I ON THE PRIMITIVE TERM OF LOGISTICf IN this article I propose to establish a theorem belonging to logistic concerning some connexions, not widely known, ...

This paper is an edited form of a letter written by the two authors (in the name of Tarski) to Wolfram Schwabhäuser around 1978. It contains extended remarks about Tarski's system of foundations for Euclidean geometry, in particular its distinctive features, its historical evolution, the history of specific axioms, the questions of independence of axioms and primitive notions, and versions of the system suitable for the development of 1-dimensional geometry.

In this article the author first described the developments which brought to focus the importance of consistency proofs for mathematics, and which led Hilbert to promote the science of metamathemat-ics. Further comments and remarks concern the (partly analogous) beginnings of the work on the decision problem, Gödel?s theorems and related matters, and general metamathematics. An appendix summarizes a text by the author on completeness and categoricity.

This book is well known for its proof that many mathematical systems - including lattice theory and closure algebras - are undecidable. It consists of three treatises from one of the greatest logicians of all time: "A General Method in Proofs of Undecidability," "Undecidability and Essential Undecidability in Mathematics," and "Undecidability of the Elementary Theory of Groups.".

Published with the aid of a grant from the National Endowment for the Humanities. Contains the only complete English-language text of The Concept of Truth in Formalized Languages. Tarski made extensive corrections and revisions of the original translations for this edition, along with new historical remarks. It includes a new preface and a new analytical index for use by philosophers and linguists as well as by historians of mathematics and philosophy.

The logical theory which is called thecalculus of (binary) relations, and which will constitute the subject of this paper, has had a strange and rather capricious line of historical development. Although some scattered remarks regarding the concept of relations are to be found already in the writings of medieval logicians, it is only within the last hundred years that this topic has become the subject of systematic investigation. The first beginnings of the contemporary theory of relations are to be found (...) in the writings of A. De Morgan, who carried out extensive investigations in this domain in the fifties of the Nineteenth Century. De Morgan clearly realized the inadequacy of traditional logic for the expression and justification, not merely of the more intricate arguments of mathematics and the sciences, but even of simple arguments occurring in every-day life; witness his famous aphorism, that all the logic of Aristotle does not permit us, from the fact that a horse is an animal, to conclude that the head of a horse is the head of an animal. In his effort to break the bonds of traditional logic and to expand the limits of logical inquiry, he directed his attention to the general concept of relations and fully recognized its significance. Nevertheless, De Morgan cannot be regarded as the creator of the modern theory of relations, since he did not possess an adequate apparatus for treating the subject in which he was interested, and was apparently unable to create such an apparatus. His investigations on relations show a lack of clarity and rigor which perhaps accounts for the neglect into which they fell in the following years. (shrink)

Published with the aid of a grant from the National Endowment for the Humanities. Contains the only complete English-language text of The Concept of Truth in Formalized Languages. Tarski made extensive corrections and revisions of the original translations for this edition, along with new historical remarks. It includes a new preface and a new analytical index for use by philosophers and linguists as well as by historians of mathematics and philosophy.

This article presents Tarski's Address at the Princeton Bicentennial Conference on Problems of Mathematics, together with a separate summary. Two accounts of the discussion which followed are also included. The central topic of the Address and of the discussion is decision problems. The introductory note gives information about the Conference, about the background of the subjects discussed in the Address, and about subsequent developments to these subjects.

We are inclined to believe that, by means of an argument entirely analogous to that which leads to the Richard antinomy, the notion of definability as applied to entities discussed in a formal system can easily be shown not to be itself definable in this system. It will be seen from this discussion that actually the situation is not quite so simple as it would appear at first glance. Our discussion will have a rather sketchy and informal character.

Le but principal de la communication est d’esquisser les traits essentiels de la méthode appliquée dans les sciences déductives.1. A quoi tend la méthode déductive? Termes primitifs et définis ; axiomes et théorèmes. Les sciences antérieures à une science donnée. La méthode déductive considérée comme propriété caractéristique des mathématiques.2. Liberté dans le choix des termes primitifs et des axiomes ; notion d’équivalence de deux systèmes de termes ou de propositions.Postulats d’indépendance des termes primitifs et des axiomes.3. Postulats de la formalisation (...) des définitions et des démonstrations. Règles de définition et de démonstration ; les démonstrations complètes ; les sciences déductives formalisées. Le rôle de la logique mathématique au point de vue des nouvelles exigences méthodologiques.4. Les notions de science non-contradictoire et complète. L’importance théorique de ces notions ; leur rapport avec le problème de la vérité des mathématiques. Pourquoi les notions en question n’exercent pas en pratique une grande influence sur la construction des sciences mathématiques.5. Certaines propriétés des sciences mathématiques découlant d’une application conséquente de la méthode déductive.ion du sens des termes primitifs dans la construction de la science. Modèles et interprétations du système d’axiomes. Concept de système formel. Valeur de la méthode déductive du point de vue de l’économie de la pensée. (shrink)

Cette Introduction à la logique constitue le cours de base des étudiants en sciences et en philosophie. La présentation de la logique formelle dans son contexte historique est accompagnée d'un exposé très clair des notions syntaxiques et sémantiques nécessaires au calcul des énoncés et au calcul des prédicats du premier ordre... Les méthodes formelles les plus classiques (tables de vérité, arbres de consistance, dérivations) sont explicitées au travers de nombreux exercices d'application accompagnés de leurs corrigés. La seconde partie de l'ouvrage (...) fournit au lecteur les contenus théoriques nécessaires à la compréhension des débats qui ont animés la philosophie de la logique et son dialogue avec les mathématiques depuis la fin du XIX, siècle. L'axiomatisation du calcul des énoncés (présentation de So), les notions de complétude sémantique et syntaxique comme le concept de décidabilité sont introduits. Un chapitre est consacré à l'axiomatisation du calcul des prédicats du premier ordre et permet de familiariser le lecteur à la métathéorie. La présentation des résultats fondateurs (théorème de consistance, théorème de Tarski, théorème de complétude) jette les bases de la théorie des modèles. Enfin, l'ouvrage s'achève par les fondations de la logique mathématique (système Se de la théorie des ensembles et système Sn de la théorie des nombres naturels) et par l'exposé des célèbres théorèmes de Gôdel... (shrink)

Alfred Tarski’s theory of truth, to which we will also refer as the classical theory,1 has a conspicuous place in mathematics as well as in general philosophy. The place in philosophy appears the more prominent of the two, although it is still somewhat unsettled, and perhaps even controversial.

This article presents Tarski's Address at the Princeton Bicentennial Conference on Problems of Mathematics, together with a separate summary. Two accounts of the discussion which followed are also included. The central topic of the Address and of the discussion is decision problems. The introductory note gives information about the Conference, about the background of the subjects discussed in the Address, and about subsequent developments to these subjects.