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  • 1.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    A Note on How to Extend Gentzen’s Second Consistency Proof to a Proof of Normalization for First Order Arithmetic2015In: 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.

  • 2.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    A Short Scientific Autobiography2015In: Dag Prawitz on Proofs and Meaning / [ed] Heinrich Wansing, Springer, 2015, p. 33-64Conference 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.

  • 3.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    An Approach to General Proof Theory and a Conjecture of a Kind of Completeness of Intuitionistic Logic Revisited2014In: 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.

  • 4.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    Assertions in the Context of Inference2010In: Judgements and Propositions : Logical, Linguistic, and Cognitive Issues / [ed] Sebastian Bab, Klaus Robering, Berlin: Logos Verlag Berlin, 2010, p. 89-98Chapter in book (Refereed)
  • 5.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    Classical versus intuitionistic logic2015In: 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)
  • 6.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    Explaining Deductive Inference2015In: 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.

  • 7.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    Inference and knowledge2009In: The logica yearbook 2008 / [ed] Michal Pelis, London: College Publications, 2009, p. 175-192Chapter in book (Other academic)
  • 8.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    Lewis Carrolls berättelse om Akilles och Sköldpaddan eller Om giltigheten hos deduktiva slutledningar2016In: Kungl. Vitterhets Historie och Antikvitets Akademiens årsbok, Stockholm: Kungl. Vitterhets Historie och Antikvitets Akademien, 2016, p. 47-62Chapter in book (Other academic)
  • 9.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    Logical Determinism and the Principle of Bivalence2009In: Philosophical Probings: Essays on von Wright's Later Work / [ed] Frederick Stoutland, København: Automatic Press , 2009, p. 111-135Chapter in book (Other academic)
  • 10.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    Natural deduction: a proof-theoretical study2006 (ed. Dover ed.)Book (Refereed)
  • 11.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    Natural deduction: a proof-theoretical study1965Doctoral thesis, monograph (Other academic)
  • 12.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    On the relation between Heyting's and Gentzen's approaches to meaning2016In: 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.

  • 13.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    Pragmatist and Verifactionist Theories of Meaning2007In: The Philosophy of Michael Dummett: The Library of Living Philosophers vol XXXI, Open Court, Chicago , 2007, p. 455-481Chapter in book (Other academic)
  • 14.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    Proofs and Perfect Syllogisms2011In: Logic and Knowledge / [ed] Cellucci C. - Grozhols E. - Ippoliti E., Newcastle Upon Tyne: Cambridge Scholars Publishing, 2011, 1, p. 385-402Chapter in book (Refereed)
  • 15.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    Proofs Verifying Programs and Programs Producing Proofs: A Conceptual Analysis2008In: Deduction Computation Experiment: Exploring the Effectiveness of Proof, Springer-Verlag Italia, Milano , 2008, p. 81-94Chapter in book (Refereed)
  • 16.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    The epistemic significance of valid inference2012In: Synthese, ISSN 0039-7857, E-ISSN 1573-0964, Vol. 187, no 3, p. 887-898Article in journal (Refereed)
    Abstract [en]

    The traditional picture of logic takes it for granted that valid arguments have a fundamental epistemic significance, but neither model theory nor traditional proof theory dealing with formal system has been able to give an account of this significance. Since valid arguments as usually understood do not in general have any epistemic significance, the problem is to explain how and why we can nevertheless use them sometimes to acquire knowledge. It is suggested that we should distinguish between arguments and acts of inferences and that we have to reconsider the latter notion to arrive at the desired explanation. More precisely, the notions should be developed so that the following relationship holds: one gets in possession of a ground for a conclusion by inferring it from premisses for which one already has grounds, provided that the inference in question is valid. The paper proposes explications of the concepts of ground and deductively valid inference so that this relationship holds as a conceptual truth. Logical validity of inference is seen as a special case of deductive validity, but does not add anything as far as epistemic significance is concerned-it resides already in the deductively valid inferences.

  • 17.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    The Fundamental Problem of General Proof Theory2019In: 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)
    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.

  • 18.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    The Seeming Interdependence Between the Concepts of Valid Inference and Proof2019In: Topoi: An International Review of Philosophy, ISSN 0167-7411, E-ISSN 1572-8749, Vol. 38, no 3, p. 493-503Article in journal (Refereed)
    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.

  • 19.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    The status of mathematical knowledge2014In: From a Heuristic Point of View: Essays in Honour of Carlo Cellucci / [ed] Emiliano Ippoliti, Cesare Cozzo, Newcastle upon Tyne: Cambridge Scholars Publishing, 2014, p. 73-90Chapter in book (Other academic)
  • 20.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    To explain deduction2018In: 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. 

  • 21.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    Truth and Proof in Intuitionism2012In: Epistemology versus Ontology: Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf / [ed] Peter Dubjer, Sten Lindström, Erik Palmgren, Göran Sundholm, Dordrecht: Springer, 2012, p. 45-67Chapter in book (Refereed)
    Abstract [en]

    Is logic in its essence epistemological or ontological? This question was the starting point of Per Martin-Löf’s lecture at the conference at which the contributions to this volume were presented. In this essay I shall limit myself to the more specific question whether the concepts of truth and proof are epistemological or ontological. That proof is an epistemic concept is of course normally not in doubt, whereas opinions differ concerning truth.

  • 22.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    Truth as an epistemic notion2012In: Topoi: An International Review of Philosophy, ISSN 0167-7411, E-ISSN 1572-8749, Vol. 31, no 1, p. 9-16Article in journal (Refereed)
    Abstract [en]

    What is the appropriate notion of truth for sentences whose meanings are understood in epistemic terms such as proof or ground for an assertion? It seems that the truth of such sentences has to be identified with the existence of proofs or grounds, and the main issue is whether this existence is to be understood in a temporal sense as meaning that we have actually found a proof or a ground, or if it could be taken in an abstract, tenseless sense. Would the latter alternative amount to realism with respect to proofs or grounds in a way that would be contrary to the supposedly anti-realistic standpoint underlying the epistemic understanding of linguistic expressions? Before discussing this question, I shall consider reasons for construing linguistic meaning epistemically and relations between such reasons and reasons for taking an anti-realist point of view towards the discourse in question.

  • 23.
    Prawitz, Dag
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    Validity of Inferences2013In: Reference, Rationality, and Phenomenology: Themes from Føllesdal / [ed] Michael Frauchiger, Frankfurt: Ontos Verlag, 2013, p. 179-204Chapter in book (Refereed)
    Abstract [en]

    Deductive inference gives us conclusive grounds for beliefs and assertions, and may even compel us, as is generally acknowledged since the time of Plato and Aristotle. But there is no generally accepted account of what is to be required of an inference in order that it is to have this justificatory and compelling power. That the inference is valid is clearly not a sufficient condition, if validity of an inference is defined as truth preservaton for all variations of the content of the non-logical constants involved. It has been suggested that the agent has to be required to know the inference to be valid, but it may be argued along the lines of Lewis Carroll's regress that neither is this a sufficient condition. In any case it is too stringent, since justification would never get off the ground if we first had to show that the inferences used in a justification are valid. The paper suggests that we have to rethink the notion of inference to account for how we acquire knowledge by making inferences. An inference is first of all an act in which an agent operates on given grounds for the premisses in order to get a ground for the conclusion. It can be defined as valid if the operation does yield a ground for the conclusion when applied to grounds for the premisses. Getting in possession of a ground for the conclusion by performing the act, the agent becomes justified in holding the conclusion true, and if aware of this possession, she even gets compelled to do so.

1 - 23 of 23
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