Alfred Tarski

Review of Douglas Patterson. Alfred Tarski: Philosophy of Language and Logic.

In 1924 Banach and Tarski demonstrated the existence of a paradoxical decomposition of the 3-ball B, i.e., a piecewise isometry from B onto two copies of B. This article answers a question of de Groot from 1958 by showing that there is a paradoxical decomposition of B in which the pieces move continuously while remaining disjoint to yield two copies of B. More generally, we show that if n ≥ 2, any two bounded sets in.

The aim of this paper is to investigate whether Tarski’s truth definition explains the notion of truth as correspondence with reality and whether it is really a semantic definition of truth. I defend the view that, although Tarski succeeded in the task of constructing correct and adequate definitions, according to his own criteria of correctness and adequateness, the definitions obtained by his method neither are explanations of the main point of a correspondence theory of truth, the relationship between language and (...) reality on which depends the truth of a sentence, nor can be considered genuinely semantic. (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)

Partly due to the influence of Tarski's work, it is commonly assumed that any good theory of truth implies biconditionals of the sort mentioned in Convention T: instances of the T-Schema "s is true in L if and only if p" where the sentence substituted for "p" is equivalent in meaning to s. I argue that we must take care to distinguish the claim that implying such instances is sufficient for adequacy in an account of truth from the claim that (...) doing so is necessary. The claim that doing so is sufficient is a common component of deflationary theories of truth, while the claim that it is necessary, though often assumed, must be denied by proponents of rival inflationary theories of truth. I discuss the clarification of the debate between these views of truth that results from distinguishing the necessity and the sufficiency claims, and examine the prospects for its resolution. (shrink)

[ES] La teoría de modelos parte generalmente de la definición de consecuencia lógica ofrecida por Tarski en 1936. John Etchemendy asevera que esta definición contiene una falacia si se le toma como definición genuina de este concepto. Esta aseveración ha desatado una polémica interesante. El presente ensayo resume los puntos principales en discusión y sugiere en su conclusión que el rechazo de la propuesta de Etchemendy se basa en un malentendido de la intención de su crítica, cuyo objeto no es, (...) propiamente dicho, la definición de Tarski, sino que ésta ofrece un fundamento insuficiente para la teoría de modelos. [EN] Model Theory generally assumes the definition of logical consequence proffered by Tarski in 1936. John Etchemendy suggests that Tarski’s definition contains a fallacy, from which an interesting controversy ensued about the definition and its meaning. The main points in discussion are summarized and it is suggested that the controversy is based on a misunderstanding of the intentions of Etchemendy’s criticism: the target is not so much Tarski’s definition, but that it offers an insufficient foundation for Model Theory. (shrink)

In earlier work I claimed that when Tarski wrote his seminal 1936 paper on logical consequence, he had in mind a now nonstandard convention, that he also used in his 1937 logic manual, requiring the domain of quantification of the different interpretations of a first-order mathematical language to covary with changes in the interpretation of a non-logical “domain predicate”. Recently Paolo Mancosu has rejected this claim, holding that it can be established on the basis of a passage from Tarski’s manual (...) that he did not employ that convention. I show that Mancosu misinterprets the passage in question and that detailed examination of the surrounding text actually confirms my earlier claim. (shrink)

This paper was written with two aims in mind. A large part of it is just an exposition of Tarski's theory of truth. Philosophers do not agree on how Tarski's theory is related to their investigations. Some of them doubt whether that theory has any relevance to philosophical issues and in particular whether it can be applied in dealing with the problems of philosophy (theory) of science.In this paper I argue that Tarski's chief concern was the following question. Suppose a (...) language L belongs to the class of languages for which, in full accordance with some formal conditions set in advance, we are able to define the class of all the semantic interpretations the language may acquire. Every interpretation of L can be viewed as a certain structure to which the expressions of the language may refer. Suppose that a specific interpretation of the language L was singled out as the intended one. Suppose, moreover, that the intended interpretation can be characterized in a metalanguage L +. If the above assumptions are satisfied, can the notion of truth for L be defined in the metalanguage L + and, if it can, how can this be done? (shrink)

In his published work and even more in conversations, Tarski emphasized what he thought were important philosophical aspects of his work. The English translation of his more philosophical papers [56m] was dedicated to his teacher Tadeusz Kotarbinski, and in informal discussions of philosophy he often referred to the influence of Kotarbinski. Also, the influence of Leiniewski, his dissertation adviser, is evident in his early papers. Moreover, some of his important papers of the 1930s were initially given to philosophical audiences. For (...) example, the famous monograph on the concepotf truth ([33”], [35b]) was first given as twol ectures to the Logic Section of the Philosophical Society in Warsaw in 1930. Second, his paper [33], which introduced the concepts of co-consistency and co-completeness as well as the rule of infinite induction: was first given at the Second Conference of the Polish Philosophical Society in Warsaw in 1927. Also [35c] was based upon an address given in 1934 to the conference for the Unity of Science in Prague; C361 and [36a] summarize an address given at the International Congress of Scientific Philosophy in Paris in 1935. The article [44a] was published in a philosophical journal and widely reprinted in philosophical texts. This list is of course not exhaustiveb ut only representative of Tarski’s philosophical interactions as reflected in lectures given to philosophical audiences, which were later embodied in substantial papers. After 1945 almost all of Tarski’s publications and presentations are mathematical in character with one or two minor exceptions. This division, occurring about 1945, does not, however, indicate al oss of interest in philosophical questionbsu t is a result of Tarski’s moving to the Department of Mathematics at Berkeley. There he assumed an important role in the development of logic within mathematics in the United States. (shrink)

Kit Fine has argued that the Tarski Semantics for the language of first order logic is inadequate. A semantic theory for FOL is inadequate if there are formulae of FOL whose meanings or satisfaction conditions it cannot compositionally account for. It is argued here that Fine’s case against Tarski rests on a mistake.

Many of Tarski’s better known papers are either about or include lengthy discussions of how to properly define various concepts: truth, logical consequence, semantic concepts, or definability. In general, these papers identify two primary conditions for successful definitions: formal correctness and material adequacy. Material adequacy requires that the concept expressed by the formal definition capture the intuitive content of truth. Our primary interest in this paper is to better understand Tarski’s thinking about material adequacy, and whether components of his view (...) developed over time. More precisely, we are concerned with how Tarski’s understanding of the content of the common-sense, every-day usage of truth may have developed over time. We distinguish this concern from the character of the extensional criterion of adequacy Tarski proposes: that a materially adequate definition must entail all instances of Convention T. We will develop our reading of Tarski as follows: first, we will review the “Polemical Remarks,” focusing primarily on §§14 and 17, and Tarski’s references to Naess’ empirical research. Next, we will provide a summary and discussion of Naess’ work, especially his findings with respect to Tarski’s definition of truth and his research that suggests there is no single common or everyday concept of truth. Third, we will consider several possible objections to our interpretation of the Tarski–Naess dialectic. We will conclude that Tarski’s conception of material adequacy developed over time, potentially because of what he had learned through his interactions with Naess. (shrink)

All arithmetical identities involving 1, addition, multiplication and exponentiation will be true in a 2-element model of Tarski's system if a certain sequence of natural numbers is not bounded. That sequence can be bounded only if the set of Fermat's prime numbers is finite.

Tarski’s Convention T is often taken to claim that it is both sufficient and necessary for adequacy in a definition of truth that it imply instances of the T-schema where the embedded sentence translates the mentioned sentence. However, arguments against the necessity claim have recently appeared, and, furthermore, the necessity claim is actually not required for the indefinability results for which Tarski is justly famous; indeed, Tarski’s own presentation of the results in the later Undecidable Theories makes no mention of (...) an assumption to the effect that the definition of truth implies the biconditionals. This raises a question: was Tarski in fact committed to the necessity claim in the important papers of the 1930s and 40s? I argue that he was not. The discussion of this apparently esoteric interpretive issue in fact gets to the heart of many important questions about truth, and in the final sections of the paper I discuss the importance of the T-biconditionals in the theory of meaning and the relation of deflationary and inflationary theories of truth to the semantic paradoxes. (shrink)

Alfred Tarski (1944) wrote that "the condition of the 'essential richness' of the metalanguage proves to be, not only necessary, but also sufficient for the construction of a satisfactory definition of truth." But it has remained unclear what Tarski meant by an 'essentially richer' metalanguage. Moreover, DeVidi and Solomon (1999) have argued in this Journal that there is nothing that Tarski could have meant by that phrase which would make his pronouncement true. We develop an answer to the historical question (...) of what Tarski meant by 'essentially richer' and pinpoint the general result that stands behind his essential richness claim. In defense of Tarski, we then show that each of the several arguments of DeVidi and Solomon are either moot or mistaken. One of the fruits of our investigation is the reclamation of what Tarski took to be his central result on truth. This is a reclamation since: (i) if one does not understand 'essential richness', one does not know what that result is, and (ii) we must unearth a heretofore unrecognized change that occurs in Tarski's view - an alteration of his main thesis in light of a failing he discovered in it. (shrink)

In a paper ‘Is it True What They Say About Tarski?’, Dr Susan Haack explicitly denies the truth of an implicit assertion of mine, by writing ‘… Tarski does not present his theory as a correspondence theory’. My reply is to quote two brief passages from Tarski's work.

This paper describes Tarski’s project of rehabilitating the notion of truth, previously considered dubious by many philosophers. The project was realized by providing a formal truth definition, which does not employ any problematic concept.

Alfred Tarski seems to endorse a partial conception of truth, the T-schema, which he believes might be clarified by the application of empirical methods, specifically citing the experimental results of Arne Næss (1938a). The aim of this paper is to argue that Næss’ empirical work confirmed Tarski’s semantic conception of truth, among others. In the first part, I lay out the case for believing that Tarski’s T-schema, while not the formal and generalizable Convention-T, provides a partial account of truth that (...) may be buttressed by an examination of the ordinary person’s views of truth. Then, I address a concern raised by Tarski’s contemporaries who saw Næss’ results as refuting Tarski’s semantic conception. Following that, I summarize Næss’ results. Finally, I will contend with a few objections that suggest a strict interpretation of Næss’ results might recommend an overturning of Tarski’s theory. (shrink)

Greg Frost-Arnold’s book is a highly elegant edition and commentary of Carnap’s notes, claiming just as much as he is warranted on the basis of the manuscript and other relevant texts, and formulating his scholarly assumptions very carefully. Along the way he tries to unify the three historiographical strategies: narrative, argumentative and micro-historical.

In his 1936 paper, "The Concept of Logical Consequence," Alfred Tarski set out the first formulation of the model-theoretic definition of logical consequence. This definition has hence become a central canon of contemporary logical inquiry. However, it has come under recent attack. In this dissertation, I present a defense of the standard model-theoretic analyses of the concepts of validity and consequence against various objections. ;In the first chapter, it is argued that the model-theoretic analysis of natural language argumentation is based (...) on determining relationships between structural properties of mathematical models and those of the language under study. This use of model theoretic structures involves the application of mathematical models to the description of a natural language; and as such, the determination of the analysis can best be decided in the context of discussing the analysis, qua mathematical analysis. ;In the second chapter, I discuss the distinction between representational semantics and interpretational semantics. I argue that Etchemendy presents us with a false dilemma: the semantics underlying the model-theoretic analyses of the logical properties is neither a representational semantics, nor an interpretational semantics. I then formulate a precise and defensible version of Tarski's Thesis. ;In chapter 3, I take up the two main lines of argument directed against Tarski's Thesis. First, I argue that neither Etchemendy's under-generational and over-generational examples, nor McGee's reliability problem argue against the formulation of Tarski's Thesis given in this work. Second, I argue that the claims brought out in Etchemendy's discussion of the Reduction Principle and in McGee's discussion of the Contingency Problem are mistaken, and that the conclusions they draw from these claims are unwarranted. ;Finally, chapter 4 takes up historical issues surrounding both the discrepancies between Tarski's original formulation of the model-theoretic definitions and their present day incarnations, and Etchemendy's attribution to Tarski of "Tarski's Fallacy." I argue that the historical account given Tarski's 1936 paper by Etchemendy is mistaken, and that only the most narrow and unkind reading supports the fallacy attribution. (shrink)

In his 1936 article, "Uber den Begriff der logischen Folgerung," Tarski claims to have provided precise definitions of the concepts of logical truth and logical consequence that remain "close in essentials" to the "common" or "everyday" concepts. In this dissertation, I examine Tarski's account of these notions, as well as the standard model theoretic definitions that are based on that analysis. I argue that Tarski's analysis, and hence in turn its model theoretic heirs, are fundamentally mistaken, that they do not (...) capture the essential features of our ordinary understanding of those notions. I claim that the account's misdirection is, in a sense, quite obvious, but that it has been disguised in various different ways. First is a fallacious argument that seems to have been offered by Tarski in support of his analysis, second is the conflation of the model theoretic definitions with what I call "representational semantics," and third is a confusion of the principle underlying Tarski's account with a completely uncontroversial principle concerning the closure of logical truth under logical consequence. (shrink)

Among the influential contributions of Alfred Tarski to logic and philosophy, and close in importance to his widely applied and discussed definition of truth, one finds his definition of logical consequence for formal languages. Like his definition of truth, Tarski's definition of logical consequence has been widely and fruitfully applied. Unlike the definition of truth, that of logical consequence has been rarely discussed philosophically. The main aim of this dissertation is to offer a thorough discussion of some philosophical issues arising (...) from the neglected definition. ;The point of departure is an interpretative reading of the relevant writings by Tarski. This reading yields historically interesting data that refute other recent readings. It also provides philosophical illumination, by clarifying how Tarski thought that his definition satisfied some pretheoretic desiderata. ;Next it is discussed whether Tarski's definition satisfies other pretheoretic desiderata, not stated by Tarski and in some cases not endorsed by him, but which are desiderata nonetheless from reasonable standpoints. These include the desiderata that all instances of logical consequence in Tarski's defined sense be instances of analytical implication, and that they be instances of valid implication. The conclusion of this part of the dissertation is that, under reasonable assumptions regarding the field of applicability of the definition and the concepts used in it, there are reasons to believe that these desiderata are met, and none to believe they are not. ;Finally, the problem of logical constants is examined. The concept of a "logical constant" is the only concept not taken from logic and mathematics that appears in Tarski's definition. The problem of logical constants is that of replacing the primitive 'logical constant' with a defined, better understood concept, and in a such a way that the resulting defined concept of logical consequence meets the previously discussed desiderata. The main conclusion of this part is that some attempts to do this replace the obscure with the more obscure, while other attempts that seem to be in the right spirit have shortcomings of their own. (shrink)