In this book the authors reconcile two different viewpoints of the foundations of mathematics, namely mathematical logic and category theory. In Part I, they show that typed lambda-calculi, a formulation of higher order logic, and cartesian closed categories are essentially the same. In Part II, it is demonstrated that another formulation of higher order logic is closely related to topos theory. Part III is devoted to recursive functions. Numerous applications of the close relationship between traditional logic and the algebraic language (...) of category theory are given. The authors have included an introduction to category theory and develop the necessary logic as required, making the book essentially self-contained. Detailed historical references are provided throughout, and each section concludes with a set of exercises. Thus it is well-suited for graduate courses and research in mathematics and logic. Researchers in theoretical computer science, artificial intelligence and mathematical linguistics will also find this an accessible introduction to a subject of increasing application to these disciplines. (shrink)

In this paper we consider the relations existing between four deductive systems that have been called categorial grammars and have relevant connections with linguistic investigations: the syntactic calculus, bilinear logic, compact bilinear logic and Curry''s semantic calculus.

One way of coping with agreement of features in French is to perform two parallel computations, one in the free pregroup of syntactic types, the other in that of feature types. Technically speaking, this amounts to working in the direct product of two free pregroups.

Grammar can be formulated as a kind of substructural propositional logic. In support of this claim, we survey bare Gentzen style deductive systems and two kinds of non-commutative linear logic: intuitionistic and compact bilinear logic. We also glance at their categorical refinements.

We study the monoid of primitive recursive functions and investigate a onestep construction of a kind of exact completion, which resembles that of the familiar category of modest sets, except that the partial equivalence relations which serve as objects are recursively enumerable. As usual, these constructions involve the splitting of symmetric idempotents.

This article was written jointly by a philosopher and a mathematician. It has two aims: to acquaint mathematicians with some of the philosophical questions at the foundations of their subject and to familiarize philosophers with some of the answers to these questions which have recently been obtained by mathematicians. In particular, we argue that, if these recent findings are borne in mind, four different basic philosophical positions, logicism, formalism, platonism and intuitionism, if stated with some moderation, are in fact reconcilable, (...) although with some reservations in the case of logicism, provided one adopts a nominalistic interpretation of Plato's ideal objects. This eclectic view has been asserted by Lambek and Scott (LS 1986) on fairly technical grounds, but the present argument is meant to be accessible to a wider audience and to provide some new insights. (shrink)

This book is dedicated to the life and work of the mathematician Joachim Lambek. The editors gather together noted experts to discuss the state of the art of various of Lambek’s works in logic, category theory, and linguistics and to celebrate his contributions to those areas over the course of his multifaceted career. After early work in combinatorics and elementary number theory, Lambek became a distinguished algebraist. In the 1960s, he began to work in category theory, (...) categorical algebra, logic, proof theory, and foundations of computability. In a parallel development, beginning in the late 1950s and for the rest of his career, Lambek also worked extensively in mathematical linguistics and computational approaches to natural languages. He and his collaborators perfected production and type grammars for numerous natural languages. Lambek grammars form an early noncommutative precursor to Girard’s linear logic. In a surprising development, he introduced a novel and deeper algebraic framework for analyzing natural language, along with algebraic, higher category, and proof-theoretic semantics. This book is of interest to mathematicians, logicians, linguists, and computer scientists. (shrink)

This work is devoted to the relations between Lambek’s Syntactic Calculus and noncommutative variants of Girard’s Linear Logic; in particular the paper will consider: the geometrical representation of the laws of LC by means of proof-nets; the discovery - due to such a geometrical representation - of some laws of LC not yet considered; the discussion of possible linguistic uses of these new laws.

Bringing together ethnographic exposition with philosophical concepts and arguments and effectively transcending subdisciplinary boundaries between cultural and ...

This book suggests a bold comparative approach to broad cultural differences between Africa and Melanesia. Its theme is personhood, understood in terms of what anthropologists call embodiment. These concepts are applied to questions ranging from the meanings of spirit possession, to the logics of witchcraft and kinship relations, the use of rituals in healing, and even the impact of capitalism. Questioning common assumptions about the huge differences among these discrete areas, the contributions document surprising continuities.

The Tanner Lectures are a collection of educational and scientific discussions relating to human values. Conducted by leaders in their fields, the lectures are presented at renowned institutions around the world, including the Universities of Oxford, Harvard, and Yale. In January 2019, University of Toronto's Michael Lambek, professor, former Canada Research Chair, and member of the Royal Society of Canada, delivered the Tanner Lecture at the University of Michigan's Department of Philosophy on the topic of "Concepts and Persons." As (...) well as tracing his career in social and cultural anthropology, Lambek's Tanner Lecture spoke on the intersection of anthropology and philosophy as a means of articulating the moral basis of human action. By elucidating where anthropology and philosophy might intersect, Lamberk's lecture is a profound examination of the human condition, and is beautifully captured in this publication. Concepts and Persons recounts the lecture as delivered at the prestigious event, the commentary of three distinguished respondents, and Lambek's own response to that commentary. The book's presentation of the lecture also includes a rich and layered set of notes that augment the lecture significantly, as well as additional clarification and thought that has developed since the event. (shrink)

'Philosophische Anthropologie' meint in dieser Studie nicht eine philosophische Subdisziplin, sondern eine besondere Theorierichtung in der deutschsprachigen Philosophie des 20. Jahrhunderts, die mit den Namen Max Scheler, Helmuth Plessner, Erich Rothacker, Arnold Gehlen, Adolf Portmann u.v.a. mehr verbunden ist. Der erste Teil erzählt die verwickelte, teils abenteuerliche Entstehungs-, Aufstiegs- und Entfaltungsgeschichte dieser Denkergruppe von 1919 bis 1975 - einschließlich ihrer beachtlichen Wirkungsgeschichte in verschiedenen Disziplinen wie der Soziologie, Psychologie, Biologie und der Philosophie selbst. Im zweiten Teil wird der philosophische Identitätskern (...) dieser Philosophischen Anthropologie als Denkansatz präzisiert. (shrink)

Anthropology has recently seen a lively interest in the subject of ethics and comparative notions of morality and freedom. This masterclass brings together four of the most eminent anthropologists working in this field--Michael Lambek, Veena Das, Didier Fassin, and Webb Keane--to discuss, via lectures and responses, important topics facing anthropological ethics and the theoretical debates that surround it. The authors explore the ways we understand morality across many different cultural settings, asking questions such as: How do we recognize the (...) ethical in different ethnographic worlds? What constitutes agency and awareness in everyday life? What might an anthropology of ordinary ethics look like? And what happens when ethics approaches the political in both Western and non-Western societies. Contrasting perspectives and methods- and yet in complimentary ways- this masterclass will serve as an essential guide for how an anthropology of ethics can be formulated in the twenty-first century. (shrink)

Synthetic biology is an emerging technology that aims to design and engineer DNA and molecular structures of single cell organisms. Existing organisms can be altered, novel organisms can be created. In doing so, synthetic biology makes use of specific technoscientific understandings of living beings. This volume sets out to explore and assess synthetic biology and its notions of life from philosophical, ethical, social, and legal perspectives. Contents Concepts, Metaphors, Worldviews.- Public Good and Private Ownership.Social and Legal Ramifications.- Opportunities, Risks, Governance. (...) Target Groups Scholars of sociology, philosophy, theory of science, history of science, biology, engineering Politicians and policymakers The Editor Dr. Joachim Boldt is Deputy Director at the Institut für Ethik und Geschichte der Medizin in Freiburg, Germany. (shrink)

'Philosophische Anthropologie' meint in dieser Studie nicht eine philosophische Subdisziplin, sondern eine besondere Theorierichtung in der deutschsprachigen Philosophie des 20. Jahrhunderts, die mit den Namen Max Scheler, Helmuth Plessner, Erich Rothacker, Arnold Gehlen, Adolf Portmann u.v.a. mehr verbunden ist. Der erste Teil erzählt die verwickelte, teils abenteuerliche Entstehungs-, Aufstiegs- und Entfaltungsgeschichte dieser Denkergruppe von 1919 bis 1975 - einschließlich ihrer beachtlichen Wirkungsgeschichte in verschiedenen Disziplinen wie der Soziologie, Psychologie, Biologie und der Philosophie selbst. Im zweiten Teil wird der philosophische Identitätskern (...) dieser Philosophischen Anthropologie als Denkansatz präzisiert. (shrink)

The Historical Dictionary of Philosophy, the _Historisches Wörterbuch der Philosophie,_ is distinguished by its particular presentation of philosophical terms, ideas and concepts. Rather than providing mere defintions or descriptive and analytical explanantions the _HWPh_ strictly applies the critical method of history of concepts developed by the eminent German scholar and philosopher Joachim Ritter. By means of precise and detailed references it documents the origin, first occurrence, the historical evolution and the changes of meaning of each concept, from Ancient Greek (...) to contemporary philosophy. For the reader this presentation is of unique value: it makes traceable the importance of terms and concepts at certain periods or for a particular philosopher, as well as its changes and development of meaning. Voulmes 1–12 of the _HWPh_ comprises more than 17.000 text columns on 3.670 philosophical terms. The dictionary does not include articles on persons. Volume 13 includes, besides of a comprehensive introductory essay, three different indexes: - the Subject Index classifies the articles by disciplinary and systematic categories - the Main Index includes all philosophical keywords with more than 30.000 references to their occurrences in other articles and contexts - the Author Index lists all authors and their contributions The included CD-ROM allows full text research of the _HWPh's_ entire content. (shrink)

Written over a thirty-year span, Michael Lambek’s essays in this collection point with definitive force toward a single central truth: ethics is intrinsic to social life. As he shows through rich ethnographic accounts and multiple theoretical traditions, our human condition is at heart an ethical one—we may not always be good or just, but we are always subject to their criteria. Detailing Lambek’s trajectory as one anthropologist thinking deeply throughout a career on the nature of ethical life, the (...) essays accumulate into a vibrant demonstration of the relevance of ethics as a practice and its crucial importance to ethnography, social theory, and philosophy. Organized chronologically, the essays begin among Malagasy speakers on the island of Mayotte and in northwest Madagascar. Building from ethnographic accounts there, they synthesize Aristotelian notions of practical judgment and virtuous action with Wittgensteinian notions of the ordinariness of ethical life and the importance of language, everyday speech, and ritual in order to understand how ethics are lived. They illustrate the multiple ways in which ethics informs personhood, character, and practice; explore the centrality of judgment, action, and irony to ethical life; and consider the relation of virtue to value. The result is a fully fleshed-out picture of ethics as a deeply rooted aspect of the human experience. (shrink)

This chapter contains sections titled: Introduction What is Chemistry about? Is Chemistry Reducible to Physics? Are There Fundamental Limits to Chemical Knowledge? Is Chemical Research Ethically Neutral? Conclusion References.

What are consequential world problems? As “grand societal challenges”, one might define them as problems that affect a large number of people, perhaps even the entire planet, including problems such as climate change, distributive justice, world peace, world nutrition, clean air and clean water, access to education, and many more. The “Sustainable Development Goals”, compiled by the United Nations, represent a collection of such global problems. From my point of view, these problems can be seen as complex. Such complex problems (...) are characterized by the complexity, connectivity, dynamics, intransparency, and polytely of their underlying systems. These attributes require special competencies for dealing with the uncertainties of the given domains, e.g., critical thinking. My position is that it is not IQ, but complex problem-solving competencies for dealing with complex and dynamic situations, that is important for handling consequential global problems. These problems require system competencies, i.e., competencies that go beyond analytical intelligence, and comprise systems understanding as well as systems control. Complex problem-solving is more than analytic intelligence. (shrink)

The discussion in Perspectives on Psychological Science about criteria for scholarly merit shows a potential bias of quantitative measurements compared with informed judgments of scholarly merits. This comment argues for a selection procedure that is open for qualitative arguments.

Gottesbeweise gehören zu den großen Themen der abendländischen Philosophie. Im 20. Jahrhundert sind sie mit Hilfe der modernen Logik neu formuliert worden und auch in der analytischen Philosophie werden Gottesbeweise seit Jahrzehnten kontrovers diskutiert. Offenkundig ist die Frage nach der Existenz Gottes im nachmetaphysischen Zeitalter aktueller denn je. Der Band versammelt die großen Gottesbeweise des Mittelalters und der Neuzeit ebenso wie die klassischen Einwände von Hume und Kant. Die sprachanalytische Debatte wird ausführlich dokumentiert und ein eigener Teil ist Kurt Gödel (...) gewidmet, dessen ontologischer Beweis bis heute nicht stichhaltig widerlegt worden ist. Umfassende einleitende Essays führen in die Problematik der Gottesbeweise ein und bieten gut verständliche Rekonstruktionen der jeweiligen Beweisversuche. (shrink)

This article presents and discusses different kinds of data documents, including data sets, data studies, data papers and data journals. It provides descriptive and bibliometric data on different kinds of data documents and discusses the theoretical and philosophical problems by classifying documents according to the DIKW model (data documents, information documents, knowl­edge documents and wisdom documents). Data documents are, on the one hand, an established category today, even with its own data citation index (DCI). On the other hand, data documents (...) have blurred boundaries in relation to other kinds of documents and seem sometimes to be understood from the problematic philosophical assumption that a datum can be understood as “a single, fixed truth, valid for everyone, everywhere, at all times”. (shrink)

It is now—at least loosely—acknowledged that most health and clinical outcomes are influenced by different interacting causes. Surprisingly, medical research studies are nearly universally designed to study—usually in a binary way—the effect of a single cause. Recent experiences during the coronavirus disease 2019 pandemic brought to the forefront that most of our challenges in medicine and healthcare deal with systemic, that is, interdependent and interconnected problems. Understanding these problems defy simplistic dichotomous research methodologies. These insights demand a shift in our (...) thinking from ‘cause and effect’ to ‘causes and effects’ since this transcends the classical way of Cartesian reductionist thinking. We require a shift to a ‘causes and effects’ frame so we can choose the research methodology that reflects the relationships between variables of interest—one-to-one, one-to-many, many-to-one or many-to-many. One-to-one (or cause and effect) relationships are amenable to the traditional randomized control trial design, while all others require systemic designs to understand ‘causes and effects’. Researchers urgently need to re-evaluate their science models and embrace research designs that allow an exploration of the clinically obvious multiple ‘causes and effects’ on health and disease. Clinical examples highlight the application of various systemic research methodologies and demonstrate how ‘causes and effects’ explain the heterogeneity of clinical outcomes. This shift in scientific thinking will allow us to find the necessary personalized or precise clinical interventions that address the underlying reasons for the variability of clinical outcomes and will contribute to greater health equity. (shrink)

Algorithmic predictions are promising for insurance companies to develop personalized risk models for determining premiums. In this context, issues of fairness, discrimination, and social injustice might arise: Algorithms for estimating the risk based on personal data may be biased towards specific social groups, leading to systematic disadvantages for those groups. Personalized premiums may thus lead to discrimination and social injustice. It is well known from many application fields that such biases occur frequently and naturally when prediction models are applied to (...) people unless special efforts are made to avoid them. Insurance is no exception. In this paper, we provide a thorough analysis of algorithmic fairness in the case of insurance premiums. We ask what “fairness” might mean in this context and how the fairness of a premium system can be measured. For this, we apply the established fairness frameworks of the fair machine learning literature to the case of insurance premiums and show which of the existing fairness criteria can be applied to assess the fairness of insurance premiums. We argue that two of the often-discussed group fairness criteria, independence (also called statistical parity or demographic parity) and separation (also known as equalized odds), are not normatively appropriate for insurance premiums. Instead, we propose the sufficiency criterion (also known as well-calibration) as a morally defensible alternative that allows us to test for systematic biases in premiums towards certain groups based on the risk they bring to the pool. In addition, we clarify the connection between group fairness and different degrees of personalization. Our findings enable insurers to assess the fairness properties of their risk models, helping them avoid reputation damage resulting from potentially unfair and discriminatory premium systems. (shrink)

Prolegomena to Pure Logic (1900) is the definitive statement of Husserl’s early logic. But what does it say that logic is? I argue that Husserl in the Prolegomena thinks logic is its own discipline, namely the “doctrine of science” (Wissenschaftslehre), but has two conflicting ideas of what that is. One idea—expressed by the book’s general argument, and which I call Husserl’s Austrian Semanticism about logic—is that the Wissenschaftslehre is the positive science explaining what science is (which turns out just to (...) be the study of meaning) plus the dependent art that, applying the science, teaches us how to scientifically know. The other idea—expressed by the book’s opening chapter, and which I call Husserl’s German Idealism about logic—is that the Wissenschaftslehre is the purely reflective self-knowing of science, independent of science’s positive expansion. These two ideas are incompatible. Thus, the Prolegomena is ambivalent on what logic is. But since the ambivalence only deepens the significance of Husserl’s early logic, the ambivalence should be embraced. (shrink)

National governments have failed spectacularly to mitigate anthropogenic climate change and a sustainable approach to mitigation remains out of sight. This circumstance alone demonstrates t...

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