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)

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)

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.

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.

This chapter contains section titled: Some Context The Traditional Interpretation Disarming the Evidence for the Traditional Interpretation Evidence that Hume Considers Inductive Inference Justified The Traditional Interpretation Revisited Hume's Epistemic Options Applications to Extended Objects and Belief in God Limitations on Enumerative Induction Acknowledgments References.

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)

The distinguished philosopher Louis Loeb examines the epistemological framework of Scottish philosopher David Hume, as employed in his celebrated work A Treatise of Human Nature. Loeb's project is to advance an integrated interpretation of Hume's accounts of belief and justification. His thesis is that Hume, in his Treatise, has a "stability-based" theory of justification which posits that his belief is justified if it is the result of a belief producing mechanism that engenders stable beliefs. But Loeb argues (...) that the striking corollary to this theory is that no belief generating mechanism is fully stable - or fully justified - for a fully reflective person. This carefully argued and original interpretation of Hume makes sense of seemingly contradictory ideas and will provoke serious discussion among Hume scholars. (shrink)

This chapter contains sections titled: Recurrence‐Awareness Recurrence‐Evidence Recurrence‐Significance Recurrence‐Time Recurrence‐Coherence Conclusion.

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 ...

Since the mid-1970s, scholars have recognized that the skeptical interpretation of Hume's central argument about induction is problematic. The science of human nature presupposes that inductive inference is justified and there are endorsements of induction throughout "Treatise" Book I. The recent suggestion that I.iii.6 is confined to the psychology of inductive inference cannot account for the epistemic flavor of its claims that neither a genuine demonstration nor a non-question-begging inductive argument can establish the uniformity principle. For Hume, that inductive inference (...) is justified is part of the data to be explained. Bad argument is therefore excluded as the cause of inductive inference; and there is no good argument to cause it. Does this reinstate the problem of induction, undermining Hume's own assumption that induction is justified? It does so only if justification must derive from "reason", from the availability of a cogent argument. Hume rejects this internalist thesis; induction's favorable epistemic status derives from features of custom, the mechanism that generates inductive beliefs. Hume is attracted to this externalist posture because it provides a direct explanation of the epistemic achievements of children and non-human animals — creatures that must rely on custom unsupplemented by argument. (shrink)

Do newborns think-do they know that 'three' is greater than 'two'? Do they prefer 'right' to 'wrong'? What about emotions--do newborns recognize happiness or anger? If they do, then how are our inborn thoughts and feelings encoded in our bodies? Could they persist after we die? Going all the way back to ancient Greece, human nature and the mind-body link are the topics of age-old scholarly debates. But laypeople also have strong opinions about such matters. Most people believe, for example, (...) that newborn babies don't know the difference between right and wrong-such knowledge, they insist, can only be learned. For emotions, they presume the opposite-that our capacity to feel fear, for example, is both inborn and embodied. These beliefs are stories we tell ourselves about what we know and who we are. They reflect and influence our understanding of ourselves and others and they guide every aspect of our lives. In a twist that could have come out of a Greek tragedy, Berent proposes that our errors are our fate. These mistakes emanate from the very principles that make our minds tick: our blindness to human nature is rooted in human nature itself. An intellectual journey that draws on philosophy, anthropology, linguistics, cognitive science, and Berent's own cutting-edge research, The Blind Storyteller grapples with a host of provocative questions, from why we are so infatuated with our brains to what happens when we die. The end result is a startling new perspective on our humanity. (shrink)

In his Treatise, Hume confronted the tensions between his project of uncovering the causal operations of the human mind and the extreme skeptical tendencies of his system. Louis Loeb argues that Hume overreaches, and he advances a controversial interpretation of Hume's epistemological framework that shows how Hume could have avoided the more destructive positions in his work.

In seinen Analysen des Bildbewusstseins begreift Edmund Husserl das Bild als eine Herausforderung phänomenologischen Fragens. Diese Herausforderung begründet sich durch seine spezifische Gegenstands- und Erfahrungsstruktur, die das Bild als ein besonderes unter den Phänomenen auszeichnet, als ein Phänomen, das einen Unterschied zwischen wirklicher und vorgestellter Welt einführt und hierin Differenzbewusstsein konstituiert. Vor allem mit Blick auf das Thema der Zeit erweist sich die Auseinandersetzung mit der differentiellen Natur des Bildes als folgenreich. Iris Laners systematische Studie widmet sich der Aufgabe, (...) den heuristischen Wert der philosophischen Auseinandersetzung mit dem Bild für die Beantwortung der komplexen Frage nach der Zeit in und nach der Phänomenologie zu bestimmen. Damit unterstreicht sie nicht nur die Bedeutung des Bildes als Phänomen, sondern erörtert dieses auch als grundlegendes Konzept des Denkens. (shrink)

Hume's claim that a state is a belief is often intertwined—though without his remarking on this fact—with epistemic approval of the state. This requires explanation. Beliefs, in Hume's view, are steady dispositions (not lively ideas), nature's provision for a steady influence on the will and action. Hume's epistemic distinctions call attention to circumstances in which the presence of conflicting beliefs undermine a belief's influence and thereby its natural function. On one version of this interpretation, to say that a belief is (...) justified, ceteris paribus, is to say that for all that has been shown the belief would be steady in its influence under suitable reflection. On a second version, it is to say that prima facie justification is an intrinsic property of the state, in virtue of its steadiness. These versions generate different understandings of the relationship between Parts iii and iv of Book I of the Treatise. (shrink)

Hume's claim that a state is a belief is often intertwined—though without his remarking on this fact—with epistemic approval of the state. This requires explanation. Beliefs, in Hume's view, are steady dispositions (not lively ideas), nature's provision for a steady influence on the will and action. Hume's epistemic distinctions call attention to circumstances in which the presence of conflicting beliefs undermine a belief's influence and thereby its natural function. On one version of this interpretation, to say that a belief is (...) justified, ceteris paribus, is to say that for all that has been shown the belief would be steady in its influence under suitable reflection. On a second version, it is to say that prima facie justification is an intrinsic property of the state, in virtue of its steadiness. These versions generate different understandings of the relationship between Parts iii and iv of Book I of the Treatise. (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)

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

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

I feel there are demons around.' Everyone is thinking about Julius King. For comfortable, long-married Hilda and Rupert, he is a mystery. For Morgan, Hilda's tormented sister, he is an obsession. For Morgan's abandoned husband, Tallis, he is the source of ruin. For Simon and Axel, deeply in love, he stirs up jealousy and unease. What is Julius thinking about? He's thinking about Hilda, Rupert, Morgan, Tallis, Simon and Axel, and they will not all survive his malevolent attention. WITH AN (...) INTRODUCTION BY GARTH GREENWELL VINTAGE CLASSICS MURDOCH: Funny, subversive, fearless and fiercely intelligent, Iris Murdoch was one of the great writers of the twentieth century. To celebrate her centenary Vintage Classics presents special editions of her greatest and most timeless novels. (shrink)

The decline of religion and ever increasing influence of science pose acute ethical issues for us all. Can we reject the literal truth of the Gospels yet still retain a Christian morality? Can we defend any 'moral values' against the constant encroachments of technology? Indeed, are we in danger of losing most of the qualities which make us truly human? Here, drawing on a novelist's insight into art, literature and abnormal psychology, Iris Murdoch conducts an ongoing debate with major (...) writers, thinkers and theologians—from Augustine to Wittgenstein, Shakespeare to Sartre, Plato to Derrida—to provide fresh and compelling answers to these crucial questions. (shrink)

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)

Muḥammad Mahdī ibn Abī Z̲arr Narāqī, -1794 or 1795. Jāmiʻ al-saʻādāt - Criticism and interpretation; Islamic ethics.

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.

The goal of discourse-based decision-making is to free negotiations from self-interested, strategic interactions. In this paper it is argued that the absence of interests may lead to both efficiency losses and redistribution between participants of the discourse and outsiders. The latter effect is the stronger, the more personal relationships between discourse participants become. Therefore, it is essential to validate the recommendations arrived at in discourses in democratic decision-making processes that are driven by competition between different points of views and interests.

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)

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)

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.

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.