Dag Prawitz on Proofs and Meaning (Outstanding Contributions to Logic Book 7)

This volume is dedicated to Prof. Dag Prawitz and his outstanding contributions to philosophical and mathematical logic. Prawitz's eminent contributions to structural proof theory, or general proof theory, as he calls it, and inference-based meaning theories have been extremely influential in the development of modern proof theory and anti-realistic semantics. In particular, Prawitz is the main author on natural deduction in addition to Gerhard Gentzen, who defined natural deduction in his PhD thesis published in 1934. The book opens with an introductory paper that surveys Prawitz's numerous contributions to proof theory and proof-theoretic semantics and puts his work into a somewhat broader perspective, both historically and systematically. Chapters include either in-depth studies of certain aspects of Dag Prawitz's work or address open research problems that are concerned with core issues in structural proof theory and range from philosophical essays to papers of a mathematical nature. Investigations into the necessity of thought and the theory of grounds and computational justifications as well as an examination of Prawitz's conception of the validity of inferences in the light of three “dogmas of proof-theoretic semantics” are included. More formal papers deal with the constructive behaviour of fragments of classical logic and fragments of the modal logic S4 among other topics. In addition, there are chapters about inversion principles, normalization of p roofs, and the notion of proof-theoretic harmony and other areas of a more mathematical persuasion. Dag Prawitz also writes a chapter in which he explains his current views on the epistemic dimension of proofs and addresses the question why some inferences succeed in conferring evidence on their conclusions when applied to premises for which one already possesses evidence. This Volume Is Dedicated To Prof. Dag Prawitz And His Outstanding Contributions To Philosophical And Mathematical Logic. Prawitz's Eminent Contributions To Structural Proof Theory, Or General Proof Theory, As He Calls It, And Inference-based Meaning Theories Have Been Extremely Influential In The Development Of Modern Proof Theory And Anti-realistic Semantics. In Particular, Prawitz Is The Main Author On Natural Deduction In Addition To Gerhard Gentzen, Who Defined Natural Deduction In His Phd Thesis Published In 1934. The Book Opens With An Introductory Paper That Surveys Prawitz's Numerous Contributions To Proof Theory And Proof-theoretic Semantics And Puts His Work Into A Somewhat Broader Perspective, Both Historically And Systematically. Chapters Include Either In-depth Studies Of Certain Aspects Of Dag Prawitz's Work Or Address Open Research Problems That Are Concerned With Core Issues In Structural Proof Theory And Range From Philosophical Essays To Papers Of A Mathematical Nature. Investigations Into The Necessity Of Thought And The Theory Of Grounds And Computational Justifications As Well As An Examination Of Prawitz's Conception Of The Validity Of Inferences In The Light Of Three “dogmas Of Proof-theoretic Semantics” Are Included. More Formal Papers Deal With The Constructive Behaviour Of Fragments Of Classical Logic And Fragments Of The Modal Logic S4 Among Other Topics. In Addition, There Are Chapters About Inversion Principles, Normalization Of Proofs, And The Notion Of Proof-theoretic Harmony And Other Areas Of A More Mathematical Persuasion. Dag Prawitz Also Writes A Chapter In Which He Explains His Current Views On The Epistemic Dimension Of Proofs And Addresses The Question Why Some Inferences Succeed In Conferring Evidence On Their Conclusions When Applied To Premises For Which One Already Possesses Evidence. Prawitz, Proofs, And Meaning; Wansing, Heinrich -- A Short Scientific Autobiography; Prawitz, Dag -- Explaining Deductive Inference; Prawitz, Dag -- Necessity Of Thought; Cozzo, Cesare -- On The Motives For Proof Theory; Detlefsen, Michael -- Inferential Semantics; Došen, Kosta -- Cut Elimination, Substitution And Normalization; Dyckhoff, Roy -- Inversion Principles And Introduction Rules; Milne, Peter -- Intuitionistic Existential Instantiation And Epsilon Symbol; Mints, Grigori -- Meaning In Use; Negri, Sara And Von Plato, Jan -- Fusing Quantifiers And Connectives: Is Intuitionistic Logic Different?; Pagin, Peter -- On Constructive Fragments Of Classical Logic; Pereira; Luiz Carlos And Haeusler, Edward Hermann -- General-elimination Harmony And Higher-level Rules; Read, Stephen -- Hypothesis-discharging Rules In Atomic Bases; Sandqvist, Tor -- Harmony In Proof-theoretic Semantics: A Reductive Analysis; Schroeder-heister, Peter -- First-order Logic Without Bound Variables: Compositional Semantics; Tait, William W -- On Gentzen’s Structural Completeness Proof; Tennant, Neil -- A Notion Of C-justification For Empirical Statements; Usberti, Gabriele. Edited By Heinrich Wansing. This volume is dedicated to Prof. Dag Prawitz and his outstanding contributions to philosophical and mathematical logic. Prawitz's eminent contributions to structural proof theory, or general proof theory, as he calls it, and inference-based meaning theories have been extremely influential in the development of modern proof theory and anti-realistic semantics. In particular, Prawitz is the main author on natural deduction in addition to Gerhard Gentzen, who defined natural deduction in his PhD thesis published in 1934. The book opens with an introductory paper that surveys Prawitz's numerous contributions to proof theory and proof-theoretic semantics and puts his work into a somewhat broader perspective, both historically and systematically. Chapters include either in-depth studies of certain aspects of Dag Prawitz's work or address open research problems that are concerned with core issues in structural proof theory and range from philosophical essays to papers of a mathematical nature. Investigations into the necessity of thought and the theory of grounds and computational justifications as well as an examination of Prawitz's conception of the validity of inferences in the light of three ĺldogmas of proof-theoretic semanticsĺl are included. More formal papers deal with the constructive behaviour of fragments of classical logic and fragments of the modal logic S4 among other topics. In addition, there are chapters about inversion principles, normalization of proofs, and the notion of proof-theoretic harmony and other areas of a more mathematical persuasion. Dag Prawitz also writes a chapter in which he explains his current views on the epistemic dimension of proofs and addresses the question why some inferences succeed in conferring evidence on their conclusions when applied to premises for which one already possesses evidence Front Matter....Pages i-xiii Prawitz, Proofs, and Meaning....Pages 1-32 A Short Scientific Autobiography....Pages 33-64 Explaining Deductive Inference....Pages 65-100 Necessity of Thought....Pages 101-120 On the Motives for Proof Theory....Pages 121-145 Inferential Semantics....Pages 147-162 Cut Elimination, Substitution and Normalisation....Pages 163-187 Inversion Principles and Introduction Rules....Pages 189-224 Intuitionistic Existential Instantiation and Epsilon Symbol....Pages 225-238 Meaning in Use....Pages 239-257 Fusing Quantifiers and Connectives: Is Intuitionistic Logic Different?....Pages 259-280 On Constructive Fragments of Classical Logic....Pages 281-292 General-Elimination Harmony and Higher-Level Rules....Pages 293-312 Hypothesis-Discharging Rules in Atomic Bases....Pages 313-328 Harmony in Proof-Theoretic Semantics: A Reductive Analysis....Pages 329-358 First-Order Logic Without Bound Variables: Compositional Semantics....Pages 359-384 On Gentzen’s Structural Completeness Proof....Pages 385-414 A Notion of C -Justification for Empirical Statements....Pages 415-450 Dag Prawitz’s Published Books and Papers, by Year (Selected)....Pages 451-458 6.2 Propositions, Names and Inferences6.3 Two Dogmas of Semantics; 6.4 Concerning the First Dogma; 6.5 Concerning the Second Dogma; 6.6 A Third Dogma of Semantics; 6.7 Inferences as Operations on Grounds; References; 7 Cut Elimination, Substitution and Normalisation; 7.1 Introduction; 7.2 Technical Background; 7.2.1 Natural Deduction, in Logistic Style; 7.2.2 Reductions of Lambda Terms; 7.2.3 Sequent Calculus; 7.3 Translation from Sequent Calculus into Natural Deduction; 7.4 Translation from Natural Deduction to Sequent Calculus; 7.5 Substitution; 7.6 Cut Reduction; 7.7 Completeness 3.2 A Fundamental Problem3.2.1 The Form of an Explanation; 3.2.2 Can Legitimacy be Explained in Terms of Validity?; 3.2.3 The Intertwining of Evidence and Proofs; 3.3 Meaning and Evidence; 3.4 Three Proposals for How Linguistic Meaning Can be Explained in Epistemic Terms; 3.4.1 Radical Inferentialism; 3.4.2 Proof-Theoretic Semantics; 3.4.3 Intuitionistic Constructions; 3.5 Evidence and Grounds; 3.5.1 Evidence Represented in the Form of Grounds; 3.5.2 The Language of Grounds; 3.5.3 Operations on Grounds; 3.6 Deductively Valid Inferences; 3.7 Concluding Remarks; References 4 Necessity of Thought4.1 Inference and Consequence; 4.2 The Fundamental Task; 4.3 Grounds; 4.4 Recapitulation; 4.5 Mistakes; 4.6 Epistemic Contexts; References; 5 On the Motives for Proof Theory; 5.1 Introduction; 5.1.1 Different Conceptions of Axiomatization; 5.1.2 Different Standards of Justification; 5.2 Background; 5.2.1 Conservative Views; 5.2.2 More Liberal Views; 5.3 The RRT; 5.4 Frege's Futility Argument; 5.5 Hilbert's `Direct' Approach to Consistency; 5.6 Freedom and the Direct Approach; 5.7 Conclusion; References; 6 Inferential Semantics; 6.1 Introduction 2.4 Towards the Doctoral Thesis--Normalizations of Natural Deductions2.5 Docent at Stockholm and Lund--Visiting Professor in US; 2.6 Professor of Philosophy at Oslo University--Fellow at Oxford; 2.7 Back to Stockholm--Chairman of the Department; 2.8 New Tasks; 2.9 Philosophy in Spare Time; 2.10 Retirement; 3 Explaining Deductive Inference; 3.1 The Phenomenon of Deductive Inference; 3.1.1 A First Characterization of Inferences--Inferential Transitions; 3.1.2 Inference Acts, Inference Figures, and Inference Schemata; 3.1.3 The Aim of Inference; 3.1.4 Deductive Proofs Preface; Contents; About the Contributors; 1 Prawitz, Proofs, and Meaning; 1.1 Introduction; 1.2 General Proof Theory; 1.3 Natural Deduction; 1.4 The Relation Between Proofs and Meaning; 1.4.1 Natural Deduction Rules and the Meaning of the Logical Operations; 1.4.2 Proof-Theoretic Validity Concepts; 1.4.3 Proof-Theoretic Notions of Functional Completeness; 1.4.4 Some Other Philosophical Aspects; 1.5 Conclusion; References; 2 A Short Scientific Autobiography; 2.1 Childhood and School; 2.2 Undergraduate Studies; 2.3 Mechanical Proof Procedures