Substructural logics : a primer / Francesco Paoli.

Format:

Book

Author/Creator:

Paoli, Francesco.

Contributor:

Alumni and Friends Memorial Book Fund.

Series:

Trends in logic ; v. 13.

Trends in logic ; v. 13

Language:

English

Subjects (All):

Logic.

Physical Description:

xiii, 304 pages : illustrations ; 25 cm.

Place of Publication:

Dordrecht ; Boston, MA : Kluwer Academic, [2002]

Summary:

The aim of the present book is to give a comprehensive account of the state of the art of substructural logics, focusing both on their proof theory (especially on sequent calculi and their generalizations) and on their semantics (both algebraic and relational. It is for graduate students in either philosophy, mathematics, theoretical computer science or theoretical linguistics as well as specialists and researchers.

Contents:

Part I The philosophy of substructural logics

Chapter 1. The role of structural rules in sequent calculi 3

1. The "inferential approach" to logical calculus 3

1.1 Structural rules, operational rules, and meaning 5

1.2 Discovering the effects of structural rules 11

2. Reasons for dropping structural rules 15

2.1 Reasons for dropping structural rules altogether 15

2.2 Reasons for dropping (or eliminating) the cut rule 17

2.3 Reasons for dropping the weakening rules 21

2.4 Reasons for dropping the contraction rules 25

2.5 Reasons for dropping the exchange rules 28

2.6 Reasons for dropping the associativity of comma 30

3. Ways of reading a sequent 30

3.1 The truth-based reading 31

3.2 The proof-based reading 31

3.3 The informational reading 32

3.4 The "Hobbesian" reading 34

Part II The proof theory of substructural logics

Chapter 2. Basic proof systems for substructural logics 41

1. Some basic definitions and notational conventions 42

2. Sequent calculi 44

2.1 The calculus LL 44

2.2 Adding the empty sequent: the dialethic route 49

2.3 Adding the lattice-theoretical constants: the bounded route 49

2.4 Adding contraction: the relevant route 50

2.5 Adding weakening: the affine route 55

2.6 Adding restricted structural rules 57

2.7 Adding the exponentials 65

3. Hilbert-style calculi 68

3.1 Presentation of the systems 68

3.2 Derivability and theories 73

3.3 Lindenbaum-style constructions 81

4. Equivalence of the two approaches 83

Chapter 3. Cut elimination and the decision problem 87

1. Cut elimination 87

1.1 Cut elimination for LK 87

1.2 Cut elimination for calculi without the contraction rules 94

1.3 Cut elimination for calculi without the weakening rules 97

1.4 Cases where cut elimination fails 99

2. The decision problem 100

2.1 Gentzen's method for establishing the decidability of LK 101

2.2 A decision method for contraction-free systems 105

2.3 A decision method for weakening-free systems 106

2.4 Other decidability (and undecidability) results 111

Chapter 4. Other formalisms 115

1. Generalizations of sequent calculi 116

1.1 N-sided sequents 116

1.2 Hypersequents 121

1.3 Dunn-Mints calculi 127

1.4 Display calculi 130

1.5 A comparison of these frameworks 136

2. Proofnets 137

3. Resolution calculi 145

3.1 Classical resolution 146

3.2 Relevant resolution 149

3.3 Resolution systems for other logics 153

Part III The algebra of substructural logics

Chapter 5. Algebraic structures 159

1. *-autonomous lattices 161

1.1 Definitions and elementary properties 161

1.2 Notable *-autonomous lattices 165

1.3 Homomorphisms, l-filters, l-ideals, congruences 171

1.4 Principal, prime and regular l-ideals 181

1.5 Representation theory 186

2. Classical residuated lattices 187

2.1 Maximal, prime, and primary l-ideals 188

2.2 Subdirectly irreducible c.r. lattices 190

2.3 Weakly simple, simple and semisimple c.r. lattices 192

Part IV The semantics of substructural logics

Chapter 6. Algebraic semantics 201

1. Algebraic soundness and completeness theorems 202

1.1 Calculi without exponentials 203

1.2 Calculi with exponentials 209

2. Totally ordered models and the single model property 213

3. Applications 219

Chapter 7. Relational semantics 221

1. Semantics for distributive logics 222

1.1 Routley-Meyer semantics: definitions and results 223

1.2 Applications 235

2. Semantics for nondistributive logics 239

2.1 General phase structures 240

2.2 General phase semantics 250

2.3 The exponentials 252

2.4 Applications 254

Appendix A Basic glossary of algebra and graph theory 257

Appendix B Other substructural logics 271

1. Lambek calculus 271

2. Ono's subintuitionistic logics 277

3. Basic logic 281.

Notes:

Includes bibliographical references (pages [289]-299) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

1402006055

OCLC:

49860219