Introduction Patients require an accurate knowledge about placebos and their possible effects to ensure consent for placebo-controlled clinical trials is adequately informed. However, few previous studies have explored patients’ baseline levels of understanding and knowledge about placebos. The present online survey aimed to assess knowledge about placebos among patients with a history of back pain. Design A 15-item questionnaire was constructed to measure knowledge about placebos. Additional questions assessed sociodemographic characteristics, duration and severity of back pain, and previous experience of (...) receiving placebos. Setting Participants recruited from community settings completed the study online. Results 210 participants completed the questionnaire. 86.7% had back pain in the past 6 months, 44.3% currently had back pain. 4.3% had received a placebo intervention as part of a clinical trial and 68.1% had previously read or heard information about placebos. Overall knowledge of placebos was high, with participants on average answering 12.07 of 15 questions about placebos correctly. However, few participants correctly answered questions about the nocebo effect and the impact of the colour of a placebo pill. Conclusions The findings identified key gaps in knowledge about placebos. The lack of understanding of the nocebo effect in particular has implications for the informed consent of trial participants. Research ethics committees and investigators should prioritise amending informed consent procedures to incorporate the fact that participants in the placebo arm might experience adverse side effects. (shrink)

The fact that Gilbert Ryle and J.L. Austin seem to disagree about the ordinary use of words such as ‘voluntary’, ‘involuntary’, ‘voluntarily’, and ‘involuntarily’ has been taken to cast doubt on the methods of ordinary language philosophy. As Benson Mates puts the worry, ‘if agreement about usage cannot be reached within so restricted a sample as the class of Oxford Professors of Philosophy, what are the prospects when the sample is enlarged?’ (Mates, Inquiry 1:161–171, 1958, p. 165). In this chapter, (...) we evaluate Mates’s criticism alongside Ryle’s and Austin’s specific claims about the ordinary use of these words, assessing these claims against actual examples of ordinary use drawn from the British National Corpus (BNC). Our evaluation consists in applying a combination of methods: first aggregating judgments about a large set of samples drawn from the corpus, and then using a clustering algorithm to uncover connections between different types of use. In applying these methods, we show where and to what extent Ryle’s and Austin’s accounts of the use of the target terms are accurate as well as where they miss important aspects of ordinary use, and we demonstrate the usefulness of this new combination of methods. At the heart of our approach is a commitment to the idea that systematically looking at actual uses of expressions is an essential component of any approach to ordinary language philosophy. (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.

This new edition of the classic Contemporary Theories of Knowledge has been significantly updated to include analyses of the recent literature in epistemology.

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)

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)

"The Third Edition of Philosophical Foundations of Adult Education presents seven theoretical approaches to adult education: liberal, progressive, behaviorist, humanist, radical/critical, analytic, and postmodem. The book gives the historical grounding as well as the basic principles for each approach. In this edition each chapter has been revised and brought up to date. The chapter on radical adult education incorporates recent developments in radical education, phenomenology, feminist educational theory, and critical social theory. The book contains an entirely new chapter on postmodem (...) adult education.". (shrink)

"A sequel to Pollock's How to Build a Person, this volume builds upon that theoretical groundwork for the implementation of rationality through artificial ...

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)

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)

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.

Princeton University Press, 1974. This book is out of print, but can be downloaded as a pdf file (5 MB).

5.5. Every topos is linguistic: the equivalence theorem.

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)

(1984). Weather foreasting and the development of meteorological theory at the Paris Observatory, 1853–1878. Annals of Science: Vol. 41, No. 4, pp. 359-382.

IN THEIR WELL-KNOWN PAPER, Kochen and Specker (1967) introduce the concept of partial Boolean algebra (pBa) and show that certain (finitely generated) partial Boolean algebras arising in quantum theory fail to possess morphisms to any Boolean algebra (we call such pBa's intractable in the sequel). In this note we begin by discussing partial..

Traditionally, expressions in formal systems have been regarded as signifying finite inscriptions which are—at least in principle—capable of actually being written out in primitive notation. However, the fact that (first-order) formulas may be identified with natural numbers (via "Gödel numbering") and hence with finite sets makes it no longer necessary to regard formulas as inscriptions, and suggests the possibility of fashioning "languages" some of whose formulas would be naturally identified as infinite sets . A "language" of this kind is called (...) an infinitary language : in this article I discuss those infinitary languages which can be obtained in a straightforward manner from first-order languages by allowing conjunctions, disjunctions and, possibly, quantifier sequences, to be of infinite length. In the course of the discussion it will be seen that, while the expressive power of such languages far exceeds that of their finitary (first-order) counterparts, very few of them possess the "attractive" features (e.g., compactness and completeness) of the latter. Accordingly, the infinitary languages that do in fact possess these features merit special attention. (shrink)

There was a long tradition in philosophy according to which good reasoning had to be deductively valid. However, that tradition began to be questioned in the 1960’s, and is now thoroughly discredited. What caused its downfall was the recognition that many familiar kinds of reasoning are not deductively valid, but clearly confer justification on their conclusions. Here are some simple examples.

This essay is an attempt to sketch the evolution of type theory from its beginnings early in the last century to the present day. Central to the development of the type concept has been its close relationship with set theory to begin with and later its even more intimate relationship with category theory. Since it is effectively impossible to describe these relationships (especially in regard to the latter) with any pretensions to completeness within the space of a comparatively short article, (...) I have elected to offer detailed technical presentations of just a few important instances. (shrink)

In their writings on Africa and colonialism, John and Jean Comaroff have explored some of the fundamental questions of social science, delving into the nature of history and human agency, culture and consciousness, ritual and representation. How are human differences constructed and institutionalized, transformed and (sometimes) resisted? How do local cultures articulate with global forms? How is the power of some people over others built, sustained, eroded, and negated?These essays work toward an "imaginative sociology," demonstrating the techniques by which (...) social science may capture the contexts that human beings construct and inhabit. In the introduction the authors offer their most complete statement to date on historical anthropology. Standing apart from the traditional disciplines of social history and modernist social science, their work is dedicated to discovering how human worlds are made and signified, forgotten and remade. (shrink)

We investigate Hilbert’s varepsilon -calculus in the context of intuitionistic type theories, that is, within certain systems of intuitionistic higher-order logic. We determine the additional deductive strength conferred on an intuitionistic type theory by the adjunction of closed varepsilon -terms. We extend the usual topos semantics for type theories to the varepsilon -operator and prove a completeness theorem. The paper also contains a discussion of the concept of “partially defined‘ varepsilon -term. MSC: 03B15, 03B20, 03G30.

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.

In this book Pollock deals with the subject of probabilistic reasoning, making general philosophical sense of objective probabilities and exploring their ...

then E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map ν be defined on the family of (Kuratowski) finite subsets of the set E, and secondly, the argument will be constructive, i.e., (...) will involve no use of the law of excluded middle. To be precise, we will prove, in constructive (or intuitionistic) set theory3, the following.. (shrink)

Pollock describes an exciting theory of rationality and its partial implementation in OSCAR, a computer system whose descendants will literally be persons.

The years 1850–80 saw great advances in electrical technology in France. Innovations and inventions in this field came mainly but not exclusively from the artisan/craftsman sector, with some contribution from men from a more academic science research environment. By the end of the period inventors were more likely to have undergone some formal training in science, although there were still contributions from those who had not attended courses at the Grandes Écoles or the faculties of science but had benefited from (...) lower level classes. However, from whatever sector the innovations came, they were likely to have been, at some stage, submitted to the Academy for its judgement and approval. Thus the Academy was still playing something of its pre-revolutionary role and having an influence on the pace of electrical innovation in France. (shrink)