In this paper two deductive systems associated with relevance logic are studied from an algebraic point of view. One is defined by the familiar, Hilbert-style, formalization of R; the other one is a weak version of it, called WR, which appears as the semantic entailment of the Meyer-Routley-Fine semantics, and which has already been suggested by Wójcicki for other reasons. This weaker consequence is first defined indirectly, using R, but we prove that the first one turns out to be an (...) axiomatic extension of WR. Moreover we provide WR with a natural Gentzen calculus. It is proved that both deductive systems have the same associated class of algebras but different classes of models on these algebras. The notion of model used here is an abstract logic, that is, a closure operator on an abstract algebra; the abstract logics obtained in the case of WR are also the models, in a natural sense, of the given Gentzen calculus. (shrink)

This is a contribution to the discussion on the role of truth degrees in manyvalued logics from the perspective of abstract algebraic logic. It starts with some thoughts on the so-called Suszko’s Thesis (that every logic is two-valued) and on the conception of semantics that underlies it, which includes the truth-preserving notion of consequence. The alternative usage of truth values in order to define logics that preserve degrees of truth is presented and discussed. Some recent works studying these in the (...) particular cases of Łukasiewicz’s many-valued logics and of logics associated with varieties of residuated lattices are also presented. Finally the extension of this paradigm to other, more general situations is discussed, highlighting the need for philosophical or applied motivations in the selection of the truth degrees, due both to the interpretation of the idea of truth degree and to some mathematical difficulties. (shrink)

A filter of a sentential logic ? is Leibniz when it is the smallest one among all the ?-filters on the same algebra having the same Leibniz congruence. This paper studies these filters and the sentential logic ?+ defined by the class of all ?-matrices whose filter is Leibniz, which is called the strong version of ?, in the context of protoalgebraic logics with theorems. Topics studied include an enhanced Correspondence Theorem, characterizations of the weak algebraizability of ?+ and of (...) the explicit definability of Leibniz filters, and several theorems of transfer of metalogical properties from ? to ?+. For finitely equivalential logics stronger results are obtained. Besides the general theory, the paper examines the examples of modal logics, quantum logics and Łukasiewicz's finitely-valued logics. One finds that in some cases the existence of a weak and a strong version of a logic corresponds to well-known situations in the literature, such as the local and the global consequences for normal modal logics; while in others these constructions give an independent interest to the study of other lesser-known logics, such as the lattice-based many-valued logics. (shrink)

ukasiewicz''s four-valued modal logic is surveyed and analyzed, together with ukasiewicz''s motivations to develop it. A faithful interpretation of it in classical (non-modal) two-valued logic is presented, and some consequences are drawn concerning its classification and its algebraic behaviour. Some counter-intuitive aspects of this logic are discussed in the light of the presented results, ukasiewicz''s own texts, and related literature.

Łukasiewicz’s infinite-valued logic is commonly defined as the set of formulas that take the value 1 under all evaluations in the Łukasiewicz algebra on the unit real interval. In the literature a deductive system axiomatized in a Hilbert style was associated to it, and was later shown to be semantically defined from Łukasiewicz algebra by using a “truth-preserving” scheme. This deductive system is algebraizable, non-selfextensional and does not satisfy the deduction theorem. In addition, there exists no Gentzen calculus fully adequate (...) for it. Another presentation of the same deductive system can be obtained from a substructural Gentzen calculus. In this paper we use the framework of abstract algebraic logic to study a different deductive system which uses the aforementioned algebra under a scheme of “preservation of degrees of truth”. We characterize the resulting deductive system in a natural way by using the lattice filters of Wajsberg algebras, and also by using a structural Gentzen calculus, which is shown to be fully adequate for it. This logic is an interesting example for the general theory: it is selfextensional, non-protoalgebraic, and satisfies a “graded” deduction theorem. Moreover, the Gentzen system is algebraizable. The first deductive system mentioned turns out to be the extension of the second by the rule of Modus Ponens. (shrink)

This paper contains a joint study of two sentential logics that combine a many-valued character, namely tetravalence, with a modal character; one of them is normal and the other one quasinormal. The method is to study their algebraic counterparts and their abstract models with the tools of Abstract Algebraic Logic, and particularly with those of Brown and Suszko's theory of abstract logics as recently developed by Font and Jansana in their "A General Algebraic Semantics for Sentential Logics". The logics (...) studied here arise from the algebraic and lattice-theoretical properties we review of Tetravalent Modal Algebras, a class of algebras studied mainly by Loureiro, and also by Figallo, Landini and Ziliani, at the suggestion of the late Antonio Monteiro. (shrink)

A definition and some inaccurate cross-references in the paper A Survey of Abstract Algebraic Logic, which might confuse some readers, are clarified and corrected; a short discussion of the main one is included. We also update a dozen of bibliographic references.

In this paper we consider the structure of the class FGModS of full generalized models of a deductive system S from a universal-algebraic point of view, and the structure of the set of all the full generalized models of S on a fixed algebra A from the lattice-theoretical point of view; this set is represented by the lattice FACSs A of all algebraic closed-set systems C on A such that (A, C) ε FGModS. We relate some properties of these structures (...) with tipically logical properties of the sentential logic S. The main algebraic properties we consider are the closure of FGModS under substructures and under reduced products, and the property that for any A the lattice FACSs A is a complete sublattice of the lattice of all algebraic closed-set systems over A. The logical properties are the existence of a fully adequate Gentzen system for S, the Local Deduction Theorem and the Deduction Theorem for S. Some of the results are established for arbitrary deductive systems, while some are found to hold only for deductive systems in more restricted classes like the protoalgebraic or the weakly algebraizable ones. The paper ends with a section on examples and counterexamples. (shrink)

The “representation problem” in abstract algebraic logic is that of finding necessary and sufficient conditions for a structure, on a well defined abstract framework, to have the following property: that for every structural closure operator on it, every structural embedding of the expanded lattice of its closed sets into that of the closed sets of another structural closure operator on another similar structure is induced by a structural transformer between the base structures. This question arose from Blok and Jónsson abstract (...) analysis of one of Blok and Pigozzis’s characterizations of algebraizable logics. The problem, which was later on reformulated independently by Gil-Férez and by Galatos and Tsinakis, was solved by Galatos and Tsinakis in the more abstract framework of the category of modules over a complete residuated lattice, and by Galatos and Gil-Férez in the even more abstract setting of modules over a quantaloid. We solve the representation problem in Blok and Jónsson’s original context of M-sets, where M is a monoid, and characterise the corresponding M-sets both in categorical terms and in terms of their inner structure, using the notions of a graded M-set and a generalized variable introduced by Gil-Férez. (shrink)

This paper reviews the impact of Rasiowa's well-known book on the evolution of algebraic logic during the last thirty or forty years. It starts with some comments on the importance and influence of this book, highlighting some of the reasons for this influence, and some of its key points, mathematically speaking, concerning the general theory of algebraic logic, a theory nowadays called Abstract Algebraic Logic. Then, a consideration of the diverse ways in which these key points can be generalized allows (...) us to survey some issues in the development of the field in the last twenty to thirty years. The last part of the paper reviews some recent lines of research that in some way transcend Rasiowa's approach. I hope in this way to give the reader a general view of Rasiowa's key position in the evolution of Algebraic Logic during the twentieth century. (shrink)

A pair of deductive systems (S,S’) is Leibniz-linked when S’ is an extension of S and on every algebra there is a map sending each filter of S to a filter of S’ with the same Leibniz congruence. We study this generalization to arbitrary deductive systems of the notion of the strong version of a protoalgebraic deductive system, studied in earlier papers, and of some results recently found for particular non-protoalgebraic deductive systems. The necessary examples and counterexamples found in the (...) literature are described. (shrink)

In her well-known book, Rasiowa states without proof that in implicative algebras there is a one-to-one correspondence between kernels of epimorphisms and the so-called special implicative filters, and that in the logic whose algebraic counterpart is the class of implicative algebras the deductive filters coincide with the special implicative filters. We show that neither claim is true, and how to repair the situation by redefining some of the notions involved. We answer other questions concerning special implicative filters, taking the theory (...) of algebraizable logics of Blok and Pigozzi as a framework to approach the question in a systematic way. (shrink)

This paper presents a unified framework that explains and extends the already successful applications of the Leibniz operator, the Suszko operator, and the Tarski operator in recent developments in abstract algebraic logic. To this end, we refine Czelakowski’s notion of an S-compatibility operator, and introduce the notion of coherent family of S-compatibility operators, for a sentential logic S. The notion of coherence is a restricted property of commutativity with inverse images by surjective homomorphisms, which is satisfied by both the Leibniz (...) and the Suszko operators. We generalize several constructions and results already existing for the mentioned operators; in particular, the well-known classes of algebras associated with a logic through each of them, and the notions of full generalized model of a logic and a special kind of S-filters. We obtain a General Correspondence Theorem, extending the well-known one from the theory of protoalgebraic logics to arbitrary logics and to more general operators, and strengthening its formulation. We apply the general results to the Leibniz and the Suszko operators, and obtain several characterizations of the main classes of logics in the Leibniz hierarchy by the form of their full generalized models, by old and new properties of the Leibniz operator, and by the behaviour of the Suszko operator. Some of these characterizations complete or extend known ones, for some classes in the hierarchy, thus offering an integrated approach to the Leibniz hierarchy that uncovers some new, nice symmetries. (shrink)

This paper explores a notion of “the strong version” of a sentential logic S, initially defined in terms of the notion of a Leibniz filter, and shown to coincide with the logic determined by the matrices of S whose filter is the least S-filter in the algebra of the matrix. The paper makes a general study of this notion, which appears to unify under an abstract framework the relationships between many pairs of logics in the literature. The paradigmatic examples are (...) the local and the global consequences associated with a normal modal logic, and the logics preserving degrees of truth and preserving truth associated with certain substructural and many-valued logics. For protoalgebraic logics the results in the paper coincide with those obtained by two of the authors in 2001, so the main novelty of the approach is its suitability for all kinds of logics. The paper also studies three kinds of definability of the Leibniz filters, and their consequences for the determination of the strong version. In a second part of the paper several case studies are developed, comprising positive modal logic, Dunn–Belnap’s four-valued logic, the large family of substructural logics, and some relevance logics. (shrink)

This paper explores a notion of “the strong version” of a sentential logic S, initially defined in terms of the notion of a Leibniz filter, and shown to coincide with the logic determined by the matrices of S whose filter is the least S-filter in the algebra of the matrix. The paper makes a general study of this notion, which appears to unify under an abstract framework the relationships between many pairs of logics in the literature. The paradigmatic examples are (...) the local and the global consequences associated with a normal modal logic, and the logics preserving degrees of truth and preserving truth associated with certain substructural and many-valued logics. For protoalgebraic logics the results in the paper coincide with those obtained by two of the authors in 2001, so the main novelty of the approach is its suitability for all kinds of logics. The paper also studies three kinds of definability of the Leibniz filters, and their consequences for the determination of the strong version. In a second part of the paper several case studies are developed, comprising positive modal logic, Dunn–Belnap’s four-valued logic, the large family of substructural logics, and some relevance logics. (shrink)

This paper studies, with techniques ofAlgebraic Logic, the effects of putting a bound on the cardinality of the set of side formulas in the Deduction Theorem, viewed as a Gentzen-style rule, and of adding additional assumptions inside the formulas present in Modus Ponens, viewed as a Hilbert-style rule. As a result, a denumerable collection of new Gentzen systems and two new sentential logics have been isolated. These logics are weaker than the positive implicative logic. We have determined their algebraic models (...) and the relationships between them, and have classified them according to several standard criteria of Abstract Algebraic Logic. One of the logics is protoalgebraic but neither equivalential nor weakly algebraizable, a rare situation where very few natural examples were hitherto known. In passing we have found new, alternative presentations of positive implicative logic, both in Hilbert style and in Gentzen style, and have characterized it in terms of the restricted Deduction Theorem: it is the weakest logic satisfying Modus Ponens and the Deduction Theorem restricted to at most 2 side formulas. The algebraic part of the work has lead to the class of quasi-Hilbert algebras, a quasi-variety of implicative algebras introduced by Pla and Verdú in 1980, which is larger than the variety of Hilbert algebras. Its algebraic properties reflect those of the corresponding logics and Gentzen systems. (shrink)

The Intimate Resistance is a keen, deeply beautiful reflection on the human condition. The author explains howwe ourselves can warm, protect and guide those around us.

La filosofía fue entre nosotros, durante largo tiempo, un “juego social”, en las últimas décadas se ha venido transformando en (pseudo)campo, y en la actualidad está expuesta a sucumbir a la heteronomía, decayendo a la degradada situación de “espacio de servicios”. Así la doctrina de Bourdieu esclarece el surgimiento, la situación actual, y en cierto modo también la peripecia futura de la filosofía en España. Y al mismo tiempo consigue explicar porque, en este preciso momento, es plausible el pesimista diagnóstico (...) de que el peligro de regresión a la heteronomía compromete la existencia del incipiente campo filosófico español. Simplemente ocurre que en las situaciones de crisis las percepciones de los agentes adquieren una relevancia de la que suelen carecer en etapas más apacibles. En los períodos de desilusión los agentes se vuelven lúcidos, en una palabra, porque entonces resulta más difícil “seguir el juego” que el campo les exige. Ya que tomar parte en dicho juego no es más, según Bourdieu, que una forma de ilusión. (shrink)

Daniel Bensaïd was a Marxist philosopher and author of an extensive body of works about political strategy. His writings combine a diversity of singular influences, such as Marx, Lenin, Trotsky and Che Guevara on the one hand, and Benjamin, Péguy and Blanqui on the other. In his work, religious heresies, Marranos, moles and emblematic figures of the resistance to oppression such as Joan of Arc meet with the classic figures of Marxism. The non-linear concept of time and messianic reason support (...) a strategic reading of history and an understanding of political commitment, following Goldmann’s interpretation of Pascal’s Wager as a wager of uncertain outcome. (shrink)

First, this article summarizes the way Jan Patocˇka thinks and updates the classic theme of philosophy as care for the soul. Then goes on to show Jacques Derrida’s significant and unexpected interest in the Czech thinker, and the manner as sharp and respectful as he reads it. Derrida argues that the core of Patocˇka’s approach, under the term “responsibility”, coincides in some sense with that of Emmanuel Levinas; nucleus that has to do, in both authors, with the essence of “religion”.

The centenary of the birth of Ludwig Wittgenstein provided an opportunity for recovering some of the great Wittgensteinian subjects, for re-examining them and for discussing their implications and relevance. This volume is the result of the interchange that took place in Girona among well-known scholars of Wittgenstein's work in different countries. The eleven contributions are organized into three main subjects: on Wittgenstein's method, on knowledge and meaning, on language and use. This volume is not only the result of the different (...) and mature reflexions of its authors, but in two cases it has also unfortunately become their last contribution to the subject. (shrink)

The centenary of the birth of Ludwig Wittgenstein (1889-1951) provided an opportunity for recovering some of the great Wittgensteinian subjects, for re-examining them and for discussing their implications and relevance. This volume is the result of the interchange that took place in Girona (Spain) among well-known scholars of Wittgenstein's work in different countries. The eleven contributions are organized into three main subjects: on Wittgenstein's method (B. McGuinness, E. Tugendhat and J.M. Terricabras), on knowledge and meaning (G.E.M. Anscombe, R. Bambrough, N. (...) Malcolm, and P.T. Geach), on language and use (D. Pears, E.v. Savigny, J. Bouveresse and J. Ferrater Mora). This volume is not only the result of the different and mature reflexions of its authors, but in two cases (Malcolm and Ferrater Mora) it has also unfortunately become their last contribution to the subject. (shrink)

En la gran producció filosòfica d’Eusebi Colomer, s’hi poden distingir tres grans centres d’interès particularment destacats. En primer lloc, la seva dedicació a l’estudi del pensament de Ramon Llull i de Nicolau de Cusa. La seva tesi doctoral va mostrar les influències lul·lianes en l’obra de Nicolau de Cusa a partir de les anotacions autògrafes en què el jove Nicolau es feia ressò de les coses que més li interessaven de Llull. El segon gran centre d’interès el van constituir, naturalment, (...) els grans corrents de la filosofia alemanya moderna i contemporània. Des dels anys d’estudiant s’havia familiaritzat particularment amb l’idealisme, la fenomenologia i la filosofia de l’existència. Finalment, el seu tercer centre d’interès va ser el diàleg filosòfico-teològic. Colomer lamentava profundament el divorci produït entre la filosofia i la teologia. Adaptant una coneguda formulació de Kant, pensava que la teologia sense filosofia és cega, però que la filosofia sense teologia és buida. Format en la tradició metafísica espiritualista i defensor de l’humanisme cristià, Colomer es trobava, però, a les antípodes de qualsevol sectarisme o dogmatisme. La seva formació, el seu tarannà i, segurament, els seus enormes coneixements de la història de la filosofia, li havien conferit una gran capacitat per al diàleg. Ell no veia el diàleg com un recurs o com una tàctica, sinó com un element constitutiu de l’ésser humà. \n. (shrink)

Quins són els valors que regeixen la nostra vida? I per què vivim amb aquests valors i no amb uns altres? «Els valors van cap a on anem nosaltres perquè van amb nosaltres. ¬I mentre fem camí o mentre el busquem, els valors ens alimenten», diu el filòsof.

This paper has a very limited scope: it just presents some eIements which may introduce us into a wider investigation on vague predicates. Thus, webegin by pointing out our general conception of Iogic (as a descriptive science of linguistic phenomena). After that, we build up our approach by considering characteristic functions and by trying to make useful the model of logical extension which applies to classical quantifiers.

Bensaïd’s interest in Marranism is part of his broader interest in Jewish mysticism, read in a profane and secularised way, and of his search for new theoretical paths with which to renew revolutionary Marxist theory. ‘Marrano’ refers to the Spanish–Portuguese Jews who were forcibly converted to Christianity in the fifteenth century and who were suspected of judaising in secret. The term has been increasingly used by many authors, including Bensaïd, in a broad sense, often as a metaphor that goes beyond (...) the study of actual Marranos to acquire a broader meaning. Bensaïd uses Marranism to think about some aspects of political strategy and, at the same time, strategically uses Marranism to approach certain debates. The Marrano is the metaphorical and metonymic figure through which he tries to think a new internationalism that simultaneously transcends both an abstract universalism that legitimates inequalities and oppression and an anti-universalist communitarian withdrawal. (shrink)

En este trabajo se analizan las cláusulas de liquidez constitucional (CLC), tal como han sido caracterizadas por José María Sauca. Sobre los rasgos estructurales, se sostiene, primero, que es difícil reconducir los ejemplos de CLC a un solo modelo, por cuanto, una vez introducida la distinción entre reglas prescriptivas y constitutivas, caben seis posibles reconstrucciones; segundo, que el principio de prevalencia, lejos de suponer que no hay contradicciones entre las normas que puedan emanar de las CLC y la normativa constitucional, (...) las presuponen; tercero, que mantener que las CLC implican una transferencia de soberanía solo puede obedecer a un uso ambiguo del término “soberanía”. Por lo que hace a las funciones, y en contra de lo sostenido por Sauca, se muestra que las CLC pueden conducir a mutaciones constitucionales y es factible considerarlas mecanismos de última palabra. (shrink)

The shortcomings of Merleau-Ponty’s crucial concept of “institution” led him into a turning point where the way out he selected was actually much inferior, as regards the prospects of fulfillment that were latent in his thought, to the alternative path he did not take.