Truth And Assertibility

by Nik Weaver

Overview

The book is a research monograph on the notions of truth and assertibility as they relate to the foundations of mathematics. It is aimed at a general mathematical and philosophical audience. The central novelty is an axiomatic treatment of the concept of assertibility. This provides us with a device that can be used to handle difficulties that have plagued philosophical logic for over a century. Two examples relate to Frege's formulation of second-order logic and Tarski's characterization of truth predicates for formal languages. Both are widely recognized as fundamental advances, but both are also seen as being seriously flawed: Frege's system, as Russell showed, is inconsistent, and Tarski's definition fails to capture the compositionality of truth. A formal assertibility predicate can be used to repair both problems. The repairs are technically interesting and conceptually compelling. The approach in this book will be of interest not only for the uses the author has put it to, but also as a flexible tool that may have many more applications in logic and the foundations of mathematics.

Product Details

ISBN-13:	9789814619950
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	05/06/2015
Pages:	204
Product dimensions:	6.00(w) x 9.00(h) x 0.50(d)

Table of Contents

Preface vii

1 Truth 1

1.1 The liar paradox 1

1.2 Sentences and propositions 3

1.3 The T-scheme 8

1.4 Numerical truth 15

1.5 Arithmetic 22

1.6 Arithmetic mod n 26

1.7 Truth and meaning 27

2 Concepts 31

2.1 Predicates and concepts 31

2.2 Russell's paradox 34

2.3 Interpreted languages 38

2.4 Global truth 42

2.5 Defining truth 47

2.6 The revenge problem 54

2.7 Second-order logic 58

3 Deduction 61

3.1 Natural deduction 61

3.2 The completeness theorem 66

3.3 Peano arithmetic 70

3.4 The first incompleteness theorem 72

3.5 The second incompleteness theorem 76

3.6 Arithmetic with truth 79

3.7 Formal truth predicates 83

4 Assertibility 89

4.1 Assertibility as a concept 89

4.2 Proofs and meaning 94

4.3 Existence 99

4.4 A versus A(A) 104

4.5 Axiomatizing assertibility 110

4.6 Applications 117

5 Systems 123

5.1 Pure assertibility 123

5.2 Arithmetical assertibility 127

5.3 Problems of rational agency 132

5.4 Weak interpretation 136

5.5 Conceptual assertibility 143

5.6 Second-order logic 149

6 Surveyability 151

6.1 The iterative conception 151

6.2 Indefinite extensibility 155

6.3 Surveyable concepts 159

6.4 Abstract objects 164

6.5 Conclusions 168

Notes 175

Bibliography 181

Notation Index 187

Subject Index 189