Proofs, Reasoning and the Metamorphosis of Logic

Revised version of a conference given under title "From Natural Deduction to the nature of reasoning", at the colloquium Natural Deduction organized by Luiz Carlos Pereira and dedicated to the work of Dag Prawitz, Rio de Janeiro, Brazil, July 2001.With the “mathematical watershed”, Logic had been transformed into a foundational theory for mathematics, a theory of truth and proofs - far away from its philosophical status of theory of the intellectual process of reasoning. With the recent substitution of the traditional proofs-as-discourses paradigm by the proofs-as-programs one, Logic is now becomming a foundational theory for computing. One could interpret this new watershed as being “yet another technological drift”, bringing Logic always closer to applied ingeneering, always further from the human intellectual process of reasoning. This article promote the dual point of view: enlightened by the contemporary analysis of the dynamic of proofs, which bring us to a new understanding of the semantic counterpart of processes operationality (including the links between semantic dereliction due to inconsistency and computational exuberance), Logic has never appeared so close to being, finally, the theory of reasoning.A partir de son "tournant mathématique", la logique se convertit en une théorie des fondements des mathématiques, une théorie de la vérité et des preuves - loin de son statut philosophique de theorie du processus intellectuel de raisonnement. Avec le remplacement récent du paradigme traditionnel des preuves-comme-discours par celui des preuves-comme-programes, la Logique est à présent devenue une théorie des fondements du calcul. On peut voir dans ce nouveau tournant comme un glissement supplémentaire qui conduirait la logique toujours plus près de l'ingénierie appliquée, toujours plus loin de son statut historique de théorie des processus de raisonnement. Cet article défend le point de vue dual: sous la lumière des analyses contemporaines de la dynamique des preuves, qui conduisent à une compréhension originale de la contrepartie sémantique de l'opérationalité des processus (y compris les liens entre déreliction sémantique due à l'incohérence et exubérance calculatoire), la logique n'a finalement jamais semblé aussi proche de devenir la théorie du raisonnement

Nature et logique, de G. Gentzen à J-Y. Girard

Version ancienne d'un article dont la version modifiée a finalement été acceptée pour publication en 2014 dans Logique et AnalyseJ.-Y. Girard's conception of logic originates in the critique of logical naturality made by G. Gentzen around 1930. Recently, Girard radicalized this critique and proposed a program for new foundations for logic, as a product of a general theory of interaction. The present paper is an attempt to show that this program tends to reunify the idea of natural logic and the idea of a logic of nature.La conception de la logique de J.-Y. Girard s'enracine dans la critique de la naturalité logique opérée par G. Gentzen dans les années 1930. Récemment, Girard a radicalisé cette critique et proposé une entreprise de refondation de la logique comme produit d'une théorie générale de l'interaction. La présente étude vise à montrer que cette entreprise va dans le sens d'une réunification de l'idée de logique naturelle et de celle de logique de la nature

Computational isomorphisms in classical logic: (Extended Abstract)

We prove that any pair of derivations, without structural rules, of F ) G and G ) F , where F , G are rst-order formulas `without any qualities', in a constrained classical sequent calculus LK p , denes a computational isomorphism up to an equivalence on derivations based upon reversibility properties of logical rules. This result gives a rationale behind the success of Girard's denotational semantics for classical logic, in which all standard `linear' boolean equations are satised. 1 Introduction 1.1 A patch of paradise to be broadened In recent work [1] devoted to the proof theory of classical logic, we embarked on the project of overcoming the obstacles that prevent cut from being a decent binary operation on the set of classical sequent derivations. To clarify what we mean by decency, let us have a look at the world of simply typed -calculus, which, seen from a normalization-as-computation point of view, is something close to a patch of paradise. danos@logique..

Apresentação

Introduction to O que nos faz pensar 39.Apresentação à O que nos faz pensar 39

Linear logic and elementary time

AbstractA subsystem of linear logic, elementary linear logic, is defined and shown to represent exactly elementary recursive functions. Its choicest part consists in reducing the deductive power of the exponential, also known as the “bang,” which, in linear logic, is in charge of controlling duplication in the cut-elimination process

Computational isomorphisms in classical logic

All standard ‘linear’ boolean equations are shown to be computationally realized within a suitable classical sequent calculus LKη p. Specifically, LKη p can be equipped with a cut-elimination compatible equivalence on derivations based upon reversibility properties of logical rules. So that any pair of derivations, without structural rules, of F ⇒ G and G ⇒ F, where F, G are first-order formulas ‘without any qualities’, defines a computational isomorphism

Logique et métaphysique

Conférence invitée (à l'initiative de l'Universidade Federal do Rio Grance do Norte; voyage et séjour pris en charge par le colloque)International audienc

: Introduction

Cet article a été publié par erreur sous le nom "Introduction". Il s'agit en fait d'un article de synthèse sur les métamorphoses contemporaines de la logiqueInternational audienc

On the decidability of monadic first order logic in sequent calculus

In this article, a syntactical proof of decidability ofmonadic first-order logic (and of its completeness for finite models) is given. Theproof is obtained  by adapting to the case of monadic logic, the proof given byKetonen/Schütte for first-order logic completeness (method of “construction of therefutation tree”, which actually describes, for the propositional fragment, adecision algorithm).  In the general case (i.e. not restricted to monadic logic), the treatment of existential quantifiers imposes an enumeration of all the terms ofthe language, treatment that prevents the algorithm’s termination, and thus the decision. In the monadic case, however, any formula can be put in a specificcanonical form, a result due to H. Behmann (1922), canonical form which implies thatonly a bounded set of terms have to be taken in consideration. The treatment of existential quantifiers can thus be done with a finite number of terms.Decidability (and completeness for finite models) of the monadic case follows. --- Original in French

Nature et logique de G. Gentzen à J.-Y. Girard

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