Most of the Bolzano literature is exegetical, neglecting, unfortunately, the great potential of his logic as the beginning of a PROGRAMME. Specifically, his unorthodox construai of the consequence relation as triadic, and his account of logical form are promising beginnings which even as they stand shed light on question of relevance, the ancient problems of enthymemes and others. Instead of developing these suggestions, Bolzano scholars have been occupied with elucidating the ontology of sentences in themselves, and related (...) topics. I argue, and believe to be in agreement with Bolzano, that the nature of sentences is fully explained by the relations that hold between them, just as money has no nature or essence beyond the transactions it makes possible. It follows that the development of his logic would contribute at least as much to the understanding of sentences than any exegesis. (shrink)

Most of the Bolzano literature is exegetical, neglecting, unfortunately, the great potential of his logic as the beginning of a PROGRAMME. Specifically, his unorthodox construai of the consequence relation as triadic, and his account of logical form are promising beginnings which even as they stand shed light on question of relevance, the ancient problems of enthymemes and others. Instead of developing these suggestions, Bolzano scholars have been occupied with elucidating the ontology of sentences in themselves, and related (...) topics. I argue, and believe to be in agreement with Bolzano, that the nature of sentences is fully explained by the relations that hold between them, just as money has no nature or essence beyond the transactions it makes possible. It follows that the development of his logic would contribute at least as much to the understanding of sentences than any exegesis. (shrink)

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.

In this paper I argue that Bolzano's concept of deducibility and Tarski's concept of logical consequence differ with respect to their philosophical intent. I distinguish between epistemic and ontic approaches to logic, and argue that Bolzano's deducibility presupposes an epistemic approach, while Tarski's logical consequence presupposes an ontic approach.

This book with an introduction by Witold Marciszewski, views the history of philosophy and logic from 1837 to 1939 from the perspective of the cradle of modern exact philosophy - Central Europe. In a series of case studies, it illuminates the developments in this region, most notably in Austria and Poland, examining thinkers such as Bolzano, Brentano, Meinong, Husserl, Twardowski, Lesniewski, and Tarski, as well as the logicians like Frege and Russell with whom they bore a close resemblance. (...) The book challenges established views about the history of philosophy and logic in Europe, and shows the vitality of the Central European tradition. (shrink)

Inhaltsverzeichnis/Table of Contents: Vorbemerkung/Preface. Dagfin FØLLESDAL: Bolzano's Legacy. Jan BERG: Bolzano, the Prescient Encyclopedist. Jan SEBESTIK: Bolzano, Exner and the Origins of Analytical Philosophy. Paul RUSNOCK: Bolzano and the Traditions of Analysis. Peter SIMONS: Bolzano on Collections. Ali BEHBOUD: Remarks on Bolzano's Collections. Mark SIEBEL: Variation, Derivability and Necessity. Edgar MORSCHER: Bolzano's Method of Variation: Three Puzzles. Rolf GEORGE: Bolzano's Programme andObjects. Mark TEXTOR: Bolzano's Sententialism. Wolfgang KÜNNE: Propositions in (...) class='Hi'>Bolzano and Frege. Michael DUMMETT: Comments on Wolfgang Künne's Paper. Carsten Uwe GIESKE: Bolzano's Notion of Testifying. (shrink)

In a series of publications beginning in the 1980s, John Etchemendy has argued that the standard semantical account of logical consequence, due in its essentials to Alfred Tarski, is fundamentally mistaken. He argues that, while Tarski's definition requires us to classify the terms of a language as logical or non-logical, no such division is guaranteed to deliver the correct extension of our pre-theoretical or intuitive consequence relation. In addition, and perhaps more importantly, Tarski's account is claimed to (...) be incapable of explaining an essential modal/epistemological feature of consequence, namely, its necessity and apriority. Bernard Bolzano (1781-1848) is widely recognized as having anticipated Tarski's definition in his Wissenschaftslehre (or Theory of Science ) of 1837. Because of the similarities between his account and Tarski's, Etchemendy's arguments have also been extended to cover Bolzano. The purpose of this article is to consider Bolzano's theory in the light of these criticisms. We argue that, due to important differences between Bolzano's and Tarski's theories, Etchemendy's objections do not apply immediately to Bolzano's account of consequence. Moreover, Bolzano's writings contain the elements of a detailed philosophical response to Etchemendy. (shrink)

Depuis 1967, Donald Davidson defend l'idée qu'une théorie de la signification pour une langue naturelle doit prendre la forme d'une théorie tarskienne de la vérité. Je me propose ici d'exposer les grandes lignes de l a conception davidsonienne de la sémantique des langues naturelles et de chercher à préciser en quel sens une theorie tarskienne de la vérité pour une langue L constitue, selon Davidson, une théorié de la signification pour L. Je ferai pour cela abstraction des obstacles qu'il pourrait (...) y avoir à la réalisation du programme de Davidson. (shrink)

Contemporary historians of logic tend to credit Bernard Bolzano with the invention of the semantic notion of consequence, a full century before Tarski. Nevertheless, Bolzano's work played no significant rôle in the genesis of modern logical semantics. The purpose of this paper is to point out three highly original, and still quite relevant themes in Bolzano's work, being a systematic study of possible types of inference, of consistency, as well as their meta-theory. There are certain analogies (...) with Tarski's concerns here, although the main thrust seems to be different, both philosophically and technically. Thus, if only obliquely, we also provide some additional historical perspective on Tarski's achievement. (shrink)

Une théorie davidsonienne de la signification pour une langue L prend la forme d'une theorie tarskienne de la véeritée-dans-L. Une telle théeorie sera absolument radicale s'il est possible d'éetablir qu'elle est tarskienne, c'est-à-dire conforme à la convention T de Tarski, en n'utilisant que des donnéees empiriques dont la description ne fait intervenir aucun concept linguistique, tandis qu'elle sera relativement radicale s'il est possible d'éetablir qu'elle est tarskienne en n'utilisant que des donnéees empiriques dont la description ne fait intervenir aucun (...) concept linguistique propre à L. (shrink)

Contemporary historians of logic tend to credit Bernard Bolzano with the invention of the semantic notion, of consequence, a full century before Tarski. Nevertheless, Bolzano's work played no significant rôle in the genesis of modern logical semantics. The purpose of this paper is to point out three highly original, and still quite relevant themes in Bolzano's work, being a systematic study of possible types of inference, of consistency, as well as their meta-theory. There are certain analogies (...) with Tarski's concerns here, although the main thrust seems to be different, both philosophically and technically. Thus, if only obliquely, we also provide some additional historical perspective on Tarski's achievement. (shrink)

Parmi les travaux contemporains en sémantique du langage ordinaire, le "programme de Davidson " se distingue par sa portée philosophique : la théorie de la vérité que propose Davidson à titre de cadre sémantique s'épanouit en effet en une véritable philosophie du langage, connue sous le nom d'interprétation radicale. Cet ouvrage tente une évaluation critique du programme de Davidson, à partir d'une question à la fois historique et conceptuelle : quels ont les rapports entre le projet d'une théorie (...) de la vérité "à la Tarski ", et celui d'une sémantique formelle du langage ordinaire? (shrink)

The standard semantic definition of consequence with respect to a selected set X of symbols, in terms of truth preservation under replacement (Bolzano) or reinterpretation (Tarski) of symbols outside X, yields a function mapping X to a consequence relation ⇒x. We investigate a function going in the other direction, thus extracting the constants of a given consequence relation, and we show that this function (a) retrieves the usual logical constants from the usual logical consequence relations, and (b) is (...) an inverse to—more precisely, forms a Galois connection with—the Bolzano-Tarski function. (shrink)

Il n’est pas facile de voir quel est l’objectif de ce livre. Au chapitreI, Rivenc annonce que ce qui l’intéresse est le lien chez Davidson entre le format d’une théorie de la signification pour les langues naturelles et le thème de l’interprétation radicale qui serait à l’œuvre dans tout échange linguistique. En fait, Rivenc ne dit à peu près rien sur ce lien. Son livre consiste plutôt en une suite de critiques disparates du programme de Davidson, qu’il emprunte à (...) différents commentateurs de celui-ci. L’auteur omet de présenter les réponses à ces critiques qu’ont données les sympathisants de l’approche davidsonienne, et n’essaie pas de tirer des leçons plus générales de ces critiques concernant la sémantique des langues naturelles. (shrink)

Anstelle einer Einleitung: ein Thema und eine Sichtweise desselben -- Analytische Philsophie zwischen Sprache, Logik und Mathematik -- Die analytische Philosophie und das Phänomenon der Komplementarität -- Kant, Bolzano und Peirce: die Unterschedung des Analytischen und Synthetischen, oder: von der Erkenntnistheorie zur Semantik und Zeichentheorie -- Ernst Cassirer und die Entwicklung von Analyse und Synthese seit Descartes und Leibniz -- Bertrand Russell (1872-1970) -- Die naturalisierte Erkenntnistheorie zwischen Wiener Kries und Pragmatismus: Willard Van Orward [sic] Quine -- Richard Rorty: (...) der Spiegel der Natur -- Die analytische Philosophie, der Mensch, die Kunst und das Denken der Maschinen. (shrink)

Prior to Kripke's seminal work on the semantics of modal logic, McKinsey offered an alternative interpretation of the necessity operator, inspired by the Bolzano-Tarski notion of logical truth. According to this interpretation, `it is necessary that A' is true just in case every sentence with the same logical form as A is true. In our paper, we investigate this interpretation of the modal operator, resolving some technical questions, and relating it to the logical interpretation of modality and some (...) views in modal metaphysics. In particular, we present an hitherto unpublished solution to problems 41 and 42 from Friedman's 102 problems, which uses a different method of proof from the solution presented in the paper of Tadeusz Prucnal. (shrink)

I discuss Prawitz’s claim that a non-reliabilist answer to the question “What is a proof?” compels us to reject the standard Bolzano-Tarski account of validity, andto account for the meaning of a sentence in broadly verificationist terms. I sketch what I take to be a possible way of resisting Prawitz’s claim---one that concedes the anti-reliabilist assumption from which Prawitz’s argument proceeds.

Syntax and semantics in Łukasiewicz infinite-valued sentential logic Ł are harmonized by revising the Bolzano-Tarski paradigm of “semantic consequence,” according to which, θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} follows from Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta $$\end{document} iff every valuation v that satisfies all formulas in Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta $$\end{document} also satisfies θ.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta.$$\end{document} For (...) θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} to be a consequence of Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta $$\end{document}, we also require that any infinitesimal perturbation of v that preserves the truth of all formulas of Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta $$\end{document} also preserves the truth of θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}. An elementary characterization of Łukasiewicz implication shows that the Łukasiewicz axiom →Y)→→X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \rightarrow Y ) \rightarrow \rightarrow X )$$\end{document} guarantees the continuity and the piecewise linearity of the implication operation →\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rightarrow $$\end{document}, an appropriate fault-tolerance property of any logic of [0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{[0,1]}\,}}$$\end{document}-valued observables. The directional derivability of the functions coded by all ψ∈Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi \in \Theta $$\end{document} and by θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} then provides a quantitative formulation of our refinement of Bolzano-Tarski consequence, which turns out to coincide with the time-honored syntactic Ł-consequence. (shrink)

Boolean logic deals with {0, 1}-observables and yes–no events, as many-valued logic does for continuous ones. Since every measurement has an error, continuity ensures that small measurement errors on elementary observables have small effects on compound observables. Continuity is irrelevant for {0, 1}-observables. Functional completeness no longer holds when n-ary connectives are understood as [0, 1]-valued maps defined on [0, 1] n . So one must envisage suitable selection criteria for [0, 1]-connectives. Łukasiewicz implication has a well known characterization as (...) the only continuous connective \ satisfying the following conditions: \= y\Rightarrow}\) and \ . Then syntactic consequence can be defined purely algorithmically using the Łukasiewicz axioms and Modus Ponens. As discussed in this paper, to recover a strongly complete semantics one may use differential valuations. (shrink)

Bernard Bolzano presents in his work Wissenschaftslehre a definition of derivability among sentences. Tarski publishes his well-known definition of logical consequence almost one hundred years later. This article intends to prove that the Bolzano´s definition is fully satisfactory compared to the Tarski´s one, even though no attention was drawn to Bolzano´s definition at that time.

Bolzano reduced inferential validity of the inference (from premise judgements to conclusion judgment) to the holding of logical consequence between the propositions (in themselves) that serve as contents of the respective judgements. This explicit reduction of inferential validity among judgements to logical consequence among propositions (or, alternatively, to logical truth of certain implicational propositions) has been largely taken over by current logical theory, say, by Wittgenstein’s Tractatus, by Hilbert and Ackermann, by Quine, and by Tarski also. Frege, though, (...) stands out among those who did not adopt such an account. Under the Bolzano reduction also some blind inferences, with no epistemic support, are deemed valid, which is unacceptable. The Completeness Theorem reduces inferential validity to the logical truth of a certain well-formed formula, whence the unacceptable blindness phenomena are retained. Accordingly, the interest of the Completeness Theorem for logic, construed as the theory of inference, is nugatory. (shrink)

The aim of this paper is to show that the account of objective truth taken for granted by logicians at least since the publication in 1933 of Tarski’s “The Concept of Truth in Formalized Languages” arose out of a tradition of philosophical thinking initiated by Bolzano and Brentano. The paper shows more specifically that certain investigations of states of affairs and other objectual correlates of judging acts, investigations carried out by Austrian and Polish philosophers around the turn of (...) the century, formed part of the background of views that led to standard current accounts of the objectivity of truth. It thus lends support to speculations on the role of Brentano and his heirs in contemporary logical philosophy advanced by Jan Wolenski in his masterpiece of 1989 on the Logic and philosophy in the Lvov-Warsaw School. (shrink)

Here is an account of logical consequence inspired by Bolzano and Tarski. Logical validity is a property of arguments. An argument is a pair of a set of interpreted sentences (the premises) and an interpreted sentence (the conclusion). Whether an argument is logically valid depends only on its logical form. The logical form of an argument is fixed by the syntax of its constituent sentences, the meanings of their logical constituents and the syntactic differences between their non-logical constituents, (...) treated as variables. A constituent of a sentence is logical just if it is formal in meaning, in the sense roughly that its application is invariant under permutations of individuals.1 Thus ‘=’ is a logical constant because no permutation maps two individuals to one or one to two; ‘∈’ is not a logical constant because some permutations interchange the null set and its singleton. Truth functions, the usual quantifiers and bound variables also count as logical constants. An argument is logically valid if and only if the conclusion is true under every assignment of semantic values to variables (including all non-logical expressions) under which all its premises are true. A sentence is logically true if and only if the argument with no premises of which it is the conclusion is logically valid, that is, if and only if the sentence is true under every assignment of semantic values to variables. An interpretation assigns values to all variables. (shrink)

Here is an account of logical consequence inspired by Bolzano and Tarski. Logical validity is a property of arguments. An argument is a pair of a set of interpreted sentences (the premises) and an interpreted sentence (the conclusion). Whether an argument is logically valid depends only on its logical form. The logical form of an argument is fixed by the syntax of its constituent sentences, the meanings of their logical constituents and the syntactic differences between their non-logical constituents, (...) treated as variables. A constituent of a sentence is logical just if it is formal in meaning, in the sense roughly that its application is invariant under permutations of individuals.1 Thus ‘=’ is a logical constant because no permutation maps two individuals to one or one to two; ‘∈’ is not a logical constant because some permutations interchange the null set and its singleton. Truth functions, the usual quantifiers and bound variables also count as logical constants. An argument is logically valid if and only if the conclusion is true under every assignment of semantic values to variables (including all non-logical expressions) under which all its premises are true. A sentence is logically true if and only if the argument with no premises of which it is the conclusion is logically valid, that is, if and only if the sentence is true under every assignment of semantic values to variables. An interpretation assigns values to all variables. (shrink)

Fondée sous les auspices du père de notre modernité philosophique Descartes, puis consolidée par des penseurs aussi importants que Leibniz, Bolzano ou Husserl, la mathesis universalis paraît représenter à elle seule l'ambitieux programme du « rationalisme classique ». Des philosophes tels que Husserl, Russell, Heidegger ou Cassirer ont pu s'accorder en ce point. Le développement de la « science moderne » aurait porté ce grand « rêve dogmatique » pour mener vers son terme le destin de la métaphysique (...) occidentale. Pourtant les recherches historiques récentes ont montré que l'idée de « mathématique universelle » existait bien avant Descartes, que ce dernier ne revendiquait d'ailleurs aucune rupture sur ce point et que sa réflexion se situait même assez clairement dans l'héritage des Anciens. Comment dès lors justifier que les Anciens, avec lesquels le programme des Classiques était censé rompre, aient pu déjà se préoccuper de « mathématique universelle »? Plus simplement encore, de quoi se préoccupaient donc ces philosophes sous ce concept? Le regain d'intérêt pour la mathesis universalis à la fin du XIXe siècle n'avait-il pas conduit paradoxalement à la perte de son sens comme problème? Cette étude a pour but de suivre ces questions jusqu'à leur origine et de montrer leur importance dans le dialogue entre mathématique et philosophie. Pages de début Remerciements Introduction La constitution de la « mathématique universelle » comme problème philosophique I. Aristote II. « Mathématique universelle » et théories mathématiques : Aristote, Euclide, Epinomis III. Le moment néo-platonicien Vers la science de l'ordre et de la mesure IV. La renaissance de la mathématique universelle V. La mathesis universalis cartésienne Conclusion Annexe I. La quaestio de scientia mathematica communi Annexe II. Essai bibliographique sur la mathesis universalis chez Descartes etLeibniz Bibliographie Index nominum Pages de fin. (shrink)

We trace two logical ideas further back than they have previously been traced. One is the idea of using diagrams to prove that certain logical premises do—or don’t—have certain logical consequences. This idea is usually credited to Venn, and before him Euler, and before him Leibniz. We find the idea correctly and vigorously used by Abū al-Barakāt in 12th century Baghdad. The second is the idea that in formal logic, P logically entails Q if and only if every model of (...) P is a model of Q. This idea is usually credited to Tarski, and before him Bolzano. But again we find Abū al-Barakāt already exploiting the idea for logical calculations. Abū al-Barakāt’s work follows on from related but inchoate research of Ibn Sīnā in eleventh century Persia. We briefly trace the notion of model-theoretical consequence back through Paul the Persian and in some form back to Aristotle himself. (shrink)

Une théorie de la science qui permette une « étude systématique des formes » est l’ἐπιστήμη ζητοῦμένη de Cavaillès : ce dernier ne la trouve ni dans l’analytique kantienne, ni dans la théorie de la science de Bolzano ; pas plus que chez Frege, Carnap ou Tarski ; et ni dans la théorie de la démonstration, ni dans la mathesis husserlienne. Cet article défend l’idée qu’il aurait pu la trouver, ou du moins s’en approcher, s’il avait accordé davantage (...) d’attention à une figure de la logique qu’il a délaissée : celle qu’incarnent Russell et son héritier Wittgenstein. L’analyse de la notion même de forme, que celle-ci désigne un complexe symbolique, une structure mathématique ou un schéma démonstratif, y aurait également gagné. (shrink)

L’article présente le concept de déductibilité que Bolzano introduit dans sa Wissenscahftslehre, indique quelques traits caractéristiques en vertu desquels ce concept diffère de plusieurs conceptions contemporaines de la conséquence et examine l’affirmation selon laquelle il présente une forte similarité avec la conception de Tarski et la logique de la pertinence.The article presents the concept of deducibility which Bolzano introduced in his Wissenschaftslehre, points out some of the characteristic features in virtue of which it differs from many modern (...) conceptions of consequence and examines the claims that it displays a strong similarity to Tarski’s account and relevance logic. (shrink)

This is a survey paper of approaches to the concept of logical implication. Roughly stated the main motivation of these approaches is to provide a necessary and sufficient condition for a set of propositions to logically imply a single proposition. In regard to their affinities these approaches are grouped into two: the transformational conception and the informational conception. Some approaches in each conception are philosophical and some are mathematical in character, their common assumption being that they reflect a previous intuition (...) – in some cases – already at work in classical mathematics. The first part of the paper is devoted to approaches within the transformational conception of logical implication emphasizing the contributions of Bolzano, Russell, Tarski, Quine and current model – theory. The second part of the paper discusses approaches within the informational conception of logical implication. Carnap and Bar Hillel’s extrinsic approach and Corcoran’s intrinsic approach developed in his information-theoretic logic are respectively compared and some of their philosophical underpinnings are brought to light. The paper concludes with some final remarks concerning the ontic question of the nature of logical implication and the epistemic question of the human access to this relation in both conceptions of logical implication. The article also attempts to connect the views of these representative thinkers with the current philosophical debate on the nature of logical consequence and the role of set-theoretic model theory in philosophy of logic. (shrink)

The last chapter is an interview Ernest Lepore, Director of the Rutgers Center for Cognitive Science and a friend of the Davidson family, has conducted with the author in 1988. In this interview, the author speaks of his childhood, his student years at Harvard, and his service in the navy in the Second World War. He describes his academic career, which took him from Queens College to Stanford, Princeton, and Rockefeller University, and illuminates his personal and philosophical relationships with contemporary (...) philosophers and logicians such as Quine, Dummett, Carnap, and Tarski. He finally clarifies Lepore's questions regarding the development of and relations between his philosophical programmes in the philosophy of action and the philosophy of language. (shrink)

What Gödel referred to as “outer” consistency is contrasted with the “inner” consistency of arithmetic from a constructivist point of view. In the settheoretic setting of Peano arithmetic, the diagonal procedure leads out of the realm of natural numbers. It is shown that Hilbert’s programme of arithmetization points rather to an “internalisation” of consistency. The programme was continued by Herbrand, Gödel and Tarski. Tarski’s method of quantifier elimination and Gödel’s Dialectica interpretation are part and parcel of (...) Hilbert’s finitist ideal which is achieved by going back to Kronecker’s programme of a general arithmetic of forms or homogeneous polynomials. The paper can be seen as a historical complement to our result on “The Internal Consistency of Arithmetic with Infinite Descent” . An internal consistency proof for arithmetic means that transfinite induction is not needed and that arithmetic can be shown to be consistent within the bounds of arithmetic, that is with the help of Fermat’s infinite descent and Kronecker’s general or polynomial arithmetic, thus returning into arithmetic without the detour of Cantor’s transfinite arithmetic of ideal elements. (shrink)

The relevance of metamathematical researches for philosophy of math- ematics is an indubitable matter. In the paper I shall speak about impli- cations of metamathematics for general philosophy, especially for classical epistemological problems. Let us start with a historical observation con- cerning Hilbert's programme, the rst research programme in metamathe- matics as a separate study of formal systems. This programme was strongly in uence by epistemological considerations. In fact, Hilbert wanted to se- cure all classical mathematics against (...) inconsistencies and this aim had be achieved with the help of nitary consistency proof. Hilbert's claim is like the Cartesian project of reduction of all concepts to clarae et distinctae ideae. The epistemological commitment of original Hilbert's programme may be treated as a prelim- inary heuristic motivation for looking for epistemological implications of metamathematical results. It seems reasonable to examine in this respect three so-called limitative theorems: the rst Godel's incompleteness theo- rem , the second Godel's incom- pleteness theorem consistency of S, cannot be proved in S, providing that S is consistent) and Tarski's undenability theorem ; \S" stands for formal system containing elementary number theory. From the above-mentioned theorems we obtain an important, from philosophical point of view, conclusion: semantics of S cannot be expressed in S. Now, I shall present three applications of limitative theorems to the analysis of classical epistemological problems. (shrink)

The paper examines some doctrines of the Davidsonian Programme of truth conditional Semantics that relates truth to meaning using Tarski’s T-Convention, in relation to its efficacy in a semantic valuation of the EkeGusii proverb: Nda ’indongi ereta morogi ereta moibi which exemplifies a kind of complex sentence that a given system of Semantics is meant to account for. The coverage of Davidsonian truth-conditional notion of T-convention and that of compositionality are considered to have only a partial reach in (...) accounting for the meaning of the proverb by not incorporating pragmatic aspects. The failure of T-convention is not alleviated by the adoption of radical interpretation as posited by Davidson but is extended to consider aspects of pragmatic enrichment and dynamic Semantics. (shrink)

This work aims to make tense logic a more robust tool for ontologists, philosophers, knowledge engineers and programmers by outlining a fusion of tense logic and ontology of time. In order to make tense logic better understandable, the central formal primitives of standard tense logic are derived as theorems from an informal and intuitive ontology of time. In order to make formulation of temporal propositions easier, temporal operators that were introduced by Georg Henrik von Wright are developed, and mapped to (...) the ontology of time. (shrink)

Bertrand Russell was one of the protagonists of the programme of reducing “disagreeable” concepts to philosophically more respectable ones. Throughout his life he was engaged in eliminating or paraphrasing away a copious variety of allegedly dubious concepts: propositions, definite descriptions, knowing subjects, and points, among others. The critical aim of this paper is to show that Russell’s construction of points, which has been considered as a paradigm of a logical construction überhaupt, fails for principal mathematical reasons. Russell could have (...) known this, if he had taken into account some pertinent results due to Hausdorff or Tarski. Its constructive aim is to show that one can save Russell’s thesis – that points can be defined in terms of events or regions – by using the conceptual resources of modern pointless topology. (shrink)

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.

La teoría de números, o la aritmética pura, concierne a los números naturales mismos, no a la notación usada, y en particular no a los numerales. La teoría de ristras, o la sintaxis pura, concierne a los numerales como ristras de caracteres «no-interpretados», al margen de los números que puedan denotar cuando son usados. La teoría de los números es puramente aritmética, la teoría de ristras es puramente sintáctica… en tanto se considere el universo del discurso solo. La aritmética semántica (...) es un amplio tema que se inicia cuando se menciona a los numerales y éstos se mencionan como nombres de números. La aritmética semántica da lugar a muchos fascinantes y sorprendentes algoritmos y procesos de decisión; revela en forma vívida el importe experiencial de las proposiciones matemáticas y el poder predictivo del conocimiento matemático; aporta una interesante perspectiva para los estudios filosóficos, históricos y pedagógicos, acerca del crecimiento del conocimiento científico y del papel del discurso metalingüístico en el pensamiento científico. (shrink)

Einleitungsband. 1. T. Biographie -- 2. T. Bolzano-Bibliographie und Editionsprinzipien der Gesamtausgabe. (v. <1-2>). Supplement <1-2> -- Reihe I, Schriften -- Bd. 2. Erbauungsreden für Akademiker -- Bd. 6. Lehrbuch der Religionswissenschaft, Erster Teil. (2 v.) -- Bd. 7. Lehrbuch der Religionswissenschaft, Zweiter Teil. (2 v.) -- Bd. 8. Lehrbuch der Religionswissenschaft, Dritter Teil. (v. <1-4 >) -- Bd. 11. Wissenschaftslehre (3 v.) -- Bd. 12. Wissenschaftslehre. (3 v.) -- Bd. 13. Wissenschaftslehre. (3 v.) -- Bd. 14. Wissenschaftslehre. (v. (...) <1-3>) -- Bd. 16. Vermischte Schriften, 1839-1840. (2 v.) -- Bd. 18. Mathematisch-physikalische und philosophische Schriften, 1842-1843 -- Bd. 19. Über die Perfectibilität des Katholizismus. (2 v.) --. (shrink)

Bernard Bolzano (1781-1848) a passe toute sa vie en Boheme, qui faisait encore partie de l'Empire autrichien. Apres des etudes de philosophie, mathematique et theologie, il est devenu pretre et professeur de Science de la religion a l'Universite de Prague. Heritier de l'Aufklarung, il a consacre sa vie a la reforme de la semi-feodale societe autrichienne et a la reforme des sciences a priori: logique, mathematique et theologie. Ses critiques de la constitution et de l'ordre existant lui valurent d'etre (...) destitue de sa chaire en 1819 et de passer le reste de sa vie en exil interieur entoure d'un petit cercle d'amis, ecrivant ses grands traites: le Traite de science de la religion (1834), la Theorie de la science (1837), la Theorie des grandeurs ainsi que l'utopie Du meilleur Etat et les Paradoxes de l'infini (1851). Le texte De la methode mathematique, extrait de l'Introduction a la Theorie des grandeurs, resume quelques unes de ses plus importantes innovations en logique et presente sa philosophie des mathematiques, concue en opposition a Kant. Bolzano l'a choisi pour l'envoyer a Franz Exner, nomme professeur de philosophie a l'Universite de Prague en 1831. L'echange epistolaire qui s'en est suivi tourne autour des deux theses controversees de la logique de Bolzano: sa conception des objets logiques en soi, independants de la pensee et de la langue, et son concept d'intuition. Dans cette controverse s'est joue le sort de la philosophie autrichienne. Bolzano n'a pas reussi a convaincre Exner, qui lui oppose avec tenacite les idees de Herbart. (shrink)

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.