Anssi Korhonen Logic as the Universal Science: Bertrand Russell’s Early Conception of Logic and Its Philosophical Context Philosophical Studies from the University of Helsinki 18 Filosofisia tutkimuksia Helsingin yliopistosta Filosofiska studier från Helsingfors universitet Philosophical Studies from the University of Helsinki Publishers: Department of Philosophy Department of Social and Moral Philosophy P.O. Box 9 (Siltavuorenpenger 20 A) 00014 University of Helsinki Finland Editors: Marjaana Kopperi Panu Raatikainen Petri Ylikoski Bernt Österman Anssi Korhonen Logic as the Universal Science: Bertrand Russell’s Early Conception of Logic and Its Philosophical Context ISBN 978-952-10-4406-9 (paperback) ISBN 978-952-10-4407-6 (pdf, http://ethesis.helsinki.fi) ISSN 1458-8331 Vantaa 2007 Dark Oy Acknowledgements I owe very special thanks to Professor Gabriel Sandu for his continuous and very concrete support of this project, which has been rather slow in unfolding. I wish to thank my supervisor, Professor Leila Haaparanta, who introduced me to the serious study of the history of modern logic and analytic philosophy. Professor Nicholas Griffin, Professor André Maury and Dr. Patrick Sibelius provided invaluable comments on earlier drafts of the manuscript. I owe special thanks to the wonderful, and wonderfully heterogeneous, collective that is constituted by the members of the Russell-l discussion forum. I also wish to extend my thanks to Dr. Panu Raatikainen, Mr. Simo Rinkinen and Mr. Max Weiss for discussions, comments and concrete advice. I owe very special thanks to my dear friends and colleagues, Dr. Markku Keinänen and Mr. Pekka Mäkelä, for co-operation, support and innumerable other things. I am indebted to my colleagues and room- mates, Dr. Pauliina Remes and Mr. Fredrik Westerlund, for providing a perfect atmosphere in which to fight the last battle against the recalcitrant manuscript. The Head of our Department, Dr. Thomas Wallgren, provided very concrete help during the final stage of this study. In preparing the manuscript for print, I have received excellent editorial help from Mrs. Auli Kaipainen. Last, but in many ways first, I wish to thank Niina for her unfailing support, encouragement, sympathy and patience. The financial support provided for this work by the Finnish Cultural Foundation, Emil Aaltosen Säätiö, The Finnish Academy and University of Helsinki is gratefully acknowledged. I dedicate this work to the memory of my parents. Helsinki, November 2007 Anssi Korhonen Acknowledgements 5 Contents 7 Introduction 15 1 Preliminary Remarks on Russell’s Early Logicism 27 1.0 Introduction 27 1.1 Different Logicisms 30 1.2 Analyticity and Syntheticity 32 1.2.1 Preliminary Remarks 32 1.2.2 Kantian Analyticity 33 1.2.3 Logical Empiricism and Analyticity 35 1.2.4 Analyticity in Frege and Russell 36 1.3 The Pursuit of Rigour 42 1.3.1 The Mathematical Context 42 1.3.2 Why Rigour? 45 1.3.3 Epistemic Logicism 48 1.3.4 What is Really Involved in Rigour 59 1.4 Conclusions: Russell and Kant 63 2 Kant on Formal-logical and Mathematical Cognition 71 2.0 Introduction 71 2.1 Kant’s Programme for the Philosophy of Mathematics 73 2.1.1 Preliminary Remarks 73 2.1.2 The Leibnizian Background 74 2.1.3 Kant on Analytic and Synthetic Judgments 78 2.1.4 The Reasons behind Kant’s Innovations 80 2.1.4.1 Concepts and Constructions 80 2.1.4.2 The Containment Model for Concepts 83 2.1.4.3 A Comparison with Frege 86 8 Contents 2.1.4.4 Kant on Philosophical and Mathematical Method 88 2.1.4.5 Summary 90 2.2 Constructibility and Transcendental Aesthetic 92 2.3 The Semantics of Geometry Explained 94 2.3.1 Constructions in Geometry 94 2.3.2 Geometry and Space 101 2.4 Constructions in Arithmetic 105 2.5 The Applicability of Mathematics 108 2.6 Pure and Applied Mathematics 109 2.7 The Apriority of Mathematics According to Kant 112 2.8 Conclusions 117 3 Russell on Kant 121 3.0 Introduction 121 3.1 Russell on the Nature of the Mathematical Method 124 3.1.1 “The Most Important Year in My Intellectual Life” 124 3.1.2 Russell and Leibniz 125 3.1.3 Russell and Peano 127 3.1.4 The Concept of Deductive Rigour 132 3.1.4.1 General Remarks 132 3.1.4.2 Pasch on Rigorous Reasoning 135 3.1.4.3 The Logicization of MathematicalProof 143 3.1.5 Russell on Rigorous Reasoning 145 3.1.5.1 Self-Evidence and Rigour 145 3.1.5.2 Different Sources of Self-Evidence 147 3.1.5.3 Logical Self-Evidence 150 3.1.5.4 Poincaré on Intuition and Self-Evidence 149 Contents 9 3.2 Kant and Misplaced Rigorization 156 3.2.1 Russell and Kant on Mathematical Reasoning 156 3.2.2 Some Remarks on the Standard View 160 3.3 Russell’s Criticisms of Kant 166 3.3.1 Russell on Intuitions 166 3.3.2 Russell’s Kantian Background 169 3.3.3 Quantity in the Principles 173 3.3.4 Propositional Functions in the Principles 177 3.3.5 Against Russell: the Notion of Intuition Again 180 3.4 Summary 186 3.5. The Role of Logicism 189 3.5.1 Hylton on the Role of Logicism 189 3.5.2 Criticism of Hylton’s Reconstruction 193 3.6 Russell’s Case against Kant 203 3.6.1 The Standard Picture of TranscendentalIdealism 203 3.6.2 The Implications of the Standard Picture 207 3.6.2.1 Kant’s “Subjectivism” 207 3.6.2.2 Moore against Kant 208 3.6.3 The Relativized Model of the Apriori 212 3.6.3.1 Preliminary Remarks 212 3.6.3.2 Three Direct Arguments Againstthe R-Model 213 3.6.3.3 Three Indirect Arguments againstthe R-Model 222 3.6.3.3.1 The Consequences of the R-Model 222 3.6.3.3.2The Argument from Necessity 224 3.6.3.3.3 Another Argument from Necessity 258 3.6.3.3.4The Argument from Truthand the Argumentfrom Universality 259 10 Contents 4 Logic as the Universal Science I: the van Heijenoort Interpretation and Russell’s Conception of Logic 265 4.0 Introduction 265 4.1 “Logic as Calculus and Logic as Language” 266 4.2 van Heijenoort’s Distinction 271 4.2.1 The Technical Core of the Model-Theoretic Conception 271 4.2.2Two Conceptions of Generality 276 4.2.3 The Technical Core of the Universalist Conception 280 4.3 The Philosophical Implications of the Universalist Conception 283 4.3.1 “No Metaperspective” 283 4.3.2 Two Senses of “Interpretation” 287 4.3.3A Flowchart 292 4.4 Russell’s Notion of Proposition 299 4.4.1 Preliminary Remarks 299 4.4.2 Moore’s Theory of Judgment 302 4.4.3 Predication 309 4.4.3.1 Moore on Predication 309 4.4.3.2 Russell’s Criticisms of Moore 311 4.4.4 Moorean and Peanist Elements in Russell’sTheory of Propositions 314 4.4.5 The Notion of Term 315 4.4.6 The Problem of Unity 318 4.4.6.1 A Fregean Perspective on Predication 318 4.4.6.2 Comparing Frege and Russellon Predication 321 4.4.6.3 Problems with Russell’s Account of the Problem of Unity 325 4.4.6.4 Propositions as Facts 327 4.4.6.5 A Way Out for Russell? 330 4.4.7 Russell’s Notion of Assertion 333 4.4.8 Peano’s Logic 338 Contents 11 4.4.8.1 Why Peano is Superior to Moore 338 4.4.8.2 Predication and Propositional Functions 341 4.4.9 The Theory of Denoting 345 4.4.9.1 Introducing Denoting Concepts 345 4.4.9.2 Why Denoting Concepts are Needed 348 4.4.9.3 Denoting Concepts and PropositionalFunctions 352 4.4.10 Russell’s Analysis of Generality 355 4.4.10.1 General Remarks 355 4.4.10.2 Formal Implications 357 4.4.10.3 The Propositions of Logic Again 359 4.4.11 Comparing Russell and Frege on the Constitution of Propositions 363 4.4.12 Russell’s Account of Variables 369 4.4.13 Conclusions 374 4.5 Russell’s Version of the Universalist Conception of Logic 375 4.5.1 Preliminary Remarks 375 4.5.2 Russell’s Alleged Anti-Semanticism 380 4.5.2.1 The Fixed Content Argumentand the Argumentfor Uniqueness 380 4.5.2.2 What is wrong with the UniquenessArgument 384 4.5.2.2.1 Did Russell Have a “Calculusof Logic”? 387 4.5.2.2.2 Further Remarks on Russell’s Conception of Calculus 391 4.5.2.2.3Reasoning about Reasoning 392 4.5.2.2.4 The Justification of Logic 395 4.5.2.3 What is wrong with the FixedContent Argument 400 4.5.2.3.1Justification and 12 Contents SemanticExplanation 400 4.5.2.3.2Frege on the Semantic Justificationof Logic 403 4.5.2.3.3Calculus for Logic and the Science of Logic 405 4.5.2.3.4Russellian Metatheory? 407 4.5.3 The True Source of Russell’s Anti-Semanticism 414 4.5.4Russell on Alternative Interpretations 417 4.5.5 “Interpretation” and Semantics 427 4.5.6 Generality and Quantification 433 4.5.6.1 Unrestricted Generality 433 4.5.6.2 Hylton on Russell on Generality 435 4.5.6.3 Criticism of Hylton’s Reading 437 4.5.7 Russell’s Concept of Truth 441 4.5.7.1 Preliminary Remarks 441 4.5.7.2 An Analogy with Frege 442 4.5.7.3 Frege’s Version of the Argumentagainst Truthdefinitions 448 4.5.7.4 The Metaphysics of Truth 450 4.5.7.5 Truth-Primitivism and Truth-Attributions 453 4.5.7.6 Use of the Truth-Predicate 455 4.6 Russell’s Conception of Mathematical Theories 459 4.6.1 General Remarks 459 4.6.2 The Frege-Hilbert Controversy 460 4.6.3The Russell-Poincaré Controversy 465 4.6.4 Philosophical and Mathematical Definitions 470 4.6.5 What did Russell Really say about MathematicalTheories 472 4.6.6Russell and Abstract Axiomatics 475 4.7 Conclusion 479 Contents 13 5 Logic as the Universal Science II: Logic as a Synthetic Apriori Science 481 5.0 Introduction 481 5.1 Hylton on Russell’s Commitment to the Universalist Conception 483 5.2 Kant on Formal Logic 496 5.2.1 Preliminary Remarks 496 5.2.2 Formality and the Analyticity-Constraint on Kant 499 5.2.3 Analyticity and Apriority 507 5.3 The Bolzanian Account of Logic 509 5.3.1 Preliminary Remarks 509 5.3.2 The Basic Assumptions behind the BolzanianAccount 511 5.3.3 Russell’s Version of the Bolzanian Account 517 5.3.3.1 Russell