The paper surveys different notions of implicit definition. In particular, we offer an examination of a kind of definition commonly used in formal axiomatics, which in general terms is understood as providing a definition of the primitive terminology of an axiomatic theory. We argue that such “structural definitions” can be semantically understood in two different ways, namely as specifications of the meaning of the primitive terms of a theory and as definitions of higher-order mathematical concepts or structures. We analyze these (...) two conceptions of structural definition both in the history of modern axiomatics and in contemporary philosophical debates. Based on that, we give a systematic assessment of the underlying semantics of these two ways of understanding the definiens of such definitions, by considering alternative model-theoretic and inferential accounts of meaning. (shrink)

A theory of magnitudes involves criteria for their equivalence, comparison and addition. In this article we examine these aspects from an abstract viewpoint, by focusing on the so-called De Zolt’s postulate in the theory of equivalence of plane polygons (“If a polygon is divided into polygonal parts in any given way, then the union of all but one of these parts is not equivalent to the given polygon”). We formulate an abstract version of this postulate and derive it from some (...) selected principles for magnitudes. We also formulate and derive an abstract version of Euclid’s Common Notion 5 (“The whole is greater than the part”), and analyze its logical relation to the former proposition. These results prove to be relevant for the clarification of some key conceptual aspects of Hilbert’s proof of De Zolt’s postulate, in his classicalFoundations of Geometry(1899). Furthermore, our abstract treatment of this central proposition provides interesting insights for the development of a well-behaved theory ofcompatiblemagnitudes. (shrink)

El artículo examina una de las contribuciones más importantes a los fundamentos de la geometría euclídea elemental lograda por David Hilbert en su obra Fundamentos de la geometría, a saber: la reconstrucción de la teoría euclídea de las proporciones y de los triángulos semejantes. Se argumenta que dicha reconstrucción no sólo estuvo motivada por la identificación de Hilbert de suposiciones implícitas en la teoría de Euclides, sino que además estuvo esencialmente ligada a la preocupación por la 'pureza del método'. Más (...) aún, se afirma que, en este caso específico, el requerimiento de Hilbert por la pureza del método posee un carácter general o fundacional, esto es, no se refiere a la demostración de un teorema en particular, sino más bien es planteado respecto de la construcción axiomática de la teoría misma. (shrink)

The paper outlines an interpretation of one of the most important and original contributions of David Hilbert’s monograph Foundations of Geometry , namely his internal arithmetization of geometry. It is claimed that Hilbert’s profound interest in the problem of the introduction of numbers into geometry responded to certain epistemological aims and methodological concerns that were fundamental to his early axiomatic investigations into the foundations of elementary geometry. In particular, it is shown that a central concern that motivated Hilbert’s axiomatic investigations (...) from very early on was the aim of providing an independent basis for geometry. Accordingly, these concerns about an independent grounding for elementary geometry determined very clear methodological constraints in the process of embedding it into a formal axiomatic system. It is argued that Hilbert not only sought to show that geometry could be considered a pure mathematical theory, once it was presented as a formal axiomatic system; he also aimed at showing that in the construction of such an axiomatic system one could proceed purely geometrically, avoiding concept formations borrowed from other mathematical disciplines like arithmetic or analysis. (shrink)

This paper provides a detailed study of David Hilbert’s axiomatization of the theory of plane area, in the classical monograph Foundation of Geometry. On the one hand, we offer a precise contextualization of this theory by considering it against its nineteenth-century geometrical background. Specifically, we examine some crucial steps in the emergence of the modern theory of geometrical equivalence. On the other hand, we analyze from a more conceptual perspective the significance of Hilbert’s theory of area for the foundational program (...) pursued in Foundations. We argue that this theory played a fundamental role in the general attempt to provide a new independent basis for Euclidean geometry. Furthermore, we contend that our examination proves relevant for understanding the requirement of “purity of the method” in the tradition of modern synthetic geometry. (shrink)

El artículo documenta y analiza las vicisitudes en torno a la incorporación de Hilbert de su famoso axioma de completitud, en el sistema axiomático para la geometría euclídea. Esta tarea es emprendida sobre la base del material que aportan sus notas manuscritas para clases, correspondientes al período 1894–1905. Se argumenta que este análisis histórico y conceptual no sólo permite ganar claridad respecto de cómo Hilbert concibió originalmentela naturaleza y función del axioma de completitud en su versión geométrica, sino que además (...) permite disipar equívocos en cuanto a la relación de este axioma con la propiedad metalógica de completitud de un sistema axiomático, tal como fue concebida por Hilbert en esta etapa inicial.The paper reports and analyzes the vicissitudes around Hilbert’s inclusion of his famous axiom of completeness, into his axiomatic system for Euclidean geometry. This task is undertaken on the basis of his unpublished notes for lecture courses, corresponding to the period 1894–1905. It is argued that this historical and conceptual analysis not only sheds new light on how Hilbert conceived originally the nature of his geometrical axiom of completeness, but also it allows to clarify some misunderstandings regarding this axiom and the metalogical property of completeness of an axiomatic system, as it was understoodby Hilbert in this initial stage. (shrink)

This paper analyzes the theory of area developed by Euclid in the Elements and its modern reinterpretation in Hilbert’s influential monograph Foundations of Geometry. Particular attention is bestowed upon the role that two specific principles play in these theories, namely the famous common notion 5 and the geometrical proposition known as De Zolt’s postulate. On the one hand, we argue that an adequate elucidation of how these two principles are conceptually related in the theories of Euclid and Hilbert is highly (...) relevant for a better understanding of the respective geometrical practices. On the other hand, we claim that these conceptual relations unveil interesting issues between the two main contemporary approaches to the study of area of plane rectilinear figures, i.e., the geometrical approach consisting in the geometrical theory of equivalence and the metrical approach based on the notion of measure of area. Finally, in an appendix logical relations among equivalence, comparison and addition of magnitudes are examined schematically in an abstract setting. (shrink)

El artículo presenta una interpretación del abordaje axiomático temprano a la geometría de David Hilbert, i.e., el desarrollado entre 1891 y 1905. Se sostiene que diversos aspectos de este abordaje, a primera vista problemáticos, se pueden comprender mejor si se contrastan con una de sus influencias más importantes en este periodo: la teoría pictórica [Bildtheorie] de Heinrich Hertz. En particular se argumenta que un análisis de la concepción axiomática de Hilbert a la luz de la teoría de Hertz permite aliviar (...) ciertas tensiones en la concepción hilbertiana; más precisamente, las tensiones que surgen de mantener, al mismo tiempo, una posición axiomática abstracta o formal y una concepción empirista de la geometría, que la considera una ciencia natural. (shrink)

Theoria, Volume 88, Issue 3, Page 629-652, June 2022.

Sobre la base que aportan las notas manuscritas de David Hilbert para cursos sobre geometría, el artículo procura contextualizar y analizar una de las contribuciones más importantes y novedosas de su célebre monografía Fundamentos de la geometría, a saber: el cálculo de segmentos lineales. Se argumenta que, además de ser un resultado matemático importante, Hilbert depositó en su aritmética de segmentos un destacado significado epistemológico y metodológico. En particular, se afirma que para Hilbert este resultado representaba un claro ejemplo de (...) uno de los rasgos más fructíferos y atractivos de su nuevo método axiomático formal, o sea, la capacidad de descubrir y exhibir conexiones estructurales o internas entre diferentes teorías matemáticas. On the basis of a set of unpublished notes for lecture courses on geometry, the paper seeks to contextualize and analyze one of the most important and original contributions of David Hilbert's celebrated monograph Foundations of Geometry, namely its arithmetic of line segments. It is argued that Hilbert attributed to his arithmetic of segments an important epistemological and methodological meaning, in addition to its relevance as an original mathematical result. In particular, it is claimed that for Hilbert his arithmetic of segments represented a clear example of one of the most fruitful and attractive traits of his new formal axiomatic method, i.e., the power to discover and exhibit inner or structural connections among different mathematical theories. (shrink)

El artículo ofrece una interpretación de las contribuciones de David Hilbert a la teoría de las magnitudes poligonales, desarrollada en su influyente monografía Fundamentos de la geometría, publicada en 1899. Se argumenta que la construcción de esta parte central de la geometría euclidiana represento para Hilbert un desafío muy significativo, en razón de su objetivo general de proporcionar una axiomatización estrictamente sintética de esta teoría geométrica; es decir, en virtud de sus conocidos requerimientos metodológicos y epistemológicos de construir la geometría (...) elemental evitando toda apelación esencial a presupuestos numéricos. Se sostiene además que dicho desafío consistió en ofrecer un nuevo fundamento lógicamente sólido para la teoría del área poligonal, pero que al mismo tiempo pueda ser considerado como puramente geométrico. The paper offers an interpretation of David Hilbert's contributions to the theory of polygonal magnitudes, developed in his influential monograph Foundations of Geometry, published in 1899. It is argued that the construction of this central part of Euclidean geometry presented an appealing challenge to Hilbert's general aim of providing a purely synthetic axiomatization of this geometrical theory; in other words, to his epistemological and methodological concerns of constructing elementary geometry without resorting to any kind of numerical assumption. Moreover, it is claimed this challenge consisted in offering a new foundation for the theory of polygonal area, which could be considered at the same time as logically sound and purely geometrical. (shrink)

El artículo presenta y analiza un conjunto de notas manuscritas de clases para cursos sobre geometría, dictados por David Hilbert entre 1891 y 1905. Se argumenta que en estos cursos el autor elabora la concepción de la geometría que subyace a sus investigaciones axiomáticas en Fundamentos de la geometría . Por un lado, afirmo que lo que caracteriza esta concepción de la geometría es: i) una posición axiomática abstracta o formal; ii) una posición empirista respecto del origen de la geometría (...) y de su lugar dentro de las distintas teorías matemáticas. Por otro lado, sostengo que el papel que Hilbert le confiere a la intuición geométrica en el proceso de axiomatización de esta teoría, permite apreciar claramente su oposición respecto de las posiciones formalistas con las que habitualmente es identificado. (shrink)

This paper aims to offer an interpretation of the Transcendental Deduction of the Categories which puts together two of its most distinctive and fundamental traits: the constant reference to the temporal character of the human consciousness and the use of the analytic method of exposition. We will defend the thesis that the connection between both traits is essential, i. e., it will be argued that it is precisely the usage of the analytic method what confers to the pure consciousness of (...) time a significant role in the structure of the argument. With this thesis an alternative reading of the A-Deduction is pursued, one which moves away from the predominant one which claims that Kant's persistent allusion in it to the pure time representation - and to the synthesis which make it possible - is just a direct consequence of its psychological character.El objetivo del artículo es ofrecer una interpretación de la Deducción Transcendental de las Categorías que ponga en relación dos de sus aspectos más distintivos y fundamentales: la referencia permanente al carácter temporal de la conciencia humana, y la utilización del método analítico de exposición. Se defenderá la tesis de que la conexión entre ambos aspectos es esencial, en tanto que es precisamente la adopción del método analítico lo que determina que se le confiera a la conciencia pura del tiempo un rol fundamental en la estructura del argumento. Con esta tesis se intenta ofrecer una lectura alternativa de la Deducción A, que se diferencie de la habitual según la cual la apelación constante de Kant a la representación pura del tiempo - y a las síntesis que lo hacen posible - es una consecuencia directa del carácter psicologista que la distingue. (shrink)

A theory of magnitudes involves criteria for their comparison, equivalence and addition. We examine these aspects from an abstract viewpoint, stressing independence and definability. These considerations are triggered by the so-called De Zolt’s principle in the theory of equivalence of plane polygons.

Eduardo Arroyo, Armelle Auris, François Boissonnet, Paul HENRY, Pierre-Jean LABARRIERE, MATIEU, Gilles AILLAUD, Gérad THALMANN, Nicole MATHIEU, Jacques VIMARD, Michel QUAREZ, Leonardo CREMONINI, L'atelier d'Eduardo Arryo: Le carré blanc du ring, Rue Descartes, No. 7, Logiques de l'éthique (Juin 1993), pp. 149-168.

This volume collects a number of important and revealing interviews with Richard Rorty, spanning more than two decades of his public intellectual commentary, engagement, and criticism. In colloquial language, Rorty discusses the relevance and nonrelevance of philosophy to American political and public life. The collection also provides a candid set of insights into Rorty's political beliefs and his commitment to the labor and union traditions in this country. Finally, the interviews reveal Rorty to be a deeply engaged social thinker and (...) observer. (shrink)

This paper focuses on the extension of AGM that allows change for a belief base by a set of sentences instead of a single sentence. In [FH94], Fuhrmann and Hansson presented an axiomatic for Multiple Contraction and a construction based on the AGM Partial Meet Contraction. We propose for their model another way to construct functions: Multiple Kernel Contraction, that is a modification of Kernel Contraction, proposed by Hansson [Han94] to construct classical AGM contractions and belief base contractions. This construction (...) works out the unsolved problem pointed out by Hansson in [Han99, pp. 369]. (shrink)

Adding a transparent truth predicate to a language completely governed by classical logic is not possible. The trouble, as is well-known, comes from paradoxes such as the Liar and Curry. Recently, Cobreros, Egré, Ripley and van Rooij have put forward an approach based on a non-transitive notion of consequence which is suitable to deal with semantic paradoxes while having a transparent truth predicate together with classical logic. Nevertheless, there are some interesting issues concerning the set of metainferences validated by this (...) logic. In this paper, we show that this logic, once it is adequately understood, is weaker than classical logic. Moreover, the logic is in a way similar to the paraconsistent logic LP. (shrink)

Until now, North American and European philosophies have been engaged in debates about the possibility of a postmetaphysical philosophy and the consequences of the linguistic turn for the assessment of modernity; they have done so, however, without departing from the narrow horizons of their respective nationalistic perspectives. In this incisive critique, Dussel demonstrates how most of thse philosophies have either failed to give historically faithful analyses of the genesis of the "myth" of modernity, or have never engaged in a serious (...) questioning of their own Eurocentric presuppositions. He shows how North American and European philosophers have presupposed a no-longer-acceptable philosophy of history that has led them to fall into a "developmental fallacy," the belief that there is a linear sequence that moves from the premodern, underdeveloped, or on the way to industrialization, to the modern, developed, and industrialized. (shrink)

In this article, we will present a number of technical results concerning Classical Logic, ST and related systems. Our main contribution consists in offering a novel identity criterion for logics in general and, therefore, for Classical Logic. In particular, we will firstly generalize the ST phenomenon, thereby obtaining a recursively defined hierarchy of strict-tolerant systems. Secondly, we will prove that the logics in this hierarchy are progressively more classical, although not entirely classical. We will claim that a logic is to (...) be identified with an infinite sequence of consequence relations holding between increasingly complex relata: formulae, inferences, metainferences, and so on. As a result, the present proposal allows not only to differentiate Classical Logic from ST, but also from other systems sharing with it their valid metainferences. Finally, we show how these results have interesting consequences for some topics in the philosophical logic literature, among them for the debate around Logical Pluralism. The reason being that the discussion concerning this topic is usually carried out employing a rivalry criterion for logics that will need to be modified in light of the present investigation, according to which two logics can be non-identical even if they share the same valid inferences. (shrink)

This book explains how the logic of theory change employs formal models in the investigation of changes in belief states and databases. The topics covered include equivalent characterizations of AGM operations, extended representations of the belief states, change operators not included in the original framework, iterated change, applications of the model, its connections with other formal frameworks, and criticism of the model.

In this paper we develop a general representation theory for MV-algebras. We furnish the appropriate categorical background to study this problem. Our guide line is the theory of classifying topoi of coherent extensions of universal algebra theories. Our main result corresponds, in the case of MV-algebras and MV-chains, to the representation of commutative rings with unit as rings of global sections of sheaves of local rings. We prove that any MV-algebra is isomorphic to the MV-algebra of all global sections of (...) a sheaf of MV-chains on a compact topological space. This result is intimately related to McNaughton’s theorem, and we explain why our representation theorem can be viewed as a vast generalization of McNaughton’s theorem. In spite of the language used in this abstract, we have written this paper in the hope that it can be read by experts in MV-algebras but not in sheaf theory, and conversely. (shrink)

In the last 2 decades, the literature has documented the upsurge of community-driven processes of consumer-producer cooperation, which are alternative to the dominant food system. These organizational arrangements have been conceptualized differently, witnessing the growing importance of local communities in generating place-based solutions to the demand for organic, local, and sustainable food. Relying on a practice theory approach, this article delves into two key inquiries: first, what motivates individuals to become part of Civic Food Networks (CFNs) and how does this (...) influence the variety of food entering their households? And second, how does participation in a CFN impact people’s daily food practices, with a particular focus on diet diversification? We employ the term “civic food networks (CFN)” to underscore the significance of grassroots initiatives in the broader context of local food policies. This term also highlights the connections to other movements known for their potential for transformation, such as the solidarity economy, transition towns, and de-growth movements. Our analysis draws on a comprehensive qualitative approach, which includes participant observation, in-depth interviews, and a focus group involving members of the CSA _Naturalmente in Trentino_, situated in the North-Eastern part of Italy. The findings reveal that altering dietary habits poses various challenges both at the individual and institutional levels. This process involves the stabilization of ingrained habits, adjustments to daily routines, and the reinforcement of shared norms. Furthermore, it underscores the critical role of food policies in promoting sustainable eating practices. (shrink)

In “A new proof of the completeness of the Lukasiewicz axioms” Chang proved that any totally ordered MV-algebra A was isomorphic to the segment \}\) of a totally ordered l-group with strong unit A *. This was done by the simple intuitive idea of putting denumerable copies of A on top of each other. Moreover, he also show that any such group G can be recovered from its segment since \^*}\), establishing an equivalence of categories. In “Interpretation of AF C (...) *-algebras in Lukasiewicz sentential calculus” Mundici extended this result to arbitrary MV-algebras and l-groups with strong unit. He takes the representation of A as a sub-direct product of chains A i, and observes that \ where \. Then he let A * be the l-subgroup generated by A inside \. He proves that this idea works, and establish an equivalence of categories in a rather elaborate way by means of his concept of good sequences and its complicated arithmetics. In this note, essentially self-contained except for Chang’s result, we give a simple proof of this equivalence taking advantage directly of the arithmetics of the the product l-group \, avoiding entirely the notion of good sequence. (shrink)

In this paper we develop a general representation theory for MV-algebras. We furnish the appropriate categorical background to study this problem. Our guide line is the theory of classifying topoi of coherent extensions of universal algebra theories. Our main result corresponds, in the case of MV-algebras and MV-chains, to the representation of commutative rings with unit as rings of global sections of sheaves of local rings. We prove that any MV-algebra is isomorphic to the MV-algebra of all global sections of (...) a sheaf of MV-chains on a compact topological space. This result is intimately related to McNaughton’s theorem, and we explain why our representation theorem can be viewed as a vast generalization of McNaughton’s theorem. In spite of the language used in this abstract, we have written this paper in the hope that it can be read by experts in MV-algebras but not in sheaf theory, and conversely. (shrink)

We study the algebraizability of the logics constructed using literal-paraconsistent and literal-paracomplete matrices described by Lewin and Mikenberg in [11], proving that they are all algebraizable in the sense of Blok and Pigozzi in [3] but not finitely algebraizable. A characterization of the finitely algebraizable logics defined by LPP-matrices is given.We also make an algebraic study of the equivalent algebraic semantics of the logics associated to the matrices ℳ32,2, ℳ32,1, ℳ31,1, ℳ31,3, and ℳ4 appearing in [11] proving that they are (...) not varieties and finding the free algebra over one generator. (shrink)

Entrevista com o Prof. Eduardo Jardim, Revisor técnico da edição brasileira de "Pensar sem corrimão", mais recente livro de Hannah Arendt no Brasil.

In some recent articles, Cobreros, Egré, Ripley, & van Rooij have defended the idea that abandoning transitivity may lead to a solution to the trouble caused by semantic paradoxes. For that purpose, they develop the Strict-Tolerant approach, which leads them to entertain a nontransitive theory of truth, where the structural rule of Cut is not generally valid. However, that Cut fails in general in the target theory of truth does not mean that there are not certain safe instances of Cut (...) involving semantic notions. In this article we intend to meet the challenge of answering how to regain all the safe instances of Cut, in the language of the theory, making essential use of a unary recovery operator. To fulfill this goal, we will work within the so-called Goodship Project, which suggests that in order to have nontrivial naïve theories it is sufficient to formulate the corresponding self-referential sentences with suitable biconditionals. Nevertheless, a secondary aim of this article is to propose a novel way to carry this project out, showing that the biconditionals in question can be totally classical. In the context of this article, these biconditionals will be essentially used in expressing the self-referential sentences and, thus, as a collateral result of our work we will prove that none of the recoveries expected of the target theory can be nontrivially achieved if self-reference is expressed through identities. (shrink)

Who Comes After the Subject offers the most comprehensive overview to date of contemporary French thinking on the question of the "subject." Nineteen philosophers and critics offer diverse perspectives on the subject as it has manifested itself in our modern discourses: the subject of philosophy, of the State, of history, of psychoanalysis. Each contribution asks What has become of the subject? or What has the subject become? in the wake of its critiques and deconstructions .

O artigo busca abordar o pensamento de Auerbach quanto ao romance do século XX, especificamente o que ele chama de romance realista. Realismo, representação, historicismo são alguns dos conceitos-chave para a compreensão do pensamento auerbachiano e cuja apresentação serve como abertura do artigo. A seguir há uma investigação das características do romance entreguerras tal como Auerbach o pensa. Finalmente faço a relação entre o projeto filológico e o tratamento que o filólogo dá ao romance do século XX.

En este artículo se analiza la posible influencia del estoicismo en algunos aspectos precisos de la ética kantiana. Se intenta defender la tesis de que, aunque Kant critica muchas veces la ética estoica y en especial su eudemonismo, también adopta en su ética algunas de las posiciones estoicas y las incorpora en ciertos aspectos clave de su propia Doctrina de la virtud. En especial, se intenta mostrar cómo adopta Kant el ideal del sabio estoico y una de sus características más (...) notables: la apatía. (shrink)

In this paper we discuss the extent to which the very existence of substructural logics puts the Tarskian conception of logical systems in jeopardy. In order to do this, we highlight the importance of the presence of different levels of entailment in a given logic, looking not only at inferences between collections of formulae but also at inferences between collections of inferences—and more. We discuss appropriate refinements or modifications of the usual Tarskian identity criterion for logical systems, and propose an (...) alternative of our own. After that, we consider a number of objections to our account and evaluate a substantially different approach to the same problem. (shrink)