The Boundary Stones of Thought : An Essay in the Philosophy of Logic

__The Boundary Stones of Thought__ seeks to defend classical logic from a number of attacks of a broadly anti-realist character. Ian Rumfitt is sympathetic to many of the premisses underlying these attacks. Indeed, he regards some of them as effective challenges to certain principles of classical semantics, notably the Principle of Bivalence. He argues, though, that they are ineffective against classical logic itself. The book starts by considering the general problem of how conflicts over logical laws may be rationally discussed and adjudicated. This leads to a consideration of the nature of logic: Rumfitt identifies the particular features that mark out logical consequence from other consequence relations, and he advances a new argument for the ancient thesis that there is a modal element in the notion of logical consequence. He develops a theory of that modal element in terms of perhaps incomplete possibilities, rather than fully determinate possible worlds. Some prima facie powerful arguments against the validity of certain classical logical laws are then analysed in the light of this account of logic. Throughout, care is taken to separate lines of anti-classical argument that, although distinct, are often run together or confused.The analysis yields, as by-products, semantic theories for a number of problematical areas of discourse. These areas include our talk about sub-atomic particles, about the infinite, about infinitesimals, about sets, and vague discourse. Rumfitt concludes by defending his stance of accepting classical logic while rejecting Bivalence, against Aristotle's argument that a classical logician is committed to Bivalence. The ultimate aim is to liberate classical logic from the dead hand of classical semantics. Cover 1 The Boundary Stones of Thought 4 Copyright 5 Dedication 6 Contents 14 1. Introduction 18 1.1 Disputes Over Logical Laws 18 1.2 The Scope of This Book, and the Nature of Disputes Over Basic Logical Laws 31 1.3 The Argument of the Present Book 38 Part I The Nature of Logic 46 2. Logical Laws 48 2.1 Consequence 48 2.2 Inference and Deduction 51 2.3 The Varieties of Deduction and of Implication Relations 55 2.4 Implications and Possibilities 63 2.5 The Role of Logic 69 2.6 Knowledge by Deduction 73 3. Logical Necessity 83 3.1 Logical Consequence Redux 83 3.2 The Controversy Over Logical Necessity 85 3.3 Notions of Necessity 91 3.4 Logical Necessity versus Apriority and Metaphysical Necessity 98 3.5 Logical and Metaphysical Necessity: The Paradox Resolved 105 Part II Five Attacks on Classical Logic 110 4. The Argument of Dummett’s ‘Truth’ 112 4.1 The Argument against Classical Logic in Dummett’s ‘Truth’ 112 4.2 The Exclusionary Theory of Conceptual Content 116 4.3 Where the Argument of ‘Truth’ Fails 122 4.4 Exclusion and Truth 128 4.5 An Exclusionary Semantics for the Language of the Propositional Calculus 134 4.6 The Choice of Logic within an Exclusionary Semantics 139 5. The Verificationist Attack on Classical Logic 142 5.1 The Strong Verificationist Attack on Classical Logic 142 5.2 How to Be a Strong Verificationist 146 5.3 A Renewed Threat to Classical Logic 155 5.4 Why One Should Not Be a Strong Verificationist 160 5.5 Dubious Grounds: McDowell’s Challenge to Classical Logic 164 6. Possibilities 170 6.1 Moderate Modal Realism and Possible Worlds 170 6.2 Reasons for Seeking an Unworldly Theory 174 6.3 Previous Unworldly Theories 177 6.4 The Structure of the Space of Possibilities; Truth-Ground Semantics 179 6.5 Distribution and Quantum Mechanics 184 6.6 Distribution and Regularity 198 7. Challenges from the Infinite and from the Infinitesimal 201 7.1 The Semantics and Logic of Negation 202 7.2 Statements With and Without Backs 210 7.3 The Intuitionists on Infinity 214 7.4 A Consolation Prize for the Intuitionist: Smooth Infinitesimal Analysis 227 7.5 Logic and Metaphysics 234 8. The Challenge from Vagueness 237 8.1 The Paradox of the Heap 237 8.2 Intuitionism as the Logic of Vagueness 240 8.3 A Semantics for Vague Predicates that Validates Intuitionistic Logic 245 8.4 Paradigms and Poles 252 8.5 A Semantics for Polar Predicates that Validates Classical Logic 259 8.6 The Sorites Revisited 267 8.7 Vagueness and Distribution 272 9. On the Use of Classical Logic in Set Theory 280 9.1 What is Mathematics About? 281 9.2 Attempts to Attain Categoricity 287 9.3 The Iterative Conception of Sets, and the Threat to Classical Logic 293 9.4 Attempts to Justify Classical Logic under the Iterative Conception 298 9.5 Classical Logic Justified via a Negative Translation 303 9.6 Classical Logic in Set Theories Weaker than ZF 306 9.7 ‘As Far as Possible’ versus ‘As Far as Necessary’ 316 10. Conclusion 319 10.1 The Simple and Revised Arguments for Bivalence 319 10.2 Where the Arguments Go Wrong 325 10.3 Classical Logic versus Classical Semantics 334 References 338 Index 354 The Boundary Stones of Thought seeks to defend classical logic from a number of attacks of a broadly anti-realist character. Ian Rumfitt is sympathetic to many of the premisses underlying these attacks. Indeed, he regards some of them as effective challenges to certain principles of classical semantics, notably the Principle of Bivalence. He argues, though, that they are ineffective against classical logic itself. The book starts by considering the general problem of how conflicts over logical laws may be rationally discussed and adjudicated. This leads to a consideration of the nature of logic: Rumfitt identifies the particular features that mark out logical consequence from other consequence relations, and he advances a new argument for the ancient thesis that there is a modal element in the notion of logical consequence. He develops a theory of that modal element in terms of perhaps incomplete possibilities, rather than fully determinate possible worlds. Some prima facie powerful arguments against the validity of certain classical logical laws are then analysed in the light of this account of logic. Throughout, care is taken to separate lines of anti-classical argument that, although distinct, are often run together or confused. The analysis yields, as by-products, semantic theories for a number of problematical areas of discourse. These areas include our talk about sub-atomic particles, about the infinite, about infinitesimals, about sets, and vague discourse. Rumfitt concludes by defending his stance of accepting classical logic while rejecting Bivalence, against Aristotle's argument that a classical logician is committed to Bivalence. The ultimate aim is to liberate classical logic from the dead hand of classical semantics. Classical Logic Has Been Attacked By Adherents Of Rival, Anti-realist Logical Systems: Ian Rumfitt Comes To Its Defence. He Considers The Nature Of Logic, And How To Arbitrate Between Different Logics. He Argues That Classical Logic May Dispense With The Principle Of Bivalence, And May Thus Be Liberated From The Dead Hand Of Classical Semantics. The Nature Of Logic. Logical Laws ; Logical Necessity -- Five Attacks On Classical Logic. The Argument Of Dummett's 'truth' ; The Verificationist Attack On Classical Logic ; Possibilities ; Challenges From The Infinite And From The Infinitesimal ; The Challenge From Vagueness ; On The Use Of Classical Logic In Set Theory ; Conclusion. Ian Rumfitt. Includes Bibliographical References (pages 321-336) And Index.