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Prawitz, Dag
Publications (10 of 23) Show all publications
Prawitz, D. (2019). The Fundamental Problem of General Proof Theory. Studia Logica: An International Journal for Symbolic Logic, 107(1), 11-29
Open this publication in new window or tab >>The Fundamental Problem of General Proof Theory
2019 (English)In: Studia Logica: An International Journal for Symbolic Logic, ISSN 0039-3215, E-ISSN 1572-8730, Vol. 107, no 1, p. 11-29Article in journal (Refereed) Published
Abstract [en]

I see the question what it is that makes an inference valid and thereby gives a proof its epistemic power as the most fundamental problem of general proof theory. It has been surprisingly neglected in logic and philosophy of mathematics with two exceptions: Gentzen's remarks about what justifies the rules of his system of natural deduction and proposals in the intuitionistic tradition about what a proof is. They are reviewed in the paper and I discuss to what extent they succeed in answering what a proof is. Gentzen's ideas are shown to give rise to a new notion of valid argument. At the end of the paper I summarize and briefly discuss an approach to the problem that I have proposed earlier.

Keywords
Proof theory, Proof, Valid inference, Valid argument, Gentzen's naturaldeduction, Intuitionism
National Category
Mathematics Computer and Information Sciences Philosophy, Ethics and Religion
Identifiers
urn:nbn:se:su:diva-166562 (URN)10.1007/s11225-018-9785-9 (DOI)000458570100003 ()
Available from: 2019-03-04 Created: 2019-03-04 Last updated: 2022-03-23Bibliographically approved
Prawitz, D. (2019). The Seeming Interdependence Between the Concepts of Valid Inference and Proof. Topoi: An International Review of Philosophy, 38(3), 493-503
Open this publication in new window or tab >>The Seeming Interdependence Between the Concepts of Valid Inference and Proof
2019 (English)In: Topoi: An International Review of Philosophy, ISSN 0167-7411, E-ISSN 1572-8749, Vol. 38, no 3, p. 493-503Article in journal (Refereed) Published
Abstract [en]

We may try to explain proofs as chains of valid inference, but the concept of validity needed in such an explanation cannot be the traditional one. For an inference to be legitimate in a proof it must have sufficient epistemic power, so that the proof really justifies its final conclusion. However, the epistemic concepts used to account for this power are in their turn usually explained in terms of the concept of proof. To get out of this circle we may consider an idea within intuitionism about what it is to justify the assertion of a proposition. It depends on Heyting's view of the meaning of a proposition, but does not presuppose the concept of inference or of proof as chains of inferences. I discuss this idea and what is required in order to use it for an adequate notion of valid inference.

Keywords
Valid inference, Proof, Epistemic ground, Intuitionism
National Category
Philosophy, Ethics and Religion
Identifiers
urn:nbn:se:su:diva-173091 (URN)10.1007/s11245-017-9506-4 (DOI)000481853300002 ()
Available from: 2019-10-07 Created: 2019-10-07 Last updated: 2024-01-08Bibliographically approved
Prawitz, D. (2018). To explain deduction. In: Michael Frauchiger (Ed.), Truth, Meaning, Justification, and Reality: Themes from Dummett (pp. 103-122). Berlin: Walter de Gruyter
Open this publication in new window or tab >>To explain deduction
2018 (English)In: Truth, Meaning, Justification, and Reality: Themes from Dummett / [ed] Michael Frauchiger, Berlin: Walter de Gruyter, 2018, p. 103-122Chapter in book (Refereed)
Abstract [en]

The Justification of Deduction is the title of one of Michael Dummett’s essays. It names also an important theme in his writings to which he returned in the book The Logical Basis of Metaphysics. In the essay he distinguishes differ-ent levels of justification of increasing philosophical depth. At the third and deepest level, the focus is on explaining deduction rather than on justifying it. The task is to explain how deduction can be both legitimate and useful in giving us knowledge. I suggest that it can be described as essentially being the task to say what it is that gives a deduction its epistemic force. It is a fact that deduc-tion has such force, consisting in its capacity to provide grounds for assertions and thereby extend our knowledge, but it is a fact that has to be explained. What is it that gives a deduction this capacity? This task is more challenging than is usually assumed. Obviously, it is not the validity of its inferences, as this is usually understood, which gives a deduction its epistemic force. Truth condi-tional theory of meaning does not seem to have any satisfactory solution to offer, and I argue that nor have inferential theories of meaning, which take the meaning of sentences to be determined by inference rules accepted in a lan-guage. In the last part of the paper, I sketch a different approach to the problem. The main idea is here to give the concept of inference a richer content, so that to perform an inference is not only to make a speech act in which a conclusion is claimed to be supported by a number of premisses, but is in addition to operate on the grounds for the premisses with the aim of getting a ground for the con-clusion. I suggest that it is thanks to such operations that deductions provide grounds for their final conclusions. 

Place, publisher, year, edition, pages
Berlin: Walter de Gruyter, 2018
Series
Launer Library of Analytical Philosophy, ISSN 2198-2155 ; 4
National Category
Philosophy, Ethics and Religion
Research subject
Theoretical Philosophy
Identifiers
urn:nbn:se:su:diva-159433 (URN)10.1515/9783110459135-007 (DOI)978-3-11-045839-8 (ISBN)9783110458312 (ISBN)
Available from: 2018-08-29 Created: 2018-08-29 Last updated: 2023-03-08Bibliographically approved
Prawitz, D. (2016). Lewis Carrolls berättelse om Akilles och Sköldpaddan eller Om giltigheten hos deduktiva slutledningar. In: Kungl. Vitterhets Historie och Antikvitets Akademiens årsbok: (pp. 47-62). Stockholm: Kungl. Vitterhets Historie och Antikvitets Akademien
Open this publication in new window or tab >>Lewis Carrolls berättelse om Akilles och Sköldpaddan eller Om giltigheten hos deduktiva slutledningar
2016 (Swedish)In: Kungl. Vitterhets Historie och Antikvitets Akademiens årsbok, Stockholm: Kungl. Vitterhets Historie och Antikvitets Akademien, 2016, p. 47-62Chapter in book (Other academic)
Place, publisher, year, edition, pages
Stockholm: Kungl. Vitterhets Historie och Antikvitets Akademien, 2016
Series
Kungl. Vitterhets Historie och Antikvitets Akademiens årsbok, ISSN 0083-6796 ; 2016
National Category
Philosophy
Research subject
Theoretical Philosophy
Identifiers
urn:nbn:se:su:diva-145470 (URN)978-91-7402-452-4 (ISBN)
Available from: 2017-08-04 Created: 2017-08-04 Last updated: 2022-02-28Bibliographically approved
Prawitz, D. (2016). On the relation between Heyting's and Gentzen's approaches to meaning. In: Thomas Piecha, Peter Schroeder-Heister (Ed.), Advances in proof-theoretic semantics: (pp. 5-25). Cham: Springer
Open this publication in new window or tab >>On the relation between Heyting's and Gentzen's approaches to meaning
2016 (English)In: Advances in proof-theoretic semantics / [ed] Thomas Piecha, Peter Schroeder-Heister, Cham: Springer, 2016, p. 5-25Chapter in book (Refereed)
Abstract [en]

Proof-theoretic semantics explains meaning in terms of proofs. Two different concepts of proof are in question here. One has its roots in Heyting’s explanation of a mathematical proposition as the expression of the intention of a construction, and the other in Gentzen’s ideas about how the rules of Natural Deduction are justified in terms of the meaning of sentences. These two approaches to meaning give rise to two different concepts of proof, which have been developed much further, but the relation between them, the topic of this paper, has not been much studied so far. The recursive definition of proof given by the so-called BHK-interpretation is here used as an explication of Heyting’s idea. Gentzen’s approach has been developed as ideas about what it is that makes a piece of reasoning valid. It has resulted in a notion of valid argument, of which there are different variants. The differences turn out to be crucial when comparing valid arguments and BHK-proofs. It will be seen that for one variant, the existence of a valid argument can be proved to be extensionally equivalent to the existence of a BHK-proof, while for other variants, attempts at similar proofs break down at different points.

Place, publisher, year, edition, pages
Cham: Springer, 2016
Series
Trends in logic, ISSN 1572-6126 ; 43
Keywords
Proof, Valid argument, Meaning, Semantics, Heyting, Gentzen
National Category
Philosophy
Research subject
Theoretical Philosophy
Identifiers
urn:nbn:se:su:diva-125726 (URN)10.1007/978-3-319-22686-6_2 (DOI)978-3-319-22685-9 (ISBN)
Available from: 2016-01-17 Created: 2016-01-17 Last updated: 2022-02-23Bibliographically approved
Prawitz, D. (2015). A Note on How to Extend Gentzen’s Second Consistency Proof to a Proof of Normalization for First Order Arithmetic. In: Reinhard Kahle, Michael Rathjen (Ed.), Gentzen's Centenary: The Quest for Consistency (pp. 131-176). Springer
Open this publication in new window or tab >>A Note on How to Extend Gentzen’s Second Consistency Proof to a Proof of Normalization for First Order Arithmetic
2015 (English)In: Gentzen's Centenary: The Quest for Consistency / [ed] Reinhard Kahle, Michael Rathjen, Springer, 2015, p. 131-176Chapter in book (Refereed)
Abstract [en]

The purpose of this note is to show that the normalization theorem can be proved for first order Peano arithmetic by adapting to natural deduction the method used in Gentzen’s second consistency proof. Gentzen explained the intuitive idea behind his proof by informally arguing for the possibility of a normalization theorem of natural deduction, but what he actually proved was a special case of the Hauptsatz for a sequent calculus formalization of arithmetic. To transfer Gentzen’s method to natural deduction, I shall assign his ordinals to notations for natural deductions that use an explicit operation of substitution. The idea is first worked out for predicate logic. The main problems reside there and consist in finding a normalization strategy that harmonizes with the ordinal assignment. The result for predicate logic is then extended to arithmetic without effort, and thereby full normalization of natural deductions in first order arithmetic is achieved.

Place, publisher, year, edition, pages
Springer, 2015
National Category
Philosophy
Research subject
Theoretical Philosophy
Identifiers
urn:nbn:se:su:diva-125728 (URN)10.1007/978-3-319-10103-3_6 (DOI)978-3-319-10102-6 (ISBN)978-3-319-10103-3 (ISBN)
Available from: 2016-01-17 Created: 2016-01-17 Last updated: 2022-02-23Bibliographically approved
Prawitz, D. (2015). A Short Scientific Autobiography. In: Heinrich Wansing (Ed.), Dag Prawitz on Proofs and Meaning: . Paper presented at Dag Prawitz on Proofs and Meaning Workshop, Bochum, Germany, September 10-11, 2012 (pp. 33-64). Springer
Open this publication in new window or tab >>A Short Scientific Autobiography
2015 (English)In: Dag Prawitz on Proofs and Meaning / [ed] Heinrich Wansing, Springer, 2015, p. 33-64Conference paper, Published paper (Refereed)
Abstract [en]

Being born in 1936 in Stockholm, I have memories from the time of the Second World War. But Sweden was not involved, and my childhood was peaceful. One notable effect of the war was that even in the centre of Stockholm, where I grew up, there was very little automobile traffic. Goods were often transported by horse-drawn wagons. At the age of six we children could play in the streets and run to the nearby parks without the company of any adults.

Place, publisher, year, edition, pages
Springer, 2015
Series
Outstanding Contributions to Logic, ISSN 2211-2758, E-ISSN 2211-2766 ; 7
National Category
Philosophy
Research subject
Theoretical Philosophy
Identifiers
urn:nbn:se:su:diva-125732 (URN)10.1007/978-3-319-11041-7_2 (DOI)000357742900002 ()978-3-319-11040-0 (ISBN)978-3-319-11041-7 (ISBN)
Conference
Dag Prawitz on Proofs and Meaning Workshop, Bochum, Germany, September 10-11, 2012
Available from: 2016-01-17 Created: 2016-01-17 Last updated: 2022-02-23Bibliographically approved
Prawitz, D. (2015). Classical versus intuitionistic logic. In: Edward Hermann Haeusler, Wagner de Campos Sanz, Bruno Lopes (Ed.), Why is this a Proof?: Festschrift for Luiz Carlos Pereira (pp. 15-32). College Publications
Open this publication in new window or tab >>Classical versus intuitionistic logic
2015 (English)In: Why is this a Proof?: Festschrift for Luiz Carlos Pereira / [ed] Edward Hermann Haeusler, Wagner de Campos Sanz, Bruno Lopes, College Publications, 2015, p. 15-32Chapter in book (Refereed)
Place, publisher, year, edition, pages
College Publications, 2015
Series
Tributes ; 27
National Category
Philosophy
Research subject
Theoretical Philosophy
Identifiers
urn:nbn:se:su:diva-129288 (URN)9781848901728 (ISBN)
Available from: 2016-04-20 Created: 2016-04-20 Last updated: 2022-02-23Bibliographically approved
Prawitz, D. (2015). Explaining Deductive Inference. In: Heinrich Wansing (Ed.), Dag Prawitz on Proofs and Meaning : (pp. 65-100). Cham: Springer
Open this publication in new window or tab >>Explaining Deductive Inference
2015 (English)In: Dag Prawitz on Proofs and Meaning / [ed] Heinrich Wansing, Cham: Springer, 2015, p. 65-100Chapter in book (Refereed)
Abstract [en]

We naturally take for granted that by performing inferences we can obtain evidence or grounds for assertions that we make. But logic should explain how this comes about. Why do some inferences give us grounds for their conclusions? Not all inferences have that power. My first aim here is to draw attention to this fundamental but quite neglected question. It seems not to be easily answered without reconsidering or reconstructing the main concepts involved, that is, the concepts of ground and inference. Secondly, I suggest such a reconstruction, the main idea of which is that to make an inference is not only to assert a conclusion claiming that it is supported by a number of premisses, but is also to operate on the grounds that one assumes or takes oneself to have for the premisses. An inference is thus individuated not only by its premisses and conclusion but also by a particular operation. A valid inference can then be defined as one where the involved operation results in a ground for the conclusion when applied to grounds for the premisses. It then becomes a conceptual truth that a valid inference does give a ground for the conclusion provided that one has grounds for the premisses.

Place, publisher, year, edition, pages
Cham: Springer, 2015
Series
Outstanding Contributions to Logic, ISSN 2211-2758 ; 7
Keywords
Inference, Deduction, Proof, Ground, Meaning, Logical validity Inferentialism, Intuitionism, Proof-theoretic semantics
National Category
Philosophy
Research subject
Theoretical Philosophy
Identifiers
urn:nbn:se:su:diva-125733 (URN)10.1007/978-3-319-11041-7_3 (DOI)000357742900003 ()978-3-319-11040-0 (ISBN)978-3-319-11041-7 (ISBN)
Available from: 2016-01-17 Created: 2016-01-17 Last updated: 2022-02-23Bibliographically approved
Prawitz, D. (2014). An Approach to General Proof Theory and a Conjecture of a Kind of Completeness of Intuitionistic Logic Revisited. In: Luiz Carlos Pereira, Edward Hermann Haeusler, Valeria de Paiva (Ed.), Advances in Natural Deduction: A celebration of Dag Prawitz’s work (pp. 269-279). Dordrecht: Springer Netherlands
Open this publication in new window or tab >>An Approach to General Proof Theory and a Conjecture of a Kind of Completeness of Intuitionistic Logic Revisited
2014 (English)In: Advances in Natural Deduction: A celebration of Dag Prawitz’s work / [ed] Luiz Carlos Pereira, Edward Hermann Haeusler, Valeria de Paiva, Dordrecht: Springer Netherlands, 2014, p. 269-279Chapter in book (Refereed)
Abstract [en]

Thirty years ago I formulated a conjecture about a kind of completeness of intuitionistic logic. The framework in which the conjecture was formulated had the form of a semantic approach to a general proof theory (presented at the 4th World Congress of Logic, Methodology and Philosophy of Science at Bucharest 1971 [6]). In the present chapter, I shall reconsider this 30-year old conjecture, which still remains unsettled, but which I continue to think of as a plausible and important supposition. Reconsidering the conjecture, I shall also reconsider and revise the semantic approach in which the conjecture was formulated.

Place, publisher, year, edition, pages
Dordrecht: Springer Netherlands, 2014
Series
Trends in Logic, ISSN 1572-6126 ; 39
National Category
Philosophy
Research subject
Theoretical Philosophy
Identifiers
urn:nbn:se:su:diva-112077 (URN)10.1007/978-94-007-7548-0_12 (DOI)000355750700013 ()978-94-007-7547-3 (ISBN)978-94-007-7548-0 (ISBN)
Available from: 2015-01-09 Created: 2015-01-09 Last updated: 2022-02-23Bibliographically approved
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