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, ...

In this manuscript, published here for the first time, Tarski explores the concept of logical notion. He draws on Klein's Erlanger Programm to locate the logical notions of ordinary geometry as those invariant under all transformations of space. Generalizing, he explicates the concept of logical notion of an arbitrary discipline.

This classic undergraduate treatment examines the deductive method in its first part and explores applications of logic and methodology in constructing mathematical theories in its second part. Exercises appear throughout.

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)

We provide for the first time an exact translation into English of the Polish version of Alfred Tarski's classic 1936 paper, whose title we translate as ?On the Concept of Following Logically?. We also provide in footnotes an exact translation of all respects in which the German version, used as the basis of the previously published and rather inexact English translation, differs from the Polish. Although the two versions are basically identical, to an extent that is even uncanny, we note (...) more than 400 differences. Several dozen of these are substantive differences due to revisions by Tarski to the Polish version which he did not incorporate in the German version. With respect to these revisions the Polish version should be regarded as more authoritative than the German. Hence scholars limited to an English translation should use ours. (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.

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.

Provability, Computability and Reflection.

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.

Sur la décomposition des ensembles de points en parties respectivement congruentes.

Der Laie spricht manchmal die Ansicht aus, die Mathematik ware heutzutage schon eine tote Wissenschaft: nachdem sie einen ungemein hohen Grad der Entwicklung erreicht hat, sei sie in ihrer steinernen Vollkommenheit erstarrt. Dies ist ein vollig irriges Bild der Situation: nur wenige Wissenschaftsgebiete befinden sich heute in der Phase einer solch intensiven Entwicklung wie die Mathematik. Diese Entwicklung ist dabei auBerordentlich vie1seitig: die Mathematik erweitert ihre Domane nach allen moglichen Richtungen, sie wachst in die Rohe, in die Weite und in (...) die Tiefe. Sie wachst in die Rohe, da auf dem Boden ihrer alten Theorien, denen eine jahrhundert-, ja sogar jahrtausend- lange Entwicklung zugrunde liegt, immer wieder neue Prob1eme auftauchen, immer scharfere und vollkommenere Resultate er- zielt werden; in die Weite, da ihre Methoden andere Wissen- schaftszweige durchdringen, ihr Untersuchungsbereich immer umfangreichere Gebiete von Erscheinungen umfaBt und immer neue Theorien in den groBen Kreis mathematischer Disziplinen einbezogen werden; und schlieBlich in die Tiefe, da ihre Grund- lagen immer mehr gefestigt, die bei ihrem Aufbau angewandten Methoden immer vollkommener werden und ihre Prinzipien an Dauerhaftigkeit gewinnen. In dem vorliegenden Buch wiinschte ich dem Leser, der fiir die gegenwartige Mathematik Interesse aufweist, aber ihr fern- steht, mindestens einen ganz allgemeinen Begriff von dieser dritten Entwicklungslinie der Mathematik, d. i. von ihrer Ent- wicklung in die Tiefe, zu geben. (shrink)

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)

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 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.