Alfred Tarski

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)

KennedyJuliette.* * _ Gödel, Tarski and the Lure of Natural Language: Logical Entanglement, Formalism Freeness. _Cambridge University Press, 2021. Pp. xii + 187. ISBN: 978-1-107-01257-8 ; 978-0-511-99839-3 ; 978-1-00902851-6, 978-1-00902823-3. doi.org/10.1017/9780511998393.

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)

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.

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)

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

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.

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)

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

[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.

This is a concise, advanced introduction to current philosophical debates about truth. A blend of philosophical and technical material, the book is organized around, but not limited to, the tendency known as deflationism, according to which there is not much to say about the nature of truth. In clear language, Burgess and Burgess cover a wide range of issues, including the nature of truth, the status of truth-value gaps, the relationship between truth and meaning, relativism and pluralism about truth, and (...) semantic paradoxes from Alfred Tarski to Saul Kripke and beyond. Following a brief introduction that reviews the most influential traditional and contemporary theories of truth, short chapters cover Tarski, deflationism, indeterminacy, realism, antirealism, Kripke, and the possible insolubility of semantic paradoxes. The book provides a rich picture of contemporary philosophical theorizing about truth, one that will be essential reading for philosophy students as well as philosophers specializing in other areas. (shrink)

Down through the ages, logic has adopted many strange and awkward technical terms: assertoric, prove, proof, model, constant, variable, particular, major, minor, and so on. But truth-value is a not a typical example. Every proposition, even if false, no matter how worthless, has a truth-value:even “one plus two equals four” and “one is not one”. In fact, every two false propositions have the same truth-value—no matter how different they might be, even if one is self-contradictory and one is consistent. It (...) is not such a big surprise that every true proposition, no matter how worthless, has a truth-value. Every proposition, whether valuable or worthless, has a truth-value.But it is a little strange that every two true propositions, no matter how different they might be, have the same truth-value. The Pythagorean Theorem has the same truth-value as the proposition that one is one. It is a stretch to think of truth-values as values in any of the normal senses of the word ‘value’. (shrink)

“Etchemendy’s Critique to Formalism”. John Etchemendy claims that,given the failure of the Tarskian intuitive notion of logical consequence, there isno reason to consider formality as a necessary condition for this relationship.This paper critiques this argument. First, it seeks to show that Etchemendy’scritique to Tarskian analysis assumes two requisites of elucidatory success thatcannot be held together reasonably. Secondly, it shows that, once the previousassumption is rejected, two arguments in favour of the extensional adequacyof the former argument actually support formalism. Finally, this (...) paper reviewssome well known pragmatic considerations in favour of formalism. (shrink)

Tarski avoids the liar paradox by relativizing truth and falsehood to particular languages and forbidding the predication to sentences in a language of truth or falsehood by any sentences belonging to the same language. The Tarski truth-schemata stratify an object-language and indefinitely ascending hierarchy of meta-languages in which the truth or falsehood of sentences in a language can only be asserted or denied in a higher-order meta-language. However, Tarski’s statement of the truth-schemata themselves involve general truth functions, and in particular (...) the biconditional, defined in terms of truth conditions involving truth values standardly displayed in a truth table. Consistently with his semantic program, all such truth values should also be relativized to particular languages for Tarski. The objection thus points toward the more interesting problem of Tarski’s concept of the exact status of truth predications in a general logic of sentential connectives. Tarski’s three-part solution to the circularity objection which he anticipates is discussed and refuted in detail. (shrink)

The subject of the paper is the issue whether the property of truth as ascribed to sentences and propositions is a real one, i.e. capable of being ascribed to entities such as physical objects, their properties, relations between them, sets of objects, properties of the sets of objects, etc. The point of departure for the analysis is Alfred Tarski’s definition of true sentence in a language L. His definition does not imply anything about the ontology of the property of truth, (...) but it implies instead that for each language there is a multitude of such properties . This claim, however, does have ontological consequences. For this reason a number of the so-called deflationary theories of truth have been proposed. According to them, the expression “property of truth” stands for a heterogeneous set of properties that make a sentence true in a given language, context, etc. The author of the paper argues that deflationary theories eliminate the notion of truth but, together with it, they banish also some other cognitive values which seems contrary to the original intentions of their proponents. The author concludes that the property of truth cannot be captured with the desirable philosophical generality if one holds on to the idea of truth as a property of sentences. Instead he proposes to interpret truth as a property of propositions. In this case it seems more reasonable to speak both of a property of truth and of a norm of truth. As he points out, however, the reasoning along these lines encounters some serious stumble blocks; he concludes that a further account of the predicate involves a shift from propositions to judgments, which in turn implicates one in a temporal structure of judgment-making. He asserts that such an intrusion of metaphysical or transcendental notions simply replaces one problem by an another instead of providing a genuine solution to the original one. He illustrates this fact by invoking Edmund Husserl’s position in Die formale und die transcendentale Logik and in his Erfahrung und Urteil. Since some of Husserl’s notions are incurably mysterious, the author ends up with rather skeptical conclusion that no account of the intuitive bond between the property and the normative character of truth is possible within inherited philosophical frameworks. Key words TRUTH, PROPOSITION, NORM OF TRUTH, PROPERTY OF TRUTH. (shrink)

I argue that recent defenses of the view that in 1936 Tarski required all interpretations of a language to share one same domain of quantification are based on misinterpretations of Tarski’s texts. In particular, I rebut some criticisms of my earlier attack on the fixed-domain exegesis and I offer a more detailed report of the textual evidence on the issue than in my earlier work. I also offer new considerations on subsisting issues of interpretation concerning Tarski’s views on the logical (...) correctness of certain omega-arguments, on the Tarskian proof that Etchemendy took to be modal and fallacious, and on Tarski’s appeals to the “common concept of consequence”. (shrink)

The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...) and Skolem. Itinerary V surveys the work in logic connected to the Hilbert school, and itinerary V deals specifically with consistency proofs and metamathematics, including the incompleteness theorems. Itinerary VII traces the development of intuitionistic and many-valued logics. Itinerary VIII surveys the development of semantical notions from the early work on axiomatics up to Tarski's work on truth. (shrink)

This article discusses two coextensive concepts of logical consequence that are implicit in the two fundamental logical practices of establishing validity and invalidity for premise-conclusion arguments. The premises and conclusion of an argument have information content (they ?say? something), and they have subject matter (they are ?about? something). The asymmetry between establishing validity and establishing invalidity has long been noted: validity is established through an information-processing procedure exhibiting a step-by-step deduction of the conclusion from the premise-set. Invalidity is established by (...) exhibiting a countermodel satisfying the premises but not the conclusion. The process of establishing validity focuses on information content; the process of establishing invalidity focuses on subject matter. Corcoran's information-theoretic concept of logical consequence corresponds to the former. Tarski's model-theoretic concept of logical consequence formulated in his famous 1936 no-countermodels definition corresponds to the latter. Both are found to be indispensable for understanding the rationale of the deductive method and each complements the other. This study discusses the ontic question of the nature of logical consequence and the epistemic question of the human capabilities presupposed by practical applications of these two concepts as they make validity and invalidity accessible to human knowledge. (shrink)

Theories of linguistic meaning have been a major influence in twentieth century philosophy. This is due, in part, to the assumption that meaning is the crucial and interesting thing about language. To know the meaning of an expression is to understand it, and since understanding is central to philosophy in many different ways, it should be no surprise that the notion of meaning has often taken center stage. The aim of this paper is to briefly explore some influential views concerning (...) linguistic meaning. The final objective will be to demonstrate some alternatives which are open to theory with respect to this notion,for there are those who have wanted to ban talk of meaning from serious scientific discourse. The point is that many of the disadvantages of traditional notions of meaning are avoidable¾in particular, they are avoidable along a path which starts from Frege and moves on via Tarski and Davidson. (shrink)

Alfred Tarski was a nominalist. But he published almost nothing on his nominalist views, and until recently the only sources scholars had for studying Tarski’s nominalism were conversational reports from his friends and colleagues. However, a recently-discovered archival resource provides the most detailed information yet about Tarski’s nominalism. Tarski spent the academic year 1940-41 at Harvard, along with many of the leading lights of scientific philosophy: Carnap, Quine, Hempel, Goodman, and (for the fall semester) Russell. This group met frequently to (...) discuss logical and philosophical topics of shared interest. At these meetings, Carnap took dictation notes, which are now stored in the Archives of Scientific Philosophy. Interestingly, and somewhat surprisingly, the plurality of notes covers a proposal Tarski presents for a nominalist language of unified science. This chapter addresses the following questions about this project. What, precisely, is Tarski’s nominalist position? What rationales are given for Tarski’s nominalist stance—and are these rationales defensible? Finally, how is Tarskian nominalism of 1941 related to current nominalist projects? (shrink)