This article presents a weak system of intuitionistic second-order arithmetic, WKV, a subsystem of the one in S.C. Kleene, R.E. Vesley [The Foundations of Intuitionistic Mathematics: Especially in Relation to Recursive Functions, North-Holland Publishing Company, Amsterdam, 1965]. It is then shown that some statements of real analysis, like a version of the Heine–Borel Theorem, and some statements of logic, e.g. compactness of classical proposition calculus, are equivalent to the Fan Theorem in this system.

In his famous paper Der Wahrheitsbegriff in den formalisierten Sprachen (Polish edition: Nakładem/Prace Towarzystwa Naukowego Warszawskiego, wydzial, III, 1933), Alfred Tarski constructs a materially adequate and formally correct definition of the term “true sentence” for certain kinds of formalised languages. In the case of other formalised languages, he shows that such a construction is impossible but that the term “true sentence” can nevertheless be consistently postulated. In the Postscript that Tarski added to a later version of this paper (Studia Philosophica, (...) 1, 1935), he does not explicitly include limits for the kinds of language for which such a construction is possible. This absence of such limits has been interpreted as an implied claim that such a definition of the term “true sentence” can be constructed for every language. This has far-reaching consequences, not least for the widely held belief that Tarski changed from an universalistic to an anti-universalistic standpoint. We will claim that the consequence of anti-universalism is unwarranted, given that it can be argued that the Postscript is not in conflict with the existence of limits outside of which a definition of “true sentence” cannot be constructed. Moreover, by a discussion of transfinite type theory, we will also be able to accommodate other of the changes made in Tarski’s Postscript within a type-theoretical framework. The awareness of transfinite type theory afforded by this discussion will lead, in turn, to an account of Tarski’s Postscript that shows a gradual change in his logical work, rather than any of the more radical transitions which the Postscript has been claimed to reflect. (shrink)

We shift attention from the development of model theory for demarcated languages to the development of this theory for fragments of a language. Although it is often assumed that model theory for demarcated languages is not compatible with a universalist conception of logic, no one has denied that model theory for fragments of a language can be compatible with that conception. It thus seems unwarranted to ignore the universalist tradition in the search for the origins and development of model theory. (...) This point is illustrated by Carnap’s early semantics and model theory, which he developed within a type theoretical framework and which stand out both for their universalistic treatment and for certain idiosyncratic technicalities by which the construction is supported. One special property is that individuals are context relative in Carnap’s system. This leads to a model theory in which the model domains are more flexible than has been suggested in the literature. (shrink)

G. Schiemer has recently ascribed to Carnap the so-called domains-as-fields conception of models, which he subsequently used to defend Carnap’s treatment of extremal axioms against J. Hintikka’s criticism that the number of tuples in a relation, and not the domain of discourse, is optimised in Carnap’s treatment. We will argue by a careful textual analysis, however, that this domains-as-fields conception cannot be applied to Carnap’s early semantics, because it includes a notion of submodel and subrelation that is not only absent (...) from Carnap’s work at that time, but even contradicts it. As a consequence, Schiemer’s defense of Carnap’s extremal axioms against Hintikka’s criticism fails. We will reconcile Carnap’s treatment of extremal axioms and Hintikka’s observation by taking into account the practice of axiomatics in the early twentieth century. If one realises that, in Carnap’s time, a predicate for the domain of discourse was often introduced in the formal theory, and that Carnap defined such predicates from the basic relations of an axiom system, the apparent disagreement between optimising relations and optimising domains disappears. (shrink)

We give an overview of the role of equicontinuity of sequences of real-valued functions on [0,1] and related notions in classical mathematics, intuitionistic mathematics, Bishop’s constructive mathematics, and Russian recursive mathematics. We then study the logical strength of theorems concerning these notions within the programme of Constructive Reverse Mathematics. It appears that many of these theorems, like a version of Ascoli’s Lemma, are equivalent to fan-theoretic principles.

It is often argued that by assuming the existence of a universal language, one prohibits oneself from conducting semantical investigations. It could thus be thought that Tarski’s stance towards a universal language in his fruitful Wahrheitsbegriff differs essentially from Carnap’s in the latter’s less successful Untersuchungen zur allgemeinen Axiomatik. Yet this is not the case. Rather, these two works differ in whether or not the studied fragments of the universal language are languages themselves, i.e., whether or not they are closed (...) under derivation rules. In Carnap’s case, axiom systems are not closed under derivation rules, which enables him to adopt a substitutional concept of models. His approach is directly rooted in the tradition of formal axiomatics, we argue, and in this contrary to Tarski’s. In comparing these works by Carnap and Tarski, our aim will be to qualify the connection between Tarski’s approach and the tradition of formal axiomatics, which has been overemphasized in the literature. (shrink)

In the 18th and 19th centuries two transitions took place in the development of mathematical analysis: a shift from the geometric approach to the formula-centered approach, followed by a shift from the formula-centered approach to the concept-centered approach. We identify, on the basis of Bolzano's Purely Analytic Proof [Bolzano 1817], the ways in which Bolzano's approach can be said to be concept-centered. Moreover, we conclude that Bolzano's attitude towards the geometric approach on the one hand and the formula-centered approach on (...) the other were of a different nature; the for­mer being one of rejection, the latter of non-participation. Bolzano supports his concept-centered methodology by philosophical views, which were partially shared by mathematicians with a formula-centered approach to analysis. (shrink)

It is shown that—over Bishop's constructive mathematics—the indecomposability of ℝ is equivalent to the statement that all functions from a complete metric space into a metric space are sequentially nondiscontinuous. Furthermore we prove that the indecomposability of ℝ \ {0} is equivalent to the negation of the disjunctive version of Markov's Principle. These results contribute to the programme of Constructive Reverse Mathematics.

In 1970 Vesley proposed a substitute of Kripke's Scheme. In this paper it is shown that —over Bishop's constructive mathematics— the indecomposability of negative dense subsets of ℝ is equivalent to a weakening of Vesley's proposal. This result supports the idea that full Kripke's Scheme might not be necessary for most of intuitionistic mathematics. At the same time it contributes to the programme of Constructive Reverse Mathematics and gives a new answer to a 1997 question of Van Dalen.

The paper [Tarski: Les fondements de la géométrie des corps, Annales de la Société Polonaise de Mathématiques, pp. 29—34, 1929] is in many ways remarkable. We address three historico-philosophical issues that force themselves upon the reader. First we argue that in this paper Tarski did not live up to his own methodological ideals, but displayed instead a much more pragmatic approach. Second we show that Leśniewski's philosophy and systems do not play the significant role that one may be tempted to (...) assign to them at first glance. Especially the role of background logic must be at least partially allocated to Russell's systems of Principia mathematica. This analysis leads us, third, to a threefold distinction of the technical ways in which the domain of discourse comes to be embodied in a theory. Having all of this in place, we discuss why we have to reject the argument in [Gruszczyński and Pietruszczak: Full development of Tarski's Geometry of Solids, The Bulletin of Symbolic Logic, vol. 4, no. 4, pp. 481—540] according to which Tarski has made a certain mistake. (shrink)

The glueing of (sequentially, pointwise, or uniformly) continuous functions that coincide on the intersection of their closed domains is examined in the light of Bishop-style constructive analysis. This requires us to pay attention to the way that the two domains intersect.

Iris Marion Young (1949-2006) was one of the most influential and innovative political theorists of her generation who had a significant impact on a wide range of topics such as democratic theory, feminist theory, and justice. She bridged many longstanding divides among political theorists, engaging in Continental and critical theory, but also insisting on the importance of normative argument: her corpus stands as a testament to the fruitfulness of engaging in both abstract theory and the 'real world' of everyday (...) politics. This volume spans the several decades of her work, illustrating her intellectual development over time through three major areas of innovation: Gender: Maintaining that gender is both conceptually and politically meaningful, Young theorized gender in terms of structures that, in combination, position different people we call "women" in different ways, such that some women have some structures in common, without all women sharing all gendered structures in common. Justice: Young's early writings on a critical theory of justice evolved in her later and posthumously published works where she developed an account of justice that brought together her theorization of structure with her concern to respond to contemporary claims of injustice. The Politics of Difference: Young rejected universal and abstract theories of justice and maintained that justice instead required attending to the experiences of people marked by difference. (shrink)

In a typical classroom interaction, the teacher asks questions, students answer, and the teacher—knowing the answer—evaluates the responses. This structure might cultivate a view of knowledge as objective, uncontested, and immutable. In response to criticisms of such models, teachers increasingly encourage students’ self-expression, communicating the validity of multiple solutions and perspectives. Stressing that there is "no one right answer,” might be important in countering absolutist and encouraging relativist understandings of knowledge. Cultivating more qualified and critical perspectives that seek to use (...) criteria to judge an answer's quality and identify better answers from among a set of "right answers" may be more difficult to achieve, and might explain the prevalence of multiplist and paucity of evaluativist positions among youth. We suggest that teacher training that emphasizes pedagogy apart from subject-matter learning might contribute to practices that encourage individual expression without providing the domain-specific tools or values of evaluating knowledge claims. (shrink)

Moral realism, the view that there are moral facts that are independent of our beliefs about them, has many defenders. But much less has been said about realism concerning other sorts of value. One of these, gastronomic realism is likely to seem implausible on its face. This paper argues, however, that much of the reasoning used to defend moral realism is about as well suited for defending gastronomic realism. Although these considerations do not directly undermine moral realism, they do suggest (...) that the two views should stand or fall together. And they rob moral realists of one ad hominem argument that often emerges in their debate with irrealists, that the irrealists cannot justify their widespread practice of taking their own moral values seriously. 2012 APA, all rights reserved). (shrink)

Critical Concepts for the Creative Humanities is the first book to outline and define the specific and evolving field of the creative humanities and provides the field's nascent bibliography.

The idea of perfection.--On God and Good.--The sovereignty of good over other concepts.

In this classic work of feminist political thought, Iris Marion Young challenges the prevailing reduction of social justice to distributive justice.

Iris Murdoch was one of the great philosophers and novelists of the twentieth century and The Sovereignty of Good is her most important and enduring philosophical work. She argues that philosophy has focused, mistakenly, on what it is right to do rather than good to be and that only by restoring the notion of ‘vision’ to moral thinking can this distortion be corrected. This brilliant work shows why Iris Murdoch remains essential reading: a vivid and uncompromising style, a (...) commitment to forceful argument, and a courage to go against the grain. With a foreword by Mary Midgley. (shrink)

Despite their success in describing and predicting cognitive behavior, the plausibility of so-called ‘rational explanations’ is often contested on the grounds of computational intractability. Several cognitive scientists have argued that such intractability is an orthogonal pseudoproblem, however, since rational explanations account for the ‘why’ of cognition but are agnostic about the ‘how’. Their central premise is that humans do not actually perform the rational calculations posited by their models, but only act as if they do. Whether or not the problem (...) of intractability is solved by recourse to ‘as if’ explanations critically depends, inter alia, on the semantics of the ‘as if’ connective. We examine the five most sensible explications in the literature, and conclude that none of them circumvents the problem. As a result, rational ‘as if’ explanations must obey the minimal computational constraint of tractability. (shrink)

Iris Murdoch once observed: 'philosophy is often a matter of finding occasions on which to say the obvious'. What was obvious to Murdoch, and to all those who read her work, is that Good transcends everything - even God. Throughout her distinguished and prolific writing career, she explored questions of Good and Bad, myth and morality. The framework for Murdoch's questions - and her own conclusions - can be found here.

Iris Murdoch was one of the great philosophers and novelists of the twentieth century and The Sovereignty of Good is her most important and enduring philosophical work. She argues that philosophy has focused, mistakenly, on what it is right to do rather than good to be and that only by restoring the notion of ‘vision’ to moral thinking can this distortion be corrected. This brilliant work shows why Iris Murdoch remains essential reading: a vivid and uncompromising style, a (...) commitment to forceful argument, and a courage to go against the grain. With a foreword by Mary Midgley. (shrink)

Iris Murdoch once observed: 'philosophy is often a matter of finding occasions on which to say the obvious'. What was obvious to Murdoch, and to all those who read her work, is that Good transcends everything - even God. Throughout her distinguished and prolific writing career, she explored questions of Good and Bad, myth and morality. The framework for Murdoch's questions - and her own conclusions - can be found here.

This latest work from one of the world's leading political philosophers will appeal to audiences from a variety of fields, including philosophy, political science, women's studies, ethnic studies, sociology, and communications studies.

In her long-awaited Responsibility for Justice, Young discusses our responsibilities to address "structural" injustices in which we among many are implicated, often by virtue of participating in a market, such as buying goods produced in sweatshops, or participating in booming housing markets that leave many homeless.

The acclaimed author of The Good Apprentice draws on the entire history of philosophy--and particularly on Plato and Kant--to formulate her own model of morality and demonstrate how thoroughly it is bound up with our daily lives. "An utterly absorbing book".--The Wall Street Journal.

A collection of the author's most influential essays and short works includes her critique of existentialism, her two dialogues on art and religion, key texts on the continuing importance of the sublime, the concept of love, and more.

Iris Murdoch's moral philosophy has long influenced contemporary ethics, yet it has not, in general, received the kind of sustained critical attention that it deserves. "Existentialists and Mystics" and "Metaphysics as a Guide to Morals" provide new access to most of Murdoch's philosophical writings and make possible a deeper appreciation of her contribution to current thought. After assessing the recent critical reception of Murdoch's thought, this review places her moral philosophy in the context of contemporary trends in ethics by (...) tracing her influence on the work of Charles Taylor, highlights the distinctive features of her moral philosophy, and suggests future directions for Murdochian ethics. (shrink)

The essay theorizes the responsibilities moral agents may be said to have in relation to global structural social processes that have unjust consequences. How ought moral agents, whether individual or institutional, conceptualize their responsibilities in relation to global injustice? I propose a model of responsibility from social connection as an interpretation of obligations of justice arising from structural social processes. I use the example of justice in transnational processes of production, distribution and marketing of clothing to illustrate operations of structural (...) social processes that extend widely across regions of the world. The social connection model of responsibility says that all agents who contribute by their actions to the structural processes that produce injustice have responsibilities to work to remedy these injustices. I distinguish this model from a more standard model of responsibility, which I call a liability model. I specify five features of the social connection model of responsibility that distinguish it from the liability model: it does not isolate perpetrators; it judges background conditions of action; it is more forward looking than backward looking; its responsibility is essentially shared; and it can be discharged only through collective action. The final section of the essay begins to articulate parameters of reasoning that agents can use for thinking about their own action in relation to structural injustice. a Footnotesa Thanks to David Alexander, Daniel Drezner, David Owen, and Ellen Frankel Paul for comments on an earlier version of this essay. Thanks to David Newstone for research assistance. (shrink)

Feminist social theory and female body experience are the twin themes of Iris Marion Young's twelve outstanding essays written over the past decade and brought together here. Her contributions to social theory raise critical questions about women and citizenship, the relations of capitalism and women's oppression, and the differences between a feminist theory that emphasizes women's difference and one that assumes a gender-neutral humanity. Loosely following a phenomenological method of description, Young's essays on female embodiment discuss female movement, pregnancy, (...) clothing, and the breasted body. In an introduction that situates her work in the context of shifts in feminist theory and politics over the past decade, Young emphasizes the rootedness of her theorizing in a dedicated and seasoned political activism. (shrink)

Experimental moral philosophy emerged as a methodology in the last decade of the twentieth century, as a branch of the larger experimental philosophy (X-Phi) approach. Experimental moral philosophy is the empirical study of moral intuitions, judgments, and behaviors. Like other forms of experimental philosophy, it involves gathering data using experimental methods and using these data to substantiate, undermine, or revise philosophical theories. In this case, the theories in question concern the nature of moral reasoning and judgment; the extent and sources (...) of moral obligations; the nature of a good person and a good life; even the scope and nature of moral theory itself. This entry begins with a brief look at the historical uses of empirical data in moral theory and goes on to ask what, if anything, is distinctive about experimental moral philosophy—how should we distinguish it from related work in empirical moral psychology? After discussing some strategies for answering this question, the entry examines two of the main projects within experimental moral philosophy, and then discusses some of the most prominent areas of research within the field. As we will see, in some cases experimental moral philosophy has opened up new avenues of investigation, while in other cases it has influenced longstanding debates within moral theory. (shrink)

In this essay I follow one argument strand from Linda Singer's Erotic Welfare. How can we have a forward-looking and affirmative ideal of sexual freedom when the AIDS panic has altered the sexual landscape and instigated new justifications for oppressive sexual disciplines? How can we be sexual subjects when processes of commodification and disciplinary practices have constrained sexual expression while proliferating sexual fetishes? These are some of the questions this book formulates, without answering.

Written over a span of more than two decades, the essays by Iris Marion Young collected in this volume describe diverse aspects of women's lived body experience in modern Western societies. Drawing on the ideas of several twentieth century continental philosophers--including Simone de Beauvoir, Martin Heidegger, Luce Irigaray, Julia Kristeva, and Maurice Merleau-Ponty--Young constructs rigorous analytic categories for interpreting embodied subjectivity. The essays combine theoretical description of experience with normative evaluation of the unjust constraints on their freedom and opportunity (...) that continue to burden many women. The lead essay rethinks the purpose of the category of "gender" for feminist theory, after important debates have questioned its usefulness. Other essays include reflection on the meaning of being at home and the need for privacy in old age residences as well as essays that analyze aspects of the experience of women and girls that have received little attention even in feminist theory--such as the sexuality of breasts, or menstruation as punctuation in a woman's life story. Young describes the phenomenology of moving in a pregnant body and the tactile pleasures of clothing. While academically rigorous, the essays are also written with engaging style, incorporating vivid imagery and autobiographical narrative. On Female Body Experience raises issues and takes positions that speak to scholars and students in philosophy, sociology, geography, medicine, nursing, and education. (shrink)

Many cognitive scientists, having discovered that some computational-level characterization f of a cognitive capacity φ is intractable, invoke heuristics as algorithmic-level explanations of how cognizers compute f. We argue that such explanations are actually dysfunctional, and rebut five possible objections. We then propose computational-level theory revision as a principled and workable alternative.

This paper calls attention to a philosophical presupposition, coined here the continuity thesis which underlies and unites the different, often conflicting, hypotheses in the origin of life field. This presupposition, a necessary condition for any scientific investigation of the origin of life problem, has two components. First, it contends that there is no unbridgeable gap between inorganic matter and life. Second, it regards the emergence of life as a highly probable process. Examining several current origin-of-life theories. I indicate the implicit (...) or explicit role played by the continuity thesis in each of them. In addition, I identify the rivals of the thesis within the scientific community — the almost miracle camp. Though adopting the anti-vitalistic aspect of the continuity thesis, this camp regards the emergence of life as involving highly improbable events. Since it seems that the chemistry of the prebiotic stages and of molecular self-organization processes rules out the possibility that life is the result of a happy accident, I claim that the almost miracle view implies in fact, a creationist position. (shrink)

Some philosophical positions maintain that some aspect of reality depends on human practices, cognitive attitudes or sentiments. This paper presents a framework for understanding such positions in a way that renders them immune to a number of natural but allegedly devastating objections.