Our concern is the completeness problem for spi-logics, that is, sets of implications between strictly positive formulas built from propositional variables, conjunction and modal diamond operators. Originated in logic, algebra and computer science, spi-logics have two natural semantics: meet-semilattices with monotone operators providing Birkhoff-style calculi and first-order relational structures (aka Kripke frames) often used as the intended structures in applications. Here we lay foundations for a completeness theory that aims to answer the question whether the two semantics define the same (...) consequence relations for a given spi-logic. (shrink)

We prove that the two-variable fragment of first-order intuitionistic logic is undecidable, even without constants and equality. We also show that the two-variable fragment of a quantified modal logic L with expanding first-order domains is undecidable whenever there is a Kripke frame for L with a point having infinitely many successors (such are, in particular, the first-order extensions of practically all standard modal logics like K, K4, GL, S4, S5, K4.1, S4.2, GL.3, etc.). For many quantified modal logics, including those (...) in the standard nomenclature above, even the monadic two-variable fragments turn out to be undecidable. (shrink)

We solve a major open problem concerning algorithmic properties of products of ‘transitive’ modal logics by showing that products and commutators of such standard logics as K4, S4, S4.1, K4.3, GL, or Grz are undecidable and do not have the finite model property. More generally, we prove that no Kripke complete extension of the commutator [K4,K4] with product frames of arbitrary finite or infinite depth (with respect to both accessibility relations) can be decidable. In particular, if.

We show the first examples of recursively enumerable (even decidable) two-dimensional products of finitely axiomatisable modal logics that are not finitely axiomatisable. In particular, we show that any axiomatisation of some bimodal logics that are determined by classes of product frames with linearly ordered first components must be infinite in two senses: It should contain infinitely many propositional variables, and formulas of arbitrarily large modal nesting-depth.

Built on the foundations laid by Peirce, Schröder, and others in the 19th century, the modern development of relation algebras started with the work of Tarski and his colleagues [21, 22]. They showed that relation algebras can capture strong first‐order theories like ZFC, and so their equational theory is undecidable. The less expressive class WA of weakly associative relation algebras was introduced by Maddux [7]. Németi [16] showed that WA's have a decidable universal theory. There has been extensive research on (...) increasing the expressive power of WA by adding new operations [1, 4, 11, 13, 20]. Extensions of this kind usually also have decidable universal theories. Here we give an example – extending WA's with set‐theoretic projection elements – where this is not the case. These “logical” connectives are set‐theoretic counterparts of the axiomatic quasi‐projections that have been investigated in the representation theory of relation algebras [22, 6, 19]. We prove that the quasi‐equational theory of the extended class PWA is not recursively enumerable. By adding the difference operator D one can turn WA and PWA to discriminator classes where each universal formula is equivalent to some equation. Hence our result implies that the projections turn the decidable equational theory of “WA + D ” to non‐recursively enumerable (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

Our concern is the axiomatisation problem for modal and algebraic logics that correspond to various fragments of two-variable first-order logic with counting quantifiers. In particular, we consider modal products with Diff, the propositional unimodal logic of the difference operator. We show that the two-dimensional product logic $Diff \times Diff$ is non-finitely axiomatisable, but can be axiomatised by infinitely many Sahlqvist axioms. We also show that its ‘square’ version (the modal counterpart of the substitution and equality free fragment of two-variable first-order (...) logic with counting to two) is non-finitely axiomatisable over $Diff \times Diff$ , but can be axiomatised by adding infinitely many Sahlqvist axioms. These are the first examples of products of finitely axiomatisable modal logics that are not finitely axiomatisable, but axiomatisable by explicit infinite sets of canonical axioms. (shrink)

There are two known general results on the finite model property of commutators [L0,L1]. If L is finitely axiomatizable by modal formulas having universal Horn first-order correspondents, then both [L,K] and [L,S5] are determined by classes of frames that admit filtration, and so they have the fmp. On the negative side, if both L0 and L1 are determined by transitive frames and have frames of arbitrarily large depth, then [L0,L1] does not have the fmp. In this paper we show that (...) commutators with a “weakly connected” component often lack the fmp. Our results imply that the above positive result does not generalize to universally axiomatizable component logics, and even commutators without “transitive” components such as [K3,K] can lack the fmp. We also generalize the above negative result to cases where one of the component logics has frames of depth one only, such as [S4.3,S5] and the decidable product logic S4.3×S5. We also show cases when already half of commutativity is enough to force infinite frames. (shrink)

We show that—unlike products of ‘transitive’ modal logics which are usually undecidable—their ‘expanding domain’ relativisations can be decidable, though not in primitive recursive time. In particular, we prove the decidability and the finite expanding product model property of bimodal logics interpreted in two-dimensional structures where one component—call it the ‘flow of time’—is • a finite linear order or a finite transitive tree and the other is composed of structures like • transitive trees/partial orders/quasi-orders/linear orders or only finite such structures expanding (...) over time. The decidability proof is based on Kruskal’s tree theorem, and the proof of non-primitive recursiveness is by reduction of the reachability problem for lossy channel systems. The result is used to show that the dynamic topological logic interpreted in topological spaces with continuous functions is decidable if the number of function iterations is assumed to be finite. (shrink)

We solve a major open problem concerning algorithmic properties of products of ‘transitive’ modal logics by showing that products and commutators of such standard logics asK4,S4,S4.1,K4.3,GL, orGrzare undecidable and do not have the finite model property. More generally, we prove that no Kripke complete extension of the commutator [K4, K4] with product frames of arbitrary finite or infinite depth (with respect to both accessibility relations) can be decidable. In particular, ifl1andl2are classes of transitive frames such that their depth cannot be (...) bounded by any fixedn< ω, then the logic of the class {5ℑ1× ℑ2∣ ℑ1∈l1, ℑ2, ∈l2} is undecidable. (On the contrary, the product of, say,K4and the logic of all transitive Kripke frames of depth ≤n, for some fixedn< ω, is decidable.) The complexity of these undecidable logics ranges from r.e. to co-r.e. and Π11-complete. As a consequence, we give the first known examples of Kripke incomplete commutators of Kripke complete logics. (shrink)

The paper sets out to offer an alternative to the function/argument approach to the most essential aspects of natural language meanings. That is, we question the assumption that semantic completeness (of, e.g., propositions) or incompleteness (of, e.g., predicates) exactly replicate the corresponding grammatical concepts (of, e.g., sentences and verbs, respectively). We argue that even if one gives up this assumption, it is still possible to keep the compositionality of the semantic interpretation of simple predicate/argument structures. In our opinion, compositionality presupposes (...) that we are able to compare arbitrary meanings in term of information content. This is why our proposal relies on an ‘intrinsically’ type free algebraic semantic theory. The basic entities in our models are neither individuals, nor eventualities, nor their properties, but ‘pieces of evidence’ for believing in the ‘truth’ or ‘existence’ or ‘identity’ of any kind of phenomenon. Our formal language contains a single binary non-associative constructor used for creating structured complex terms representing arbitrary phenomena. We give a finite Hilbert-style axiomatisation and a decision algorithm for the entailment problem of the suggested system. (shrink)

The practice of relational leadership is essential for dealing with the increasingly urgent and complex social, economic and environmental issues that characterize sustainability. Despite growing attention to both relational leadership and leadership for sustainability, an ethical understanding of both is limited. This is problematic as both sustainability and relational leadership are rife with moral implications. This paper conceptually explores how the moral theory of ‘ethics of care’ can help to illuminate the ethical dimensions of relational leadership for sustainability. In doing (...) so, the implications of ethics of care more broadly for the practice of relational leadership development are elaborated. From a caring perspective, a ‘relational stance’ or logic of effectiveness can be fostered through engaging in a reflective process of moral education through conversation. In starting this dialogue, we can begin to build capacity for relational leadership for sustainability and, thus, support the development of individual well-being and organizational and societal flourishing. (shrink)

The general methodology of "algebraizing" logics is used here for combining different logics. The combination of logics is represented as taking the colimit of the constituent logics in the category of algebraizable logics. The cocompleteness of this category as well as its isomorphism to the corresponding category of certain first-order theories are proved.

We prove that everyn-modal logic betweenKnandS5nis undecidable, whenever n ≥ 3. We also show that each of these logics is non-finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov–Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a reduction of the representation problem of finite relation (...) algebras. (shrink)

It is shown that the many-dimensional modal logic K n , determined by products of n-many Kripke frames, is not finitely axiomatisable in the n-modal language, for any $n > 2$ . On the other hand, K n is determined by a class of frames satisfying a single first-order sentence.

We prove that everyn-modal logic betweenKnandS5nis undecidable, whenever n ≥ 3. We also show that each of these logics is non-finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov–Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a reduction of the representation problem of finite relation (...) algebras. (shrink)

One of the basic theorems in universal algebra is Birkhoff's variety theorem: the smallest equationally axiomatizable class containing a class K of algebras coincides with the class obtained by taking homomorphic images of subalgebras of direct products of elements of K. G. Gratzer asked whether the variety theorem is equivalent to the Axiom of Choice. In 1980, two of the present authors proved that Birkhoff's theorem can already be derived in ZF. Surprisingly, the Axiom of Foundation plays a crucial role (...) here: we show that Birkhoff's theorem cannot be derived in ZF + AC {Foundation}. even if we add Foundation for Finite Sets. We also prove that the variety theorem is equivalent to a purely set-theoretical statement, the Collection Principle. This principle is independent of $ZF \{\operatorname{Foundation}$ . The second part of the paper deals with further connections between axioms of ZF-set theory and theorems of universal algebra. (shrink)

Contemporary yoga is popularly represented in various media by a fit, white woman. Yoga Journal is a magazine recognized by many as an industry cornerstone and an institution in and of itself. It represents the distinctive face of yoga. By analyzing the visual and textual content of the Yoga Journal magazine covers, from its first issue in 1975 to issue 313, we describe the produced and consumed portrait of yoga. By focusing on the cover themes, together with the objects and (...) persons depicted, we recognize three phases of development that have contributed to the understanding of the changing image of yoga in the media. While, in the initial phase, yoga was represented as a mystical and mysterious spiritual discipline that originated in the Orientalized East, in the later phase, it was depicted first as a global and universal phenomenon and then as a mainstream everyday fitness regime. Yoga’s depiction in the last phase is compatible with the scholarly representation of contemporary yoga as posture and body centered, but not with its earlier depictions. We found that in its mature phase, the yoga body, which is criticized for its lack of inclusiveness, emerged to become omnipresent. We conclude that the newly formed face of yoga is problematic, both for its female readers, who are encouraged to conform to a unique body type, and also for the yoga community, which encounters an appearance-based restriction of access. (shrink)

We prove that everyn-modal logic betweenKnandS5nis undecidable, whenever n ≥ 3. We also show that each of these logics is non-finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov–Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a reduction of the (undecidable) representation problem of finite (...) relation algebras. (shrink)

We prove that every n-modal logic between K n and S5 n is undecidable, whenever n ≥ 3. We also show that each of these logics is non- finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov-Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a (...) reduction of the (undecidable) representation problem of finite relation algebras. (shrink)

We give an overview of decidability results for modal logics having a binary modality. We put an emphasis on the demonstration of proof-techniques, and hope that this will also help in finding the borderlines between decidable and undecidable fragments of usual first-order logic.

We prove that the equational theory of a semigroups becomes undecidable if we add a semilattice structure with a ‘touch of symmetric difference’. As a corollary we obtain that the variety of all Boolean algebras with an associative binary operator has a ‘hereditarily’ undecidable equational theory. Our results have implications in logic, e.g. they imply undecidability of modal logics extending the Lambek Calculus and undecidability of Arrow Logics with an associative arrow modality.

En este trabajo abordo el problema de la herencia nietzscheana en el pensamiento de Jacques Derrida como base para plantear la cuestión de la huella de éste último en la filosofía actual. Para ello, me aproximo a la relación entre vida e ideal de vida, tal como la pensaron ambos. Partiendo de este aspecto, planteo el problema de la relación entre vida y obra, la cuestión de la identidad personal, la valoración del pensamiento como actividad crítica y la cuestión de (...) la posibilidad de lo imposible, que resulta central en el pensamiento derridiano. A través de todo ello, pretendo poner de manifiesto la enorme proyección de futuro que a mi juicio tiene la obra de Derrida. (shrink)

Según plantea Heidegger en Ser y tiempo, la pregunta por el ser no es fruto de la espontaneidad de la llamada conciencia natural, aunque tampoco en la racionalidad científica pueda encontrar los elementos adecuados para su formulación. Sabemos, en todo caso, que la ciencia puede contribuir a la respuesta, aunque no ayude a comprender lo que el hecho mismo de esa enunciación implica. La tesis que se defiende en este trabajo es que la correlación entre la pregunta por el ser (...) y la pregunta por el tiempo, posee unas dimensiones objetiva, subjetiva e histórica, estas tres dimensiones todavía no han sido, a mi entender, suficientemente exploradas. La conclusión a que se llega puede resumirse afirmando que en cada una de ellas acaba implicado el problema de la autenticidad, un problema que no puede ser tratado de forma adecuada desde la concepción heideggeriana del tiempo desarrollada en Ser y tiempo. According to what Heidegger exposed in Being and Time, the question of being is not the result of the spontaneity of what we call natural consciousness, although we cannot find in scientific rationality the right items for its formulation. We know, in any case, that science can contribute to the answer, but does not help us understand what the fact of this statement implies. The thesis defended in this work is that the correlation between the question of being and the question of the time has an objective, subjective and historical dimension, not yet fully explored. The conclusion reached is that in all of them is involved the problem of authenticity, that can’t be addressed adequately from Heidegger's conception of time exposed in Being and Time. (shrink)

La experiencia de la finitud no otorga, en sí misma, un sentido a la vida, pero sí introduce en ella algo que la hace intensamente humana. La finitud de la existencia queda marcada por la certeza de la presencia ineludible de la muerte. La angustia que produce la certeza de la inevitabilidad de la propia muerte es la forma más radical en que se nos manifiesta la naturaleza del Ser, no la ausencia de éste. La referencia a las ideas de (...) Derrida, Heidegger y Jankélévitch, nos permite perfilar en este trabajo una línea de reflexión relevante sobre un eterno problema filosófico: la significación de la muerte y su relación con el sentido de la vida. El modesto objetivo que persigo con este trabajo es mostrar, no sólo que sigue teniendo sentido reflexionar sobre uno de los temas ineludibles en la historia de la filosofía, sino también poner de manifiesto la pertinencia de la reflexión filosófica, aun cuando abordamos a través de ella asuntos que de antemano sabemos que jamás se podrán resolver de forma definitiva. (shrink)

El objetivo primordial de este artículo es poner en valor las contribuciones de Jacques Derrida a la filosofía del derecho. En esa línea, una de las tareas más importantes que se abordan en él es mostrar que la labor de deconstrucción, se refiera ésta a cualquier ámbito filosófico o jurídico, es ante todo poner en evidencia los límites, carencias e incoherencias internas que se dan en dicho ámbito. Además de ello, se expone en este trabajo cómo Jacques Derrida, el creador (...) de este método de análisis, al aclarar el sentido y operatividad de la deconstrucción nos explica que ésta se plantea la tarea de desmontar lo que se presenta como construido con una inconmovible solidez. Se pone asimismo de manifiesto en este artículo que la deconstrucción puede contribuir a la concreción de la justicia, en diversos contextos, que van desde el mismo proceso judicial a la dimensión colectiva de las acciones que conducen a la materialización de un orden justo. (shrink)

El objetivo de este trabajo es ofrecer al lector un recorrido por las más signifi cativas interpretaciones del mito de Prometeo, intentando, a la luz del contenido de las mismas, contribuir al esclarecimiento de la relación entre ley moral y ley política. En particular, se trata de poner de relieve cómo hay en la actitud de Prometeo algo que delata la presencia de una conciencia, fuertemente individualizada, cuyo dictado le conduce a asumir el choque con el poder en su máxima (...) expresión. Por otro lado, se analizan las diferentes interpretaciones del concepto griego de Ley, estudiando cómo la máxima expresión de la misma es deudora de la idea de destino. En base a la Ley, se establece un orden común, que enlaza a dioses y hombres, si bien no con los mismos grados de sujeción. (shrink)

Más allá del principio del poder es el título de la contribución de Derrida al homenaje a Foucault, que realizó la Universidad de New York tras la muerte del filósofo. Acerca de su contenido trata este pequeño ensayo. En su intervención, Derrida pone en relación las ideas de Freud y las de Foucault, con las suyas propias, a propósito de la relación entre saber, poder y sexualidad.Su objetivo final es hacer hablar, pero al mismo tiempo acallar la voluntad de destrucción, (...) encontrando la forma de imponer la vida a la muerte, aún en la misma muerte y más allá de la muerte. Este es el imposible que ha de imponerse a la voluntad de destrucción del otro, a la persistente búsqueda de su absoluta sumisión aniquiladora, que conllevaría a su vez el aniquilamiento del verdugo. (shrink)

Resumen: El pasado 6 de junio se cumplieron 30 años de la muerte del pensador Vladimir Jankélévitch, acaecida en Paris en 1985. Se trata de un lapso temporal considerable, casi un tercio de siglo, que nos permite realizar un balance de la actualidad y previsible pervivencia en el futuro de su obra. En el presente trabajo pretendo realizar un breve recorrido por algunos de los temas filosóficos que abordó en sus obras más relevantes, reivindicando la vigencia de su pensamiento en (...) este siglo XXI, que va abriéndose paso entre profundos dramas humanos. En cierta forma, tan complejas circunstancias constituyen otras tantas proyecciones de las que Jankélévitch conoció a lo largo de su extraordinaria peripecia vital. Con todo ello como telón de fondo, focalizaré mi atención en cinco grandes asuntos: el estatuto del saber filosófico, la ética como base de la relación con el otro, el amor, la muerte y la búsqueda de la pureza.: On June 6 the 30th anniversary of the death of Vladimir Jankélévitch, happened in Paris in 1985. After a considerable time lag, almost a third of a century, allows us to take stock of the present and foreseeable future survival of his work. In this paper we try to provide a brief overview of some of the philosophical issues addressed in his most important works, vindicating their importance thought this century, which are presented through deep human dramas. In a way, such complex circumstances are projections of those that Jankélévitch met throughout his extraordinary life. With all this as background, I will focus my attention on five major issues: the status of philosophical knowledge, ethics as the basis for the relationship with the other, love, death and the search for purity. (shrink)