We discuss Lorenzen’s consistency proof for ramified type theory without reducibility, published in 1951, in its historical context and highlight Lorenzen’s contribution to the development of modern proof theory, notably by the introduction of the ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-rule.

We give an overview of recent results in ordinal analysis. Therefore, we discuss the different frameworks used in mathematical proof-theory, namely "subsystem of analysis" including "reverse mathematics", "Kripke-Platek set theory", "explicit mathematics", "theories of inductive definitions", "constructive set theory", and "Martin-Löf's type theory".

In Against Purity, Alexis Shotwell proposes a powerful new conception of social movements as custodians for the past and incubators for liberated futures. Against Purity undertakes an analysis that draws on theories of race, disability, gender, and animal ethics as a foundation for an innovative approach to the politics and ethics of responding to systemic problems.

In “The Book of the Crown (_Kitāb al-Iklīl_) of pseudo-Rhazes” Oliver Kahl and Henrietta Sharp Cockrell offer a facsimile edition, with annotated English translation and introductory study, of a unique and highly unusual medieval Arabic medical text.

In this programmatic paper we renew the well-known question “What is a proof?”. Starting from the challenge of the mathematical community by computer assisted theorem provers we discuss in the first part how the experiences from examinations of proofs can help to sharpen the question. In the second part we have a look to the new challenge given by “big proofs”.

This volume reflects the breadth of Alexy's philosophy, identifies new areas of inquiry and offers a new impetus to the discourse theory of law.

Gerhard Gentzen has been described as logic’s lost genius, whom Gödel called a better logician than himself. This work comprises articles by leading proof theorists, attesting to Gentzen’s enduring legacy to mathematical logic and beyond. The contributions range from philosophical reflections and re-evaluations of Gentzen’s original consistency proofs to the most recent developments in proof theory. Gentzen founded modern proof theory. His sequent calculus and natural deduction system beautifully explain the deep symmetries of logic. They underlie modern developments in computer (...) science such as automated theorem proving and type theory. (shrink)

This paper explores the hypothesis that all comparative sentences— nominal, verbal, and adjectival—contain instances of a single morpheme that compositionally introduces degrees. This morpheme, sometimes pronounced much, semantically contributes a structure-preserving map from entities, events, or states, to their measures along various dimensions. A major goal of the paper is to argue that the differences in dimensionality observed across domains are a consequence of what is measured, as opposed to which expression introduces the measurement. The resulting theory has a number (...) of interesting properties. It characterizes the notion of ‘measurement’ uniformly across comparative constructions, in terms of non-trivial structure preservation. It unifies the distinctions between mass/count nouns and atelic/telic verb phrases with that between gradable and non-gradable adjectives. Finally, it affords a uniform characterization of semantically anomalous comparisons across categories. (shrink)

This book reimagines the compositional semantics of comparative sentences using words such as more, as, too, and others. The book's central thesis entails a rejection of a fundamental assumption of degree semantic frameworks: that gradable adjectives like tall lexicalize functions from individuals to degrees, i.e., measure functions. I argue that comparative expressions in English themselves introduce “measure functions”; this is the case whether that morphology targets adjectives, as in *taller* or *more intelligent*; nouns, as in *more coffee*, *more coffees*; verbs, (...) such as *run more*, *jump more*; or expressions of other categories. Furthermore, she suggests that expressions that comfortably and meaningfully appear in the comparative form should be distinguished from those that do not in terms of a general notion of "measurability": a measurable predicate has a domain of application with non-trivial structure. This notion unifies the independently motivated distinctions between, for example, gradable and non-gradable adjectives, mass and count nouns, singular and plural noun phrases, and telic and atelic verb phrases. Based on careful examination of the distribution of dimensions for comparison within the class of measurable predicates, I tie the selection of measure functions to the specific nature and structure of the domain entities targeted for measurement. The book ultimately explores how, precisely, we should understand semantic theories that invoke the "nature" of domain entities: does the theory depend for its explanation on features of metaphysical reality, or something else? Such questions are especially pertinent in light of a growing body of research in cognitive science exploring the understanding and acquisition of comparative sentences. (shrink)

This is a concise, advanced introduction to current philosophical debates about truth. A blend of philosophical and technical material, the book is organized around, but not limited to, the tendency known as deflationism, according to which there is not much to say about the nature of truth. In clear language, Burgess and Burgess cover a wide range of issues, including the nature of truth, the status of truth-value gaps, the relationship between truth and meaning, relativism and pluralism about truth, and (...) semantic paradoxes from Alfred Tarski to Saul Kripke and beyond. Following a brief introduction that reviews the most influential traditional and contemporary theories of truth, short chapters cover Tarski, deflationism, indeterminacy, realism, antirealism, Kripke, and the possible insolubility of semantic paradoxes. The book provides a rich picture of contemporary philosophical theorizing about truth, one that will be essential reading for philosophy students as well as philosophers specializing in other areas. (shrink)

Mario Bunge forcefully argues for Dual Axiomatics, i.e., an axiomatic method applied to natural sciences which explicitly takes into account semantic aspects of the concepts involved in an axiomatization. In this paper we will discuss how dual axiomatics is equally important in mathematics; both historically in Hilbert and Bernays’s conception as well as today in a set-theoretical environment.

In this paper, we discuss the question how Peano’s Arithmetic reached the place it occupies today in Mathematics. We compare Peano’s approach with Dedekind’s account of the subject. Then we highlight the role of Hilbert and Bernays in subsequent developments.

Es ist bekannt, dass Kurt Gödel im „mathematischen Realismus“ seine philosophische Grundposition wiedergegeben sah. Zielsetzung dieses Essays ist es zu zeigen, dass sich auch umgekehrt ein mathematischer Realismus praktisch zwangsläufig als eine Konsequenz aus dem Unvollständigkeitssatz ergibt. Im Weiteren zeigt sich damit eine unüberwindliche Hürde für eine reduktionistische Mathematik.

"Draws on philosophers, political theorists, activists, and poets to explain how unspoken and unspeakable knowledge is important to racial and gender formation; offers a usable conception of implicit understanding"--Provided by publishers.

Reconstructing the philosophical project of William James, Alexis Dianda deploys a concept of experience that avoids both foundationalist epistemology and an account of the subject rooted in immediately given objects of consciousness. In doing so, Dianda rethinks the role of experience as well as the aims and resources of pragmatic philosophy.

Which concepts should we use to think and talk about the world and to do all of the other things that mental and linguistic representation facilitates? This is the guiding question of the field that we call ‘conceptual ethics’. Conceptual ethics is not often discussed as its own systematic branch of normative theory. A case can nevertheless be made that the field is already quite active, with contributions coming in from areas as diverse as fundamental metaphysics and social/political philosophy. In (...) this pair of papers, we try to unify the field, reflecting on its basic nature, structure, and methodology. (shrink)

In seiner Zeit am Institute of Advanced Study wurde Kurt Gödel ein enger Freund von Albert Einstein und hat sich insbesondere mit der Relativitätstheorie beschäftigt. Ein Ergebnis dieser Untersuchung war die Entdeckung des sogenannten Gödel-Universums (Genau genommen handelt es sich um eine ganze Klasse von Universen; die Singularform steht hier als Kollektivum.), einem Modell der Einsteinschen Feldgleichungen der allgemeinen Relativitätstheorie, in dem sich keine absolute Zeit definieren lässt – theoretisch wären in einem solchen Universum Zeitreisen möglich.

During his time at the Institute for Advanced Study, Kurt Gödel became a close friend of Albert Einstein, and in particular studied the theory of relativity. One result of this study was the discovery of the so-called Gödel universe (strictly speaking, it is about a whole class of universes; the singular form stands here as a collective noun.), a model of Einstein’s field equations of general relativity in which no absolute time can be defined. Theoretically, time travel would be possible (...) in such a universe. (shrink)

Which concepts should we use to think and talk about the world and to do all of the other things that mental and linguistic representation facilitates? This is the guiding question of the field that we call ‘conceptual ethics’. Conceptual ethics is not often discussed as its own systematic branch of normative theory. A case can nevertheless be made that the field is already quite active, with contributions coming in from areas as diverse as fundamental metaphysics and social/political philosophy. In (...) this pair of papers, we try to unify the field, reflecting on its basic nature, structure, and methodology. (shrink)

Robert Alexy develops his influential theory of legal reasoning exploring the nature of legal argumentation and its relation to practical reasoning. In doing so he sheds light on fundamental questions of law and rationality, which are as crucial to practising lawyers and law students as they are to scholars of legal theory.

Conceptual engineering and conceptual ethics are branches of philosophy concerned with questions about how to assess and ameliorate our representational devices (such as concepts and words). It's a part of philosophy concerned with questions about which concepts we should use (and why), how concepts can be improved, when concepts should be abandoned, and how proposals for amelioration can be implemented. Central parts of the history of philosophy have engaged with these issues, but the focus of this volume is on applications (...) to work in contemporary philosophy of language and mind, epistemology, gender and race theory, ethics, philosophy of science, and philosophical logic. This is the first volume devoted entirely to conceptual engineering and conceptual ethics. The volume explores the possibilities, benefits, problems, and applications of conceptual engineering and conceptual ethics. It consists of twenty chapters written by leading philosophers. (shrink)

Alexis de Samos impute à des prostituées athéniennes accompagnatrices de Périclès l’installation d’un culte d’Aphrodite à Samos en 439 av. J.-C. Après analyse du passage et de son contexte, il est proposé qu’Alexis a écrit une œuvre nationaliste, partisane et critique de Périclès et d’Aspasie, et que le sanctuaire a été fondé, sans l’intervention de prostituées, après une victoire navale, en remerciement à Aphrodite, déesse de la navigation en mer.Alexis, the Prostitutes and Aphrodite in Samos. According to (...) Alexis of Samos, Athenian prostitutes founded a cult of Aphrodite in Samos in 439 BC. A reappraisal of the information and its context leads to the hypothesis that Alexis wrote his Samian Chronicles criticising Perikles and Aspasia, and, therefore, that prostitutes were not involved in founding the cult, which was established after a naval victory and in honor of Aphrodite, goddess of seafaring. (shrink)

AbstractWhile the literature on oncofertility decision-making was central to the bioethics debate on social egg freezing when the practice emerged in the late 2000s, there has been little discussion juxtaposing the two forms of egg freezing since. This article offers a new perspective on this debate by comparing empirical qualitative data of two previously conducted studies on medical and social egg freezing. We re-analysed the interview data of the two studies and did a thematic analysis combined with interdisciplinary collaborative auditing (...) for empirical ethics projects. Despite their different contexts, major similarities in women’s decision-making and reasoning were found. We developed two main common themes. Firstly, women felt a clear need to plan for future options. Secondly, they manipulated decision-times by postponing definitive decisions and making micro-decisions. The comparison highlights that the passage of time and the preservation of future choice seems to permeate all aspects of the patient experiences in both studies. As a result of considering real-world lived experiences, we suggest that there are many overlaps in women’s reasoning about egg freezing regardless of why they are making a decision to freeze. These overlaps are morally relevant and thus need to be further integrated into the existing arguments that have been canvassed in the flourishing egg freezing and fertility preservation debates across the field, and in policy and practice globally. (shrink)

Many academics and researchers who publish scholarly articles on Plato’s philosophy of education claim that the ultimate educational goal for Plato is simply the acquisition of virtues. While such a claim may not be entirely incorrect, it is nevertheless substantially wanting; for although the acquisition of virtue is no doubt paramount, for Plato it primarily serves as a means to another end. In this paper, I aim to show that, for Plato, the final summit of all educational enterprise is not (...) really to become virtuous but rather to attain the state of becoming like God, and that is, homoiōsis theōi. -/- . (shrink)

At the heart of this book is the age-old question of how law and morality are related. The legal positivist, insisting on the separation of the two, explicates the concept of law independently of morality. The author challenges this view, arguing that there are, first, conceptually necessary connections between law and morality and, second, normative reasons for including moral elements in the concept of law. While the conceptual argument alone is too limited to establish a sufficiently strong connection between law (...) and morality, and the normative argument alone fails to address the nature of law, the two arguments together support a nonpositivistic concept of law, toppling legal positivism qua comprehensive theory of law. (shrink)

Which concepts should we use to think and talk about the world, and to do all of the other things that mental and linguistic representation facilitates? This is the guiding question of the field that we call ‘conceptual ethics’. Conceptual ethics is not often discussed as its own systematic branch of normative theory. A case can nevertheless be made that the field is already quite active, with contributions coming in from areas as diverse as fundamental metaphysics and social/political philosophy. In (...) this pair of papers, we try to unify the field, reflecting on its basic nature, structure, and methodology. (shrink)

In this study, we use a mixed methods research design to investigate how national cultural forces may impede or enhance the positive impact of females’ economic and political empowerment on increasing gender diversity of corporate boards. Using both a longitudinal correlation-based methodology and a configurational approach with fuzzy-set qualitative comparative analysis, we integrate theoretical mechanisms from gender schema and institutional theories to develop a mid-range theory about how female empowerment and national culture shape gender diversity on corporate boards around the (...) world. With our configurational approach, we conceptually and empirically model the complexity that is associated with the simultaneous interdependencies, both complementary and substitutive ones, between female empowerment processes and various cultural dimensions. Our findings contribute unique insights to research focused on board gender diversity as well as provide information for firm decision makers and policymakers about possible solutions for addressing the continuing issue of the underrepresentation of women on corporate boards. (shrink)

Solidarity is one of the central concepts in social theory and has gained much attention due to the multiple challenges that the EU has been facing the last decade and due to the most recent COVID-19 pandemic. Although the debate on the nature and conditions of solidarity has been revitalized, there remains a large variety in how to conceptualize solidarity. In contrast to other approaches, we do not conceive solidarity as normative concept, but as descriptive–analytical one. Therefore, we provide a (...) theory-based definition that is prone to capture the empirical dimensions of solidarity. Accounting for the dynamic and interactive character of solidarity as subject to permanent societal and political renegotiation, we conceptualize solidarity from a discourse perspective and follow a multilevel design breaking down the understandings of solidarity on different levels. This approach contributes to the research of solidarity that is interested to capturing the ‘real world’ dimensions of solidarity. (shrink)

Alexis Tadié | : Dans cet essai, on analyse la relation entre fait et fiction dans le contexte du développement de la forme romanesque. Je suggère que l’on peut approcher la nature du récit en examinant les manières dont les fictions et les histoires fictionnelles en sont venues à constituer des artefacts philosophiques complexes, la question du genre littéraire étant subordonnée à celle de fictionnalité. | : In this essay, I would like to analyse the relationship between fiction and (...) fact within the context of the development of the novel. I suggest that we may approach narratives by looking at the ways in which fiction and fictional stories came to constitute complex philosophical artefacts — the question of genre being dependent on the idea of fictionality. (shrink)

This article provides the proof-theoretic analysis of the transfinitely iterated fixed point theories $\widehat{ID}_\alpha and \widehat{ID}_{<\alpha};$ the exact proof-theoretic ordinals of these systems are presented.

In this paper we consider some of the problems inherent in the attempt to define and circumscribe an exclusively ‘educational love’, as presented by Joris Vlieghe and Piotr Zamojski in a recent paper for this journal. In seeking to move beyond the confusing interpersonal relations involved in student-centred discourses on teaching, the authors aim to articulate an ‘educational love’ that is more oriented towards subject matter than the student subject. In the process, the concept of love itself becomes increasingly abstract (...) and open to interpretation. Further, the attempt to reconcile these elements in an uncomplicated ‘love for the object’, following the example of Alain Badiou's St Paul, constitutes an injustice to the other subjects—the students—involved in the teaching situation. We consider the ways in which such abstractions might sit with student teachers, to demonstrate how and why the intersubjective dimension—with some of the possible emotive messiness that is implied therein—must not fade from view, if we are to avoid conceiving of education as something that can be purified of individual interest, motivation and desire. (shrink)