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.

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)

We analyze Zorn's Lemma and some of its consequences for Boolean algebras in a constructive setting. We show that Zorn's Lemma is persistent in the sense that, if it holds in the underlying set theory, in a properly stated form it continues to hold in all intuitionistic type theories of a certain natural kind. (Observe that the axiom of choice cannot be persistent in this sense since it implies the law of excluded middle.) We also establish the persistence of some (...) familiar results in the theory of (complete) Boolean algebras--notably, the proposition that every complete Boolean algebra is an absolute subretract. This (almost) resolves a question of Banaschewski and Bhutani as to whether the Sikorski extension theorem for Boolean algebras is persistent. (shrink)

We analyze Zorn's Lemma and some of its consequences for Boolean algebras in a constructive setting. We show that Zorn's Lemma is persistent in the sense that, if it holds in the underlying set theory, in a properly stated form it continues to hold in all intuitionistic type theories of a certain natural kind. We also establish the persistence of some familiar results in the theory of Boolean algebras--notably, the proposition that every complete Boolean algebra is an absolute subretract. This (...) resolves a question of Banaschewski and Bhutani as to whether the Sikorski extension theorem for Boolean algebras is persistent. (shrink)

Much philosophical attention has been devoted to “The Land Ethic,” especially by Anglo-American philosophers, but little has been paid to A Sand County Almanac as a whole. Read through the lens of continental philosophy, A Sand County Almanac promulgates an evolutionary-ecological world view and effects a personal self- and a species-specific Self-transformation in its audience. It’s author, Aldo Leopold, realizes these aims through descriptive reflection that has something in common with phenomenology-although Leopold was by no stretch of the imagination a (...) phenomenologist. Consideration of human-animal intersubjectivity, thematized in A Sand County Almanac, brings to light the moral problem of hunting and killing animal subjects. Leopold does not confront that problem, but it is confronted and resolved by Jose Ortega y Gassett, Henry Beston, and Paul Shepard in terms of an appropriate human relationship with wild-animal Others. Comparison with the genuinely Other-based Leopold-Ortega-Beston-Shepard wild-animal ethic shows the purportedly Other-based humanand possibly animal ethic of Emmanuel Levinas actually to be Same-based after all. (shrink)

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)

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.

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)

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.

Through case studies that highlight the type of information that is seldom reported in the news, Faces of Environmental Racism exposes the type and magnitude of environmental racism, both domestic and international. The essays explore the justice of current environmental practices, asking such questions as whether cost-benefit analysis is an appropriate analytic technique and whether there are alternate routes to sustainable development in the South.

Abstract:Vulnerability is an important criterion to assess the ethical justification of the inclusion of participants in research trials. Currently, vulnerability is often understood as an attribute inherent to a participant by nature of a diagnosed condition. Accordingly, a common ethical concern relates to the participant’s decisionmaking capacity and ability to provide free and informed consent. We propose an expanded view of vulnerability that moves beyond a focus on consent and the intrinsic attributes of participants. We offer specific suggestions for how (...) relational aspects and the dynamic features of vulnerability could be more fully captured in current discussions and research practices. (shrink)

A key idea in both Frege's development of arithmetic in theGrundlagen[7] and Zermelo's 1904 proof [10] of the well-ordering theorem is that of a “type reducing” correspondence between second-level and first-level entities. In Frege's construction, the correspondence obtains betweenconceptandnumber, in Zermelo's (through the axiom of choice), betweensetandmember. In this paper, a formulation is given and a detailed investigation undertaken of a system ℱ of many-sorted first-order logic (first outlined in the Appendix to [6]) in which this notion of type reducing (...) correspondence is accorded a central role and which enables Frege's and Zermelo's constructions to be presented in such a way as to reveal their essential similarity. By adapting Bourbaki's version of Zermelo's proof of the well-ordering theorem, we show that, within ℱ, any correspondencecbetween second-level entities (here calledconcepts) and first-level ones (here calledobjects) induces a well-ordering relationW(c) in a canonical manner. We shall see that, whencis the “Fregean” correspondence between concepts and cardinal numbers,W(c) is (the well-ordering of) the ordinalω+ 1, and whencis a “Zermelian” choice function on concepts,W(c) is a well-ordering of the universal concept embracing all objects.In ℱ an important role is played by the notion ofextensionof a concept. To each conceptXwe assume there is assigned an objecte(X) in such a way that, for any conceptsX, Ysatisfying a certain predicateE, we havee(X) =e(Y) iff the same objects fall underXandY. (shrink)

Through case studies that highlight the type of information that is seldom reported in the news, Faces of Environmental Racism exposes the type and magnitude of environmental racism, both domestic and international. The essays explore the justice of current environmental practices, asking such questions as whether cost-benefit analysis is an appropriate analytic technique and whether there are alternate routes to sustainable development in the South.

The term conceptual simulation refers to a type of everyday reasoning strategy commonly called “what if” reasoning. It has been suggested in a number of contexts that this type of reasoning plays an important role in scientific discovery; however, little direct evidence exists to support this claim. This article proposes that conceptual simulation is likely to be used in situations of informational uncertainty, and may be used to help scientists resolve that uncertainty. We conducted two studies to investigate the relationship (...) between conceptual simulation and informational uncertainty. Study 1 was an in vivo study of expert scientists; the results suggest that scientists do use conceptual simulation in situations of informational uncertainty, and that they use conceptual simulation to make inferences from their data using the analogical reasoning process of alignment by similarity detection. Study 2 experimentally manipulated experts' level of uncertainty and provides further support for the hypothesis that conceptual simulation is more likely to be used in situations of informational uncertainty. Finally, we discuss the relationship between conceptual simulation and other types of reasoning using qualitative mental models. (shrink)