Inλ-calculus, the strategy of leftmost reduction (“call-by-name”) is known to have good mathematical properties; in particular, it always terminates when applied to a normalizable term. On the other hand, with this strategy, the argument of a function is re-evaluated at each time it is used.To avoid this drawback, we define the notion of “storage operator”, for each data type. IfT is a storage operator for integers, for example, let us replace the evaluation, by leftmost reduction, ofϕτ (whereτ is an integer, (...) andϕ anyλ-term) by the evaluation oftτϕ. Then, this computation is the same as the following: first computeτ up to some reduced formτ 0, and then applyϕ toτ 0. So, we have simulated “call-by-value” evaluation within the strategy of leftmost reduction.The main theorem of the paper (Corollary of Theorem 4.1) shows that, in a second orderλ-calculus, using Gödel's translation of classical intuitionistic logic, we can find a very simple type (or specification) for storage operators. Thus, it gives a way to get such operators, which is to prove this type in second order intuitionistic predicate calculus. (shrink)

Le problème de la vérité a toujours préoccupé l'esprit humain; il hante pour ainsi dire l'homme d'aujourd'hui. Dans les brèves réflexions qui vont suivre, nous nous proposons de présenter une approximation tres limitée de ce problème tel qu'il se pose à la philosophic traditionnelle, à savoir: Le caractère historique et relatifde la connaissance humaine est-il conciliable avec I'affirmation de vérites immuables?

The relation between morality and rationality (in the sense of rational choice and rational behaviour) is a prominent theme in (the tradition of) moral philosophy. D. Gauthier’s account of this relation is an extraordinarily impressive one. He attempts to demonstrate a general coincidence (at different levels) between rationality and morality. His approach is discussed in what follows, and it will be shown that most of his ‘coincidence claims’ are exaggerated.

, which uses the intuitionistic propositional calculus, with the only connective →. It is very important, because the well known Curry-Howard correspondence between proofs and programs was originally discovered with it, and because it enjoys the normalization property: every typed term is strongly normalizable. It was extended to second order intuitionistic logic, in 1970, by J.-Y. Girard [4], under the name of system F, still with the normalization property.More recently, in 1990, the Curry-Howard correspondence was extended to classical logic, following (...) Felleisen and Griffin [6] who discovered that the law of Peirce corresponds to control instructions in functional programming languages. It is interesting to notice that, as early as 1972, Clint and Hoare [1] had made an analogous remark for the law of excluded middle and controlled jump instructions in imperative languages.There are now many type systems which are based on classical logic; among the best known are the system LC of J.-Y. Girard [5] and the λμ-calculus of M. Parigot [11]. We shall use below a system closely related to the latter, called the λ c -calculus [8, 9]. Both systems use classical second order logic and have the normalization property.In the sequel, we shall extend the λ c -calculus to the Zermelo-Frænkel set theory. The main problem is due to the axiom of extensionality. To overcome this difficulty, we first give the axioms of ZF in a suitable (equivalent) form, which we call ZF ɛ. (shrink)

We describe here a simple method in order to obtain programs from proofs in second-order classical logic. Then we extend to classical logic the results about storage operators proved by Krivine for intuitionistic logic. This work generalizes previous results of Parigot.

Introduction. Il est bien connu que la correspondance de Curry-Howard permet d'associer un programme, sous la forme d'un λ-terme, à toute preuve intuitionniste, formalisée dans le calcul des prédicats du second ordre. Cette correspondance a été étendue, assez récemment, à la logique classique moyennant une extension convenable du λ-calcul. Chaque théorème formalisé en logique du second ordre correspond donc à une spécification de programme.Il se pose alors le problème, en général tout à fait non trivial, de trouver la spécification associée (...) à un théorème donné; autrement dit, de déterminer le comportement opérationnel commun aux λ-termes associés aux diverses démonstrations formelles du théorème considéré.Cette question est résolue ici pour le théorème de complétude de la logique classique.La première étape consiste à formaliser convenablement ce théorème en logique du second ordre. Ce travail est fait complètement dans la section 1. Il a comme sous-produit, peut-être inattendu, de montrer que ce théorème est prouvable en logique intuitionniste du second ordre. Ceci, toutefois, à condition d'introduire une légère variante de la notion de modèle, en admettant un modèle supplémentaire trivial, où toute formule est satisfaite.On notera, à ce sujet, que des preuves intuitionnistes du théorème de complétude de la logique intuitionniste, utilisant des variantes de la notion de modele de Kripke, ont été données par H. Friedman [7] et W. Veldman [8]. On remarquera également qu'un argument de G. Kreisel [2] montre que le théorème de complétude habituel de la logique intuitionniste n'a pas de preuve intuitionniste. (shrink)

Artificial intelligency can bring speed and reliability to drug discovery process. It represents an additional intelligence, which in any case can replace the strategic and logic creative insight of the medicinal chemist who remains the architect and molecule master designer. In terms of drug design, artificial intelligency, deep learning machines, and other revolutionary technologies will match with the medicinal chemist’s natural intelligency, but for sure never go beyond. This manuscript tries to assess the impact of the artificial intelligency on drug (...) discovery today. (shrink)

Aristotle’s Prior Analytics identifies deductions simpliciter with inferential necessity, so that a deduced conclusion is necessarily inferred from some premises. Modern logical reconstructions claim that inferential necessity in Aristotle corresponds to logical validity. However, this logical reconstruction fails on two accounts. First, logical validity does not highlight Aristotle’s distinction between inferential necessity and predicative necessity, meaning that the inferential necessity of a deduction is not of the same kind as the predicative necessity of a non‑deductive argument. Second, logical validity does (...) not explain the relevance of Aristotle’s distinction between complete and incomplete deductions. Logicians speak of complete deduction by adding the term “obvious” or “transparent” to logical validity, and then criticize Aristotle’s view for being unclear. However, Aristotle’s position is not confronted with this difficulty. There is nothing to add to inferential necessity, which already means complete deducibility, as opposed to incomplete deducibility, deemed to be potentially complete. Accordingly, the Prior Analytics reduces the incomplete deductions to the complete deductions in order to prove the potential, inferential necessity of the incomplete deductions. Logical validity would have been faithful to Aristotle’s text, if it had been possible to coin a notion of potential validity, distinct from both validity and invalidity. (shrink)

This paper shows that Aristotle's De Interpretatione does not separate syntax from semantics. Linguistic sentences are not syntactic entities, and non-linguistic meanings are not semantic propositions expressed by linguistic sentences. In fact, Aristotle resorts to a mental conception of meaning, distinguishing linguistic meanings in a given language from non-linguistic mental contents in relation to actual things: while the former are not the same for all, the latter are shared by everyone. Aristotle is not a modern logician, like Boole, Frege, or (...) Russell, in so far as a mental conception of meaning does not reveal an abstract semantics for a syntactic language. (shrink)

L’une des conséquences indirectes de la rupture de l’unité confessionnelle aux temps modernes, fut une exploration accélérée du passé chrétien. Pour cela, une masse régulièrement accrue de documents venait nourrir deux représentations antagonistes de l’histoire de l’Eglise, lesquelles étaient défendues par des travaux d’érudition au service de deux conceptions de l’histoire, qui étaient aussi bien deux théologies de l’Eglise. Cependant, l’exigence de proximité des sources devait induire un mode particulier d’écriture de l’histoire qui transcenderait les frontières confessionnelles, tout en maintenant (...) des positions différentes. La prise en compte des documents allait notamment renforcer une certaine critique, pour laquelle surgissent notamment les noms de Mabillon et de Le Nain de Tillemont, même si les positions continueraient de se différencier tout au long du XVIIe siècle.One of the indirect consequences of the break up of confessional unity in modern times was an accelerated exploration of the Christian past. In the process, a constantly increasing mass of documents came to fuel two opposing representations of the history of the Church. These representations were supported by works of erudition in keeping with two different conceptions of history, which were also two theologies of the Church. The use of new sources became required, however, and led it to a particular way of writing history that would transcend confessional boundaries, while maintaining different positions. Taking these documents into account strengthened a critical current that gave rise to Mabillon and Le Nain de Tillemont, even though the positions continued to differ throughout 17th century. (shrink)

Trade in high-value-added toxins for therapeutic and biological use is expanding. These toxins are generally derived from microalgae belonging to the dinoflagellate family. Due to the difficulties to grow these sensitive planktonic species and to the complexity of methods used to synthesize these molecules, which are generally complex chemical structures, biotoxin manufacturers called on artificial intelligence technologies. Manufacturing processes have been greatly improved through the development of specific learning neural networks, applied to each phases of biotoxin production: photo-bioreactors operating at (...) optimal yied; new chemical synthesis research processes; toxin biosynthetic research pathways offering short-cut possibilities. (shrink)

On ne compte plus les articles, les revues, les livres, les colloques, les émissions de radio et de télévision qui traitent des questions éthiques. Cet engouement a profité naturellement à la théologie morale. On pourrait même dire qu'elle est la seule discipline théologique à avoir acquis un droit de parole dans la cité séculière. Pour les meilleures raisons, et aussi les moins bonnes, l'attention des chrétiens, des spécialistes et de ceux qui cherchent simplement un sens à leur vie, ne retient (...) qu'elle dans le message délivré par l'Eglise. Etonnant retournement! La conscience contemporaine s'inquiète, en effet, des développements technologiques qui précèdent toute recherche de sens et récusent toute limitation. Le progrès tient lieu de règle. Pour la première fois depuis longtemps, cette même conscience admet que les problèmes posés à nos sociétés, analysés jusque-là en termes strictement techniques, sont en réalité des défis moraux. En réponse, il convient de repenser la morale chrétienne comme une éthique de construction de soi, comme un style de vie. La tâche est immense. La première étape nous impose de visiter notre mémoire. Ce dictionnaire n'a d'autre ambition que de fournir les repères facilitant cet inventaire. Il pare au plus pressé : les interrogations actuelles portent davantage sur les fondements de la morale catholique et sur ses applications personnelles. Ce livre limite donc son propos à la morale fondamentale, personnelle et familiale. Edité pour la première fois en 1991, ce dictionnaire s'est rapidement épuisé. Des mots nouveaux font leur apparition dans cette seconde édition, tandis que d'autres ont vu leur présentation renouvelée. Une nouvelle introduction met en perspective ce travail vis-à-vis du Catéchisme de l'Eglise catholique et de Veritatis Splendor. (shrink)

Long known solely as fascism’s precursor, Joseph de Maistre re-emerges in this volume as a versatile thinker with a colossally diverse posterity whose continuing relevance in Europe is ensured by his theorization of the encounter between tradition and modernity.

In this article first appeared in 1997, Jean-Louis Chrétien examines the meaning of prayer in Kierkegaard’s writings and existence. By focusing on the difficulties of this task and with meticulous attention to the vast work of the Danish philosopher, Chrétien describes prayer as a tense and agonistic experience, akin to the evangelical struggle between Jacob and the angel. Just like in his well-known phenomenological analysis, “The Wounded Word: Phenomenology of Prayer”, the author identifies in prayer a paradoxical articulation (...) of struggle and peace, gift and endeavour, speaking and listening, through which the human being is profoundly transformed and finally learns to truly say “thank you”. (shrink)