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)

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.

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)

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)

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 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.

This volume will thus appeal to advanced students and scholars not just in the history of early modern philosophy but in epistemology and other core areas of ...

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)

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)

Experimental moral philosophy began to emerge as a methodology inthe last decade of the twentieth century, a branch of the largerexperimental philosophy approach. From the beginning,it has been embroiled in controversy on a number of fronts. Somedoubt that it is philosophy at all. Others acknowledge that it isphilosophy but think that it has produced modest results at best andconfusion at worst. Still others think it represents an important advance., Before the research program can be evaluated, we should have someconception of (...) its scope. But controversy surrounds questionsabout its boundaries as well. Uncontroversially, the distinctionbetween experimental and non-experimental philosophy is not identicalto the distinction between a posteriori and a prioriphilosophy. Experimental evidence is a proper subset of empiricalevidence, which is itself a subset of a posteriorievidence. Can any more be said?, L1: A child raised in a particular linguistic community almostinevitably ends up speaking an idiolect of the local language despitelack of sufficient explicit instruction, lack of extensive negativefeedback for mistakes, and grammatical mistakes by caretakers., character, moral | cognitivism vs. non-cognitivism, moral | emotion | ethics: virtue | intuition | moral anti-realism | moral psychology: empirical approaches | moral realism | moral relativism | well-being, Copyright © 2014 byMark Alfano Don Loeb, View this site from another server:. (shrink)

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

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)

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)

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 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.

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.

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.

A large campaign has sought to destigmatize psychiatric disorders by disseminating the view that they are in fact brain disorders. But when psychiatric disorders are associated with neurobiological correlates, laypeople's attitudes toward patients are harsher, and the prognoses seem poorer. Here, we ask whether these misconceptions could result from the essentialist presumption that brain disorders are innate. To this end, we invited laypeople to reason about psychiatric disorders that are diagnosed by either a brain or a behavioral test that were (...) strictly matched for their informative value. Participants viewed disorders as more likely to be innate and immutable when the diagnosis was supported by a brain test as compared to a behavioral test. These results show for the first time that people spontaneously essentialize psychiatric conditions that are linked to the brain, even when the brain probe offers no additional diagnostic or genetic information. This bias suggests that people consider the biological essence of living things as materially embodied. (shrink)

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 essays in this special issue shed new light on the transnational movement and exchange of researchers, data, theories, and scientific objects in the anthropological and genetic study of human populations in the twentieth century. Historians have long stressed how the study of race and human populations in this period served to create a national identity for emerging nation states. More recently, historical narratives of anthropology and human genetics have emphasized the global scale of research networks in these sciences. This (...) issue explores the specific routes, crossings, and interactions between national and international contexts prompted by the study of races and populations. The essays reveal not only how... (shrink)

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.

Personalised learning systems—systems that predict learning needs to tailor education to the unique learning needs of individual students—are gaining rapid popularity. Praise for educational technology is often focused on how technology will benefit school systems, but there is a lack of understanding of how it will affect the student and the learning process. By uncovering what the meaning of ‘personal’ is in educational philosophy and as embodied in the technology, we illustrate that these two understandings are different regarding the autonomy (...) of the student. Personalised learning technology, therefore, bears the risk of failing to achieve its educational ideal of what personalisation should be. We also illustrate how personalised learning technology effects student autonomy by requiring the intensive tracking of the learning process, exposing them to privacy and data protection risks. We do not claim that education does not need technology, but we want to illustrate the importance of values as drivers of innovation. (shrink)

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)

When Black Feminist Thought by Patricia Hill Collins was published in 1990, reviewers called it "remarkable", "rich and valuable", and proclaimed, "with the publication of this book, Black feminism has moved to a new level". Now, in Fighting Words, Collins expands and extends the discussion of the "outsider within" presented in her earlier work, investigating how effectively Black feminist thought confronts the injustices African American women currently face. Collins takes on a broad range of issues -- poverty, mothering, white supremacy (...) and Afrocentrism, the resegregation of American society by race and class, the ideas of Sojourner Truth and how they can serve as a springboard for more liberating social theory. Contrasting social theories that support unjust power relations of race, class, gender, and nation with those that challenge inequalities, Collins investigates why some ideas are granted the status of "theory" while others remain "thought". "It is not that elites produce theory while everyone else produces mere thought", she writes. "Rather, elites possess the power to legitimate the knowledge that they define as theory as being universal, normative, and ideal". Collins argues that because African American women and other historically oppressed groups seek economic and social justice, their social theories may emphasize themes and work from assumptions that are different from those of mainstream American society, generating new angles of vision on injustice. Collins also puts such oppositional social theory to the test: while the words of these theories may challenge injustice, do the ideas make a difference in the lives of the people they claim to represent? Throughout,Collins provides an essential understanding of how "outsiders" resist mainstream perspectives, and what the mainstream can learn from such "outsiders". Historically situated yet transcending the specific, Fighting Words provides a new interpretive framework for both thinking through and overcoming social injustice. (shrink)