The boundary stones of thought : an essay in the philosophy of logic / Ian Rumfitt.

Format:

Book

Author/Creator:

Rumfitt, Ian, author.

Language:

English

Subjects (All):

Logic.

Physical Description:

xiv, 345 pages ; 24 cm

Edition:

First edition.

Place of Publication:

Oxford : Clarendon Press/Oxford University Press, 2015.

Summary:

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. Book jacket.

Contents:

1 Introduction 1

1.1 Disputes Over Logical Laws 1

1.2 The Scope of This Book, and the Nature of Disputes Over Basic Logical Laws 14

1.3 The Argument of the Present Book 21

Part I The Nature of Logic

2 Logical Laws 31

2.1 Consequence 31

2.2 Inference and Deduction 34

2.3 The Varieties of Deduction and of Implication Relations 38

2.4 Implications and Possibilities 46

2.5 The Role of Logic 52

2.6 Knowledge by Deduction 56

3 Logical Necessity 66

3.1 Logical Consequence Redux 66

3.2 The Controversy Over Logical Necessity 68

3.3 Notions of Necessity 74

3.4 Logical Necessity versus Apriority and Metaphysical Necessity 81

3.5 Logical and Metaphysical Necessity: The Paradox Resolved 88

Part II Five Attacks on Classical Logic

4 The Argument of Dummett's 'Truth' 95

4.1 The Argument against Classical Logic in Dummett's 'Truth' 95

4.2 The Exclusionary Theory of Conceptual Content 99

4.3 Where the Argument of 'Truth' Fails 105

4.4 Exclusion and Truth 111

4.5 An Exclusionary Semantics for the Language of the Propositional Calculus 117

4.6 The Choice of Logic within an Exclusionary Semantics 122

5 The Verificationist Attack on Classical Logic 125

5.1 The Strong Verificationist Attack on Classical Logic 125

5.2 How to Be a Strong Verificationist 129

5.3 A Renewed Threat to Classical Logic 138

5.4 Why One Should Not Be a Strong Verificationist 143

5.5 Dubious Grounds: McDowell's Challenge to Classical Logic 147

6 Possibilities 153

6.1 Moderate Modal Realism and Possible Worlds 153

6.2 Reasons for Seeking an Unworldly Theory 157

6.3 Previous Unworldly Theories 160

6.4 The Structure of the Space of Possibilities; Truth-Ground Semantics 162

6.5 Distribution and Quantum Mechanics 167

6.6 Distribution and Regularity 181

7 Challenges from the Infinite and from the Infinitesimal 184

7.1 The Semantics and Logic of Negation 185

7.2 Statement's With and Without Backs 193

7.3 The Intuitionists on Infinity 197

7.4 A Consolation Prize for the Intuitionist: Smooth Infinitesimal Analysis 210

7.5 Logic and Metaphysics 217

8 The Challenge from Vagueness 220

8.1 The Paradox of the Heap 220

8.2 Intuitionism as the Logic of Vagueness 223

8.3 A Semantics for Vague Predicates that Validates Intuitionistic Logic 228

8.4 Paradigms and Poles 235

8.5 A Semantics for Polar Predicates that Validates Classical Logic 242

8.6 The Sorites Revisited 250

8.7 Vagueness and Distribution 255

9 On the Use of Classical Logic in Set Theory 263

9.1 What is Mathematics About? 264

9.2 Attempts to Attain Categoricity 270

9.3 The Iterative Conception of Sets, and the Threat to Classical Logic 276

9.4 Attempts to Justify Classical Logic under the Iterative Conception 281

9.5 Classical Logic Justified via a Negative Translation 286

9.6 Classical Logic in Set Theories Weaker than ZF 289

9.7 'As Fat as Possible' versus 'As Far as Necessary' 299

10 Conclusion 302

10.1 The Simple and Revised Arguments for Bivalence 302

10.2 Where the Arguments Go Wrong 308

10.3 Classical Logic versus Classical Semantics 317.

Notes:

Includes bibliographical references (pages 321-336) and index.

ISBN:

9780198733638

0198733631

OCLC:

909308804