‘co-produced’ by formal healthcare providers and patients together with unpaid carers and voluntary organizations. I draw on insights from political philosophy to highlight overlooked tensions between co-production and values like equality and (...) liberty. Regarding equality, I argue that co-production compounds both problems of gender inequality in the distribution of care labour and the challenges associated with securing equal access to care. Turning to liberty, I identify important commonalities between co-production and republicanism in political philosophy, given their shared insistence on common citizens’ civic virtue. Then, I use against co-production some liberal arguments against republicanism, to highlight a problem of over-demandingness. In bringing my argument to a close, however, I wish to caution against hastily rejecting co-production as a policy programme. (shrink)