Alfred Tarski

Emile Borel regards the Banach-Tarski Paradox as a reductio ad absurdum of the Axiom of Choice. Peter Forrest instead blames the assumption that physical space has a similar structure as the real numbers. This paper argues that Banach and Tarski's result is not paradoxical and that it merely illustrates a surprising feature of the continuum: dividing a spatial region into disjoint pieces need not preserve volume.

Drawing on recently published correspondence as well as on a survey of Polish and international philosophical activity published in 1937 and details concerning the publisher and bookseller Aleksander Mazzucato, I provide evidence that, contrary to some recent assertions (but in line with older bibliographical entries), Tarski's ‘Der Wahrheitsbegriff in den formalisierten Sprachen’ was not published in journal form until 1936, although preprints, lacking two corrections and a small addendum, were likely available in the late months of 1935.

Philosophical Investigations, EarlyView.

In his 1933 monograph on the concept of truth, Alfred Tarski claimed that his definition of truth satisfied “the usual conditions of methodological correctness”, which in a 1935 article he identified as consistency and back-translatability. Following the rules of defining for an axiomatized theory was supposed to ensure satisfaction of the two conditions. But Tarski neither explained the two conditions nor supplied rules of defining for any axiomatized theory. We can make explicit what Tarski understood by consistency and back-translatability, with (...) the help of (1) an account by Ajdukiewicz (1936) of the criteria underlying the practice of articulating rules of defining for axiomatized theories and (2) a critique by Frege (1903) of definitions that conjure an object into existence as that which satisfies a specified condition without first proving that exactly one object does so. I show that satisfaction of the conditions of consistency and back-translatability as thus explained is guaranteed by the rules of defining articulated by Leśniewski (1931) for an axiomatized system of propositional logic. I then construct analogous rules of defining for the theory within which Tarski developed his definition of truth. Tarski’s 32 definitions in this theory occasionally violate these rules, but the violations are easily repaired. I argue that the Leśniewski-Ajdukiewicz theory of formal correctness of definitions within which Tarski worked is superior in some respects to the widely accepted analogous theory articulated by Suppes (1957). (shrink)

Blum (Philosophical Investigations 46, 2023, 264) argues that Tarski's T‐schema and the thesis of the necessity of identity are mutually inconsistent. It is argued that his argument fails.

The idea that truth is eternal is an old one in philosophy, and I do not propose to survey its history here. Yet a sketch of the historical context is useful for my main purpose, which is to discuss the theme of truth and temporality in Martin Heidegger and Richard Rorty. Although both philosophers repudiate eternal truth, their reasons for doing so are different, and this difference reveals a probably irreconcilable opposition between Heidegger and the Pragmatist.

The resemblance of the theory of formal consequence first offered by the fourteenth-century logician John Buridan to that later offered by Alfred Tarski has long been remarked upon. But it has not yet been subjected to sustained analysis. In this paper, I provide just such an analysis. I begin by reviewing today’s classical understanding of formal consequence, then highlighting its differences from Tarski’s 1936 account. Following this, I introduce Buridan’s account, detailing its philosophical underpinnings, then its content. This then allows (...) us to separate those aspects of Tarski’s account representing genuine historical advances, unavailable to Buridan, from others merely differing from—and occasionally explicitly rejected by—Buridan’s account. (shrink)

In this work, we study the relational representation of the class of Tarski algebras endowed with a subordination, called subordination Tarski algebras. These structures were introduced in a previous paper as a generalisation of subordination Boolean algebras. We define the subordination Tarski spaces as topological spaces with a fixed basis endowed with a closed relation. We prove that there exist categorical dualities between categories whose objects are subordination Tarski algebras and categories whose objects are subordination Tarski spaces. These results extend (...) the known dualities for modal algebras. Finally, we are going to characterise some algebraic conditions written in the quasi-modal language by means of first-order conditions. (shrink)

In this article, I will offer some remarks about Tarski’s nominalism. First, I will show that, even though it was mainly developed in private conversation and lectures, Tarski did try to develop a rigorous nominalistic theory, which shows that the issue was of some importance to him. In particular, I show how Tarski’s formulation is based on the idea of a humanly understandable language and show how he tried to develop this idea throughout his career. Unfortunately, even though his formulation (...) is interesting, it seems to face an insurmountable obstacle, which I examine in detail in the article. Finally, I offer some remarks about what went wrong with Tarski’s nominalist program. (shrink)

According to Tarski's model-theoretic analysis of logical consequence, the sentence X is a logical consequence of a set of sentences Γ if and only if any model for Γ is also a model for X. Etchemendy, however, does not accept the analysis and critiques it. According to Etchemendy, Tarski’s analysis 1- involves a conceptual mistake: confusing the symptoms of logical consequence with their cause; 2- cannot properly explain the necessity of logical consequence; 3- faces the problem of overgeneration; and 4- (...) faces the problem of undergeneration. In the present article, by evaluating these critiques and examining the effectiveness of some of the answers presented in defense of Tarski's analysis, we try to show that among these critiques, only the problem of undergeneration is not acceptable. According to our common sense understanding, if an argument is valid, it is truth-preserving, and by assuming the truth of the premises, the conclusion will be true as well. But it does not mean that we can reduce the logical consequence relation to truth preservation. This flaw leads Tarski’s analysis to be an unacceptable analysis of logical consequence. (shrink)

Philosophical Investigations, EarlyView.

The determination of philosophy is the work of those who cooperate in the construction of knowledge, in its disparate fields, and at the same time preserve the very sense of indeterminacy. There are no areas of knowledge that cannot be also philosophical, nor can the themes, lines of research and styles of thought be limited a priori. The philosophy, it is said, is the search for truth. This is the most common definition, and therefore also the more covering than a (...) possible new point of view, which could have its roots in the ancient tradition. Philosophy is that search for truth which gave rise to our science and that critical attitude which it brings in doubt, limits and determines this same spirit of research. (shrink)

The paper presents the main idea of point-free theories of space based on Tarski's system of point-free geometry. First, the general idea of the so-called point-free ontology was discussed, as well as the epistemological and methodological reasons for its adoption. Next, Whitehead's method of extensive abstraction, which is the methodological basis for the construction of point-free theories of space, is presented, and the fundamental concepts of mereology are discussed. The main part of the paper is a discussion of Tarski’s geometry (...) of solids, its postulates and metatheoretical properties. The paper ends with a short description of the contribution of Polish researchers to the development of research on point-free theories of space. (shrink)

We give an extension of the Jónsson‐Tarski representation theorem for both normal and non‐normal modal algebras so that it preserves countably many infinite meets and joins. In order to extend the Jónsson‐Tarski representation to non‐normal modal algebras we consider neighborhood frames instead of Kripke frames just as Došen's duality theorem for modal algebras, and to deal with infinite meets and joins, we make use of Q‐filters, which were introduced by Rasiowa and Sikorski, instead of prime filters. By means of the (...) extended representation theorem, we show that every predicate modal logic, whether it is normal or non‐normal, has a model defined on a neighborhood frame with constant domains, and we give a completeness theorem for some predicate modal logics with respect to classes of neighborhood frames with constant domains. Similarly, we show a model existence theorem and a completeness theorem for infinitary modal logics which allow conjunctions of countably many formulas. (shrink)

Compared to the extensive amount of literature on various themes of W.V.O. Quine’s philosophy, his immanent concept of truth remains a relatively unexplored topic. This relative lack of research contributes to a persistent confusion on the deflationary and inflationary details of Quine’s truth. According to a popular reading, Quine’s disquotational definition of the truth predicate exhausts the content of truth, thus amounting to a deflationary view. Others promote opposing interpretations. I argue that by relying on Tarski’s semantic conception of truth, (...) Quine’s disquotational account inherits a commitment to classical correspondence intuitions. Based on this, Quine posits a firm constitution for truth as an intermediary between language and the world. From this constitution claim follows that the disquotational account proves incompatible with both the general deflationary thesis and, more specifically, the minimalist account, which deny any constitution for truth past what is given by the preferred deflationary schema. This reading is significant for refuting the widespread misrepresentations of Quine as a prominent deflationist. (shrink)

This paper is based on Tarski’s theory of truth. The purpose of this paper is to solve the liar paradox (and its cousins) and keep both of the deductive power of classical logic and the expressive power of the word “true” in natural language. The key of this paper lies in the distinction between the predicate usage and the operator usage of the word “true”. The truth operator is primarily used for characterizing the semantics of the language. Then, we do (...) not need the hierarchy of languages. The truth predicate is mainly used for grammatical function. Tarski’s schema of the truth predicate is not necessary in this proposal. The schema of the word "true" is the schema of the truth operator. The liar paradox (and its cousins) can be solved in this way. In the appendix, I show a consistent model for both of the truth predicate and the truth operator. (shrink)

This paper considers a formalisation of classical logic using general introduction rules and general elimination rules. It proposes a definition of ‘maximal formula’, ‘segment’ and ‘maximal segment’ suitable to the system, and gives reduction procedures for them. It is then shown that deductions in the system convert into normal form, i.e. deductions that contain neither maximal formulas nor maximal segments, and that deductions in normal form satisfy the subformula property. Tarski’s Rule is treated as a general introduction rule for implication. (...) The general introduction rule for negation has a similar form. Maximal formulas with implication or negation as main operator require reduction procedures of a more intricate kind not present in normalisation for intuitionist logic. (shrink)

Juliette Kennedy’s new book brims with intriguing ideas. I don’t understand all of them, and I’m not convinced that the ones I do understand all fit together, b.

Reasoning under uncertainty provides plenty of opportunities to err or to be accused of erring even if one is not. One can present a plausible, well-founded hypothesis, which may later be over- turned by new evidence in a totally unexpected way. In this study a drawing as test case was taken by N=259 participants. Our analysis of their solutions showed that deductive partial explanations and eliminations seem to be a good way of trying to find relevant answers to unresolved cases. (...) When an investigation stalls, one should not prematurely be fixed on the full explanation of the story of what presumably happened but make small and coherent advances in the interpretation of the details of the evidence with specific sub- hypotheses. Then again one cannot absolutely trust such deductions, as one does not know how the traces arrived there. Were they caused by the presumed incident or by random events? In a final step we integrated naïve recipes of interpretation into hierarchical levels of epistemology, so as to construct a more sophisticated heuristic. (shrink)

Several writers have assumed that when in “Outline of a Theory of Truth” I wrote that “the orthodox approach” – that is, Tarski’s account of the truth definition – admits descending chains, I was relying on a simple compactness theorem argument, and that non-standard models must result. However, I was actually relying on a paper on ‘pseudo-well-orderings’ by Harrison. The descending hierarchy of languages I define is a standard model. Yablo’s Paradox later emerged as a key to interpreting the result.

The Tarskian notion of truth-in-a-model is the paradigm formal capture of our pre-theoretical notion of truth for semantic purposes. But what exactly makes Tarski’s construction so well suited for semantics is seldom discussed. In my Semantics, Metasemantics, Aboutness (OUP 2017) I articulate a certain requirement on the successful formal modeling of truth for semantics – “locality-per-reference” – against a background discussion of metasemantics and its relation to truth-conditional semantics. It is a requirement on any formal capture of sentential truth vis-à-vis (...) the interpretation of singular terms and it is clearly met by the Tarskian notion. In this paper another such requirement is articulated – “locality-per-application” – which is an additional requirement on the formal capture of sentential truth, this time vis-à-vis the interpretation of predicates. This second requirement is also clearly met by the Tarskian notion. The two requirements taken together offer a fuller answer than has been hitherto available to the question what makes Tarski's notion of truth-in-a-model especially well suited for semantics. (shrink)

Alfred Tarski’s influence on computer science was indirect but significant in a number of directions and was in certain respects fundamental. Here surveyed is Tarski’s work on the decision procedure for algebra and geometry, the method of elimination of quantifiers, the semantics of formal languages, model-theoretic preservation theorems, and algebraic logic; various connections of each with computer science are taken up.

Logical form has always been a prime concern for philosophers belonging to the analytic tradition. For at least one century, the study of logical form has been widely adopted as a method of investigation, relying on its capacity to reveal the structure of thoughts or the constitution of facts. This book focuses on the very idea of logical form, which is directly relevant to any principled reflection on that method. Its central thesis is that there is no such thing as (...) a correct answer to the question of what is logical form: two significantly different notions of logical form are needed to fulfil two major theoretical roles that pertain respectively to logic and to semantics. This thesis has a negative and a positive side. The negative side is that a deeply rooted presumption about logical form turns out to be overly optimistic: there is no unique notion of logical form that can play both roles. The positive side is that the distinction between two notions of logical form, once properly spelled out, sheds light on some fundamental issues concerning the relation between logic and language. (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.

It is often argued that by assuming the existence of a universal language, one prohibits oneself from conducting semantical investigations. It could thus be thought that Tarski’s stance towards a universal language in his fruitful Wahrheitsbegriff differs essentially from Carnap’s in the latter’s less successful Untersuchungen zur allgemeinen Axiomatik. Yet this is not the case. Rather, these two works differ in whether or not the studied fragments of the universal language are languages themselves, i.e., whether or not they are closed (...) under derivation rules. In Carnap’s case, axiom systems are not closed under derivation rules, which enables him to adopt a substitutional concept of models. His approach is directly rooted in the tradition of formal axiomatics, we argue, and in this contrary to Tarski’s. In comparing these works by Carnap and Tarski, our aim will be to qualify the connection between Tarski’s approach and the tradition of formal axiomatics, which has been overemphasized in the literature. (shrink)

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)

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)

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.

Author's replies to an APA book symposium on "Carnap, Tarski, and Quine at Harvard.".

J.C. Beall and Greg Restall's Generalised Tarski Thesis is a generalisation of the seemingly diverse conceptions of logical consequence. However, even their apparently general account of consequence makes necessary truth-preservation a necessary condition. Sentences in the imperative mood pose a problem for any truth-preservationist account of consequence, because imperatives are not truth-apt but seem to be capable of standing in the relation of logical consequence. In this paper, I show that an imperative logic can be formulated that solves the problem (...) of imperative consequence by leading naturally to a further generalisation of the GTT. (shrink)

In his famous paper Der Wahrheitsbegriff in den formalisierten Sprachen (Polish edition: Nakładem/Prace Towarzystwa Naukowego Warszawskiego, wydzial, III, 1933), Alfred Tarski constructs a materially adequate and formally correct definition of the term “true sentence” for certain kinds of formalised languages. In the case of other formalised languages, he shows that such a construction is impossible but that the term “true sentence” can nevertheless be consistently postulated. In the Postscript that Tarski added to a later version of this paper (Studia Philosophica, (...) 1, 1935), he does not explicitly include limits for the kinds of language for which such a construction is possible. This absence of such limits has been interpreted as an implied claim that such a definition of the term “true sentence” can be constructed for every language. This has far-reaching consequences, not least for the widely held belief that Tarski changed from an universalistic to an anti-universalistic standpoint. We will claim that the consequence of anti-universalism is unwarranted, given that it can be argued that the Postscript is not in conflict with the existence of limits outside of which a definition of “true sentence” cannot be constructed. Moreover, by a discussion of transfinite type theory, we will also be able to accommodate other of the changes made in Tarski’s Postscript within a type-theoretical framework. The awareness of transfinite type theory afforded by this discussion will lead, in turn, to an account of Tarski’s Postscript that shows a gradual change in his logical work, rather than any of the more radical transitions which the Postscript has been claimed to reflect. (shrink)

In a recent article, David distinguishes between two interpretations of Tarski’s work on truth. The standard interpretation has it that Tarski gave us a definition of truth in-L within the meta-language; the non-standard interpretation, that Tarski did not give us a definition of true sentence in L, but rather a definition of truth, and Tarski does so for L within the metalanguage. The difference is crucial: for on the standard view, there are different concepts of truth, while in the alternative (...) interpretation there is just one concept. In this paper we will have a brief look at the distinction between these two interpretations and at the arguments David gives for each view. We will evaluate one of David’s arguments for the alternative view by looking at Tarski’s ‘On the concept of truth in formalized languages’, and his use of the term ‘extension’ therein, which, we shall find, yields no conclusive evidence for either position. Then we will look at how Tarski treats ‘satisfaction’, an essential concept for his definition of ‘true sentence’. It will be argued that, in light of how Tarski talks about ‘satisfaction’ in Sect. 4 of ‘CTF’ and his claims in the Postscript, the alternative view is more likely than the standard one. (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.

Tarski’s pioneering work on truth has been thought by some to motivate a robust, correspondence-style theory of truth, and by others to motivate a deflationary attitude toward truth. I argue that Tarski’s work suggests neither; if it motivates any contemporary theory of truth, it motivates conceptual primitivism, the view that truth is a fundamental, indefinable concept. After outlining conceptual primitivism and Tarski’s theory of truth, I show how the two approaches to truth share much in common. While Tarski does not (...) explicitly accept primitivism, the view is open to him, and fits better with his formal work on truth than do correspondence or deflationary theories. Primitivists, in turn, may rely on Tarski’s insights in motivating their own perspective on truth. I conclude by showing how viewing Tarski through the primitivist lens provides a fresh response to some familiar charges from Putnam and Etchemendy. (shrink)

Jamin Asay's book offers a fresh and daring perspective on the age-old question 'What is truth?', with a comprehensive articulation and defence of primitivism, the view that truth is a fundamental and indefinable concept. Often associated with Frege and the early Russell and Moore, primitivism has been largely absent from the larger conversation surrounding the nature of truth. Asay defends primitivism by drawing on a range of arguments from metaphysics, philosophy of language and philosophy of logic, and navigates between correspondence (...) theory and deflationism by reviving analytic philosophy's first theory of truth. In its exploration of the role that truth plays in our cognitive and linguistic lives, The Primitivist Theory of Truth offers an account of not just the nature of truth, but the foundational role that truth plays in our conceptual scheme. It will be valuable for students and scholars of philosophy of language and of metaphysics. (shrink)

Philosophical theorizing about truth manifests a desire to conform to the ordinary or folk notion of truth. This practice often involves attempts to accommodate some form of correspondence. We discuss this accommodation project in light of two empirical projects intended to describe the content of the ordinary conception of truth. One, due to Arne Naess, claims that the ordinary conception of truth is not correspondence. Our more recent study is consistent with Naess’ result. Our findings suggest that contextual factors and (...) respondent gender affect whether the folk accept that correspondence is sufficient for truth. These findings seem to show that the project of accommodating the ordinary notion of truth is more difficult than philosophers had anticipated because it is fragmentary. (shrink)

نگاهی بر استدلال‌های دونالد دیویدسون در باب ضرورت زبان برای اندیشه .

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

Analytic philosophy and modern logic are intimately connected, both historically and systematically. Thinkers such as Frege, Russell, and Wittgenstein were major contributors to the early development of both; and the fruitful use of modern logic in addressing philosophical problems was, and still is, definitive for large parts of the analytic tradition. More specifically, Frege's analysis of the concept of number, Russell's theory of descriptions, and Wittgenstein's notion of tautology have long been seen as paradigmatic pieces of philosophy in this tradition. (...) This close connection remained beyond what is now often called "early analytic philosophy", i.e., the tradition's first phase. In the present chapter I will consider three thinkers who played equally important and formative roles in analytic philosophy's second phase, the period from the 1920s to the 1950s: Rudolf Carnap, Kurt Gödel, and Alfred Tarski. (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)

O objetivo deste texto é discutir a tarefa filosófica de elucidação do conceito de conseqüência lógica. Primeiramente, serão eleitos dois critérios de adequação para uma elucidação desse conceito: (1) preservação da verdade nas instâncias, ou adequação material e (2) garantia da verdade da conclusão na inferência válida, ou adequação epistêmica. Em seguida serão apresentadas a proposta de Tarski (1956) e as correspondentes críticas de Etchemendy (2008). Conclui-se com comentários a respeito da natureza das investigações lógicas.