An Introduction to Substructural Logics

This is the first book to systematically survey new areas of substructural logics. This book is geared to introduce the topic to advanced students.An Introduction to Substructural Logicscovers the area of logic that is crucial to developments in computing, philosophy and linguistics.

List of Figures

ix

List of Tables

xi

Acknowledgements

xiii

Introduction

1

(6)

Introducing the Topic

1

(2)

Introducing the Book

3

(4)

I Proof Theory

7

(146)

Ifs, Ands and Ors

9

(38)

Consequences

9

(5)

Languages and Structures, Consecutions and Proofs

14

(8)

Ifs and Ands

22

(8)

Truth

30

(3)

Ands and Ors

33

(4)

Cutting and Pasting Proofs

37

(2)

Putting Things Together

39

(2)

History

41

(3)

Exercises

44

(3)

Modalities

47

(26)

Positive Modalities

47

(7)

Non-Normal Modalities

54

(5)

Negative Modalities

59

(10)

History

69

(1)

Exercises

70

(3)

Hilbert Systems

73

(16)

Mass-produced Hilbert Systems

74

(10)

Hand Tuning

84

(2)

Consequence and Deduction Theorems

86

(1)

History

87

(1)

Exercises

88

(1)

Theories

89

(16)

Defining Theories

89

(3)

Constructing Prime Theories

92

(3)

Metavaluations

95

(7)

History

102

(1)

Exercises

103

(2)

Gentzen Systems

105

(22)

An Example System

105

(6)

The Core Ideas

111

(8)

The Proof

119

(3)

Examples

122

(3)

History

125

(1)

Exercises

125

(2)

Formulae as Types, Proofs as Terms

127

(26)

Defining Terms and Decorating Proofs

129

(8)

Introducing Normalisation

137

(5)

Normalisation and Cut Elimination

142

(8)

History

150

(1)

Exercises

151

(2)

II Propositional Structures

153

(80)

Defining Propositional Structures

155

(34)

Definitions

157

(13)

Example Structures

170

(11)

Soundness and Completeness

181

(4)

History

185

(1)

Exercises

186

(3)

Manipulating Propositional Structures

189

(22)

Algebra Facts

190

(4)

Filters, Ideals and Ideal Structures

194

(7)

Negation

201

(3)

Normal Modalities

204

(3)

The Kleene Star in Structures

207

(2)

History

209

(1)

Exercises

209

(2)

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An Introduction to Substructural Logics

Summary

Table of Contents

Supplemental Materials