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.

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)

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)

This paper is a contribution to Mathematical fuzzy logic, in particular to the algebraic study of t-norm based fuzzy logics. In the general framework of propositional core and Δ-core fuzzy logics we consider three properties of completeness with respect to any semantics of linearly ordered algebras. Useful algebraic characterizations of these completeness properties are obtained and their relations are studied. Moreover, we concentrate on five kinds of distinguished semantics for these logics–namely the class of algebras defined over the real unit (...) interval, the rational unit interval, the hyperreals , the strict hyperreals and finite chains, respectively–and we survey the known completeness methods and results for prominent logics. We also obtain new interesting relations between the real, rational and hyperreal semantics, and good characterizations for the completeness with respect to the semantics of finite chains. Finally, all completeness properties and distinguished semantics are also considered for the first-order versions of the logics where a number of new results are proved. (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)

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)

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)

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)

This article proposes the meeting of fuzzy logic with paraconsistency in a very precise and foundational way. Specifically, in this article we introduce expansions of the fuzzy logic MTL by means of primitive operators for consistency and inconsistency in the style of the so-called Logics of Formal Inconsistency (LFIs). The main novelty of the present approach is the definition of postulates for this type of operators over MTL-algebras, leading to the definition and axiomatization of a family of logics, expansions of (...) MTL, whose degree-preserving counterpart are paraconsistent and moreover LFIs. (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)

In this paper we study and equationally characterize the subvarieties of BL, the variety of BL-algebras, which are generated by families of single-component BL-chains, i.e. MV-chains, Product-chain or Gödel-chains. Moreover, it is proved that they form a segment of the lattice of subvarieties of BL which is bounded by the Boolean variety and the variety generated by all single-component chains, called ŁΠG.

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)

In this paper we prove strong completeness of axiomatic extensions of first-order strict core fuzzy logics with the so-called quasi-witnessed axioms with respect to quasi-witnessed models. As a consequence we obtain strong completeness of Product Predicate Logic with respect to quasi-witnessed models, already proven by M.C. Laskowski and S. Malekpour in [19]. Finally we study similar problems for expansions with Δ, define Δ-quasi-witnessed axioms and prove that any axiomatic extension of a first-order strict core fuzzy logic, expanded with Δ, and (...) Δ-quasi-witnessed axioms are complete with respect to Δ-quasi-witnessed models. (shrink)

In this paper we study intermediate logics between the logic G≤∼, the degree preserving companion of Gödel fuzzy logic with involution G∼ and classical propositional logic CPL, as well as the intermediate logics of their finite-valued counterparts G≤n∼. Although G≤∼ and G≤ are explosive w.r.t. Gödel negation ¬, they are paraconsistent w.r.t. the involutive negation ∼. We introduce the notion of saturated paraconsistency, a weaker notion than ideal paraconsistency, and we fully characterize the ideal and the saturated paraconsistent logics between (...) G≤n∼ and CPL. We also identify a large family of saturated paraconsistent logics in the family of intermediate logics for degree-preserving finite-valued Łukasiewicz logics. (shrink)

In this paper we provide a simplified, possibilistic semantics for the logics K45, i.e. a many-valued counterpart of the classical modal logic K45 over the [0, 1]-valued Gödel fuzzy logic \. More precisely, we characterize K45 as the set of valid formulae of the class of possibilistic Gödel frames \, where W is a non-empty set of worlds and \ is a possibility distribution on W. We provide decidability results as well. Moreover, we show that all the results also apply (...) to the extension of K45 with the axiom, provided that we restrict ourselves to normalised Gödel Kripke frames, i.e. frames \ where \ satisfies the normalisation condition \ = 1\). (shrink)

This conceptual paper explores and compares the ethical dimension of Islamic and so-called ethical banks. The investigation is made in two succeeding steps. First, an individual analysis as regards the respective level of correspondence between ethical principles and business practice. For the latter, a time-based perspective is adopted. Second, a side-by-side comparison of their overall “ethical coherence gap”. The results show that this gap is wider in the case of Islamic banks. The final part of the paper draws up three (...) different scenarios for future development of these financial institutions where their ethical coherence is enhanced. (shrink)

What are financial institutions' social responsibilities in developing countries? On the one hand, these institutions share the generic responsibilities of all human organizations and business enterprises. However, their specific social responsibility is the performance of the social function of financial intermediaries, which, in the case of emerging countries, consists mainly of contributing to economic growth and solving the problem of poverty. This paper describes a number of technical-economic and moral problems that take us to a consideration of the performance of (...) banking operations in microfinancing, with special reference to Latin America. The paper also provides a series of recommendations that, in addition to contributing to solving the development and poverty problems in emerging countries, help define financial institutions' social responsibility in such countries. (shrink)

What are financial institutions' social responsibilities in developing countries? On the one hand, these institutions share the generic responsibilities of all human organizations and business enterprises. However, their specific social responsibility is the performance of the social function of financial intermediaries, which, in the case of emerging countries, consists mainly of contributing to economic growth and solving the problem of poverty. This paper describes a number of technical‐economic and moral problems that take us to a consideration of the performance of (...) banking operations in microfinancing, with special reference to Latin America. The paper also provides a series of recommendations that, in addition to contributing to solving the development and poverty problems in emerging countries, help define financial institutions' social responsibility in such countries. (shrink)

The purpose of this article is to present and discuss some of the best practices of financial industry, in three emerging economies: Colombia, Ecuador, and Peru. The main thesis is that, notwithstanding the importance of certain specific deficiencies, such as an inadequate regulatory context or the lack of financial education among the population, the main factor that explains the low banking levels in emerging and developing economies, affecting mostly lower-income segments, is the use of inefficient financial service distribution models. In (...) connection with this thesis, we will try to show that traditional financial institutions, both in developing countries and in the advanced economies have a special social responsibility to help create an efficient financial system that makes saving and borrowing instruments available to the greatest possible number of citizens. (shrink)

The biblical source mentions on three occasions the nautical journeys of Solomon, king of Judah and Israel, and Hiram of Tyre to Ophir in search of riches. Throughout the study, we point out that the mention of the journeys constitutes a theological construction, composed between the twilight of the 6th century BC and the first half of the 5th century BC, to magnify, in contrast with the nautical failure of Jehoshaphat king of Judah, the figure of Solomon as the prototype (...) of the ideal monarch to whom pagan nations, represented by the Phoenicians, render servitude,. The later biblical tradition, manifested by the Greek, Latin and Arabic translation, will continue to magnify the figure of the monarch, placing the location of Ophir on an increasingly more distant horizon. (shrink)

El trabajo parte de la distinción teórica entre nación en sentido jurídico-político y nación en sentido histórico-cultural. Sobre esa base, se trata de averiguar cuál es el concepto de nación que utiliza la Constitución española. Del análisis jurídico-constitucional efectuado, se concluye que la palabra nación se utiliza en el sentido jurídico-político y se equipara a pueblo, a patria, a España o a Estado, aunque cabe hacer la distinción entre nación-constituyente y nación-constituida para encontrar el significado preciso de todos estos términos. (...) En cambio, el concepto de nación en sentido histórico-cultural es utilizado parcialmente en la Constitución —si nos fundamos en la voluntad del constituyente— bajo el término nacionalidades o regiones, aunque limitando las aspiraciones de éstas —por ello decimos parcialmente— al derecho a la autonomía y no al derecho a la soberanía. (shrink)

Globalisation has given rise to a new field of debate due to the growing mobility of the workers and the consequent cultural diversity inside companies. In this sense, a complex world like ours demands a review of the professional profile which should include the so called intercultural competences as a structuring element of companies’ policy and strategic plan. Subsequently it is suggested that, as the intercultural competences affect the cognitive, affective and behavioural aspects, they actually imply a transformation of the (...) person’s identity. (shrink)

This article aims to find out the relation between secular and atheistic spiritualties underlining the only one that seems to be able to suit the knowledge society. We analyze several texts of main classical personalities of both spiritualities, such as Feuerbach, Marx, Nietzsche and Comte Sponville, among others. Some of the essential elements of an atheistic spirituality are specifically discussed: critical spirit and rationality, centrality and sufficiency of human beings and their release from submission and exploitation, and a will without (...) limits instead of laziness. However, the main objective is to highlight that there is a culture of absolute dimension of reality beyond the religious traditions. We also intend to point out that culture, in changing societies, requires, as Corbí suggests, a new anthropology, of an animal who talks, with a dual access to reality. We also point to the existence of a new epistemology, beyond the previous mythical paradigms, allowing the merely free and symbolic use of mankind wisdoms. Finally, it is a question of survival of humankind and of its environment. (shrink)

Aquí se presenta la colección de volúmenes que recogen las ponencias y discusiones de los, hasta hoy, nueve Encuentros Internacionales organizados por el Centro de Estudio de las Tradiciones de Sabiduría de Barcelona. Un requisito principal en la realización de los Encuentros ha sido un marcado carácter interdisciplinar, interreligioso y espiritual. El objetivo general que alumbra las diferentes ediciones ha sido abordar el papel vital, para la especie y para el medio, de la cualidad humana profunda o espiritualidad en sociedades (...) de cambio continuo. Todo ello bajo la preocupación por la crisis religiosa con consecuencias sobre los valores, que se sufre en Europa, que se extiende a todos los países, a medida que se incorporan a la sociedad que vive de la innovación y el cambio continuo. A pesar de que los países de procedencia de los diferentes participantes sufren la crisis religiosa en diverso grado, gracias al intenso trabajo realizado en las diferentes ediciones, se ha conseguido aunar el interés de los participantes en torno al objetivo general. También se ha conseguido llegar a ciertos acuerdos sobre las bases teóricas desde las cuales cada uno construye sus aportaciones a los temas concretos de cada edición. Así pues, podemos decir que se ha conseguido generar un equipo estable de doce investigadores, muchos de ellos profesores universitarios de España así como de México, Costa Rica, Brasil, Colombia, que trabaja para ir construyendo una respuesta al cambio epistemológico que sufren las sociedades actuales y que crea la inadecuación de la religión. (shrink)

The framework presented by Bayefsky and Berkman is based on having clear and accurate genetic information to offer parents, for them to either decide to prepare for birth or to terminate the...

En 1935 Baltasar Samper pronunció varias conferencias sobre jazz en Barcelona. La recuperación de los textos de dichas conferencias nos sirven ahora para resaltar la dimensión didáctica del compositor en un contexto cultural marcado por el academicismo. Sin embargo, los esfuerzos de Samper por divulgar la música de jazz entre sus colegas, quienes consideraban el jazz una música estigmatizada, dará lugar a unos contenidos a la altura de los postulados del parisino Hugues Panassié. Este artículo concluye constatando la vigencia de (...) las enseñanzas samperianas sobre jazz y apuntando su continuidad en contextos didácticos actuales. (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)

The concept of the Noosphere is of great importance when looking at the values underpinning the technocratic artifacts and technocracies (human physical technological objects and knowledge processes) by which Humans relate to the Geosphere through other human beings. In this sense, the Noosphere may inform geoethics as an environmental, social, and spiritual praxis and thinking aiming at ecological justice. The concept of the Noosphere represents the coexistence and coevolution of Humans and the Geosphere, overcoming the dichotomy between instrumental materialistic and (...) intrinsic ecocentric values but considering the meaning of a constitutive dimension. Thus, the Noosphere becomes a concept for reconnection with the human community, the natural world, and the Divine, and develops into an ecological mysticism that, in turn, unfolds in resistance in hope as a kind of spiritual activism. The theoretical framework is illustrated with the case study of the Terres de L’Ebre in Catalonia (Spain). (shrink)