The logical hexagon (or hexagon of opposition) is a strange, yet beautiful, highly symmetrical mathematical figure, mysteriously intertwining fundamental logical and geometrical features. It was discovered more or less at the same time (i.e. around 1950), independently, by a few scholars. It is the successor of an equally strange (but mathematically less impressive) structure, the “logical square” (or “square of opposition”), of which it is a much more general and powerful “relative”. The discovery of the former did not raise interest, (...) neither among logicians, nor among philosophers of logic, whereas the latter played a very important theoretical role (both for logic and philosophy) for nearly two thousand years, before falling in disgrace in the first half of the twentieth century: it was, so to say, “sentenced to death” by the so-called analytical philosophers and logicians. Contrary to this, since 2004 a new, unexpected promising branch of mathematics (dealing with “oppositions”) has appeared, “oppositional geometry” (also called “n-opposition theory”, “NOT”), inside which the logical hexagon (as well as its predecessor, the logical square) is only one term of an infinite series of “logical bi-simplexes of dimension m”, itself just one term of the more general infinite series (of series) of the “logical poly-simplexes of dimension m”. In this paper we recall the main historical and the main theoretical elements of these neglected recent discoveries. After proposing some new results, among which the notion of “hybrid logical hexagon”, we show which strong reasons, inside oppositional geometry, make understand that the logical hexagon is in fact a very important and profound mathematical structure, destined to many future fruitful developments and probably bearer of a major epistemological paradigm change. (shrink)

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a (...) “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “ n -opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m ” ( m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid). (shrink)

Some Carrollian posthumous manuscripts reveal, in addition to his famous ‘logical diagrams’, two mysterious ‘logical charts’. The first chart, a strange network making out of fourteen logical sentences a large 2D ‘triangle’ containing three smaller ones, has been shown equivalent—modulo the rediscovery of a fourth smaller triangle implicit in Carroll's global picture—to a 3D tetrahedron, the four triangular faces of which are the 3+1 Carrollian complex triangles. As it happens, such an until now very mysterious 3D logical shape—slightly deformed—has been (...) rediscovered, independently from Carroll and much later, by a logician , a mathematician and a linguist studying the geometry of the ‘opposition relations’, that is, the mathematical generalisations of the ‘logical square’. We show that inside what is called equivalently ‘n-opposition theory’, ‘oppositional geometry’ or ‘logical geometry’, Carroll's first chart corresponds exactly, duly reshap.. (shrink)

En la consideración de numerosos asuntos y respecto de muy variadas exposiciones, el uso de expresiones como "filosófica" sugiere que debemos remitirnos a procederes, preguntas o exigencias especiales. Rabossi propone un modo de caracterizar el sentido con que usamos esas expresiones y, sobre esa base, concluye que la filosofía tal como se la practica desde hace doscientos años pretende ser una disciplina profesional pero no puede serlo debido a la índole de la preceptiva que la constituye. En este artículo se (...) examinan sus argumentos y se sostiene que, aunque no parecen suficientes para la conclusión a la que apuntan, hay razones para modificarlos de cierto modo que conducen a ese resultado. Whenever expressions like "X is philosophical" appear in different questions and assorted statements, the reference to special kinds of actions, questions or demands are suggested by these uses. Rabossi propounds a way for characterizing the sense of our using such expressions and, based on it, he states that philosophy, as it is accomplished from the last two centuries up to now, pretends to be a professional discipline but she cannot to be as such because the nature of its precepts. Rabossi's arguments are examined and it is maintained that although they do not seem to be sufficient for the pursued conclusion, there are reasons to modify them such a way to come to that conclusion. (shrink)

Se examinan las motivaciones teóricas, los lineamientos generales y las principales variantes de dos modos complementarios de entender la naturaleza de la lógica. El primero centrado en logros técnicos alcanzados desde fines del siglo XIX, y el segundo vinculado con la reflexión clásica sobre la naturaleza de los principios de la actividad de la razón. Se pone énfasis en la importancia del vínculo constitutivo entre las reglas y leyes lógicas y la estructura del lenguaje, en particular de aquél que subyace (...) cuando el habla se orienta por pretensiones cognoscitivas. (shrink)

Imaginando circunstancias extravagantes, pero de obediente prolijidad, Putnam ha procurado aislar objetos, capacidades o sucesos, de los que podría pensarse constituyen ejemplos de representaciones per se, o medios suficientes para que, de alguien que los posea o utilice, se diga que representa o refiere a alguna cosa.

Decostruzione o biopolitica? aims to bring to light both differences and affinities – in their problematic interplay – between deconstruction and biopolitics, thus lifting the smokescreen of French Theory as a unified field of research and revoking its role as an unambiguous and functional category of the history of philosophy. At the same time, this collected edition points towards the work that ITN has been doing for the past decade: retracing the roots of contemporary Italian philosophy and the emergence of (...) Italian biopolitics as a prominent topic in international debates. The struggle between deconstruction and biopolitics becomes then a symbol for the complex genealogies that can be re-activated towards the construction of a thought intended as a conflictual practice. (shrink)

Se examinan algunas versiones representativas de las tesis contrarias a la relación entre las estructuras del discurso y de la realidad. Se propone distinguir entre los aspectos constitutivos y no constitutivos del lenguaje y de la realidad, manifiestos en la lógica de un lenguaje común. Se sostiene que los primeros son mínimos y que las teorías lógicas específicas han de considerarse partes de teorías generales sobre el mundo, parcialmente normativas, empíricamente discutibles e históricamente cambiantes. En consecuencia, no hay motivos para (...) creer que exista un dominio universal de objetos o de propiedades y, correlativamente, una única estructura del mundo, una sola concepción aceptable o completa de este, o una única lógica universal aceptable. The article examines some representative versions of views opposing the relationship between the structures of discourse and reality. On that basis, it proposes making a distinction between constitutive and non-constitutive aspects of language and reality, manifest in the logic of a common language. It argues that the former are minimal and that the specific logical theories are to be considered part of general theories about the world, partially normative, empirically questionable and historically changing. As a result, there are not grounds to believe that there is a universal domain of objects or properties and, correspondingly, a single structure of the world, only one acceptable or complete worldview, or only one acceptable universal logic. (shrink)

Con esta ceremonia culminan los actos del otorgamiento del título de Doctor Honoris Causa de la Universidad de Buenos Aires al filósofo, maestro de maestros, Thomas Moro Simpson.De este modo nuestra Universidad, siguiendo una noble idea iniciada por los romanos mucho antes de que existieran las universidades, la idea de destacar la labor de aquellos ciudadanos que hubieran contribuido notablemente a mejorar la vida de la ciudad común, ha resuelto hacer público su reconocimiento de los excepcionales méritos intelectuales, éticos y (...) académicos de nuestro querido Tomás Simpson. He sido encargado de recordar ahora, sumariamente, ante la comunidad universitaria y, con eso, ante nuestra comunidad simpliciter, algunos de los motivos que condujeron a esta resolución. (shrink)

The fundamental link between logical and semantic inquiries is emphasized, and by relating some of Frege’s ideas on the concepts of logic and truth with others of Davidson’s on the Tarskian structure of interpretation, an attempt is made to clarify the Fregean point of view about the nature of the logical-semantic reflection.

Gregorio Klimovsky argumentó en contra de los análisis sintacticistas de la noción de lenguaje. En este trabajo se examinan sus argumentos, destacando su vínculo con las discusiones recientes sobre universalidad expresiva, y se plantean algunas réplicas atendibles. Luego se adopta un enfoque general de la cuestión que resulta incompatible con el sintacticismo estricto pero que permite una reformulación aceptable, vinculada con una idea minimalista sobre el alcance de la teoría semántica. Gregorio Klimovsky argued against syntactical analyses of the notion of (...) language. In this paper I examine his arguments, noting their links with recent discussions of universal expressibility, and raise some plausible objections. Afterwards I adopt a general approach to the issue which is incompatible with strict syntacticism but allowing an acceptable reformulation, associated with a minimalist notion of the scope of the semantic theory. (shrink)

Se distingue entre principios racionales sustantivos y metódicos, y entre criterios de aceptabilidad y criterios de cosntrucción de argumentos. Sobre esta base se mantiene que la lógica y semántica formales son componentes necesarios de la teoría de la argumentación. Tmbién se consideran dos objeciones típicas : la retendida inaplicabilidad en genral de esas técnicas debido a su radical insuficiencia, y la presunta inutilidad de la lógica formal.

In 1995 Slater argued both against Priest’s paraconsistent system LP (1979) and against paraconsistency in general, invoking the fundamental opposition relations ruling the classical logical square. Around 2002 Béziau constructed a double defence of paraconsistency (logical and philosophical), relying, in its philosophical part, on Sesmat’s (1951) and Blanche’s (1953) “logical hexagon”, a geometrical, conservative extension of the logical square, and proposing a new (tridimensional) “solid of opposition”, meant to shed new light on the point raised by Slater. By using n-opposition (...) theory (NOT) we analyse Beziau’s anti-Slater move and show both its right intuitions and its technical limits. Moreover, we suggest that Slater’s criticism is much akin to a well-known one by Suszko (1975) against the conceivability of many-valued logics. This last criticism has been addressed by Malinowski (1990) and Shramko and Wansing (2005), who developed a family of tenable logical counter-examples to it: trans-Suszkian systems are radically many-valued. This family of new logics has some strange logical features, essentially: each system has more than one consequence operator. We show that a new, deeper part of the aforementioned geometry of logical oppositions (NOT), the “logical poly-simplexes of dimension m”, generates new logical-geometrical structures, essentially many-valued, which could be a very natural (and intuitive) geometrical counterpart to the “strange”, new, non-Suszkian logics of Malinowski, Shramko and Wansing. By a similar move, the geometry of opposition therefore sheds light both on the foundations of paraconsistent logics and on those of many-valued logics. (shrink)

Developing a Center for Teaching Excellence: A Case Study Using the Integrated Readiness Matrix builds on the 2015 text, Integrating Pedagogy and Technology: Improving Teaching and Learning in Higher Education with a focus on teaching in higher education. Developing a Center for Teaching Excellence is premised on our contention in the first book that, while individual faculty members can independently begin to use the IRM to improve their pedagogical and technological skills in their content areas, an organizational structure is needed (...) to sustain ongoing improvement. In addition, while the first book provided a primer on learning theory as it relates to pedagogy, Developing a Center for Teaching Excellence plumbs this topic more deeply from the perspective of the college instructor. Further, the second book is dedicated to demonstrating how the IRM can be institutionalized as the foundation for providing the structure and support to faculty and how they can help shape centers for teaching excellence by becoming more familiar with relevant learning theories and related pedagogical and technological approaches. (shrink)

Se presentan dos variantes principales de la semántica actual (1990) que hace del concepto de condiciones veritativas de una oración la clave para desarrollar las ideas de significado y comprensión. Se examinan algunas críticas y se sostiene que no afectan esencialmente el programa. Se distingue entre las versiones al modo de Davidson de aquellas que siguen ideas de Montague y se sostiene que, cuando el fin teórico primario es explicar la comprensión lingüística, hay razones para dar preferencia a las variantes (...) del primer tipo, y que también las hay cuando se busque una semántica apropiada para caracterizar posiciones ontológicas dependientes de restricciones epistémicas; aunque las preferencias se invierten si la ontología que se quiere expresar va a formularse prescindiendo de consideraciones gnoseológicas. (shrink)

Critical review of Orayen´s book Lógica, significado y ontología, UNAM, 1989.