The year 1897 saw the publication of the first of the modern logical paradoxes. It was published by Cesare Burali-Forti, the Italian mathematician whose name it has come to bear. Burali-Forti's own formulation of the paradox was not altogether satisfactory, as he had confused well-ordered sets as defined by Cantor with what he himself called “perfectly ordered sets”. However, he soon realized his mistake, and published a note admitting the error and making the correction. He (...) concluded the note with the observation that his result could be established on the basis of the correct definition of well-ordered set as easily as for the “perfectly ordered sets” for which it had first been obtained. We shall reproduce his results in their corrected form. (shrink)

Russell’s paradox is purely logical in the following sense: a contradiction can be formally deduced from the proposition that there is a set of all non-self-membered sets, in pure first-order logic—the first-order logical form of this proposition is inconsistent. This explains why Russell’s paradox is portable—why versions of the paradox arise in contexts unrelated to set theory, from propositions with the same logical form as the claim that there is a set of all non-self-membered sets. Burali-Forti’s paradox, like (...) Russell’s paradox, is portable. I offer the following explanation for this fact: Burali-Forti’s paradox, like Russell’s, is purely logical. Concretely, I show that if we enrich the language \ of first-order logic with a well-foundedness quantifier W and adopt certain minimal inference rules for this quantifier, then a contradiction can be formally deduced from the proposition that there is a greatest ordinal. Moreover, a proposition with the same logical form as the claim that there is a greatest ordinal can be found at the heart of several other paradoxes that resemble Burali-Forti’s. The reductio of Burali-Forti can be repeated verbatim to establish the inconsistency of these other propositions. Hence, the portability of the Burali-Forti’s paradox is explained in the same way as the portability of Russell’s: both paradoxes involve an inconsistent logical form—Russell’s involves an inconsistent form expressible in \ and Burali-Forti’s involves an inconsistent form expressible in \. (shrink)

In studying the early history of mathematical logic and set theory one typically reads that Georg Cantor discovered the so-called Burali-Forti (BF) paradox sometime in 1895, and that he offered his solution to it in his famous 1899 letter to Dedekind. This account, however, leaves it something of a mystery why Cantor never discussed the paradox in his writings. Far from regarding the foundations of set theory to be shaken, he showed no apparent concern over the paradox and (...) its implications whatever. Against this account, I will argue here that in fact Cantor never saw any paradox at all, but that his conception of set at that time, and already as far back as 1883, was one in which the paradoxes cannot arise. (shrink)

Philip Jourdain put this question to Frege in a letter of 28 January 1909. Frege had, indeed, next to nothing to say about ordinals, and in this respect Bob Hale has followed the master. As I hope this chapter will show, though, the topic is worth addressing. The natural abstraction principle for ordinals combines with full, impredicative second-order logic to engender a contradiction, the so-called Burali-Forti Paradox. I shall contend that the best solution involves a retreat to a (...) predicative logic. Such a retreat has implications for other neo-Fregean theories, including the cardinal arithmetic on which Hale has focused. The discussion will touch on a topic which has been at the centre of Hale’s more recent work—namely, the interpretation of plural and higher-order quantifiers. (shrink)

After briefly reviewing the standard set-theoretic resolutions of the Burali-Forti paradox, we examine how the paradox arises in set theory formalized with plural quantifiers. A significant choice emerges between the desirable unrestricted availability of ordinals to represent well-orderings and the sensibility of attempting to refer to ‘absolutely all ordinals’ or ‘absolutely all well-orderings’. This choice is obscured by standard set theories, which rely on type distinctions which are obliterated in the setting with plurals. Zermelo's attempt ( 1930 ) (...) to secure ordinal representability of arbitrary well-orderings through relativization of quantification to set-theoretic models is reviewed and found wanting. The natural modal-structural recasting provides, it is claimed, a good repair. (shrink)

The paradox that appears under Burali-Forti’s name in many textbooks of set theory is a clever piece of reasoning leading to an unproblematic theorem. The theorem asserts that the ordinals do not form a set. For such a set would be—absurdly—an ordinal greater than any ordinal in the set of all ordinals. In this article, we argue that the paradox of Burali-Forti is first and foremost a problem about concept formation by abstraction, not about sets. We (...) contend, furthermore, that some hundred years after its discovery the paradox is still without any fully satisfactory resolution. A survey of the current literature reveals one key assumption of the paradox that has gone unquestioned, namely the assumption that ordinals are objects. Taking the lead from Russell’s no class theory, we interpret talk of ordinals as an efficient way of conveying higher-order logical truths. The resulting theory of ordinals is formally adequate to standard intuitions about ordinals, expresses a conception of ordinal number capable of resolving Burali-Forti’s paradox, and offers a novel contribution to the longstanding program of reducing mathematics to higher-order logic. (shrink)

L'historiographie des paradoxes ensemblistes attribue classiquement la "découverte" du paradoxe du plus grand ordinal au mathématicien italien Burali-Forti. Un examen attentif de ses démonstration et revirement révèle qu'il n'en est rien. Nous tenterons de dégager l'impact de la publication des Principles de Russell sur la constitution de cette version historiographique officielle, avant d'aborder la question de la définition des paradoxes et ce qui peut en motiver une conception restrictive.

El presente escrito presenta las ventajas y desventajas de la formalización matemática como una herramienta para el análisis de argumentos complejos o difusos en la filosofía. De tal forma, aquí se encuentra un recorrido histórico de algunas consideraciones del papel de las matemáticas en la búsqueda del conocimiento. Posterior a ello, se muestra cómo por medio de la teoría de conjuntos y laabstracción matemática, es posible proponer una reinterpretación de algunos textos filosóficos. Para lograr este objetivo, se presenta, a manera (...) de ejemplo, la formalización de los conceptos de “Dios”, “error” y “Creación” dados por Descartes en su libro las Meditaciones de la filosofía primera, este ejercicio permiteproponer una paradoja en donde se evidencia una posible falencia en la delimitación del concepto de Dios. Con todo ello, se demuestra por qué los enfoques matemáticos deben ser promovidos en los estudios filosóficos. (shrink)

It is common in contemporary sport to liken the notion of the team to that of the family. That is, the family is used to evoke team life. Portraying the team as a family usually implies a positive evaluation. Despite its prevalence, the team as a family equation has not been analyzed in the sport philosophy literature. Thus, the purpose of this article is twofold. First, it explores whether the team is to be equated with the family. To discuss the (...) nature of the family and the team, I draw together the work of several sport philosophers with that of Laura Wildemann Kane. Second, and considering what characterizes the team, this article investigates whether teammates have special responsibilities to one another by virtue of their relationship and the value attached to it. I propose that the team as a family equation does not align with the constitution of these two social groups and that teammates have special responsibilities to one another, but such responsibilities are different from those observed in the family. (shrink)

Despite society’s increasing sensitivity toward green production, companies often struggle to find effective communication strategies that induce consumers to buy green products or engage in other environmentally friendly behaviors. To add clarity to this situation, we investigated the effectiveness of negative versus positive message framing in promoting green products, whereby companies highlight the detrimental versus beneficial environmental consequences of choosing less versus more green options, respectively. Across four experiments, we show that negatively framed messages are more effective than positively framed (...) ones in prompting consumers to engage in pro-environmental behaviors. More importantly, we find that anticipated shame is the emotion responsible for this effect. Furthermore, both environmental concern and the type of product promoted serve as moderators; thus, the mediating role of anticipated shame is attenuated when environmental concern is low and the product is a luxury one. Finally, we discuss the theoretical and managerial implications of our work, along with its limitations and some directions for future research. (shrink)

Includes a translator’s preface, note on the text, and suggestions for further reading.

Aristotle normally used historical notations to support his arguments. This is somewhat true for all the works of the corpus, but above all for Politics: the nature, objectives, and methodology of the investigations in this treatise present the strongest links with actual and concrete data, and therefore with historia. Obviously even the Aristotle of Politics is not a historian who wants to report known historiographical traditions; however, regardless of his intentions, there is no doubt that the work in question contains (...) precious ‘fragments’ of history which, in general, confirm or supplement our knowledge. There are, however, cases in which the Aristotelian exempla end up filling in the omissions and gaps of the available sources, such as the cursory reference to the nomos of the Aphytaians, which appears in the section of book 6 dedicated to the so-called agricultural democracy. (shrink)

Pretende-se oferecer aqui uma justificativa, tal como ela parece ter sido indicada por Platão no Teeteto, para o método de Sócrates, anunciado em Apologia como uma anthropíne sophía, capaz de avaliar os diferentes tipos de conhecimento. O recurso à imagem da maiêutica, no Teeteto, traz a representação de um aspecto desse método, posterior à refutação, em que o examinador ou parteiro elevam as opiniões do interlocutor até um princípio transcendente, para mostrar-lhes ou sua vacuidade ou sua solidez. Por fim, o (...) Diálogo de Platão parece não apenas representar o método de Sócrates, mas se inspirar nele para a composição do seu discurso, que joga com a forma de fazer ver e com aquilo que se vê a partir dele. (shrink)

Recent studies have pointed to the existence of a strong relationship between memory and mental representation, while others have shown that images are open to reinterpretation and manipulation; this volume offers a historical overview of the problem as well as a review of the research in psychology and related fields.

In this article, we scrutinize views that justify exclusionary policies regarding transgender athletes based primarily on physiological criteria. We introduce and examine some elements that deserve...

El propósito central enfatiza la comprensión histórico-cultural de la diversidad y la diferencia como productoras de escenarios donde la tolerancia debe redimir la exclusión y fomentar la equidad; explica, reflexiva y críticamente, algunas ideas que subrayan la urgen te necesidad de replantear los d..

This paper argues that racism should not only be conceived as a moral concept whose main aim is to condemn severe wrongs in the domain of race. The paper advances a complementary interpretation of racism as an explanatory concept--one that plays a key role in explaining race-based social problems afflicting members of subordinate racialized groups. As an explanatory concept, the term 'racism' is used to diagnose and highlight the causes of race-related social problems. The project of diagnosing race-based social problems (...) contributes to the pragmatic anti-racist end of developing better political and policy strategies for solving these social problems. The paper defends this interpretation of racism as an explanatory concept through a critical engagement with Urquidez's moral-philosophical account of racism. (shrink)

Much media attention has been given to the potentially shifting voting patterns of the U.S. Hispanic population in the 2020 election. However, far less attention has been given to the implications...

Los convulsos acontecimientos de estos últimos años han demostrado, a escala global, cuán enajenante es nuestra experiencia presente del tiempo. Las tendencias actuales de la teoría crítica y otros enfoques histórico-sociológicos insisten en el ritmo devorador de la aceleración (P. Virilio; H. Rosa) y en la reconfiguración temporal a partir del “presentismo” (Koselleck; F. Hartog). Sin embargo, sabemos que estas cuestiones distan de ser absolutamente nuevas, y que ya preocupaban a los filósofos y a los fenomenólogos en la primera mitad (...) del siglo XX. La publicación reciente de los últimos tratados del Ereignis y de los Cuadernos negros, redactados en los años 1930 y 1940, nos permite acceder a una reflexión original sobre lo que Heidegger nombra la maquinación o fabricación [Machenschaft]. Nos gustaría analizar la manera en la que, a partir de la fabricación, se puede replantear el sentido de nuestra experiencia temporal íntima frente a la situación moderna. Más concretamente, ahondaremos en las formas en las que la primacía de la fabricación funciona como cerrazón de posibilidades, llegando a impedir que pueda ocurrir algo realmente “histórico”. (shrink)

Portuguese introduction to volume 2, issue 2.

The fallibility of deduction is the thesis that a thoughtful speaker-reasoner can wrongly believe that an inference is deductively valid. The author presents an argument to the effect that the fallibility of deduction is incompatible with the widespread view that deduction is epistemically unfruitful (the conclusion is contained in the premises, and the transition from premises to conclusion never extends knowledge). If the fallibility of deduction is a fact, the argument presented is a refutation of the doctrine of the unfruitfulness (...) of deduction. But its relevance goes further, because the argument reveals a connection between two different problems: a solution to the problem of fruitfulness (how can deductive inferences increase our knowledge?) is a starting point for a solution to the problem of fallibility (how can deductive inferences be wrong?). To solve the problem of fruitfulness, Michael Dummett proposed some general ideas concerning meaning and inference. Starting from these meaning-theoretical ideas, one can develop an explanation of the fallibility of deduction. The general explanatory strategy does not depend on the choice between a theory of meaning centered on truth, verification, inferential role, or on other key notions. (shrink)

Este artículo tiene por objeto presentar algunas notas reflexivas y descriptivas del desarrollo teórico que hemos realizado sobre el enfoque de las epistemologías de las política educativa en tanto se convierte en una herramienta para llevar a cabo la reflexividad epistemológica sobre la investigación en política educacional. Y puede emplearse tanto como vigilancia epistemológica del propio proceso de investigación como para el meta-análisis de las investigaciones, ingresando de algún modo en lo que hemos denominado el campo de los estudios epistemológicos (...) en política educacional y cuyo continente son las ciencias sociales y políticas. Considerando que la complejidad social y política de la realidad educativa conllevan a construir nuevas miradas epistemológicas y por tanto a cuestionar el objeto de estudio de la política educacional. (shrink)

En este trabajo, se presentan y analizan, a modo de debate ciertas problemáticas inherentes a la elección de los investigadores al momento de definir el eje epistemológico de su investigación: pregunta o hipótesis. Durante mucho tiempo se ha considerado que estos componentes deben estar presentes simultáneamente en un proyecto de investigación, cuando en realidad pertenecen a lógicas diversas. Aquí planteamos la necesidad de reflexionar sobre ejes epistemológicos distintos y a la vez potentes para iniciar una investigación. Pero la co-presencia de (...) ambos ejes en la investigación le quita solidez y consistencia epistemológica a un proyecto de investigación, dado que la racionalidad subyacente con que se construye cada uno de estos ejes difiere entre sí.This paper presents and analyses by way of an open debate certain issues inherent to the choice of researchers when defining the epistemological axis of his research: question or hypothesis. It has been long considered that these components must be present simultaneously in a research project, when in fact belong to different epistemological axes. Each one is different and powerful enough to start some research. But the simultaneous presence of both axes removes epistemological strength and consistency to a research project because the different rationale of each of these axes. (shrink)

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