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)

In 1939, the influential psychophysicist S. S. Stevens proposed definitional distinctions between the terms ‘number,’ ‘numerosity,’ and ‘numerousness.’ Although the definitions he proposed were adopted by syeveral psychophysicists and experimental psychologists in the 1940s and 1950s, they were almost forgotten in the subsequent decades, making room for what has been described as a “terminological chaos” in the field of numerical cognition. In this paper, I review Stevens’s distinctions to help bring order to this alleged chaos and to shed light on (...) two closely related questions: whether it is adequate to speak of a number sense and how philosophers can make sense of the claim by cognitive scientists that numbers are perceptual entities. Moreover, I offer further support to Stevens’s distinction between numerosity and numerousness by showing that they are relational properties that emerge through different relationships between agents and their environments. The final conclusion is that, by adopting Stevens’s distinctions, numbers do not need to be seen as perceptual entities, since the so-called number sense is better described as a sense of numerousness. (shrink)

One important limitation of the current renditions of interpretivism is that its emphasis on the moral dimension of sport has overlooked the aesthetic dimension lying at the core of this account of sport. The interpretivist’s failure to acknowledge and consider the aesthetic implicitly distances this realm from the moral. Marcia Muelder Eaton calls this distancing the separatist mistake. This paper argues that interpretivism presupposes not only moral but also aesthetic principles and values. What it sets out to demonstrate is that (...) interpretivism is an integralist, or nonseparatist, account of sport, one in which ethical and aesthetic values are not exclusive. Making explicit and specifying interpretivism’s combined moral-aesthetic approach to sport not only helps to better distinguish the whole range of values that make up sport as well as their interconnection but also encourages sportspeople to pursue more coherent sport and, thus, more enriching lives. (shrink)

El artículo se propone indagar en la concepción de la «realidad» que tiene la sofística, como referente fundamental para entender mejor su propuesta crítica ante el avance que en esa época está teniendo la «filosofía dogmática» de tipo metafísica. En particular centrando el análisis en la invectiva que al respecto realiza Gorgias de Leontinos en los fragmentos conservados de dos de sus obras fundamentales: el tratado De lo que no es o de la naturaleza y el discurso Encomio de Helena. (...) Aquí se considera que esta es una posible ruta que permite reconstruir parte de su sediciosa propuesta filosófica. El núcleo fundamental que impulsa la reflexión gorgiana es la infranqueable distancia que separa la concepción de una supuesta «realidad en sí» del pensamiento y lenguaje; ante esta prerrogativa es factible entender el poder que le atribuye al logos para influir en las almas y acciones de los seres humanos, pero también en la capacidad del discurso para «hacer diciendo»; en este sentido perfila su perspectiva extrema de que las palabras estrictamente humanas en su fuerza performativa hacen factible «crear la realidad». (shrink)

This volume advances a better, more historical and contextual, manner to consider not only the present, but also the future of ‘crisis’ and ‘renewal’ as key concepts of our political language as well as fundamental categories of interpretation.

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)

Nos anos 1970s, Hilary Putnam defendeu a tese que designadores de espécies naturais, como “água”, “tigre” e “ouro”, são termos indexicais que mudam de significado a cada contexto. No entanto, Tyler Burge rejeitou essa tese, e Putnam veio a adotar a posição de Burge. A rejeição de Burge está apoiada na distinção entre crenças de dicto e crenças de re. Nesse artigo veremos os pontos de contato entre as posições de Putnam e Burge, a posição de Putnam nos anos 1970s, (...) os principais argumentos de Burge contra a tese da indexicalidade dos designadores de espécies naturais e a diferença entre crenças de dicto e crenças de re. (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)

O objetivo deste artigo consiste em aprofundar a intuição foucaultiana no livro As palavras e as coisas a respeito da máthêsis universalis como ciência geral da ordem e da medida, para tratar da Idade Clássica. Essa caracterização de Michel Foucault lhe permite afastar-se da percepção desta época pela lente do racionalismo, mecanicismo e matematização da natureza. Ao refletir sobre a ordem empírica dos seres, palavras e coisas pela relação entre máthêsis, taxinomia e teoria do signo, Foucault relativiza também certa percepção (...) da fenomenologia (husserliana) segundo a qual o racionalismo seria o solo fundante da Modernidade. Palavras-chave: Máthêsis. Taxinomia. Arqueologia. Fenomenologia. Michel Foucault. Descartes. (shrink)

Comenzamos con una cuestión general: ¿puede el arte moderno ser entendido como una vía de salvación como lo señala Schopenhauer?Con el apoyo de varios textos, como los de Arthur C. Danto, Después del fin del arte: El arte contemporáneo y el linde de la historia, se contrastará lo que se considera como arte moderno contextualizándolo a la idea de Arthur Schopenhauer expresada en su libro El mundo como voluntad y representación. Arthur Danto explica y ejemplifica los conceptos moderno, contemporáneo y (...) posmoderno, que se vuelven bastante importantes pues son el apoyo para lograr contrastar el texto de Schopenhauer y saber si el arte moderno puede ser vía de salvación. (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)

This article aims to highlight the productivity of falsehood in the context of its asymmetrical position with truth. The core of the discussion can be summarized in the following statement: if only truths can be drawn from truths, it cannot be said that truth cannot be drawn from falsehood. The central thesis of the text is that the asymmetrical behavior of falsehood promotes the dissociation between the criterion (based on the conservation of truth) and the field of validity so that (...) the latter, broader than the former, incorporates the movement not only from truth to truth, but also from falsehood to falsehood and, above all, from falsehood to truth. The distinction between the symmetrical and asymmetrical fields allows understanding, for example, the emergence of certain logical fallacies, the separation between demonstration and deduction, the thesis of Popperian fallibilism, and the use of the hypothetical-deductive method. Truth does not have functions analogous to those of falsehood. (shrink)

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...

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 paper I argue that some human reproductive genome editing interventions can be therapeutic in nature, and thus that it is false that all such interventions just create healthy individuals. I do this by showing that the conditions established by a therapy definition are met by certain reproductive genome editing interventions. I then defend this position against two objections: (a) reproductive genome editing interventions do not attain one of the two conditions for something to be a therapy, and (b) (...) some reproductive genome editing interventions are therapeutic but in a nonstandard way. In the Conclusion I call for a more nuanced discussion of the nature of reproductive genome editing interventions. (shrink)

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)