We work in set theory ZF without axiom of choice. Though the Hahn-Banach theorem cannot be proved in ZF, we prove that every Gateaux-differentiable uniformly convex Banach space E satisfies the following continuous Hahn-Banach property: if p is a continuous sublinear functional on E, if F is a subspace of E, and if f: F → ℝ is a linear functional such that f ≤ p|F then there exists a linear functional g : E → (...) ℝ such that g extends f and g ≤ p. We also prove that the continuous Hahn-Banach property on a topological vector space E is equivalent to the classical geometrical forms of the Hahn-Banach theorem on E. We then prove that the axiom of Dependent choices DC is equivalent to Ekeland's variational principle, and that it implies the continuous Hahn-Banach property on Gateaux-differentiable Banach spaces. Finally, we prove that, though separable normed spaces satisfy the continuous Hahn-Banach property, they do not satisfy the whole Hahn-Banach property in ZF+DC. (shrink)

In set theory without the axiom of choice math formula, three-space type results for the Hahn-Banach property are provided. We deduce that for every Hausdorff compact scattered space K, the Banach space C of real continuous functions on K satisfies the continuous Hahn-Banach property in math formula. We also prove in math formula Rudin's theorem: “Radon measures on Hausdorff compact scattered spaces are discrete”.

The classical Hahn–Banach Theorem states that any linear bounded functional defined on a linear subspace of a normed space admits a norm-preserving linear bounded extension to the whole space. The constructive and computational content of this theorem has been studied by Bishop, Bridges, Metakides, Nerode, Shore, Kalantari Downey, Ishihara and others and it is known that the theorem does not admit a general computable version. We prove a new computable version of this theorem without unrolling the classical proof (...) of the theorem itself. More precisely, we study computability properties of the uniform extension operator which maps each functional and subspace to the set of corresponding extensions. It turns out that this operator is upper semi-computable in a well-defined sense. By applying a computable version of the Banach–Alaoglu Theorem we can show that computing a Hahn–Banach extension cannot be harder than finding a zero in a compact metric space. This allows us to conclude that the Hahn–Banach extension operator is ${\bf {\Sigma^{0}_{2}}}$ -computable while it is easy to see that it is not lower semi-computable in general. Moreover, we can derive computable versions of the Hahn–Banach Theorem for those functionals and subspaces which admit unique extensions. (shrink)

We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König's Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multivalued function Sep and a natural notion of reducibility for multivalued functions, we obtain a computational counterpart of the subsystem of second-order arithmetic WKL0. We study analogies and differences between WKL0 and (...) the class of Sep-computable multivalued functions. Extending work of Brattka, we show that a natural multivalued function associated with the Hahn-Banach Extension Theorem is Sep-complete. (shrink)

We study theorems from Functional Analysis with regard to their relationship with various weak choice principles and prove several results about them: “Every infinite‐dimensional Banach space has a well‐orderable Hamel basis” is equivalent to ; “ can be well‐ordered” implies “no infinite‐dimensional Banach space has a Hamel basis of cardinality ”, thus the latter statement is true in every Fraenkel‐Mostowski model of ; “No infinite‐dimensional Banach space has a Hamel basis of cardinality ” is not provable in (...) ; “No infinite‐dimensional Banach space has a well‐orderable Hamel basis of cardinality ” is provable in ; (the Axiom of Choice for denumerable families of non‐empty finite sets) is equivalent to “no infinite‐dimensional Banach space has a Hamel basis which can be written as a denumerable union of finite sets”; Mazur's Lemma (“If X is an infinite‐dimensional Banach space, Y is a finite‐dimensional vector subspace of X, and, then there is a unit vector such that for all and all scalars α”) is provable in ; “A real normed vector space X is finite‐dimensional if and only if its closed unit ball is compact” is provable in ; (Principle of Dependent Choices) + “ can be well‐ordered” does not imply the Hahn‐Banach Theorem () in ; and “no infinite‐dimensional Banach space has a Hamel basis of cardinality ” are independent from each other in ; “No infinite‐dimensional Banach space can be written as a denumerable union of finite‐dimensional subspaces” lies in strength between (the Axiom of Countable Choice) and ; implies “No infinite‐dimensional Banach space can be written as a denumerable union of closed proper subspaces” which in turn implies ; “Every infinite‐dimensional Banach space has a denumerable linearly independent subset” is a theorem of, but not a theorem of ; and “Every infinite‐dimensional Banach space has a linearly independent subset of cardinality ” implies “every Dedekind‐finite set is finite”. (shrink)

Sur la décomposition des ensembles de points en parties respectivement congruentes.

An alternative proof is given that to each LLM proof net there corresponds at least one LLM sequent proof. The construction is inspired by the sweepline technique from computational geometry and includes a treatment of the multiplicative constants and of proof boxes.

Shape and locomotion of tissue cells depend on the interaction of elements of the cytoskeleton, adhesion to the substrate and an intracellular hydrostatic pressure. The existence of this pressure becomes obvious from increase in cell volume on cessation of contractile forces and from observations with ultrasound acoustic microscopy. Wherever such an internal pressure is established, it is involved in generation of shape and driving force of cell locomotion. Therefore each hypothesis on cell shape and locomotion must consider this property of (...) a living cell. Apparently different types of locomotion depend on differences in substrate adhesion and/or cytoskeleton organisation. (shrink)

It is virtuous for individual and collective agents to be tolerant. However, toleration is difficult, both in practice and in conceptualization. Firstly, given that toleration can be understood in various ways (Walzer 1997, Forst 2007), it seems that to determine what is the proper conception of toleration would be controversially difficult. Here I shall suggest one particular conception of toleration is more suitable than others. This conception allows, as I shall explain, us to better understandthe difficulties of toleration. Thus, this (...) particular conception of toleration should lead us to see what is more adequate for dealing with the difficulties of toleration. To be more precise, I shall argue for a political conception of toleration, which different from the attitudinal conception of toleration as being indifferent, or the ethical conception of toleration as respect. There is the suggestion of toleration as recognition (Galeotti 2002). These alternative understandings of toleration do not provide better diagnoses of the difficulties of toleration. The political conception of toleration is intended to be grounded on some moral considerations, notpragmatic purpose. It is political in that it recognizes the fact that toleration is essentially practiced to deal with a power relationship among the parties of toleration. Where these is no such power relationship, as I shall argue, there is no issue of toleration. Secondly, this proposed conception of toleration is political in the sense that it shall not deal with differences coming from, to use John Rawls’s phrase, the fact of pluralism by adopting any comprehensive doctrine such as an ethics of respect or recognition. (shrink)

The latest volume in the Cambridge Histories of Philosophy series, The Cambridge History of Philosophy in the Nineteenth Century brings together twenty-nine leading experts in the field and covers the years 1790-1870. Their twenty-seven chapters provide a comprehensive survey of the period, organizing the material topically. After a brief editor's introduction, it begins with three chapters surveying the background of nineteenth century philosophy: followed by two on logic and mathematics, two on nature and natural science, five on mind and language, (...) including psychology, the human sciences and aesthetics, four on ethics, three on religion, seven on society, including chapters on the French Revolution, the decline of natural right, political economy, and social discontent, and three on history, dealing with historical method, speculative theories of history and the history of philosophy. The essays are framed by an editor's introduction and a bibliography. (shrink)

The debate over environmental, health, and safety regulation has reached a new crescendo in the 104th Congress. So impassioned is the debate on occasion, and so high the feelings, that even the tools of regulatory analysis have become part of the combat.To some, the term cost-benefit analysis, for example, is virtually a swearword, a nefarious tool used by big business to undermine regulations aimed at benefiting the people at large. To others, it is the mechanism for achieving more effective regulation (...) at less cost.This new book on the subject of reforming regulation is aimed at increasing the light and turning down the heat. It consists of an introduction and nine chapters written by scientists, public policy analysts, and economists on various aspects of regulation and regulatory analysis.The book advances the latest knowledge and evidence on risk analysis, cost-benefit analysis, and related techniques for improving regulation--all based on experience and data accumulated over the quarter century since regulation in the areas of environment, health, and safety became a major player on the governmental stage. The strengths and limitations of those analytical tools are an important part of the portrait that is painted, but a common thesis is that, properly used, the tools can be helpful in getting better regulatory results. (shrink)

In their article ‘Consent-GPT: is it ethical to delegate procedural consent to conversational AI?’, Allen et al 1 explore the ethical complexities involved in handing over parts of the process of obtaining medical consent to conversational Artificial Intelligence (AI) systems, that is, AI-driven large language models (LLMs) trained to interact with patients to inform them about upcoming medical procedures and assist in the process of obtaining informed consent.1 They focus specifically on challenges related to accuracy (4–5), trust (5), privacy (5), (...) click-through consent (5) and responsibility (5–6), alongside some pragmatic considerations (6). While the authors competently navigate these critical issues and present several key perspectives, we posit that their discussion on trust in what they refer to as ‘Consent-GPT’ significantly underestimates one vital factor: the interpersonal aspect of trust. Admittedly, this …. (shrink)

We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S , all definable sets of reals are (...) Lebesgue measurable, suggesting that Connes views a theory as being “virtual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace −∫ (featured on the front cover of Connes’ magnum opus) and the Hahn–Banach theorem, in Connes’ own framework. We also note an inaccuracy in Machover’s critique of infinitesimal-based pedagogy. (shrink)

Information generally comes from less than fully reliable sources. Rationality, it seems, requires that one take source reliability into account when reasoning on the basis of such information. Recently, Bovens and Hartmann (2003) proposed an account of the conjunction fallacy based on this idea. They show that, when statements in conjunction fallacy scenarios are perceived as coming from such sources, probability theory prescribes that the “fallacy” be committed in certain situations. Here, the empirical validity of their model was assessed. The (...) model predicts that statements added to standard conjunction problems will change the incidence of the fallacy. It also predicts that statements from reliable sources should yield an increase in fallacy rates (relative to unreliable sources). Neither the former (Experiment 1) nor the latter prediction (Experiment 3) was confirmed, although Experiment 2 showed that people can derive source reliability estimates from the likelihood of statements in a manner consistent with the tested model. In line with the experimental results, model fits and sensitivity analyses also provided very little evidence in favor of the model. This suggests that Bovens and Hartmann’s present model fails to explain fully people’s judgements in standard conjunction fallacy tasks. (shrink)

The Chairman of Volkswagen's Board of Management made the following presentation in London last November at a Conference on‘Business and Moral Standards in Post‐Communist Europe’, held under the auspices of the Cardinal Archbishop of Westminster and sponsored by the Sedgwick Group and KPMG Peat Marwick. Dr Hahn's lecture is reproduced with permission.

Autour du Manifeste de Vienne se trouvent reunis des textes fondateurs ecrits autour de 1929. Leurs auteurs: Carnap, Hahn, Neurath, Schlick l'ame du Cercle de Vienne, et Waismann plus proche de Wittgenstein, temoignent d'un courant philosophique constituant aujourd'hui la tradition analytique de source continentale a la fois empiriste et logique. Forme de maniere informelle a Vienne, au coeur de l'Europe, le Cercle reunissait des savants de differentes branches qui voulaient se donner une philosophie susceptible d' unifier leurs vues. Echo (...) aux reves d'unite de la science, cette philosophie en exprime aussi le renversement complet. Veritable Discours de la methode, mais d'esprit anti-cartesien, elle entend reconstruire la science de maniere a evincer la speculation metaphysique, le but etant qu'aucun enonce ne soit laisse a l'exterieur de cette grande architecture de symboles s'il etait pourvu de sens. La declaration de guerre contre la metaphysique est un geste de militance a une epoque riche en differents manifestes. Incomprehensible en dehors du contexte socio-politique viennois de ces annees, elle porte aussi un programme de reconstruction rationnelle qui, au dela du langage de la science, devait toucher tous les secteurs de la vie. Plus complexe qu'il n'y parait, ce programme contient, en reference a Wittgenstein, son prestigieux representant a cote d'Einstein et Russell, une mise en garde contre differentes formes d'irrationalismes, qui est d'actualite. (shrink)

Factorial surveys integrate elements of survey research and classical experiments. Using a large number of respondents in a controlled setting, FSs approximate complex and realistic judgment situations through so-called vignettes—that is, carefully designed descriptions of hypothetical people, social situations, or scenarios. Despite being rooted, and predominantly applied, in sociology, FSs are particularly promising for business and society scholars. Given the multiplicity, inherent complexity, and sometimes fuzziness of B&S research objects, conventional research methods inevitably reach their limits. This article, therefore, systematically (...) presents methodological and thematic opportunities for FS studies in B&S research. It is argued that FSs are well suited to dealing with the complex interplay of societal-, organizational-, and individual-level factors in B&S research and to studying the principles underlying human perceptions, attitudes, values, social norms, and behavior. The application of the FS method is illustrated based on a showcase example in the realm of socially responsible investments. As the literature on the conceptualization of FSs is limited, methodological challenges are addressed to guide B&S researchers past the common methodological pitfalls. (shrink)

Globalization arguably generated a governance gap that is being filled by transnational rule-making involving private actors among others. The democratic legitimacy of such new forms of governance beyond nation states is sometimes questioned. Apart from nation-centered democracies, such governance cannot build, for example, on representation and voting procedures to convey legitimacy to the generated rules. Instead, alternative elements of democracy such as deliberation and inclusion require discussion to assess new instruments of governance. The recently published standard ISO 26000 is an (...) interesting example of transnational governance. ISO 26000 was developed in a lengthy multiorganizational process for the purpose of giving guidance on the social responsibility of organizations. By assessing the specific case of ISO 26000, this study sheds light on the question of how legitimacy beyond nation-state democracy is ensured or constricted. Centering on the idea of deliberate democracy and democratic legitimacy, the study offers in-depth insights on the normative legitimacy of the development process of ISO 26000. Positioned on the interface of business studies and public policy, this article contributes to the academic literature on transnational governance and on the role of multistakeholder processes in shaping the role of business in society. (shrink)

The appeal to expert opinion is an argument form that uses the verdict of an expert to support a position or hypothesis. A previous scheme-based treatment of the argument form is formalized within a Bayesian network that is able to capture the critical aspects of the argument form, including the central considerations of the expert's expertise and trustworthiness. We propose this as an appropriate normative framework for the argument form, enabling the development and testing of quantitative predictions as to how (...) people evaluate this argument, suggesting that such an approach might be beneficial to argumentation research generally. We subsequently present two experiments as an example of the potential for future research in this vein, demonstrating that participants' quantitative ratings of the convincingness of a proposition that has been supported with an appeal to expert opinion were broadly consistent with the predictions of the Bayesian model. (shrink)

I show that de Finetti’s coherence theorem is equivalent to the Hahn-Banach theorem and discuss some consequences of this result. First, the result unites two aspects of de Finetti’s thought in a nice way: a corollary of the result is that the coherence theorem implies the existence of a fair countable lottery, which de Finetti appealed to in his arguments against countable additivity. Another corollary of the result is the existence of sets that are not Lebesgue measurable. I (...) offer a subjectivist interpretation of this corollary that is concordant with de Finetti’s views. I conclude by pointing out that my result shows that there is a sense in which de Finetti’s theory of subjective probability is necessarily nonconstructive. This raises questions about whether the coherence theorem can underwrite a legitimate theory of rational belief. (shrink)

Norms—that is, specifications of what we ought to do—play a critical role in the study of informal argumentation, as they do in studies of judgment, decision-making and reasoning more generally. Specifically, they guide a recurring theme: are people rational? Though rules and standards have been central to the study of reasoning, and behavior more generally, there has been little discussion within psychology about why (or indeed if) they should be considered normative despite the considerable philosophical literature that bears on this (...) topic. In the current paper, we ask what makes something a norm, with consideration both of norms in general and a specific example: norms for informal argumentation. We conclude that it is both possible and desirable to invoke norms for rational argument, and that a Bayesian approach provides solid normative principles with which to do so. (shrink)

Millennial employees are increasingly paying more attention to well-being in the workplace and it has become an important issue for managers. Given that millennial employees are more sensitive to ethical issues, this study began by analyzing an ethical element in the organization—the ethical climate—and explored whether millennial employees have higher affective well-being in organizations with a good ethical climate. We verified our hypotheses based on 288 valid questionnaires collected from 40 teams. The results showed that: ethical climate was a positive (...) predictor of millennial employees’ organizational citizenship behavior and affective well-being, employees’ OCB partially mediated the relationship between ethical climate and affective well-being, and an employee’s moral identity effectively moderated the relationship between ethical climate and affective well-being, although it did not play a significant moderating role between ethical climate and OCB. These findings provide empirical support for applying situational strength and social information processing theories and emphasize the importance of cultivating an ethical climate in organizations. (shrink)

A common view in both philosophy and the vision sciences is that, in human vision, wavelength information is primarily ‘for’ colouring: for seeing surfaces and various media as having colours. In this article we examine this assumption of ‘colour-for-colouring’. To motivate the need for an alternative theory, we begin with three major puzzles from neurophysiology, puzzles that are not explained by the standard theory. We then ask about the role of wavelength information in vision writ large. How might wavelength information (...) be used by any monochromat or dichromat and, finally, by a trichromatic primate with object vision? We suggest that there is no single ‘advantage’ to trichromaticity but a multiplicity, only one of which is the ability to see surfaces and so on as having categorical colours. Instead, the human trichromatic retina exemplifies a scheme for a general encoding of wavelength information given the constraints imposed by high spatial resolution object vision. Chromatic vision, like its partner, luminance vision, is primarily for seeing. Viewed this way, the ‘puzzles’ presented at the outset make perfect sense. 1 Reframing the Problem1.1 Introduction1.2 Three puzzles1.2.1 Why is trichromatic vision an anomaly in diurnal mammals?1.2.2 Why does the colour system occupy such a large and central part in human vision?1.2.3 Why are the blue cones so rare?1.3 Recasting the question2 The Costs and Benefits of Spectral Vision2.1 Spectral information and object vision2.2 Encoding the spectral dimension of light2.2.1 ‘General’ versus ‘specific’ encoding2.2.2 Spectral encoding and the monochromat2.2.3 Spectral encoding and the dichromat2.2.4 Spectral encoding and the human trichromat2.3 Three puzzles revisited2.3.1 Why is trichromatic vision an anomaly in diurnal mammals?2.3.2 Why does the colour system occupy such a large and central part in human vision?2.3.3 Why are the blue cones so rare3 Conclusion. (shrink)

Many groups of closely related species have reticulate phylogenies. Recent genomic analyses are showing this in many insects and vertebrates, as well as in microbes and plants. In microbes, lateral gene transfer is the dominant process that spoils strictly tree‐like phylogenies, but in multicellular eukaryotes hybridization and introgression among related species is probably more important. Because many species, including the ancestors of ancient major lineages, seem to evolve rapidly in adaptive radiations, some sexual compatibility may exist among them. Introgression and (...) reticulation can thereby affect all parts of the tree of life, not just the recent species at the tips. Our understanding of adaptive evolution, speciation, phylogenetics, and comparative biology must adapt to these mostly recent findings. Introgression has important practical implications as well, not least for the management of genetically modified organisms in pest and disease control. (shrink)