Consecuencia lógica: modelos conjuntistas y aspectos modales

According to Etchemendy, in attempting to offer an analysis of the modal features of the intuitive concept of logical consequence, Tarski has committed a modal fallacy. In this paper, I consider the thesis according to it is posible to analyze the modals properties of concept of logical consequence through of a generalization on set-theoretical interpretations. As is known, some philosophers have tried to argue for the transit from the general to the modal by showing that there are enough settheoretic interpretations so as to be able to represent the modal features of the intuitive concept of consequence. As is also known, those people have encountered a lot of difficulties. In the present paper, I will try to show that those problems are related not with the specific possibility of accounting for the modal features by means of a set-theoretic notion of model but with the possibility of coming up with a precise mathematical theory for the concept of interpretation, and, as such, they can be solved by way of appealing to the usual solutions to this problem

Logic in the Tractatus

I present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein’s “form-series” device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named. There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is countably infinite, then the property of being a tautology is \Pi^1_1-complete. But third, it is only granted the assumption of countability that the class of tautologies is \Sigma_1-definable in set theory. Wittgenstein famously urges that logical relationships must show themselves in the structure of signs. He also urges that the size of the universe cannot be prejudged. The results of this paper indicate that there is no single way in which logical relationships could be held to make themselves manifest in signs, which does not prejudge the number of objects

Consecuencia lógica: modelos conjuntistas y aspectos modales

According to Etchemendy, in attempting to offer an analysis of the modal features of the intuitive concept of logical consequence, Tarski has committed a modal fallacy. In this paper, I consider the thesis according to it is posible to analyze the modals properties of concept of logical consequence through of a generalization on set-theoretical interpretations. As is known, some philosophers have tried to argue for the transit from the general to the modal by showing that there are enough settheoretic interpretations so as to be able to represent the modal features of the intuitive concept of consequence. As is also known, those people have encountered a lot of difficulties. In the present paper, I will try to show that those problems are related not with the specific possibility of accounting for the modal features by means of a set-theoretic notion of model but with the possibility of coming up with a precise mathematical theory for the concept of interpretation, and, as such, they can be solved by way of appealing to the usual solutions to this problem

Le concept de conséquence logique chez Tarski et sa critique

Il est commun, dans les manuels de logique, de présenter une définition sémantique du concept de conséquence logique. Cette approche est le fruit d'une tradition qu'on peut faire remonter au moins jusqu'aux travaux d'Alfred Tarski dans les années 1920 et 1930, lequel propose une définition du concept de conséquence en termes de modèles et de satisfaction : une conclusion est une conséquence logique d'un ensemble de prémisses si et seulement si tous les modèles des prémisses sont aussi un modèle de la conclusion. Autrement dit, une conclusion est une conséquence logique d'un ensemble de prémisses si la vérité est nécessairement préservée des prémisses à la conclusion. Cette définition a le mérite, selon Tarski, de rapporter le concept de conséquence aux critères de formalité et de nécessité. John Etchemendy a remis à l'ordre du jour l'analyse du concept de conséquence logique, dans les années 1980 et 1990, par sa critique de la définition de Tarski. Ses arguments visent à identifier des problèmes de nature conceptuelle et de nature extensionnelle dans la définition tarskienne. Selon lui, la définition repose d'abord sur une confusion entre les approches représentationnelle et interprétationnelle de la sémantique. Elle échoue ensuite à caractériser adéquatement la nécessité du concept de conséquence logique. Enfin, la définition tarskienne déclare ou bien trop, ou bien trop peu d'arguments comme étant valides. Ce mémoire porte sur la définition tarskienne du concept de conséquence logique et sur la littérature critique qu'elle a suscitée, particulièrement depuis les années 1990 et les travaux de John Etchemendy. Après des présentations philosophiques détaillées de la définition de Tarski et de sa critique par Etchemendy, je tente de réhabiliter la définition tarskienne en montrant des limites de cette critique sur chacun des trois axes.\ud ______________________________________________________________________________ \ud MOTS-CLÉS DE L’AUTEUR : Conséquence logique, Alfred Tarski, John Etchemendy, définition, sémantique, théorie des modèles

Carnap's early semantics

In jüngerer Zeit hat sich ein verstärktes Interesse an den historischen und technischen Details von Carnaps Philosophie der Logik und Mathematik entwickelt. Meine Dissertation knüpft an diese Entwicklung an und untersucht dessen frühe und formative Beiträge aus den späten 1920er Jahren zu einer Theorie der formalen Semantik. Carnaps zu Lebzeiten unveröffentlichtes Manuskript Untersuchungen zur allgemeinen Axiomatik (Carnap 2000) beinhaltet ein Reihe von erstmals formal entwickelten Definitionen der Begriffe ‚Modell’, ‚Modellerweiterung’, und ‚logischer Folgerung’. Die vorliegende Dissertation entwickelt eine logische und philosophische Analyse dieser semantischen Begriffsbildungen. Darüber hinaus wird Carnaps frühe Semantik in ihrem historisch-intellektuellen Entwicklungskontext diskutiert. Der Fokus der Arbeit liegt in der Thematisierung einiger interpretatorischer Fragen zu dessen implizit gehaltenen Annahmen bezüglich der Variabilität des Diskursuniversums von Modellen sowie zur Interpretation seiner typen-theoretischen logischen Sprache. Mit Bezug auf eine Reihe von historischen Dokumenten aus Carnaps Nachlass, insbesondere zu dem geplanten zweiten Teil der Untersuchungen wird erstens gezeigt, dass dessen Verständnis von Modellen in wesentlichen Punkten heterodox gegenüber dem modernen Begriffsverständnis ist. Zweitens, dass Carnap von einer ‚nonstandard’ Interpretation der logischen Hintergrundtheorie für seine Axiomatik ausgeht. Die Konsequenzen dieser semantischen Annahmen für dessen Konzeptualisierung von metatheoretischen Begriffen werden näher diskutiert. Das erste Kapitel entwickelt eine kritische Analyse von Carnaps Versuch, die axiomatische Definition von Klassen von mathematischen Strukturen mittels des Begriffs von ‚Explizitbegriffen’ formal zu rekonstruieren. Im zweiten Kapitel werden die Implikationen von Carnaps frühem Modellbegriff für seine Theorie von Extremalaxiomen näher beleuchtet. Das letzte Kapitel bildet eine Diskussion der konkreten historischen Einflüsse, insbesondere durch den Mengentheoretiker Abraham Fraenkel, auf Carnaps formale Theorie von Minimalaxiomen.In recent years one was able to witness an intensified interest in the technical and historical details of Carnap’s philosophy of logic and mathematics. In my thesis I will take up this line and focus on his early, formative contributions to a theory of semantics around 1928. Carnap’s unpublished manuscript Untersuchungen zur allgemeinen Axiomatik (Carnap 2000) includes some of the first formal definitions of the genuinely semantic concepts of a model, model extensions, and logical consequence. In the dissertation, I provide a detailed conceptual analysis of their technical details and contextualize Carnap’s results in their historic and intellectual environment. Certain interpretative issues related to his tacit assumptions concerning the domain of a model and the semantics of type theory will be addressed. By referring to unpublished material from Carnap’s Nachlass I will present archival evidence as well as more systematic arguments to the view that Carnap holds a heterodox conception of models and a nonstandard semantics for his type-theoretic logic. Given these semantic background assumptions, their impact on Carnap’s conceptualization of certain aspects of the metatheory of axiomatic theories will be evaluated. The first chapter critically discusses Carnap’s attempt to explicate one of the crucial semantic innovations of formal axiomatics, i.e. the definition of classes of structures, via his notion of ‘Explizitbegriffe’. The second chapter analyses the impact of Carnap’s early theory of model for his theory of extremal axioms. The final chapter reviews the mathematical influences, most importantly by the set theoretician Abraham Fraenkel on Carnap’s specific formalization of minimal axioms