This book comprises all of John Bell's published and unpublished papers in the field of quantum mechanics, including two papers that appeared after the first edition was published. It also contains a preface written for the first edition, and an introduction by Alain Aspect that puts into context Bell's great contribution to the quantum philosophy debate. One of the leading expositors and interpreters of modern quantum theory, John Bell played a major role in the development of our current understanding of (...) the profound nature of quantum concepts. First edition Hb (1987): 0-521-33495-0 First edition Pb (1988): 0-521-36869-3. (shrink)

A comprehensive one-year graduate (or advanced undergraduate) course in mathematical logic and foundations of mathematics. No previous knowledge of logic is required; the book is suitable for self-study. Many exercises (with hints) are included.

The strange story of the von Neumann impossibility proof is recalled, and the even stranger story of later impossibility proofs, and how the impossible was done by de Broglie and Bohm. Morals are drawn.

The idea of a 'logic of quantum mechanics' or quantum logic was originally suggested by Birkhoff and von Neumann in their pioneering paper [1936]. Since that time there has been much argument about whether, or in what sense, quantum 'logic' can be actually considered a true logic (see, e.g. Bell and Hallett [1982], Dummett [1976], Gardner [1971]) and, if so, how it is to be distinguished from classical logic. In this paper I put forward a simple and natural semantical framework (...) for quantum logic which reveals its difference from classical logic in a strikingly intuitive way, viz. through the fact that quantum logic admits (suitably formulated versions of) the characteristic quantum-mechanical notions of superposition and incompatibility of attributes. That is, precisely the features that distinguish quantum from classical physics also serve, within this framework, to distinguish quantum from classical logic. Some light is shed on the question of whether quantum logic is a genuine logical system by introducing a natural entailment relation for quantum-logical formulas with the implication symbol. The novelty is that, although implication behaves as it should (i.e. the 'deduction theorem' holds), the order of introduction of premises is significant. The fact that a reasonable entailment relation can be formulated for quantum logic supports the view that it is a genuine logical system and not merely an algebraic formalism. (shrink)

This third edition, now available in paperback, is a follow up to the author's classic Boolean-Valued Models and Independence Proofs in Set Theory. It provides an exposition of some of the most important results in set theory obtained in the 20th century: the independence of the continuum hypothesis and the axiom of choice.

In this paper (a sequel to [4]) I put forward a "local" interpretation of mathematical concepts based on notions derived from category theory. The fundamental idea is to abandon the unique absolute universe of sets central to the orthodox set-theoretic account of the foundations of mathematics, replacing it by a plurality of local mathematical frameworks - elementary toposes - defined in category-theoretic terms.

Population-level biomedical research offers new opportunities to improve population health, but also raises new challenges to traditional systems of research governance and ethical oversight. Partly in response to these challenges, various models of public involvement in research are being introduced. Yet, the ways in which public involvement should meet governance challenges are not well understood. We conducted a qualitative study with 36 experts and stakeholders using the World Café method to identify key governance challenges and explore how public involvement can (...) meet these challenges. This brief report discusses four cross-cutting themes from the study: the need to move beyond individual consent; issues in benefit and data sharing; the challenge of delineating and understanding publics; and the goal of clarifying justifications for public involvement. The report aims to provide a starting point for making sense of the relationship between public involvement and the governance of population-level biomedical research, showing connections, potential solutions and issues arising at their intersection. We suggest that, in population-level biomedical research, there is a pressing need for a shift away from conventional governance frameworks focused on the individual and towards a focus on collectives, as well as to foreground ethical issues around social justice and develop ways to address cultural diversity, value pluralism and competing stakeholder interests. There are many unresolved questions around how this shift could be realised, but these unresolved questions should form the basis for developing justificatory accounts and frameworks for suitable collective models of public involvement in population-level biomedical research governance. (shrink)

From the early 1960s until his death, French philosopher Gilles Deleuze wrote many influential works on philosophy, literature, film, and fine art. One of Deleuze's main philosophical projects was a systematic inversion of the traditional relationship between identity and difference. This Deleuzian philosophy of difference is the subject of Jeffrey A. Bell's Philosophy at the Edge of Chaos. Bell argues that Deleuze's efforts to develop a philosophy of difference are best understood by exploring both Deleuze's claim to be a Spinozist, (...) and Nietzsche's claim to have found in Spinoza an important precursor. Beginning with an analysis of these claims, Bell shows how Deleuze extends and transforms concepts at work in Spinoza and Nietzsche to produce a philosophy of difference that promotes and, in fact, exemplifies the notions of dynamic systems and complexity theory. With these concepts at work, Deleuze constructs a philosophical approach that avoids many of the difficulties that linger in other attempts to think about difference. Bell uses close readings of Plato, Aristotle, Spinoza, Nietzsche, Heidegger, Derrida, and Whitehead to illustrate how Deleuze's philosophy is successful in this regard and to demonstrate the importance of the historical tradition for Deleuze. Far from being a philosopher who turns his back on what is taken to be a mistaken metaphysical tradition, Bell argues that Deleuze is best understood as a thinker who endeavoured to continue the work of traditional metaphysics and philosophy. (shrink)

BackgroundClinical ethics consultations can be complex interventions, involving multiple methods, stakeholders, and competing ethical values. Despite longstanding calls for rigorous evaluation in the field, progress has been limited. The Medical Research Council proposed guidelines for evaluating the effectiveness of complex interventions. The evaluation of CEC may benefit from application of the MRC framework to advance the transparency and methodological rigor of this field. A first step is to understand the outcomes measured in evaluations of CEC in healthcare settings. ObjectiveThe primary (...) objective of this review was to identify and map the outcomes reported in primary studies of CEC. The secondary objective was to provide a comprehensive overview of CEC structures, processes, and roles to enhance understanding and to inform standardization.MethodsWe searched electronic databases to identify primary studies of CEC involving patients, substitute decision-makers and/or family members, clinicians, healthcare staff and leaders. Outcomes were mapped across five conceptual domains as identified a priori based on our clinical ethics experience and preliminary literature searches and revised based on our emerging interpretation of the data. These domains included personal factors, process factors, clinical factors, quality, and resource factors. ResultsForty-eight studies were included in the review. Studies were highly heterogeneous and varied considerably regarding format and process of ethical intervention, credentials of interventionist, population of study, outcomes reported, and measures employed. In addition, few studies used validated measurement tools. The top three outcome domains that studies reported on were quality, process factors, and clinical factors. The majority of studies examined multiple outcome domains. All five outcome domains were multidimensional and included a variety of subthemes.ConclusionsThis scoping review represents the initial phase of mapping the outcomes reported in primary studies of CEC and identifying gaps in the evidence. The confirmed lack of standardization represents a hindrance to the provision of high quality intervention and CEC scientific progress. Insights gained can inform the development of a core outcome set to standardize outcome measures in CEC evaluation research and enable scientifically rigorous efficacy trials of CEC. (shrink)

This book explores and articulates the concepts of the continuous and the infinitesimal from two points of view: the philosophical and the mathematical. The first section covers the history of these ideas in philosophy. Chapter one, entitled ‘The continuous and the discrete in Ancient Greece, the Orient and the European Middle Ages,’ reviews the work of Plato, Aristotle, Epicurus, and other Ancient Greeks; the elements of early Chinese, Indian and Islamic thought; and early Europeans including Henry of Harclay, Nicholas of (...) Autrecourt, Duns Scotus, William of Ockham, Thomas Bradwardine and Nicolas Oreme. The second chapter of the book covers European thinkers of the sixteenth and seventeenth centuries: Galileo, Newton, Leibniz, Descartes, Arnauld, Fermat, and more. Chapter three, 'The age of continuity,’ discusses eighteenth century mathematicians including Euler and Carnot, and philosophers, among them Hume, Kant and Hegel. Examining the nineteenth and early twentieth centuries, the fourth chapter describes the reduction of the continuous to the discrete, citing the contributions of Bolzano, Cauchy and Reimann. Part one of the book concludes with a chapter on divergent conceptions of the continuum, with the work of nineteenth and early twentieth century philosophers and mathematicians, including Veronese, Poincaré, Brouwer, and Weyl. Part two of this book covers contemporary mathematics, discussing topology and manifolds, categories, and functors, Grothendieck topologies, sheaves, and elementary topoi. Among the theories presented in detail are non-standard analysis, constructive and intuitionist analysis, and smooth infinitesimal analysis/synthetic differential geometry. No other book so thoroughly covers the history and development of the concepts of the continuous and the infinitesimal. (shrink)

climate are both linked to an organization's ability to retain healthcare professionals and increase their level of job satisfaction, leaders have a corollary responsibility to address moral distress. We recommend that leaders should provide access to ethics education and resources, offer interventions such as ethics debriefings, establish ethics committees, and/or hire a bioethicist to develop ethics capacity and to assist with addressing healthcare provider moral distress....

This paper provides a description of the role of the clinical ethicist as it is generally experienced in Canada. It examines the activities of Canadian ethicists working in healthcare institutions and the way in which their work incorporates more than ethics case consultation. The Canadian Bioethics Society established a Taskforce on Working Conditions for Bioethics (hereafter referred to as the Taskforce), to make recommendations on a number of issues affecting ethicists and to develop a model role description. This essay carefully (...) assesses this model role description. (shrink)

The usual meaning of the word continuous is “unbroken” or “uninterrupted”: thus a continuous entity —a continuum—has no “gaps.” We commonly suppose that space and time are continuous, and certain philosophers have maintained that all natural processes occur continuously: witness, for example, Leibniz's famous apothegm natura non facit saltus—“nature makes no jump.” In mathematics the word is used in the same general sense, but has had to be furnished with increasingly precise definitions. So, for instance, in the later 18th century (...) continuity of a function was taken to mean that infinitesimal changes in the value of the argument induced infinitesimal changes in the value of the function. With the abandonment of infinitesimals in the 19th century this definition came to be replaced by one employing the more precise concept of limit. (shrink)

This paper provides a description of the role of the clinical ethicist as it is generally experienced in Canada. It examines the activities of Canadian ethicists working in healthcare institutions and the way in which their work incorporates more than ethics case consultation. The Canadian Bioethics Society established a “Taskforce on Working Conditions for Bioethics” (hereafter referred to as the Taskforce), to make recommendations on a number of issues affecting ethicists and to develop a model role description. This essay carefully (...) assesses this model role description. (shrink)

_BMC Medical Ethics_ is an open access journal publishing original peer-reviewed research articles in relation to the ethical aspects of biomedical research and clinical practice, including professional choices and conduct, medical technologies, healthcare systems and health policies. _BMC __Medical Ethics _is part of the _BMC_ series which publishes subject-specific journals focused on the needs of individual research communities across all areas of biology and medicine. We do not make editorial decisions on the basis of the interest of a study or (...) its likely impact. Studies must be scientifically valid; for research articles this includes a scientifically sound research question, the use of suitable methods and analysis, and following community-agreed standards relevant to the research field. Specific criteria for other article types can be found in the submission guidelines. _BMC series - open, inclusive and trusted_. (shrink)

Summarya) Bell tries to formulate more explicitly a notion of “local causality”: correlations between physical events in different space‐time regions should be explicable in terms of physical events in the overlap of the backward light cones. It is shown that ordinary relativistic quantum field theory is not locally causal in this sense, and cannot be embedded in a locally causal theory.b) Clauser, Home and Shimony criticize several steps in Bell's argument that any theory of local “beables” is incompatible with quantum (...) mechanics. It is contended that the Clauser‐Horne derivation of a Bell‐type inequality circumvents his weak steps. The Clauser‐Horne derivation must assume that there are no undetected correlations between choices of controllable variables in two space‐like separated regions. Methodological considerations support this assumption.c) In response to criticism by Shimony, Home, and Clauser, Bell tries to clarify the argument of “The theory of local beables”, and to defend as permissible the hypothesis of free variables.d) Bell's reply to an earlier criticism by Shimony, Clauser, and Home is answered. The convergence of Bell's position towards theirs is noted. (shrink)

Machine generated contents note: 1.What is a Concept? -- 2.Why Philosophy? -- 3.How to Become a Philosopher -- 4.Putting Philosophy in its Place -- 5.Philosophy and Science -- 6.Philosophy and Logic -- 7.Philosophy and Art.

Introduction -- 1. Problem of the New -- 2. Problem of Relations -- 3. Problem of Emergence -- 4. Problem of One and Many -- 5. Plato and the Third Man Argument -- 6. Bradley and the Problem of Relations -- 7. Moore, Russell and the Birth of Analytic Philosophy -- 8. Russell and Deleuze on Leibniz -- 9. On Problematic Fields -- 10. Kant and Problematic Ideas -- 11. Armstrong and Lewis on the Problem of One and Many -- (...) 12. Determinables and Determinates -- 13. The Limits of Representational Thought -- 14. Learning from a Cup of Coffee -- 15. Carnap and the Fate of Metaphysics -- 16. Truth and Relevance -- Conclusion. (shrink)

This Element is an exposition of second- and higher-order logic and type theory. It begins with a presentation of the syntax and semantics of classical second-order logic, pointing up the contrasts with first-order logic. This leads to a discussion of higher-order logic based on the concept of a type. The second Section contains an account of the origins and nature of type theory, and its relationship to set theory. Section 3 introduces Local Set Theory, an important form of type theory (...) based on intuitionistic logic. In Section 4 number of contemporary forms of type theory are described, all of which are based on the so-called 'doctrine of propositions as types'. We conclude with an Appendix in which the semantics for Local Set Theory - based on category theory - is outlined. (shrink)

BackgroundThere are numerous ethical challenges that can impact patients and families in the health care setting. This paper reports on the results of a study conducted with a panel of clinical bioethicists in Toronto, Ontario, Canada, the purpose of which was to identify the top ethical challenges facing patients and their families in health care. A modified Delphi study was conducted with twelve clinical bioethicist members of the Clinical Ethics Group of the University of Toronto Joint Centre for Bioethics. The (...) panel was asked the question, what do you think are the top ten ethical challenges that Canadians may face in health care? The panel was asked to rank the top ten ethical challenges throughout the Delphi process and consensus was reached after three rounds.DiscussionThe top challenge ranked by the group was disagreement between patients/families and health care professionals about treatment decisions. The second highest ranked challenge was waiting lists. The third ranked challenge was access to needed resources for the aged, chronically ill, and mentally ill.SummaryAlthough many of the challenges listed by the panel have received significant public attention, there has been very little attention paid to the top ranked challenge. We propose several steps that can be taken to help address this key challenge. (shrink)

Hermann Weyl, one of the twentieth century's greatest mathematicians, was unusual in possessing acute literary and philosophical sensibilities—sensibilities to which he gave full expression in his writings. In this paper I use quotations from these writings to provide a sketch of Weyl's philosophical orientation, following which I attempt to elucidate his views on the mathematical continuum, bringing out the central role he assigned to intuition.

The principle of set theory known as the Axiom of Choice has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid's axiom of parallels which was introduced more than two thousand years ago” (Fraenkel, Bar-Hillel & Levy 1973, §II.4). The fulsomeness of this description might lead those unfamiliar with the axiom to expect it to be as startling as, say, the Principle of the Constancy of (...) the Velocity of Light or the Heisenberg Uncertainty Principle. But in fact the Axiom of Choice as it is usually stated appears humdrum, even self-evident. For it amounts to nothing more than the claim that, given any collection of mutually disjoint nonempty sets, it is possible to assemble a new set — a transversal or choice set — containing exactly one element from each member of the given collection. Nevertheless, this seemingly innocuous principle has far-reaching mathematical consequences — many indispensable, some startling — and has come to figure prominently in discussions on the foundations of mathematics. It (or its equivalents) have been employed in countless mathematical papers, and a number of monographs have been exclusively devoted to it. (shrink)

The COVID-19 pandemic has strained healthcare resources the world over, requiring healthcare providers to make resource allocation decisions under extraordinary pressures. A year later, our understanding of COVID-19 has advanced, but our process for making ethical decisions surrounding resource allocation has not. During the first wave of the pandemic, our institution uniformly ramped-down clinical activity to accommodate the anticipated demands of COVID-19, resulting in resource waste and inefficiency. In preparation for the second wave, we sought to make such ramp down (...) decisions more prudently and ethically. We report the development of a tool that can be used to make fair and ethical decisions in times of resource scarcity. We formed an interprofessional team to develop and use this tool to ensure that a diverse range of stakeholder perspectives were represented in this development process. This team, called the clinical activity recovery team, established institutional objectives that were combined with well-established procedural values, substantive ethical principles and decision-making criteria by using a variation on the well-known accountability for reasonableness ethical framework. The result of this is a stepwise, semiquantitative, ethical decision tool that can be applied to resource allocation challenges in order to reach fair and ethically defensible decisions. This ethical decision tool can be applied in various contexts and may prove useful at both the institutional and the departmental level; indeed this is how it is applied at our centre. As the second wave of COVID-19 strains healthcare resources, this tool can help clinical leaders to make fair decisions. There are no data in this work. (shrink)

5.5. Every topos is linguistic: the equivalence theorem.

This paper treats some of the issues raised by Putnam's discussion of, and claims for, quantum logic, specifically: that its proposal is a response to experimental difficulties; that it is a reasonable replacement for classical logic because its connectives retain their classical meanings, and because it can be derived as a logic of tests. We argue that the first claim is wrong (1), and that while conjunction and disjunction can be considered to retain their classical meanings, negation crucially does not. (...) The argument is conducted via a thorough analysis of how the meet, join and complementation operations are defined in the relevant logical structures, respectively Boolean- and ortholattices (3). Since Putnam wishes to reinstate a realist interpretation of quantum mechanics, we ask how quantum logic can be a logic of realism. We show that it certainly cannot be a logic of bivalence realism (i.e., of truth and falsity), although it is consistent with some form of ontological realism (4). Finally, we show that while a reasonable explication of the idealized notion of test yields interesting mathematical structure, it by no means yields the rich ortholattice structure which Putnam (following Finkelstein) seeks. (shrink)

Logical Options introduces the extensions and alternatives to classical logic which are most discussed in the philosophical literature: many-sorted logic, second-order logic, modal logics, intuitionistic logic, three-valued logic, fuzzy logic, and free logic. Each logic is introduced with a brief description of some aspect of its philosophical significance, and wherever possible semantic and proof methods are employed to facilitate comparison of the various systems. The book is designed to be useful for philosophy students and professional philosophers who have learned some (...) classical first-order logic and would like to learn about other logics important to their philosophical work. (shrink)

This paper is an introduction to topos theory which assumes no prior knowledge of category theory. It includes a discussion of internal logic in a topos, A characterization of the category of sets, And an investigation of the notions of topology and sheaf in a topos.

The centrality of the whole/part relation in mathematics is demonstrated through the presentation and analysis of examples from algebra, geometry, functional analysis,logic, topology and category theory.