The Palgrave Centenary Companion to Principia Mathematica (History of Analytic Philosophy)

This Collection Of Fifteen New Essays Marks The Centenary Of The 1910 To 1913 Publication Of The Monumental Principia Mathematica By Alfred N. Whitehead And Bertrand Russell. The Papers Study The Influence Of Pm On The Development Of Symbolic Logic In The Twentieth Century, Russell's Philosophy Of Logic And His Program Of Reducing Mathematics To Logic, The Distinctive Theory Of Logical Types That Provides A Response To The Paradoxes Of Logic That Russell And Others Discovered Around 1900, As Well As The Details Of Some Of The Mathematical Theories In The Three Volumes Of Symbolic Proofs. Principia Mathematica : The First Hundred Years / Alasdair Urquhart -- David Hilbert And Principia Mathematica / Reinhard Kahle -- Principia Mathematica In Poland / Jan Wolenski -- From Logicism To Metatheory / Patricia Blanchette -- Russell On Real Variables And Vague Denotation / Edwin Mares -- The Logic Of Classes And The No-class Theory / Byeong-uk Yi -- Why There Is No Frege-russell Definition Of Number / Jolen Galaugher -- Principia Mathematica : Versus / Gregory Llandini -- Pm's Circumflex, Syntax And Philosophy Of Types / Kevin Klement -- Principia Mathematica, The Multiple-relation Theory Of Judgment And Molecular Facts / James Levine -- Report On Some Ramified-type Assignment Systems And Their Model-theoretic Semantics / Harold Hodes -- Outline Of A Theory Of Quantification /dustin Tucker -- Whatever Happened To Group Theory? / Nicholas Griffin -- Proofs Of The Cantor-bernstein Theorem In Principia Mathematica / Arie Hinkis -- Quantity And Number In Principia Mathematica : A Plea For An Ontological Interpretation Of The Application Constraint / Sébastien Gandon. Edited By Nicholas Griffin, Mcmaster University, Canada And Bernard Linsky, University Of Alberta, Canada. Includes Bibliographical References And Index. The Palgrave Centenary Companion to Principia Mathematica 4 Contents 6 Series Editor’s Foreword 8 Acknowledgments 11 Notes on Contributors 12 Note on Citations 15 Nicholas Griffin and Bernard Linsky: Introduction: Palgrave Centenary Companion to Principia Mathematica 16 Part I: The Influence of PM 30 1 Alasdair Urquhart: Principia Mathematica: The First 100 Years 32 1 Russell’s nightmare 32 2 PM as a treatise on logic 33 3 PM as a foundation for mathematics 36 3.1 The failure of logicism 36 3.2 Type theory as a foundation 40 4 PM and the development of logic 42 4.1 The logicist tradition 43 4.2 Conceptual transformations 44 5 Conclusion 49 Note 49 2 Reinhard Kahle: David Hilbert and Principia Mathematica 50 1 Introduction 50 2 Hilbert before 1910 50 3 1914–18: The publication of Principia and Behmann’s thesis 52 4 1917: Logicism(?) and the arrival of Bernays 56 5 1920 and beyond 59 Conclusion 61 Acknowledgments 61 Notes 62 3 Jan Woleński: Principia Mathematica in Poland 64 Introduction 64 The social history of PM in Poland (a selection) 64 The scientific reception of PM in Poland 69 3.1 Leon Chwistek 70 3.2 Stanisław Leśniewski 74 3.3 Alfred Tarski 80 3.4 Other polish contributions related to PM 82 Concluding remarks 84 Part II: Russell’s Philosophy of Logic and Logicism 86 4 Patricia Blanchette: From Logicism to Metatheory 88 1 Universalism and metatheory 90 1.1 What’s metatheory? 90 1.2 What’s universalism? 91 2 Russell and metatheory 94 3 But what then? 98 Notes 106 5 Edwin Mares: Russell on Real Variables and Vague Denotation 108 1 Introduction 108 2 Real variables in principles of mathematics 109 3 Arbitrary denotation after the Principles 110 4 Open formulas and logical form 111 5 Vague denotation 113 6 Changes in logical theory between the two editions of Principia 115 7 Arbitrary denotation in the second edition 118 8 Conclusion 121 9 Appendix: pure and applied logic 122 Notes 123 6 Byeong-uk Yi: The Logic of Classes of the No-Class Theory 125 1 Introduction 125 2 The no-class theory 129 3 Logic of classes 131 4 Predicative functions 135 5 From the extensional view of class to plural languages and logic 137 6 Revisions of the no-class theory 143 6.1 Plural notion of predicativity 143 6.2 Strong reducibility 145 6.3 Plural no-class theory 146 6.4 Predicative property theory 147 7 Concluding remarks: plurals, numbers, and classes 148 Appendix 1 152 Appendix 2 153 Acknowledgments 153 Notes 153 7 Jolen Galaugher: Why There Is No Frege–Russell Definition of Number 159 1 Introduction 159 2 Frege’s extensional definition of number 161 3 The nominal definition of the cardinals and the principle of abstraction 163 4 Classes and the contradiction 168 5 ‘Propositional Functions’ and the Contradiction 171 6 Propositional functions reconsidered 176 7 The logic of propositions 180 8 Overview 183 Notes 186 Part III: Type Theory and Ontology 190 8 Gregory Landini: Principia Mathematica: φ! versus φ 192 1 Introduction 192 2 Bumblowski’s moratorium 193 3 Against the Church orthodoxy 204 4 Experimentum crucis 213 5 Church’s r-types and the substitutional theory 230 Notes 246 9 Kevin C. Klement: PM’s Circumflex, Syntax and Philosophy of Types 247 Introduction 247 Frege’s approach 249 The λ-calculus approach 251 PM’s approach 253 4.1 The development of Russell’s views 253 4.2 PM’s propositional function nominalism 255 4.3 The role of the circumflex 259 4.4 Is the circumflex a complex term-forming operator? 264 4.5 Comparison with the other approaches 267 Appendices 269 A. Syntax of PM 269 B. Semantics of PM 273 Notes 274 10 James Levine: Principia Mathematica, the Multiple-Relation Theory of Judgment and Molecular Facts 276 1 Introduction 276 2 Proposition and sentence 281 3 The MRTJ, the ‘Systematic Ambiguity of Truth and Falsehood’, and molecular facts 290 4 A ‘Real Difficulty’ facing the MRTJ 300 5 The ‘Real Difficulty’ and the demise of TK 314 6 Individuals, Universals, and Logical Constants 319 Notes 327 11 Harold T. Hodes: Report on Some Ramified-Type Assignment Systems and Their Model-Theoretic Semantics 334 1 Types and terms 335 2 A Digression on Two Alternative Approaches 339 3 Computation: ⇒nr versus ⇒r 341 4 More on Computation 344 5 Remarks on the Source Texts 346 6 Analyses, Equivalences and Models 349 7 Truth, Satisfaction and Consequence 352 8 Indiscernibility and identity 359 Notes 360 12 Dustin Tucker: Outline of a Theory of Quantification 366 1 Ramification and paradoxes 366 1.1 A paradox 366 1.2 Ramification 367 1.3 Preliminaries 369 2 Compressed orders 370 3 Constructing the orders 371 3.1 An overview of the logic 372 3.2 Syntax 373 3.3 Semantics 374 3.4 Paradoxes 375 3.5 Truth-value gaps 376 3.6 Compressed orders formally 377 3.6.1 First attempt 378 3.6.2 Second attempt 379 4 The Appendix B paradox 381 4.1 Uncompressed ramification 382 4.2 Compressed ramification 384 4.3 Machinery 384 4.3.1 Resolution 385 4.3.2 Summary 386 4.4 Related paradoxes 387 4.4.1 The original Appendix B paradox 387 4.4.2 Sets, properties, pluralities, etc. 388 4.5 A new Ramseyan division? 389 5 Additional paradoxes 389 6 Other resolutions 390 Notes 392 Part IV: Mathematics in PM 396 13 Nicholas Griffin: Whatever Happened to Group Theory? 398 Notes 415 14 Arie Hinkis: Proofs of the Cantor–Bernstein Theorem in Principia Mathematica 420 1 Introduction 420 2 The first two versions 422 3 The impredicative proof 423 4 Without the reducibility axiom 425 5 The inductive proof 427 6 The drawings 431 7 The cardinal version 433 Notes 440 15 Sébastien Gandon: On Quantity and Number in Principia Mathematica: A Plea for an Ontological Interpretation of the Application Constraint 442 1 Dedekind, Burali-Forti and Frege on rational and real numbers 443 2 Russell’s and Whitehead’s theory of number and quantity 449 2.1 Section A 450 2.2 Section B 452 2.3 Section C 455 3 Structural versus ontological interpretations of the application constraint 456 Notes 462 Bibliography 464 Works by Whitehead and Russell 464 Works by Whitehead 464 Works by Russell 464 Works by Other Authors 466 Index 484 Note on Citations Introduction: Nicholas Griffin and Bernard Linsky PART I: THE INFLUENCE OF PM 1. Principia Mathematica: The First Hundred Years; Alasdair Urquhart 2. David Hilbert and Principia Mathematica; Reinhard Kahle: 3. Principia Mathematica in Poland; Jan Woleński PART II: RUSSELL'S PHILOSOPHY OF LOGIC AND LOGICISM 4. From Logicism to Metatheory; Patricia Blanchette 5. Russell on Real Variables and Vague Denotation; Edwin Mares 6. The Logic of Classes and the No-Class Theory; Byeong-uk Yi 7. Why There Is No Frege–Russell Definition of Number; Jolen Galaugher PART III: TYPE THEORY AND ONTOLOGY 8. Principia Mathematica: φ! versus φ; Gregory Landini 9. PM's Circumflex, Syntax and Philosophy of Types; Kevin Klement 10. Principia Mathematica, the Multiple-Relation Theory of Judgment and Molecular Facts; James Levine 11. Report on Some Ramified-Type Assignment Systems and Their Model-Theoretic Semantics; Harold Hodes 12. Outline of a Theory of Quantification; Dustin Tucker PART IV: MATHEMATICS IN PM 13. Whatever Happened to Group Theory?; Nicholas Griffin 14. Proofs of the Cantor–Bernstein Theorem in Principia Mathematica; Arie Hinkis 15. Quantity and Number in Principia Mathematica: A Plea for an Ontological Interpretation of the Application Constraint; Sébastien Gandon