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[Shapiro 1998b] and [Ketland 1999] have argued against deflationary views of truth on the ground that an adequate truth-theoretic extension of a theory is a non-conservative extension. We clarify the argument and offer an alternative interpretation of the observed non-conservativeness phenomenon, compatible both with the logical facts and the deflationist’s thesis.

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... à la neige » 101 , cela semble bien suggérer que le contenu sémantique des deux énoncés est, si ce n'est rigoureusement identique, du moins grandement similaire102 et que toute justification d'un des deux énoncés vaudra justification de l'autre. Le problème c'est que ceci n'est pas sans conséquence sur la nature des noms canoniques. ...

« Qu’est-ce que la vérité ? » À cette question, les déflationnistes aléthiques contemporains proposent une réponse originale : la propriété de vérité ne serait qu’un simple outil de décitation, indispensable pour formuler certaines généralisations mais dénué de tout pouvoir explicatif propre. Selon eux, elle ne jouerait donc pas de rôle important dans notre activité scientifique. L’objectif de cette thèse est d’évaluer la solidité de la position déflationniste en la confrontant à divers arguments avancés contre ce type de conceptions de la vérité. Après avoir précisé les doctrines centrales du déflationnisme actuel, notre travail se poursuit en deux parties, que l’on peut voir comme deux tentatives complémentaires de fournir un cadre méthodologique permettant d’examiner précisément les théories déflationnistes de la vérité. Dans un premier temps, nous analysons la thèse, souvent attribuée aux déflationnistes, selon laquelle le prédicat de vérité serait une sorte de notion logique. Dans un second temps nous examinons un célèbre argument anti-déflationniste appelé « argument de la conservativité ». Au final, si le déflationnisme ne nous paraît pas totalement désarmé face aux critiques dont il a fait l’objet, notre travail a néanmoins permis de montrer que certaines réponses majeures avancées pour sa défense ne sont plus tenables.

Any (1‐) consistent and sufficiently strong system of first‐order formal arithmetic fails to decide some independent Gödel sentence. We examine consistent first‐order extensions of such systems. Our purpose is to discover what is minimally required by way of such extension in order to be able to prove the Gödel sentence in a non‐trivial fashion. The extended methods of formal proof must capture the essentials of the so‐called ‘semantical argument’ for the truth of the Gödel sentence. We are concerned to show that the deflationist has at his disposal such extended methods—methods which make no use or mention of a truth‐predicate. This consideration leads us to reassess arguments recently advanced—one by Shapiro and another by Ketland—against the deflationist's account of truth. Their main point of agreement is this: they both adduce the Gödel phenomena as motivating a ‘thick’ notion of truth, rather than the deflationist's ‘thin’ notion. But the so‐called ‘semantical argument’, which appears to involve a ‘thick’ notion of truth, does not really have to be semantical at all. It is, rather, a reflective argument. And the reflections upon a system that are contained therein are deflationarily licit, expressible without explicit use or mention of a truth‐predicate. Thus it would appear that this anti‐deflationist objection fails to establish that there has to be more to truth than mere conformity to the disquotational T‐schema.

On the one hand, the concept of truth is a major research subject in analytic philosophy. On the other hand, mathematical logicians have developed sophisticated logical theories of truth and the paradoxes. Recent developments in logical theories of the semantical paradoxes are highly relevant for philosophical research on the notion of truth. And conversely, philosophical guidance is necessary for the development of logical theories of truth and the paradoxes. From this perspective, this volume intends to reflect and promote deeper interaction and collaboration between philosophers and logicians investigating the concept of truth than has existed so far.Aside from an extended introductory overview of recent work in the theory of truth, the volume consists of articles by leading philosophers and logicians on subjects and debates that are situated on the interface between logical and philosophical theories of truth. The volume is intended for graduate students in philosophy and in logic who want an introduction to contemporary research in this area, as well as for professional philosophers and logicians.

Over the last few decades Michael Dummett developed a rich program for assessing logic and the meaning of the terms of a language.
He is also a major exponent of Frege's version of logicism in the philosophy of mathematics. Over the last decade, Neil Tennant
developed an extensive version of logicism in Dummettian terms, and Dummett influences other contemporary logicists such as
Crispin Wright and Bob Hale. The purpose of this paper is to explore the prospects for Fregean logicism within a broadly Dummettian
framework. The conclusions are mostly negative: Dummett's views on analyticity and the logical/non-logical boundary leave
little room for logicism. Dummett's considerations concerning manifestation and separability lead to a conservative extension
requirement: if a sentence S is logically true, then there is a proof of S which uses only the introduction and elimination rules of the logical terms that occur in S. If basic arithmetic propositions are logically true - as the logicist contends - then there is tension between this conservation
requirement and the ontological commitments of arithmetic. It follows from Dummett's manifestation requirements that if a
sentence S is composed entirely of logical terminology, then there is a formal deductive system D such that S is analytic, or logically true, if and only if S is a theorem of D. There is a deep conflict between this result and the essential incompleteness, or as Dummett puts it, the indefinite extensibility,
of arithmetic truth.

Some axiomatic theories of truth and related subsystems of second-order arithmetic are surveyed and shown to be conservative over their respective base theory. In particular, it is shown by purely finitistically means that the theory PA ÷ "there is a satisfaction class" and the theory FS↾ of [2] are conservative over PA.

La vérité, dit-on, est un des buts de la science. Mais quelle est la place de la notion de vérité elle-même dans le langage de la science ? Cette notion peut-elle être suffisamment clarifiée ? Et si oui, quelle peut être sa contribution au discours scientifique, pour quels usages la notion de vérité peut-elle être mobilisée ? Ce travail cherche à répondre à ces questions. Sa thèse principale est que la notion de vérité s'apparente à une notion logique. Cette idée s'inscrit dans un courant de réflexion contemporain sur la vérité appelé "déflationnisme", mais la formulation des thèses déflationnistes que nous proposons est nouvelle, comme sont nouveaux les arguments et les idées mises en œuvre pour l'étayer. Négativement, une critique détaillée d'une tentative influente de réfutation a priori des thèses déflationnistes est présentée. Positivement, nous caractérisons d'une part une classe critique d'affirmations mettant en jeu la notion de vérité comme ensemble de moyens d'expliciter des contenus déjà implicitement acceptés, et nous introduisons d'autre part des considérations et des outils permettant de comprendre le lien entre la thèse de la logicité de la notion de vérité et les thèses déflationnistes classiques relatives aux emplois légitimes de cette notion.

To what extent can mathematical thought be analyzed in formal terms? Gödel's theorems show the inadequacy of single formal systems for this purpose, except in relatively restricted parts of mathematics. However at the same time they point to the possibility of systematically generating larger and larger systems whose acceptability is implicit in acceptance of the starting theory. The engines for that purpose are what have come to be called reflection principles . These may be iterated into the constructive transfinite, leading to what are called recursive progressions of theories . A number of informative technical results have been obtained about such progressions (cf. Feferman [1962], [1964], [1968] and Kreisel [1958], [1970]). However, for some years I had hoped to give a more realistic and perspicuous finite generation procedure. This was first done in a rather special way in Feferman [1979] for the characterization of predicativity , which may be regarded as that part of mathematical thought implicit in our acceptance of elementary number theory. What is presented here is a new and simple notion of the reflective closure of a schematic theory which can be applied quite generally.
Two examples of schematic theories in the sense used here are versions of Peano arithmetic and Zermelo set theory.

Deflationsism about truth is a pot-pourri, variously claiming that truth is redundant, or is constituted by the totality of 'T-sentences', or is a purely logical device (required solely for disquotational purposes or for re-expressing finitarily infinite conjunctions and/or disjunctions). In 1980, Hartry Field proposed what might be called a 'deflationary theory of mathematics', in which it is alleged that all uses of mathematics within science are dispensable. Field's criterion for the dispensability of mathematics turns on a property of theories, called conservativeness. I present some technical results, some of which may be found in Tarski (1936), concerning the logical properties of truth theories; in particular, concerning the conservativeness of adding a truth theory for an object level language to any theory expressed in it. It transpires that various deflationary truth theories behave somewhat differently from the standard Tarskian truth theory. These results suggest that Tarskian theories of truth are not redundant or dispensable. Finally, I hint at an analogy between the behaviour of mathematical theories and of standard (Tarskian) theories of truth with respect to their indispensability to, as Quine would put, our 'scientific world-view'.

The Logical Basis of Metaphysics, The William James lectures

- Michael Dummett

Dummett, Michael
1991 The Logical Basis of Metaphysics, The William James lectures ; 1976,
Cambridge, Mass. : Harvard University Press.

Proof and truth : Through thick and thin

1998b Proof and truth : Through thick and thin, Journal of Philosophy,
95(10), 493-521.