The abstract description of a physical system is developed, along lines originally suggested by Birkhoff and von Neumann, in terms of the complete lattice of propositions associated with that system, and the distinction between classical and quantum systems is made precise. With the help of the notion of state, a propositional system is defined: it is remarked that every irreducible propositional system (of more than three dimensions) is isomorphic to the lattice of all closed subspaces of a Hilbert (...) space constructed on some division ring with involution. The propositional system consisting of a family of separable complex Hilbert spaces is treated as a particular case which is sufficiently general to include both classical and quantum mechanics. The theory of the Galilean particle without spin is given as an illustration. Finally, the basis for the statistical interpretation of wave mechanics is developed with the help of Gleason's theorem. In an appendix, a proof of essentially the first part of Gleason's theorem is given which is a little different (perhaps more geometric) from that originally given by Gleason. (shrink)

On the basis of Mackey's axiomatic approach to quantum physics or, equivalently, of a “state-event-probability” (SEVP) structure, using a quite standard “fuzzification” procedure, a set of unsharp events (or “effects”) is constructed and the corresponding “state-effect-probability” (SEFP) structure is introduced. The introduction of some suitable axioms gives rise to a partially ordered structure of quantum Brouwer-Zadeh (BZ) poset; i.e., a poset endowed with two nonusual orthocomplementation mappings, a fuzzy-like orthocomplementation, and an intuitionistic-like orthocomplementation, whose set of sharp elements is an (...) orthomodular complete lattice. As customary, by these orthocomplementations the two modal-like necessity and possibility operators are introduced, and it is shown that Ludwig's and Jauch-Piron's approaches to quantum physics are “interpreted” in complete SEFP. As a marginal result, a standard procedure to construct a lot of unsharp realizations starting from any sharp realization of a fixed observable is given, and the relationship among sharp and corresponding unsharp realizations is studied. (shrink)

Orthomodular lattices with a two-valued Jauch–Piron state split into a generalized orthomodular lattice and its dual. GOMLs are characterized as a class of L-algebras, a quantum structure which arises in the theory of Garside groups, algebraic logic, and in connections with solutions of the quantum Yang–Baxter equation. It is proved that every GOML X embeds into a group G with a lattice structure such that the right multiplications in G are lattice automorphisms. Up to isomorphism, X (...) is uniquely determined by G, and the embedding \\) is a universal group-valued measure on X. (shrink)

Tense operators in effect algebras play a key role for the representation of the dynamics of formally described physical systems. For this, it is important to know how to construct them on a given effect algebra \( E\) and how to compute all possible pairs of tense operators on \( E\) . However, we firstly need to derive a time frame which enables these constructions and computations. Hence, we usually apply a suitable set of states of the effect algebra \( (...) E\) in question. To approximate physical reality in quantum mechanics, we use only the so-called Jauch–Piron states on \( E\) in our paper. To realize our constructions, we are restricted on lattice effect algebras only. (shrink)

Peter John Olivi’s relationship to Bonaventure is intriguing.1 Outwardly, they both appear as the leading figures of two different trends of Franciscan politics: Olivi usually being qualified as a “radical” inspiring the dissidence of the Spirituals, while Bonaventure would represent a central and balanced attitude regarding Franciscan poverty. Likewise, as far as their apocalyptical expectations are concerned, Olivi is certainly an overt and avowed Joachite, whereas Bonaventure supposedly makes a more detached use of Joachim of Fiore’s works. Recent studies of (...) Olivi’s philosophical approaches present him as anticipating themes soon to be developed by Duns Scotus rather than associate him with the Bonaventurian... (shrink)

A comprehensive formal system is developed that amalgamates the operational and the realistic approaches to quantum mechanics. In this formalism, for example, a sharp distinction is made between events, operational propositions, and the properties of physical systems.

Dans des écrits datant des années 1277-79, le théologien franciscain Pierre de Jean Olivi emploie à cinq reprise le mot averroista, pour critiquer, non pas tant la doctrine de l'unicité de l'intellect possible que ses implications anthropologiques ou angélologiques. Cette critique bien informée, convergente avec d'autres attaques franciscaines, notamment les dénonciations de Roger Bacon, démontre la réalité et l'ampleur d'une entreprise philosophique menée par des maîtres ès arts, s'inspirant d'Averroès, dans les années 1260-70 ; elle indique aussi la diversité de (...) leurs points de vue. Dès son apparition, il n'y a donc pas un, mais plusieurs 'averroïsmes' latin. (shrink)

In this paper we show a duality between two approaches to represent quantum structures abstractly and to model the logic and dynamics therein. One approach puts forward a “quantum dynamic frame” :2267–2282, 2005), a labelled transition system whose transition relations are intended to represent projections and unitaries on a Hilbert space. The other approach considers a “Piron lattice”, which characterizes the algebra of closed linear subspaces of a Hilbert space. We define categories of these two sorts of structures (...) and show a duality between them. This result establishes, on one direction of the duality, that quantum dynamic frames represent quantum structures correctly; on the other direction, it gives rise to a representation of dynamics on a Piron lattice. (shrink)

Les recherches sur le travail comme valeur au Moyen Âge se sont principalement attardées sur la malédiction prononcée lors de l'expulsion du jardin d'Eden, négligeant l'activité à laquelle se livrait Adam avant la chute (Gn 2,15). A la suite de Philon d'Alexandrie, Augustin a proposé une lecture littérale de ce premier travail, qui a eu un fort impact dans la culture médiévale, certains auteurs (franciscains) comprenant ce verset au sens d'une mission d'améliorer encore la création.

Le Liber ostensor quod adesse festinant tempora, rédigé en 1356 dans la prison du Soudan à Avignon, est le plus important des écrits conservés du visionnaire et prophète et franciscain Jean de Roquetaillade. Son édition, par une équipe dirigée par A. Vauchez, offre des matériaux très riches pour l'histoire religieuse et culturelle du XIVe siècle. On note toutefois quelques lacunes dans la bibliographie et l'identification des sources, et notamment dans la notice consacrée aux emprunts faits à Pierre de Jean Olivi. (...) Au-delà des citations explicites, on peut mettre en évidence une dépendance bien plus grande, que masque pour partie une stratégie d'écriture carcérale. (shrink)

Autour de 1100, pour les jeunes femmes de l’aristocratie éduquées dans des monastères, l’écriture de lettres d’amour fictives à leur maître de rhétorique constituait un apprentissage aussi bien littéraire qu’émotionnel. Comme le rappelle Barbara Newman, les Epistolae duorum amantium, correspondance échangée par Héloïse et Pierre Abélard durant leur liaison, doivent se comprendre dans cette lumière. Leur singularité tient largement au fait qu’Héloïse a choisi de donner corps à une figure poétique, en s’identifiant aux héroïnes tragiques d’Ovide.

In this dialogue we explain why nonrelativistic and relativistic quantum mechanics are basically the same and how the Lorentz group operates canonically only on the dynamics.

Taking into account the experimental accuracy of the body motions in the solar system, we propose a General Relativity Theory which is new but nevertheless completely metric.

Dans les démocraties libérales contemporaines, les personnes « vieillissantes » forment un groupe d’âge qui est au cœur d’importants débats éthiques, politiques et administratifs. Comment, collectivement, pourrons-nous assurer une fin de vie digne et respectueuse de leur volonté à tous les citoyens qui, en vieillissant, risquent toujours plus d’être atteints par la maladie ou la perte d’autonomie, alors que leur nombre ne cesse d’augmenter pour des raisons démographiques et en raison de l’eff..

En nous laissant guider par la notion de champ qui en faitdomine toute la physique nous définissons la nature physique d'un système, ses propriétés et ses états possibles. Nous proposons alors un cadre très générale permettant la description de telssystèmes et la construction de modèles consistants.Nous en donnons des exemples et pour illustrer les concepts de notrethéorie nous décrivons différents types d'évolutions. Pour terminer nous discutons en détails unprocessus physique, les échos de spins.

Cet article rend compte de l’expérience éthique suscitée par un terrain de doctorat en anthropologie mené auprès d’écoliers de Québec dans les années 1990. Les récits de vie réalisés à cette occasion ont pris valeur de véritables rencontres éthiques. Désirant produire un savoir sur ces jeunes qui réponde de manière satisfaisante aussi bien aux exigences de la communauté scientifique qu’à ses exigences éthiques, l’auteur fait le pari de ne pas effacer dans son écriture et dans ses conclusions « savantes » (...) le lien d’ordre éthique, porteur de multiples ambiguïtés, qui s’était noué avec certains de ces jeunes. Est-il possible de construire un savoir « scientifique » sur autrui sans pour autant détruire, nier ou masquer le lien noué avec cet autrui à l’occasion du travail de recherche? (shrink)

Rien n’est plus faux que l’image de Dante comme génie isolé, tranchant sur son époque. Il appartient au contraire à une génération d’intellectuels italiens très caractéristique: laïes, souvent actifs dans plus d’un domaine, pratiquant abondamment l’expression poétique, engagés dans l’action politique, ces savants-citoyens partagent également une forte conscience historique. Leur inventivité se manifeste, en philosophie, en médecine et en droit, autant que dans l’expression littéraire. Outre des facteurs endogènes, une clé de ce phénomène tient à la réception active du savoir (...) naturaliste élaboré à Paris, qui est lu en Italie hors du contrôle des théologiens. Les parcours parallèles d’une douzaine de personnalité hors du commun montrent la nécessité d’élargir les cadres habituels de l’histoire intellectuelle médiévale. (shrink)

Gerald Odonis' treatise on contracts, restitutions, and excommunication is one of his earliest works, composed in Toulouse ca. 1315-17. Mainly based on Peter John Olivi's De contractibus, but using a variety of other sources and offering some original arguments as well, it is remarkable for its pragmatic approach to economic phenomena. His rejection of the rational argument against usury reveals a casual use of the bull Exiit qui seminat, defining Franciscan poverty, as well as a change of assumptions in the (...) approach to economic exchange. Whereas various explanations can be provided for the provocative aspect of this youthful treatise, all in all, it can best be described as a free and uninhibited interpretation of the scholastic tradition. (shrink)

Gerald Odonis' treatise on contracts, restitutions, and excommunication is one of his earliest works, composed in Toulouse ca. 1315-17. Mainly based on Peter John Olivi's De contractibus, but using a variety of other sources and offering some original arguments as well, it is remarkable for its pragmatic approach to economic phenomena. His rejection of the rational argument against usury reveals a casual use of the bull Exiit qui seminat, defining Franciscan poverty, as well as a change of assumptions in the (...) approach to economic exchange. Whereas various explanations can be provided for the provocative aspect of this youthful treatise, all in all, it can best be described as a free and uninhibited interpretation of the scholastic tradition. (shrink)

In the foundations of quantum mechanics Gleason’s theorem dictates the uniqueness of the state transition probability via the inner product of the corresponding state vectors in Hilbert space, independent of which measurement context induces this transition. We argue that the state transition probability should not be regarded as a secondary concept which can be derived from the structure on the set of states and properties, but instead should be regarded as a primitive concept for which measurement context is crucial. Accordingly, (...) we adopt an operational approach to quantum mechanics in which a physical entity is defined by the structure of its set of states, set of properties and the possible (measurement) contexts which can be applied to this entity. We put forward some elementary definitions to derive an operational theory from this State–COntext–Property (SCOP) formalism. We show that if the SCOP satisfies a Gleason-like condition, namely that the state transition probability is independent of which measurement context induces the change of state, then the lattice of properties is orthocomplemented, which is one of the ‘quantum axioms’ used in the Piron–Solèr representation theorem for quantum systems. In this sense we obtain a possible physical meaning for the orthocomplementation widely used in quantum structures. (shrink)

The general fact of the impossibility of a bivalent, truth-functional semantics for the propositional structures determined by quantum mechanics should be more subtly demarcated according to whether the structures are taken to be orthomodular latticesP L or partial-Boolean algebrasP A; according to whether the semantic mappings are required to be truth-functional or truth-functional ; and according to whether two-or-higher dimensional Hilbert spaceP structures or three-or-higher dimensional Hilbert spaceP structures are being considered. If the quantumP structures are taken to be orthomodular (...) latticesP L, then bivalent mappings which preserve the operations and relations of aP L must be truth-functional . Then as suggested by von Neumann and Jauch-Piron and as proven in this paper, the mere presence of incompatible elements in aP L is sufficient to rule out any semantical or hidden-variable proposal which imposes this strong condition, for anytwo-or-higher dimensional Hilbert spaceP L structure. Thus from the orthomodular lattice perspective, the peculiarly non-classical feature of quantum mechanics and the peculiarly non-Boolean feature of the quantum propositional structures is the existence of incompatible magnitudes and propositions. However, the weaker truth-functionality condition can instead be imposed upon the semantic or hidden-variable mappings on theP L structures, although such mappings ignore the lattice meets and joins of incompatibles and preserve only the partial-Boolean algebra structural features of theP L structures. Or alternatively, the quantum propositional structures can be taken to be partial-Boolean algebrasP A, where bivalent mappings which preserve the operations and relations of aP A need only be truth-functional (c). In either case, the Gleason, Kochen-Specker proofs show that any semantical or hidden variable proposal which imposes this truth-functionality (c) condition is impossible for anythree-or-higher dimensional Hilbert spaceP A orP L structures. But such semantical or hidden-variable proposals are possible for any two dimensional Hilbert spaceP A orP L structures, in spite of the presence of incompatibles in these structures, in spite of the fact that Heisenberg's Uncertainty Principle applies to the incompatible elements in these structures, and in spite of the fact that these structures are non-Boolean in the Piron sense. (shrink)

This new edition of Introduction to Lattices and Order presents a radical reorganization and updating, though its primary aim is unchanged. The explosive development of theoretical computer science in recent years has, in particular, influenced the book's evolution: a fresh treatment of fixpoints testifies to this and Galois connections now feature prominently. An early presentation of concept analysis gives both a concrete foundation for the subsequent theory of complete lattices and a glimpse of a methodology for data analysis that is (...) of commercial value in social science. Classroom experience has led to numerous pedagogical improvements and many new exercises have been added. As before, exposure to elementary abstract algebra and the notation of set theory are the only prerequisites, making the book suitable for advanced undergraduates and beginning graduate students. It will also be a valuable resource for anyone who meets ordered structures. (shrink)

We study the idea of implantation of Piron's and Bell's geometrical lemmas for proving some results concerning measures on finite as well as infinite-dimensional Hilbert spaces, including also measures with infinite values. In addition, we present parabola based proofs of weak Piron's geometrical and Bell's lemmas. These approaches will not used directly Gleason's theorem, which is a highly non-trivial result.

We give a characterization of the fixed points and of the lattices of fixed points of fuzzy Galois connections. It is shown that fixed points are naturally interpreted as concepts in the sense of traditional logic.

This is also where we begin investigating lattices of logics and varieties, rather than particular examples.

We study the lattice of extensions of four-valued Belnap–Dunn logic, called super-Belnap logics by analogy with superintuitionistic logics. We describe the global structure of this lattice by splitting it into several subintervals, and prove some new completeness theorems for super-Belnap logics. The crucial technical tool for this purpose will be the so-called antiaxiomatic (or explosive) part operator. The antiaxiomatic (or explosive) extensions of Belnap–Dunn logic turn out to be of particular interest owing to their connection to graph theory: (...) the lattice of finitary antiaxiomatic extensions of Belnap–Dunn logic is isomorphic to the lattice of upsets in the homomorphism order on finite graphs (with loops allowed). In particular, there is a continuum of finitary super-Belnap logics. Moreover, a non-finitary super-Belnap logic can be constructed with the help of this isomorphism. As algebraic corollaries we obtain the existence of a continuum of antivarieties of De Morgan algebras and the existence of a prevariety of De Morgan algebras which is not a quasivariety. (shrink)

Earlier algebraic semantics for Belnapian modal logics were defined in terms of twist-structures over modal algebras. In this paper we introduce the class of BK -lattices, show that this class coincides with the abstract closure of the class of twist-structures, and it forms a variety. We prove that the lattice of subvarieties of the variety of BK -lattices is dually isomorphic to the lattice of extensions of Belnapian modal logic BK . Finally, we describe invariants determining a twist-structure (...) over a modal algebra. (shrink)

Lattice logic, bilattice logic, and paraconsistent quantum logic are investigated based on monosequent systems. Paraconsistent quantum logic is an extension of lattice logic, and bilattice logic is an extension of paraconsistent quantum logic. Monosequent system is a sequent calculus based on the restricted sequent that contains exactly one formula in both the antecedent and succedent. It is known that a completeness theorem with respect to a lattice-valued semantics holds for a monosequent system for lattice logic. A (...) completeness theorem with respect to a lattice-valued semantics is proved for paraconsistent quantum logic, and a completeness theorem with respect to a bilattice-valued semantics is proved for bilattice logic. Some syntactical properties, including cut-elimination and duality, are also investigated for the monosequent systems for these logics. (shrink)

In this paper a solution of Whitehead’s problem is presented: Starting with a purely mereological system of regions a topological space is constructed such that the class of regions is isomorphic to the Boolean lattice of regular open sets of that space. This construction may be considered as a generalized completion in analogy to the well-known Dedekind completion of the rational numbers yielding the real numbers . The argument of the paper relies on the theories of continuous lattices and (...) “pointless” topology.

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