In nine essays, each related to a different time frame, author analyzes diverse aspects of Venezuelans' cultural behavior in regard to what the author considers 'taste.' Witty and amusing, often sarcastic and colloquial, book does not lose perspective of the historical context. A curious x-ray of Venezuelan idiosyncrasies"--Handbook of Latin American Studies, v. 58.

Este libro es un homenaje a la trayectoria intelectual y académica de la profesora María Teresa López de la Vieja, Catedrática emérita de la Universidad de Salamanca. En él se trazan algunos de los caminos que, con su obra, nos invita a transitar. El volumen recoge contribuciones de colegas de varias nacionalidades y procedentes de diversos ámbitos de reflexión que le son afines: la filosofía moral y política, la literatura, la teoría de la argumentación, los estudios feministas, las éticas aplicadas, (...) la bioética. Los capítulos que componen este libro dan voz a varias generaciones de investigadoras e investigadores que han acompañado y acompañan su andadura académica y que aquí dialogan con ella haciendo propias sus preocupaciones intelectuales. Con su participación contribuyen al justo reconocimiento de su dedicación a la labor docente e investigadora. (shrink)

Este artículo examina la recepción de la cultura clásica en la prosa de Gabriela Mistral. Frente a la retórica y la idealización academicistas, Mistral busca la Antigüedad viva en los poetas y los pueblos mediterráneos, y reivindica para América la herencia grecolatina del humanismo cristiano This paper examines the reception of Classical culture in Gabriela Mistral's prose writings. Faced with the rhetoric and idealized vision of academicism, Mistral seeks Antiquity living in poets and Mediterranean peoples, and she vindicates for America (...) the Greco-Latin heritage of Christian humanism. (shrink)

El trabajo que presentamos tiene dos objetivos principales: primeramente, conocer las principales características estructurales y funcionales del funduq; en segundo lugar, observar la evolución tanto estructural como funcional que estos espacios sufrieron dentro y fuera del territorio islámico. Para ello utilizaremos la información arqueológica disponible que se complementará con algunos datos provenientes de la documentación escrita. Así conoceremos el funduq clásico, un edificio de planta cuadrada o rectangular, con patio central y uno o varios pisos, dedicados el inferior a cuadras (...) y almacenes y los superiores a alojamiento de los viajeros/mercaderes. De este edificio encontraremos dos variaciones. Por una parte, el fondaco, un espacio arrendado o donado por las autoridades musulmanas a las comunidades cristianas de mercaderes que cumplía con todas las necesidades de estos últimos (religiosas, sociales y económicas). Y por otra, la alhóndiga, una institución con unas funciones específicas de almacenamiento y recogida de impuestos de determinadas mercancías. (shrink)

Las prácticas educativas están social, político e históricamente construidas. Se construyen modelos y rituales en las prácticas educativas y se reproducen constantemente, haciendo perpetuar los elementos culturales y políticos introduciéndolos en los sujetos. Son dispositivos de poder. A su vez, son prácticas construidas desde lo singular, es decir que se estructura desde cada actor de la práctica aportándole su propia significación. Así, la práctica educativa está inserta en múltiples significaciones. En este sentido, la docencia puede entenderse como un mecanismo de (...) resistencia ante los dispositivos de poder, cuando se enmarca dentro del contexto más amplio de los llamados “cura sui” o Estéticas de la existencia concebidos por Michel Foucault. (shrink)

Residuated fuzzy logic calculi are related to continuous t-norms, which are used as truth functions for conjunction, and their residua as truth functions for implication. In these logics, a negation is also definable from the implication and the truth constant $\overline{0}$ , namely $\neg \varphi$ is $\varphi \to \overline{0}$. However, this negation behaves quite differently depending on the t-norm. For a nilpotent t-norm (a t-norm which is isomorphic to Łukasiewicz t-norm), it turns out that $\neg$ is an involutive negation. However, (...) for t-norms without non-trivial zero divisors, $\neg$ is Gödel negation. In this paper we investigate the residuated fuzzy logics arising from continuous t-norms without non-trivial zero divisors and extended with an involutive negation. (shrink)

The monoidal t-norm based logic MTL is obtained from Hájek''s Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational completeness (...) (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL. Moreover, we investigate varieties of MTL algebras whose linearly ordered countable algebras embed into algebras whose lattice reduct is the real and/or the rational interval [0,1]. These embedding properties are used to investigate finite strong standard and/or rational completeness of the corresponding logics. (shrink)

The monoidal t-norm based logic MTL is obtained from Hájek's Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational completeness (...) (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL. Moreover, we investigate varieties of MTL algebras whose linearly ordered countable algebras embed into algebras whose lattice reduct is the real and/or the rational interval [0,1]. These embedding properties are used to investigate finite strong standard and/or rational completeness of the corresponding logics. (shrink)

Fact-checking agencies assess and score the truthfulness of politicians’ claims to foster their electoral accountability. Fact-checking is sometimes presented as a quasi-scientific activity, based on reproducible verification protocols that would guarantee an unbiased assessment. We will study these verification protocols and discuss under which conditions fact-checking could achieve effective reproducibility. Through an analysis of the methodological norms in verification protocols, we will argue that achieving reproducible fact-checking may not help much in rendering politicians accountable. Political fact-checkers do not deliver either (...) reproducibility or accountability today, and there are reasons to think that traditional quality journalism may serve liberal democracies better. (shrink)

En los círculos ortodoxos kantianos suelen disculparse las incoherencias del maestro diciendo que era un “hijo de su tiempo”. Ciertamente, en su época aún se excluía en toda Europa a las mujeres de la ciudadanía activa, privándolas de ser sujetos políticos y, con ello, sujetos éticos, de derecho, e incluso, históricos. Pero también es verdad que en ese momento está emergiendo un movimiento en defensa de la igualdad de género , en el que lamentablemente Kant no participa. En el presente (...) artículo, tras un preámbulo histórico en el que se confronta al Kant crítico con sus predecesores y coetáneos en lo tocante al tema de la igualdad de las mujeres, se defiende la tesis de que la verdadera moralidad no es pensable sin una idea de universalidad real o, lo que viene a ser lo mismo, de igualdad de género, porque solo a partir de ella es pensable la individualidad y la autonomía. (shrink)

In any event, I realize fully that the parole, the speech, the "word" of a writer such as myself, has something strange and even contradictory about it, even within its own creator. At the moment when I write, let us say, La Jalousie or Glissements progressifs du plaisir, what I propose is improbable and consequently unacceptable; that is, my parole as a writer or as a cinéaste in my novels or in my films is abrupt, inexplicable, nonrecuperable for any correctly (...) organized discourse. Nevertheless, you have noticed that I speak with the same clarity as any professor, and this constitutes an extremely interesting contradiction because it goes to the very heart of the debate; order and disorder never cease to interact, to contaminate each other, to practice a sort of mutual recuperation. If, having written a novel of disorder, I don't find someone—for example, Bruce Morrissette, about La Jalousie—to prove that it has order, I'll do it myself. The principle of order is so crucial that I wish to prove that the disorder which I've created I can myself transform into order. But, as soon as I have shown that it has its order, from that moment on I've destroyed the interest of my work. I have brought about within an organized discourse, organized according to the normal logic of Cartesianism, the recuperation of something which was in fact a machine of war against order. I often run into people who say to me after a film, "Ah, it's a pity that you didn't come to explain all of that before the film. We didn't understand a thing, and it is such a fine thing that you have explained it." And I reply, "Yes, but don't trust that too much," because what I've said is not at all the film. It is even almost the opposite; it is the way in which I show myself that there is in what I created a part which is in spite of everything, explainable by established order, and a part increasingly large, because order progresses. Alain Robbe-Grillet, novelist, film maker, and essayist, is the author of Les Gommes , Le Voyeur , La Jalousie , Dans le labyrinthe , La Maison de rendez-vous , Projet pour une révolution à New York , and Topologie d'une cité fantôme . His films include: L'Année dernière à Marienbad , L'Immortelle , Trans-Europ-Express , L'Homme qui ment , L'Eden et après , Glissements progressifs du plaisir , and Le Jeu avec le feu . He has presented his views on contemporary fiction in Pour un Nouveau Roman. Bruce Morrissette, author of books on Alain Robbe-Grillet and Sunny Distinguished Service Professor in the department of Romance languages and literature at the University of Chicago, has translated "Order and Disorder in Film and Fiction." He is the author of "Post-Modern Generative Fiction: Novel and Film". (shrink)

The literary friendship between Alain Robbe-Grillet and Roland Barthes lasted 25 years. Everything attests to their deep and mutual intellectual esteem: their private correspondence, their published texts, their conversations - notably in the famous dialogue which gives its name to this work. Robbe-Grillet freely said he had very few true friends but, next to the publisher Jérôme Lindon, he always cited the name of Roland Barthes. In 1980, he wrote his own ‘I love, I don’t love’, published here (...) for the first time, thinking about his friend. In 1985, he predicted: ‘It is his work as a writer which will remain’. Ten years later, in 1995, he imagined him as an impatient, blithe novelist, merrily rewriting - ‘euphorically, with inexhaustible happiness’ - The Sorrows of Young Werther. This small collection of conversations and short texts by Robbe-Grillet is like the deferred echo of those that Roland Barthes dedicated to him in his Critical Essays in 1964. It offers fresh insight into the development of Robbe-Grillet’s own work as well as that of Barthes, and is a unique testimony to one of the most important literary friendships of our time. (shrink)

La contingencia es para Leibniz, junto con la espontaneidad y la inteligencia, un componente esencial de la definición de la verdadera libertad, que nuestro pensador califica como uno de los problemas filosóficos por excelencia: “la cuestión de saber cómo se puede salir del laberinto, la gran cuestión de lo libre y lo necesario” (Teodicea, OFC, 10, 11).Leibniz, en diálogo con sus referentes polémicos (Descartes, Spinoza, Hobbes, Arnauld, King, Bayle, Clarke, entre otros) diferenciará –siguiendo la herencia agustiniana- entre el “libre albedrío” (...) y la “verdadera libertad” o determinación racional, a la que sitúa en un camino intermedio entre el determinismo absoluto y la indiferencia de equilibrio. La libertad de los seres racionales será para Leibniz la condición sine qua non para entrar a formar parte del el reino de la gracia, un mundo moral dentro del mundo natural en el que impera el bien común. (shrink)

En el presente artículo se analiza el carácter dialéctico en el enfoque que Karl Barth hace de la justicia en su comentario a la carta a los Romanos. Se muestra de qué modo tal perspectiva pone en evidencia su inserción en la tradición reformada de Lutero y Calvino y la fuerte im- pronta kierkegaardiana de su pensamiento en el que subraya las paradojas y la forma en que son superadas en Jesucristo. En la parte final del trabajo se cuestiona la (...) falta de reflexión por parte de Barth sobre las implicaciones de la justicia para los ámbitos sociales y políticos que sí están presentes en otras de sus obras.This paper discusses the dialectical nature of Karl Barth's view on justice in his commentary of Paul's letter to the Romans. The author shows how Barth's views reveal his insertion within the reformed tradition of Luther and Calvin, and the strong Kierkegaardian influence of his thought when discussing the paradoxes present and the way they are overcome in Jesus Christ.In the last part of the paper, the author questions Barth's lack of reflection on the implications of justice for social and political spheres, which nevertheless are present in other works of this renowned theologian. (shrink)

Political fact-checkers evaluate the truthfulness of politicians’ claims. This paper contributes to an emerging scholarly debate on whether fact-checkers treat political parties differently in a systematic manner depending on their ideology (bias). We first examine the available approaches to analyze bias and then present a new approach in two steps. First, we propose a logistic regression model to analyze the outcomes of fact-checks and calculate how likely each political party will obtain a truth score. We test our model with a (...) sample of fact-checks from Newtral, a major Spanish fact-checker. Our model would signal bias under two assumptions: a) all political parties are on average equally accurate in their statements; b) the verification method gives precise instructions and is implemented systematically. We investigate this second assumption with a series of interviews with Newtral fact-checkers. We show that standard verification protocols are so loosely implemented that fact-checks reflect a set of journalistic decisions, rather than a bias in the statistical sense. We call for a more rigorous definition of verification methods as a pre-requisite for an unbiased assessment of politician’s claims. (shrink)

The aim of the paper is to analyze the expressive power of the square operator of Łukasiewicz logic: ∗x=x⊙x⁠, where ⊙ is the strong Łukasiewicz conjunction. In particular, we aim at understanding and characterizing those cases in which the square operator is enough to construct a finite MV-chain from a finite totally ordered set endowed with an involutive negation. The first of our main results shows that, indeed, the whole structure of MV-chain can be reconstructed from the involution and the (...) Łukasiewicz square operator if and only if the obtained structure has only trivial subalgebras and, equivalently, if and only if the cardinality of the starting chain is of the form n+1 where n belongs to a class of prime numbers that we fully characterize. Secondly, we axiomatize the algebraizable matrix logic whose semantics is given by the variety generated by a finite totally ordered set endowed with an involutive negation and Łukasiewicz square operator. Finally, we propose an alternative way to account for Łukasiewicz square operator on involutive Gödel chains. In this setting, we show that such an operator can be captured by a rather intuitive set of equations. (shrink)

A simple complete axiomatic system is presented for the many-valued propositional logic based on the conjunction interpreted as product, the coresponding implication (Goguen's implication) and the corresponding negation (Gödel's negation). Algebraic proof methods are used. The meaning for fuzzy logic (in the narrow sense) is shortly discussed.

Growing evidence in recent years has led to an agreement on the importance and benefits that inclusive education has for students with special educational needs (SEN). However, the extension and universalization of an inclusive approach will also be enhanced with more evidence on the benefits that inclusion has for all students, including those without SEN. Based on the existing knowledge that learning interactions among diverse students are a key component of educational inclusion, the aim of this study is to identify (...) the impact on students without SEN of being educated with students with SEN in shared, inclusive, interactive learning environments. Data were collected in three schools using a qualitative approach with a communicative orientation. Semistructured interviews were held with teachers as well as community volunteers participating in the schools. Further, focus groups were conducted with students and teachers. The results show that students without SEN benefit from participating in interactive learning activities with peers with SEN in different ways: (1) they learn to respect others, accept differences, and acknowledge different abilities, thereby creating opportunities for new friendships to develop; (2) they learn about abilities related to helping others participate and learn, to be patient and to gain the satisfaction in helping others learn and behave better; and (3) they benefit from the cognitive effort required to explain themselves and from the contributions of peers with SEN from which they can learn. (shrink)

Due to the lockdown caused by the COVID-19 pandemic, e-learning suddenly spread to different levels of education, including university. In Spain, students of sports sciences are prepared during a 4-year study program to work in different areas and with different populations. The aims of this study were to assess the effect of pandemic-driven online teaching on self-efficacy for the inclusion of people with disabilities in a group of university students enrolled in a compulsory course on adapted physical activity ; compare (...) the gains from SE before and after the APA course with a sample of students who followed the same course before the pandemic; and explore the influence on SE scores according to three demographic variables: gender, previous SE training, and previous experience with people with disabilities. The study involved a sample of 124 university students, distributed into two groups: prepandemic and pandemic. They voluntarily completed the Spanish version of the Scale of Self-Efficacy of Physical Education Teachers of Physical Education toward Children with Disabilities, obtaining pre- and postcourse measurements. Due to the sudden lockdown, two-thirds of the pandemic course was taught online, and changes in the teaching strategies and tasks had to be made. Similar gains were obtained in both groups for the three subscales of the SE scale : intellectual, physical, and visual disability. No significant differences were found for comparisons between groups and the interaction effect of the course taught, nor for the three demographic co-variables. This study shows that teaching strategies that encourage student participation and reflections on learning increase student SE, regardless of the teaching format. Moreover, the gains in SE are invariable to demographic co-variables. (shrink)

El desván de la razón, cultivo de las pasiones, identidades éticas y sociedades digitales.

This article focuses on the career of the Jesuit priest, Constantino Bayle, as a historian of Spanish Catholic missions and promoter of state-sponsored arrangements that institutionalized nationalist religious historiography. He encoded religious nationalism and racist categories in academic discourse and terminology, elevating in this way racist assumptions and renewed imperialist aspirations to the level of official historiography. The article traces Bayle’s early career as an Americanista at the Spanish Catholic periodical, Razón y Fe. Bayle was an ardent supporter of Francisco (...) Franco’s military uprising of 1936. He was an apologist for Falange Española who defended its Catholic character. Alongside other Jesuits, he was responsible for forging a Spanish school of missiology that was predicated upon the tenets of Spanish national Catholicism and that was meant to rival analogous Protestant and Roman Catholic historiographic projects. Central to this culturalist endeavor were the notions of Hispanidad and Raza Hispanica. (shrink)

In this paper we show that the subvarieties of BL, the variety of BL-algebras, generated by single BL-chains on [0, 1], determined by continous t-norms, are finitely axiomatizable. An algorithm to check the subsethood relation between these subvarieties is provided, as well as another procedure to effectively find the equations of each subvariety. From a logical point of view, the latter corresponds to find the axiomatization of every residuated many-valued calculus defined by a continuous t-norm and its residuum. Actually, the (...) paper proves the results for a more general class than t-norm BL-chains, the so-called regular BL-chains. (shrink)

In this paper we carry out an algebraic investigation of the weak nilpotent minimum logic and its t-norm based axiomatic extensions. We consider the algebraic counterpart of WNM, the variety of WNM-algebras and prove that it is locally finite, so all its subvarieties are generated by finite chains. We give criteria to compare varieties generated by finite families of WNM-chains, in particular varieties generated by standard WNM-chains, or equivalently t-norm based axiomatic extensions of WNM, and we study their standard completeness (...) properties. We also characterize the generic WNM-chains, i. e. those that generate the variety [MATHEMATICAL DOUBLE-STRUCK CAPITAL W]ℕ[MATHEMATICAL DOUBLE-STRUCK CAPITAL M], and we give finite axiomatizations for some t-norm based extensions of WNM. (shrink)

En el presente trabajo se interpretan los sentidos teológicos de los términos bíblicos "carne" y "carnalidad" mostrando su carácter ambivalente que surge de binomios cargados tanto de connotaciones negativas como positivas. Se tratará de mostrar que las primeras parten de una interpretación parcial ..

As is well known, A. C. Haddon visited Torres Straits for the first time in the\nsummer of 1888 with the purpose of studying, as a marine biologist, the fauna\nand the structure and mode of formation of the coral reefs in Torres Straits. There\nbegan Haddon’s ’conversion’ from zoology to anthropology.’ It seems that\nHaddon felt an urgent need to collect ethnographic information on the islanders\nbecause he saw they were changing and diminishing in number very quickly, and\ntherefore their customs were vanishing.\nVery soon after (...) my arrival in the Straits I found that the natives of the\nislands had of late years been greatly reduced in number, and that, with the\nexception of but one or two individuals, none of the white residents knew\nanything about the customs of the natives, and not a single person cared\nabout them personally. When I began to question the natives I discovered\nthat the young men had a very imperfect acquaintance with the old habits\nand beliefs, and that only from the older men was reliable information to be\nobtained. So it was made clear to me that if I neglected to avail myself of the\npresent opportunity of collecting information on the ethnography of the\nislanders, it was extremely probable that that knowledge would never be\ngleaned - for if no one interested himself in the matter meanwhile, it was\nalmost certain that no trustworthy information could be collected in, say, ten years’ time. This being my opinion, I felt it my duty to fill up all the\ntime not actually employed in my zoological researches in anthropological\nstudies. (shrink)

A symmetric residuated lattice is an algebra such that is a commutative integral bounded residuated lattice and the equations x=x and =xy are satisfied. The aim of the paper is to investigate the properties of the unary operation ε defined by the prescription εx=x→0. We give necessary and sufficient conditions for ε being an interior operator. Since these conditions are rather restrictive →0)=1 is satisfied) we consider when an iteration of ε is an interior operator. In particular we consider the (...) chain of varieties of symmetric residuated lattices such that the n iteration of ε is a boolean interior operator. For instance, we show that these varieties are semisimple. When n=1, we obtain the variety of symmetric stonean residuated lattices. We also characterize the subvarieties admitting representations as subdirect products of chains. These results generalize and in many cases also simplify, results existing in the literature. (shrink)

This paper aims at being a systematic investigation of different completeness properties of first-order predicate logics with truth-constants based on a large class of left-continuous t-norms . We consider standard semantics over the real unit interval but also we explore alternative semantics based on the rational unit interval and on finite chains. We prove that expansions with truth-constants are conservative and we study their real, rational and finite chain completeness properties. Particularly interesting is the case of considering canonical real and (...) rational semantics provided by the algebras where the truth-constants are interpreted as the numbers they actually name. Finally, we study completeness properties restricted to evaluated formulae of the kind , where φ has no additional truth-constants. (shrink)

In this paper we show that the subvarieties of BL, the variety of BL-algebras, generated by single BL-chains on [0, 1], determined by continous t-norms, are finitely axiomatizable. An algorithm to check the subsethood relation between these subvarieties is provided, as well as another procedure to effectively find the equations of each subvariety. From a logical point of view, the latter corresponds to find the axiomatization of every residuated many-valued calculus defined by a continuous t-norm and its residuum. Actually, the (...) paper proves the results for a more general class than t-norm BL-chains, the so-called regular BL-chains. (shrink)

Our work is a contribution to the model theory of fuzzy predicate logics. In this paper we characterize elementary equivalence between models of fuzzy predicate logic using elementary mappings. Refining the method of diagrams we give a solution to an open problem of Hájek and Cintula (J Symb Log 71(3):863–880, 2006, Conjectures 1 and 2). We investigate also the properties of elementary extensions in witnessed and quasi-witnessed theories, generalizing some results of Section 7 of Hájek and Cintula (J Symb Log (...) 71(3):863–880, 2006) and of Section 4 of Cerami and Esteva (Arch Math Log 50(5/6):625–641, 2011) to non-exhaustive models. (shrink)

IMTL logic was introduced in [12] as a generalization of the infinitely-valued logic of Lukasiewicz, and in [11] it was proved to be the logic of left-continuous t-norms with an involutive negation and their residua. The structure of such t-norms is still not known. Nevertheless, Jenei introduced in [20] a new way to obtain rotation-invariant semigroups and, in particular, IMTL-algebras and left-continuous t-norm with an involutive negation, by means of the disconnected rotation method. In order to give an algebraic interpretation (...) to this construction, we generalize the concepts of perfect, bipartite and local algebra used in the classification of MV-algebras to the wider variety of IMTL-algebras and we prove that perfect algebras are exactly those algebras obtained from a prelinear semihoop by Jenei's disconnected rotation. We also prove that the variety generated by all perfect IMTL-algebras is the variety of the IMTL-algebras that are bipartite by every maximal filter and we give equational axiomatizations for it. (shrink)

Este trabajo trata de elaborar una hipótesis sobre el papel de Leibniz en la formación de las grandes categorías de la filosofía práctica moderna, poniendo de relieve una idea de racionalidad «hermenéutica» o «imperfecta» que, partiendo del concepto leibniziano de perspectivismo, constaría de tres grandes vertientes que aquí se denominan «principios»: de contingencia, autonomía y tolerancia. Lo que a Leibniz le interesa es asegurar una libertad autodeterminista de cuño rigurosamente moderno, subrayando que el universo de la praxis humana está presidido (...) por el [actum de lo contingente o de «aquello que puede ser de otra manera». Desde su concepto de individuación no se enfatiza la autonomía racional de un yo desencarnado ni mucho menos de un «autómata previsible», sino la consciencia de una subjetividad autodiferenciada que se propone a sí misma hipótesis viables para salvar la coherencia de sus acciones. El espíritu de tolerancia constituye la conclusión ética de las dos premisas anteriores: la idea general de justicia sólo es asequible desde el peculiar punto de vista de la place d'autruy, que viene a constituir el fundamento de aquélla. (shrink)

The study of perfect, local and bipartite IMTL-algebras presented in [29] is generalized in this paper to the general non-involutive case, i.e. to MTL-algebras. To this end we describe the radical of MTL-algebras and characterize perfect MTL-algebras as those for which the quotient by the radical is isomorphic to the two-element Boolean algebra, and a special class of bipartite MTL-algebras,.

In this paper we provide a finite axiomatization (using two finitary rules only) for the propositional logic (called $L\Pi$ ) resulting from the combination of Lukasiewicz and Product Logics, together with the logic obtained by from $L \Pi$ by the adding of a constant symbol and of a defining axiom for $\frac{1}{2}$ , called $L \Pi\frac{1}{2}$ . We show that $L \Pi \frac{1}{2}$ contains all the most important propositional fuzzy logics: Lukasiewicz Logic, Product Logic, Gödel's Fuzzy Logic, Takeuti and Titani's (...) Propositional Logic, Pavelka's Rational Logic, Pavelka's Rational Product Logic, the Lukasiewicz Logic with $\Delta$ , and the Product and Gödel's Logics with $\Delta$ and involution. Standard completeness results are proved by means of investigating the algebras corresponding to $L \Pi$ and $L \Pi \frac{1}{2}$ . For these algebras, we prove a theorem of subdirect representation and we show that linearly ordered algebras can be represented as algebras on the unit interval of either a linearly ordered field, or of the ordered ring of integers, Z. (shrink)