We show that the variety of Heyting algebras has finitary unification type. We also show that the subvariety obtained by adding it De Morgan law is the biggest variety of Heyting algebras having unitary unification type. Proofs make essential use of suitable characterizations (both from the semantic and the syntactic side) of finitely presented projective algebras.

We show that some common varieties of modal K4-algebras have finitary unification type, thus providing effective best solutions for equations in free algebras. Applications to admissible inference rules are immediate.

By means of models in toposes of C-sets (where C is a small category), necessary conditions are found for the minimum quantified extension of a propositional (intermediate, modal) logic to be complete with respect to Kripke semantics; in particular, many well-known systems turn out to be incomplete.

The logicD-J of the weak exluded middle with constant domains is proved to be incomplete with respect to Kripke semantics, by introducing models in presheaves on an arbitrary category. Additional incompleteness results are obtained for the modal systems with nested domains extendingQ-S4.1.

We show that (contrary to the parallel case of intuitionistic logic, see [7], [4]) there does not exist a translation fromS42 (the propositional modal systemS4 enriched with propositional quantifiers) intoS4 that preserves provability and reduces to identity for Boolean connectives and.

Sufficient syntactic conditions for canonicity in intermediate and intuitionistic modal logics are given. We present a new technique which does not require semantic first-order reduction and which is constructive in the sense that it works in an intuitionistic metatheory through a model without points which is classically isomorphic to the usual canonical model.

In this paper we present a general theory of normal forms, based on a categorial result for the free monoid construction. We shall use the theory mainly for proposictional modal logic, although it seems to have a wider range of applications. We shall formally represent normal forms as combinatorial objects, basically labelled trees and forests. This geometric conceptualization is implicit in and our approach will extend it to other cases and make it more direct: operations of a purely geometric and (...) combinatorial nature will be introduced in order to give a mathematical description of the basic logical/algebraic constructions .We begin by recalling the above-mentioned categorial construction: we need a careful inspection of it because in the various examples considered later we plan to deduce from it in a uniform way the normal forms and the description of finitely generated free algebras. This method always works whenever we can describe the category of algebras corresponding to the logic under consideration as a T-objects category. When this simple description seems not to be available, still the general theory might be of some interest, because a description of the category of algebras as a T-objects category plus equation is possible .The central part of the paper is more advanced and specific: we show how the general approach presented here can provide some insights even in the basic case of the modal system K. Section 4 contains a contribution to the theory of normal forms, namely the description of the uniform substitution. This result will enable us to give a duality theorem for the category of finitely generated free modal algebras and in Section 5 to provide a characterization of the collections of normal forms which happen to be normal forms for a logic, thus giving a description of the lattice of modal logics.Section 6 deals with some applications: we shall show how to use normal forms in order to prove for the modal system K the definability of higher-order propositional quantifiers and of the tense operator F .As to the prerequisites, the paper is almost self-contained. The reader is only assumed to have familiarity with standard techniques in algebraic logic ). Knowledge of the basic facts about adjoint functors is required too, see e.g. McLane or the appendix. (shrink)

We investigate finitarity of unification types in locally finite varieties of Heyting algebras, giving both positive and negative results. We make essential use of finite dualities within a conceptualization for E-unification theory 733–752) relying on the algebraic notion of a projective object.

A. M. Pitts in [Pi] proved that HA op fp is a bi-Heyting category satisfying the Lawrence condition. We show that the embedding $\Phi: HA^\mathrm{op}_\mathrm{fp} \longrightarrow Sh(\mathbf{P_0,J_0})$ into the topos of sheaves, (P 0 is the category of finite rooted posets and open maps, J 0 the canonical topology on P 0 ) given by $H \longmapsto HA(H,\mathscr{D}(-)): \mathbf{P_0} \longrightarrow \text{Set}$ preserves the structure mentioned above, finite coproducts, and subobject classifier, it is also conservative. This whole structure on HA op (...) fp can be derived from that of Sh(P 0 ,J 0 ) via the embedding Φ. We also show that the equivalence relations in HA op fp are not effective in general. On the way to these results we establish a new kind of duality between HA op fp and a category of sheaves equipped with certain structure defined in terms of Ehrenfeucht games. Our methods are model-theoretic and combinatorial as opposed to proof-theoretic as in [Pi]. (shrink)

We characterize (both from a syntactic and an algebraic point of view) the normal K4-logics for which unification is filtering. We also give a sufficient semantic criterion for existence of most general unifiers, covering natural extensions of K4.2⁺ (i.e., of the modal system obtained from K4 by adding to it, as a further axiom schemata, the modal translation of the weak excluded middle principle).

The role played by continuous morphisms in propositional modal logic is investigated: it turns out that they are strictly related to filtrations and to suitable variants of the notion of a free algebra. We also employ continuous morphisms in incremental constructions of (standard) finitely generated free ????4-algebras.

Under some assumptions on an equational theory S , we give a necessary and sufficient condition so that S admits a model completion. These assumptions are often met by the equational theories arising from logic. They say that the dual of the category of finitely presented S-algebras has some categorical stucture. The results of this paper combined with those of [7] show that all the 8 theories of amalgamable varieties of Heyting algebras [12] admit a model completion. Further applications to (...) varieties of modal algebras are given in [8]. (shrink)

The quantified extension of a canonical prepositional intermediate logic is complete with respect to the generalization of Kripke semantics taking into consideration set-valued functors defined on a category.

This collection of essays turnings around aesthetic and ethical questions, and intertwining them, is intended to foster and elaborate the notion of intercultural philosophy. Without idealizing any single way of thinking or tradition, without idolizing any lazy relativism, the author wants to show how interculturality is neither a comprehensive, ultimate system of thoughts, nor a disconnected plurality of opinions. Surmounting monism without fading into dualism, he moves on leads to deal with the philosophical character of symbols, analogies, and comparisons, through (...) which we can grasp the links and relations between identity and difference. (shrink)

By using algebraic-categorical tools, we establish four criteria in order to disprove canonicity, strong completeness, w-canonicity and strong w-completeness, respectively, of an intermediate propositional logic. We then apply the second criterion in order to get the following result: all the logics defined by extra-intuitionistic one-variable schemata, except four of them, are not strongly complete. We also apply the fourth criterion in order to prove that the Gabbay-de Jongh logic D1 is not strongly w-complete.

Since the end of the 16th century, when the perfectly preserved remains of an ancient early Christian underground cemetery were discovered accidentally along the Via Salaria in Rome, Christian antiquities were studied mainly for apologetic propaganda purposes, i.e. to defend the primacy of the Church of Rome, which was faltering under the blows of the Protestant reformers. Everything changed, however, around the middle of the 19th century, thanks to Giovanni Battista de Rossi, a famous archaeologist whose 200th birthday falls this (...) year. This essay sets in the context of his biography the main coordinates of his training, his numerous and fundamental discoveries and his main publications, which brought him recognition in international cultural circles of the time as the “founder of Christian archaeology”, a science that was finally recognised as such, and no longer seen as just a learned pastime for amateur antiquarians. (shrink)

Entre phénoménologie et neurologie. Merleau-Ponty, Goldstein, SacksLa phénoménologie de Merleau-Ponty a traité des matériaux et des éléments de réflexion importants de diverses études sur des pathologies psychiques et physiques. La confrontation et l’entrelacement entre la pensée du philosophe français et certaines recherches de neuropsychologie, comme Kurt Goldstein ou Oliver Sacks, font émerger la possibilité de nouvelles approches de la dimension de la maladie, entre narration, spéculation et recherche scientifique. Un dialogue fécond entre disciplines et champs de recherche différents met en (...) lumière comment toute modalité perceptive, même quand elle se présente comme déficitaire ou déviante, constitue de fait une vraie recréation ou re-constitution du monde et non pas seulement un enregistrement. Ce processus de création continue est lié à une des questions de fond soulevée par l’expérience de la maladie: celle des modes et des conditions de la singularité, de l’identité d’un soi, à la recherche de reconnaissance et d’acceptation de la part d’un regard ou d’un geste que nous savons contribuer de manière inédite à la relation entre sujets humains.Between Phenomenology and Neurology. Merleau-Ponty, Goldstein, SacksMerleau-Ponty’s phenomenology deals with materials and elements of important reflections in diverse studies on psychological and physical pathologies. The confrontation and the connections between French philosophical thought and certain research in neuropsychology, such as that of Kurt Goldstein or Oliver Sacks, allow the possibility of new approaches to the dimension of sickness to emerge, approaches between narration, speculation, and scientific research. A fertile dialogue between disciplines and fields of different research brings to light how every perceptual modality, even when it presents itself as deficient or deviant, constitutes in fact a true re-creation or reconstitution of the world and not only a recording. This process of continuous creation is linked to one of the basic questions raised by the experience of sickness: the question of the modes and conditions of singularity, of the identity of a self in pursuit of recognition or acceptance from another’s gaze or gesture, a gaze or gesture that, in the relation between human subjects, we have the ability to contribute ever in a unique manner. (shrink)

The article deals with Chinese ink painting and some aesthetic notions, in particular those of xiang 象 and shanshui 山水 (literally: “mountains-waters”, i.e. landscape...

Monadic second order logic and linear temporal logic are two logical formalisms that can be used to describe classes of infinite words, i.e., first-order models based on the natural numbers with order, successor, and finitely many unary predicate symbols.Monadic second order logic over infinite words can alternatively be described as a first-order logic interpreted in${\cal P}\left$, the power set Boolean algebra of the natural numbers, equipped with modal operators for ‘initial’, ‘next’, and ‘future’ states. We prove that the first-order theory (...) of this structure is the model companion of a class of algebras corresponding to a version of linear temporal logic without until.The proof makes crucial use of two classical, nontrivial results from the literature, namely the completeness of LTL with respect to the natural numbers, and the correspondence between S1S-formulas and Büchi automata. (shrink)

This article deals with the Chinese ink painting tradition, as a paradigm in which art and life are coupled and intertwined. In fact, in Chinese classical aesthetics, art and life do not produce a dramatic tension, but are inscribed in a common process of naturalness or spontaneity. The painter has to learn how the breath, or vital energy, that flows in every single image-phenomenon, can be enlivened by the brush strokes. Moreover, the paper builds a dialogue between the European and (...) the Chinese main patterns. Showing alternately the continuity or the discontinuity between art and life, both traditions of thought display the possibility to learn a sort of “art of living”, through – and not despite – the dynamic de-coincidence that appears in the folds of human experience. (shrink)

A specific interest and attention to translation and to the relation between signs, natural languages, and thought spread out not only from the so called “linguistic turn” in the 20th century, but also from the cross-cultural dimension that gained prominency in the last decades. Interlinguistic translation can thus be considered and analyzed as a particular and fruitful field of study for philosophy, because in that experience we have to face the intimate relationship that intertwine form and content, the contingency of (...) expression and the ideal aspiration to a shared, universal truth. The case of Chinese language and thought provide further material to explore, in order to raise more questions about the links among grammar, morphology, system of writing, and the meanings that those structures intrinsecally brings and convey. Re-considering Hegel’s ideas about alphabetical and non-alphabetical writing, and through a peculiar case study of a poem by Wang Wei, we can learn to appreciate in a new way the alterity of Chinese characters and language. Finally, the act of translation can be seen as a model of ethical understanding, in order take into account the Other, without enclosing it into the frames that have been shaped with our own categories. (shrink)

The Collectio Avellana, a collection of late antique documents which has been thoroughly investigated, above all by ancient historians and historians of Roman law, provides a wealth of historical, political-religious and legal information for the period between the middle of the 4th and the middle of the 6th century AD. This contribution, limited to the years of the pontificate of Pope Damasus, seeks to understand whether these texts can also represent a source of topographical information for the city of Rome.

Taking into account my personal path as a philosopher and as a painter, I try to sketch the perspective on aesthetics that was opened to me by a cross-cultural encounter. The European tradition, on one side, and the Sino-Japanese tradition, on the other side, are the two mirroring currents along which I moved in order to trace a sort of “deconstruction” and a “restructuring” of artistic and philosophical vision. In my painting, I aim for a confluence of different streams of (...) thought by thinking about European informal art and landscape ink-painting of China and Japan. This confluence continues to produce a fertile dialogue, enlivening the deep resources that constitute the core of a subjectivity in process. Aesthetics converts itself into ethics: an ethical move, beyond the opposition of visible and invisible as dull substances, beyond the opposition of immanence and transcendence, is one of the inner goals of this pathway between art and philosophy. (shrink)

In this paper we study the logic of relational and partial variable sets, seen as a generalization of set-valued presheaves, allowing transition functions to be arbitrary relations or arbitrary partial functions. We find that such a logic is the usual intuitionistic and co-intuitionistic first order logic without Beck and Frobenius conditions relative to quantifiers along arbitrary terms. The important case of partial variable sets is axiomatizable by means of the substitutivity schema for equality. Furthermore, completeness, incompleteness and independence results are (...) obtained for different kinds of Beck and Frobenius conditions. (shrink)

For anyone working in aesthetics interested in understanding the richness of the Chinese aesthetic tradition this handbook is the place to start. Comprised of general introductory overviews, critical reflections and contextual analysis, it covers everything from the origins of aesthetics in China to the role of aesthetics in philosophy today. Beginning in early China (1st millennium BCE), it traces the Chinese aesthetic tradition, exploring the import of the term aesthetics into Chinese thought via Japan around the end of the 19th (...) century. It looks back to early practices of art and craftsmanship, showing how the history of Chinese thought provides a multitude of artifacts and texts that give rise to a wide range of aesthetic creations and notions. Introducing various perspectives on traditional arts in China, including painting, ceramics, calligraphy, poetry, music and theatre, it explores those aesthetic traditions not included in "canonic" art forms, such as martial arts, rock gardening, and ritual performance. Written by Chinese, European, and American theoreticians and practitioners, this authoritative research resource enhances contemporary aesthetics by revealing the possibilities of a Chinese philosophy of art. (shrink)