Dag Prawitz Research Articles

Prawitz's completeness conjecture: A reassessment

AbstractIn 1973, Dag Prawitz conjectured that the calculus of intuitionistic logic is complete with respect to his notion of validity of arguments. On the background of the recent disproof of this conjecture by Piecha, de Campos Sanz and Schroeder‐Heister, we discuss possible strategies of saving Prawitz's intentions. We argue that Prawitz's original semantics, which is based on the principal frame of all atomic systems, should be replaced with a general semantics, which also takes into account restricted frames of atomic systems. We discard the option of not considering extensions of atomic systems, but acknowledge the need to incorporate definitional atomic bases in the semantic framework. It turns out that ideas and results by Westerståhl on the Carnap categoricity of intuitionistic logic can be applied to Prawitz semantics. This implies that Prawitz semantics has a status of its own as a genuine, though incomplete, semantics of intuitionstic logic. An interesting side result is the fact that every formula satisfiable in general semantics is satisfiable in an axioms‐only frame (a frame whose atomic systems do not contain proper rules). We draw a parallel between this seemingly paradoxical result and Skolem's paradox in first‐order model theory.

Correctness of assertion and validity of inference

AbstractThis is a slightly edited transcript of a lecture given by Per Martin‐Löf on 26 October 2022 at the Rolf Schock Symposium in Stockholm. In 2020, the Rolf Schock Prize in Logic and Philosophy was awarded to Dag Prawitz and Per Martin‐Löf, and the symposium was organised in their honour. The transcript was prepared by Ansten Klev and edited by the author.

Pursuit of the concept of validity: A dialogue

AbstractThis is a dialogue between Lisa and Max on Dag Prawitz's work concerning the concept of deductive validity. Lisa first explains Prawitz's criticisms of the presently prevailing non‐epistemic analyses of validity. Then Lisa describes three different ways in which Prawitz attempted to develop an epistemic concept of validity. Max asks questions for clarification, raises some objections and compares Prawitz's three approaches with other lines of thought. Two inference rules are specially discussed: disjunction introduction and ex contradictione quodlibet. Max and Lisa view Prawitz's contribution as part of a variegated, ongoing research pursuing an explication of the concept of validity, which began with the Socratic and Platonic distinction between merely persuasive arguments and good arguments and with Aristotle's definition of syllogism. Throughout the history of philosophy, logicians have aimed at clarifying a pre‐theoretic idea of good deduction and have proposed, criticized, refined, adjusted precise theoretical concepts of validity. Prawitz has criticized current views about validity, has advanced new proposals and revised them. His work is an important chapter of the long pursuit of the concept of validity.

TOWARDS AN EVALUATION OF THE NORMALISATION THESIS ON IDENTITY OF PROOFS: THE CASE OF CHURCH-TURING THESIS AS TOUCHSTONE

Abstract This article is a methodological discussion of formal approaches to the question of identity of proofs from a philosophical standpoint. First, an introduction to the question of identity of proofs itself is given, followed by a brief reconstruction of the so-called normalisation thesis, proposed by Dag Prawitz in 1971, in which some of its core mathematical and conceptual traits are presented. After that, a comparison between the normalisation thesis and the more well-known Church-Turing thesis on computability is carried out in three main parts: the first dedicated to highlighting some of the analogies between them; the second, their most remarkable differences; and the third, to the possible relations of dependence between the two. Based on these considerations, some concluding remarks concerning the potential of the normalisation thesis and similar approaches to the question of identity of proofs are made in the last section.

البرهان غیر المباشر ومشکلة التفاف البراهین ومعیاریتها فی الاستنباط الطبیعى للمنطق الکلاسیکى

الملخص تعد مشکلة التفاف(1)detour البراهين في أنسقة الاستنباط الطبيعى مشکلة في غاية الأهمية، وهى مشکلة لا نجد مثيلا لها في الأنسقة الأکسيوماتيکية. وهى تعنى، على نحو مبسط، تکرار خطوات البرهان، وهو أمر معيب من ناحية اشتقاق البراهين. لقد أدرک جنتسن G. Gentzen صاحب أول نسق في الاستنباط الطبيعى تلک المشکلة، وهو قد استطاع حلها بالنسبة لأنسقة الاستنباط الطبيعى الحدسية دون الکلاسيکية، من خلال مبرهنته على معيارية Normalization / Haupstaz البراهين الحدسية، أى البرهنة على إمکانية حذف التفاف البراهين. ويعود الفضل إلى داج برافيتس D. Prawitz فى وضع أول برهان على معيارية براهين منطق الاستنباط الطبيعى الکلاسيکى من خلال استخدام قاعدة البرهان غير المباشر. في هذا البحث سوف نتساءل عن قدرة تلک القاعدة على ذلک حقا، وإمکانية وجود قاعدة بديلة الکلمات المفتاحية: برهان غير مباشر- التفاف البراهين- استنباط طبيعى-منطق حدسى- قانون بيرس [Abstract: The problem of the detour of proofs in natural deduction systems is a serious problem. We can not find a counterpart for this problem in axiomatic systems. It means, roughly speaking, repetition to the lines or steps of proof. This is a fault from a derivative point of view. G. Gentzen, the first logician to put a calculus for natural deduction , knew this problem very well, he even could solve it in respect to intuitive natural deductions systems but not to classical ones. He solved that problem by his Haupstaz or Normalization theorem for intuitive proofs, i.e. proving the possibility of eliminating the detours in proofs. It was Dag Prawitz who put the first proof of the normalization of proofs in natural deduction for classical systems by indirect proof rule. In this paper, I shall scrutinize the ability of that rule of doing what wanted from it and if there is an alternative one for it.]

Dag Prawitz on Proofs, Operations and Grounding

Dag Prawitz’s theory of grounds proposes a fresh approach to valid inferences. Its main aim is to clarify nature and reasons of their epistemic power. The notion of ground is taken to denote what one is in possession of when in a state of evidence, and valid inferences are described in terms of operations that make us pass from grounds we already have to new grounds. Thanks to a rigorously developed proof-as-chains conception, the ground-theoretic framework permits Prawitz to overcome some conceptual difficulties of his earlier proof-theoretic explanation. Though from different points of view, anyway, the two accounts share an issue of recognizability of relevant operational properties.

Proof-Theoretic Semantics, a Problem with Negation and Prospects for Modality

This paper discusses proof-theoretic semantics, the project of specifying the meanings of the logical constants in terms of rules of inference governing them. I concentrate on Michael Dummett's and Dag Prawitz' philosophical motivations and give precise characterisations of the crucial notions of harmony and stability, placed in the context of proving normalisation results in systems of natural deduction. I point out a problem for defining the meaning of negation in this framework and prospects for an account of the meanings of modal operators in terms of rules of inference.

SUBFORMULA AND SEPARATION PROPERTIES IN NATURAL DEDUCTION VIA SMALL KRIPKE MODELS

Various natural deduction formulations of classical, minimal, intuitionist, and intermediate propositional and first-order logics are presented and investigated with respect to satisfaction of the separation and subformula properties. The technique employed is, for the most part, semantic, based on general versions of the Lindenbaum and Lindenbaum–Henkin constructions. Careful attention is paid (i) to which properties of theories result in the presence of which rules of inference, and (ii) to restrictions on the sets of formulas to which the rules may be employed, restrictions determined by the formulas occurring as premises and conclusion of the invalid inference for which a counterexample is to be constructed. We obtain an elegant formulation of classical propositional logic with the subformula property and a singularly inelegant formulation of classical first-order logic with the subformula property, the latter, unfortunately, not a product of the strategy otherwise used throughout the article. Along the way, we arrive at an optimal strengthening of the subformula results for classical first-order logic obtained as consequences of normalization theorems by Dag Prawitz and Gunnar Stålmarck.

Antirealism and universal knowability

Truth’s universal knowability entails its discovery. This threatens antirealism, which is thought to require it. Fortunately, antirealism is not committed to it. Avoiding it requires adoption (and extension) of Dag Prawitz’s position in his long-term disagreement with Michael Dummett on the notion of provability involved in intuitionism’s identification of it with truth. Antirealism (intuitionism generalized) must accommodate a notion of lost-opportunity truth (a kind of recognition-transcendent truth), and even truth consisting in the presence of unperformable verifications. Dummett’s position cannot abide this, while Prawitz’s can. Antirealism’s epistemic notion of truth derives from general features of its meaning theory, not from a universal knowability principle.

Gentzen's Proof of Normalization for Natural Deduction

AbstractGentzen writes in the published version of his doctoral thesis Untersuchungen über das logische Schliessen (Investigations into logical reasoning) that he was able to prove the normalization theorem only for intuitionistic natural deduction, but not for classical. To cover the latter, he developed classical sequent calculus and proved a corresponding theorem, the famous cut elimination result. Its proof was organized so that a cut elimination result for an intuitionistic sequent calculus came out as a special case, namely the one in which the sequents have at most one formula in the right, succedent part. Thus, there was no need for a direct proof of normalization for intuitionistic natural deduction. The only traces of such a proof in the published thesis are some convertibilities, such as when an implication introduction is followed by an implication elimination [1934–35, II.5.13]. It remained to Dag Prawitz in 1965 to work out a proof of normalization. Another, less known proof was given also in 1965 by Andres Raggio.We found in February 2005 an early handwritten version of Gentzen's thesis, with exactly the above title, but with rather different contents: Most remarkably, it contains a detailed proof of normalization for what became the standard system of natural deduction. The manuscript is located in the Paul Bernays collection at the ETH-Zurichwith the signum Hs. 974: 271. Bernays must have gotten it well before the time of his being expelled from Göttingen on the basis of the racial laws in April 1933.

On Inversion Principles

The idea of an ‘inversion principle’, and the name itself, originated in the work of Paul Lorenzen in the 1950s, as a method to generate new admissible rules within a certain syntactic context. Some fifteen years later, the idea was taken up by Dag Prawitz to devise a strategy of normalization for natural deduction calculi (this being an analogue of Gentzen's cut-elimination theorem for sequent calculi). Later, Prawitz used the inversion principle again, attributing it with a semantic role. Still working in natural deduction calculi, he formulated a general type of schematic introduction rules to be matched – thanks to the idea supporting the inversion principle – by a corresponding general schematic Elimination rule. This was an attempt to provide a solution to the problem suggested by the often quoted note of Gentzen. According to Gentzen ‘it should be possible to display the elimination rules as unique functions of the corresponding introduction rules on the basis of certain requirements’. Many people have since worked on this topic, which can be appropriately seen as the birthplace of what are now referred to as “general elimination rules”, recently studied thoroughly by Sara Negri and Jan von Plato. In this study, we retrace the main threads of this chapter of proof-theoretical investigation, using Lorenzen's original framework as a general guide.

A new S4 classical modal logic in natural deduction

AbstractWe show, first, that the normalization procedure for S4 modal logic presented by Dag Prawitz in [5] does not work. We then develop a new natural deduction system for S4 classical modal logic that is logically equivalent to that of Prawitz, and we show that every derivation in this new system can be transformed into a normal derivation.

The Russell–Prawitz modality

In his 1903, Principles of Mathematics, Bertrand Russell mentioned possible definitions of conjunction, disjunction, negation and existential quantification in terms of implication and universal quantification, exploiting impredicative universal quantifiers over all propositions. In his 1965 Ph.D. thesis Dag Prawitz showed that these definitions hold in intuitionistic second order logic. More recently, these definitions have been used to represent logic in various impredicative type theories. This treatment of logic is distinct from the more standard Curry–Howard representation of logic in a dependent type theory.The main aim of this paper is to compare, in a purely logical, non type-theoretic setting, this Russell–Prawitz representation of intuitionistic logic with other possible representations. It turns out that associated with the Russell–Prawitz representation is a lax modal operator, which we call the Russell–Prawitz modality, and that any lax modal operator can be used to give a translation of intuitionistic logic into itself that generalises both the double negation interpretation, double negation being a paradigm example of a lax modality, and the Russell–Prawitz representation.

Harmony and Autonomy in Classical Logic

Michael Dummett and Dag Prawitz have argued that a constructivist theory of meaning depends on explicating the meaning of logical constants in terms of the theory of valid inference, imposing a constraint of harmony on acceptable connectives. They argue further that classical logic, in particular, classical negation, breaks these constraints, so that classical negation, if a cogent notion at all, has a meaning going beyond what can be exhibited in its inferential use. I argue that Dummett gives a mistaken elaboration of the notion of harmony, an idea stemming from a remark of Gerhard Gentzen"s. The introduction-rules are autonomous if they are taken fully to specify the meaning of the logical constants, and the rules are harmonious if the elimination-rule draws its conclusion from just the grounds stated in the introduction-rule. The key to harmony in classical logic then lies in strengthening the theory of the conditional so that the positive logic contains the full classical theory of the conditional. This is achieved by allowing parametric formulae in the natural deduction proofs, a form of multiple-conclusion logic.

Proofs as Acts and Proofs as Objects: Some questions for Dag Prawitz*

Distinguant la notion de construction-acte et de construction-objet en reference a Brouwer et Heyting, l'A. confronte sa propre definition de la preuve-acte et de la preuve-objet a la position anti-realiste de Prawitz concernant la philosophie de la logique et des mathematiques. L'A. examine les points suivants: 1) l'interpretation semantique de la deduction naturelle des arbres de derivation; 2) la relation entre les notions de conservation et d'harmonie; 3) la formulation des principes fondamentaux de Dummett; 4) les principes et la signification constructive de l'implication.

Philosophical and logical publications of Dag Prawitz

TheoriaVolume 64, Issue 2-3 p. 338-343 Philosophical and logical publications of Dag Prawitz First published: 11 February 2008 https://doi.org/10.1111/j.1755-2567.1998.tb00191.xAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Volume64, Issue2-3August 1998Pages 338-343 RelatedInformation

Partial realizations of Hilbert's program

§0. Introduction. What follows is a write-up of my contribution to the symposium “Hilbert's Program Sixty Years Later” which was sponsored jointly by the American Philosophical Association and the Association for Symbolic Logic. The symposium was held on December 29,1985 in Washington, D. C. The panelists were Solomon Feferman, Dag Prawitz and myself. The moderator was Wilfried Sieg. The research which I discuss here was partially supported by NSF Grant DMS-8317874.I am grateful to the organizers of this timely symposium on an important topic. As a mathematician I particularly value the opportunity to address an audience consisting largely of philosophers. It is true that I was asked to concentrate on the mathematical aspects of Hilbert's program. But since Hilbert's program is concerned solely with the foundations of mathematics, the restriction to mathematical aspects is really no restriction at all.Hilbert assigned a special role to a certain restricted kind of mathematical reasoning known as finitistic. The essence of Hilbert's program was to justify all of set-theoretical mathematics by means of a reduction to finitism. It is now well known that this task cannot be carried out. Any such possibility is refuted by Gödel's theorem. Nevertheless, recent research has revealed the feasibility of a significant partial realization of Hilbert's program. Despite Gödel's theorem, one can give a finitistic reduction for a substantial portion of infinitistic mathematics including many of the best-known nonconstructive theorems. My purpose here is to call attention to these modern developments.

Partial realizations of Hilbert's program

§0. Introduction. What follows is a write-up of my contribution to the symposium “Hilbert's Program Sixty Years Later” which was sponsored jointly by the American Philosophical Association and the Association for Symbolic Logic. The symposium was held on December 29,1985 in Washington, D. C. The panelists were Solomon Feferman, Dag Prawitz and myself. The moderator was Wilfried Sieg. The research which I discuss here was partially supported by NSF Grant DMS-8317874.I am grateful to the organizers of this timely symposium on an important topic. As a mathematician I particularly value the opportunity to address an audience consisting largely of philosophers. It is true that I was asked to concentrate on the mathematical aspects of Hilbert's program. But since Hilbert's program is concerned solely with the foundations of mathematics, the restriction to mathematical aspects is really no restriction at all.Hilbert assigned a special role to a certain restricted kind of mathematical reasoning known as finitistic. The essence of Hilbert's program was to justify all of set-theoretical mathematics by means of a reduction to finitism. It is now well known that this task cannot be carried out. Any such possibility is refuted by Gödel's theorem. Nevertheless, recent research has revealed the feasibility of a significant partial realization of Hilbert's program. Despite Gödel's theorem, one can give a finitistic reduction for a substantial portion of infinitistic mathematics including many of the best-known nonconstructive theorems. My purpose here is to call attention to these modern developments.

A History of Philosophy. Volume 2: The Modern Age to Romanticism, and: Insights and Oversights of Great Thinkers: An Evaluation of Western Philosophy (review)

BOOK REVXEWS 273 congratulated on packing so much good material into a small space. Gouinlock, it should be added, faced an extremely difficult task in finding a theme by which he could intelligibly discuss the potpourri contents of volume 2. That he succeeded is no small accomplishment. EDWARD H. MADDEN State University of New York at Buffalo and University of Kentucky Anders Wedberg. A History of Philosophy. Volume 2: The Modern Age to Romanticism. New York: Oxford Univ. Press, 1983. Pp. 227. $24.5o, cloth; $9.95, paper. Charles Hartshorne. Insights and Oversights of Great Thinkers: An Evaluation of Western Philosophy. Albany: State University of New York Press, 1983. Pp. xviii + 393. It is helpful to consider these books together in a single review. They open up some possibilities for the study of historical sources in philosophy and for presenting these sources in fresh perspective to students. Both authors take an approach based on themes and problems, not one organized primarily around the individual philosophers in chronological order. Both authors also place a premium on clarifying the philosophical issues themselves rather than upon the conventional outlines and terminology . However, there are also some definite differences between these two works which must be noted. Anders Wedberg, late professor of philosophy at Stockholm University, published his three volume History of Philosophy in Swedish in 1958-66. The present book is the first volume in this series, first published in 1959 in Swedish and translated basically by Wedberg himself, with some revisions by Dag Prawitz and others. The eight main parts deal with: the modern mechanistic world view, the Platonic-Aristotelian ideal of science, the continental systems (of Descartes, Spinoza, and Leibniz), British epistemology, empiricist critique of causation and necessity, empiricist theories of the external world, Kant on synthetic a priori judgments, and the transition from Kant to the later Germans. There is also an Appendix on Marx and Engels. Within each of the divisions there are chapters devoted to the main problems. For instance, the mechanistic world view provides a framework for treating the corpuscular theory, the dualism of matter and consciousness, and questions about space and time. Under the heading "The Revival of the Platonic-Aristotelian Ideal of Science" Wedberg seeks to reformulate his treatment of Greek philosophy (in Volume 1 of his History) in a way that is related to the so-called hypotheticodeductive method in modern science and especially to Leibniz's vision of a universal characteristic. The chief philosophers of the seventeenth and eighteenth centuries are given unequal attention. Among the continental system-builders the main focus is upon 274 JOURNAL OF THE HISTORY OF PHILOSOPHY Spinoza and Leibniz, since Descartes is treated mainly through the problem of matter and consciousness. Wedberg acknowledges the complexity and richness of Leibniz's philosophy and his supple notion of a deductive method. Similarly Locke is presented chiefly as a forerunner, as one who sets up terminology and problems for the empiricists. In Kant's case, the process of selectivity manifests itself in concentration upon the first Critique's handling of synthetic a priori judgments in the Esthetic and the Analytic. Wedberg has separate sections of criticism of the positions he has selected and formalized. The continental systems are regarded as faulted because of their reliance on the deductive methods and their search for certainty. The same search is also attributed to the empiricists, but with a difference. For they temper the all-or-nothing notion of knowledge with a practical appreciation for belief and probability. As Wedberg phrases it, "the reasons for opinions vary in strength, and I ought, in each case, to let the degree of my belief correspond to the strength of the reasons." According to the author, the blending of the absolutely certain and the probable in human knowledge constitutes an "unconscious psychological conflict" reaching down to Bertrand Russell. In assessing Kant, Wedberg distinguishes between the criticist deduction (an analysis of the notion of experience as a stated fact) and the Copernican revolution (an explanation of the mental factors that produce experience). It is asked how Kant can claim to know a correspondence between phenomena and things in themselves; how the latter act upon consciousness; how the ego...

J. I. Zucker. The adequacy problem for classical logic. Journal of philosophical logic, vol. 7 (1978), pp. 517–535. - J. I. Zucker and R. S. Tragesser. The adequacy problem for inferential logic. Journal of philosophical logic, pp. 501–516. - Dag Prawitz. Proofs and the meaning and completeness of the logical constants. Essays on mathematical and philosophical logic, Proceedings of the

J. I. Zucker. The adequacy problem for classical logic. Journal of philosophical logic, vol. 7 (1978), pp. 517–535. - J. I. Zucker and R. S. Tragesser. The adequacy problem for inferential logic. Journal of philosophical logic, pp. 501–516. - Dag Prawitz. Proofs and the meaning and completeness of the logical constants. Essays on mathematical and philosophical logic, Proceedings of the Fourth Scandinavian Logic Symposium and of the First Soviet-Finnish Logic Conference, Jyväskyla, Finland, June 29-July 6,1976, edited by Jaakko Hintikka, Ilkka Niiniluoto, and Esa Saarinen, Synthese library, vol. 122, D. Reidel Publishing Company, Dordrecht, Boston, and London, 1979, pp. 25–40. - Dag Prawitz. Meaning and proofs: on the conflict between classical and intuitionistic logic. Theoria, vol. 43 (1977), pp. 2–40. - M. A. E. Dummett. The justification of deduction. Proceedings of the British Academy, vol. 59 (for 1973, pub. 1975), pp. 201–232. - Michael Dummett. The philosophical basis of intuitionistic logic. Logic Colloquium '73, Proceedings of the Logic Colloquium, Bristol, July 1973, edited by H. E. Rose and J.C. Shepherdson, Studies in logic and the foundations of mathematics, vol. 80, North-Holland Publishing Company, Amsterdam and Oxford, and American Elsevier Publishing Company, New York, 1975, pp. 5–40. - Ian Hacking. Do-it-yourself semantics for classical sequent calculi, including ramified type theory. Logic, foundations of mathematics, and computahility theory, Part one of the proceedings of the Fifth International Congress of Logic, Methodology, and Philosophy of Science, London, Ontario, Canada–1975, edited by Robert E. Butts and Jaakko Hintikka, The University of Western Ontario series in Philosophy of science, vol. 9, D. Reidel Publishing Company, Dordrecht and Boston1977, pp. 371–390. - Ian Hacking. What is logic?The journal of philosophy, vol. 76 (1979), pp. 285–319. - Göran Sundholm. Hacking's logic. The journal of philosophy, vol. 78 (1981), pp. 160–168. - Christopher Peacocke. Hacking on logic: two comments. The journal of philosophy, vol. 78 pp. 168–175. - Volume 47 Issue 3