Table Of Content

SUB STRUCTURAL LOGICS: A PRIMER TRENDS IN LOGIC Studia Logica Library VOLUME 13 Managing Editor Ryszard W6jcicki, Institute of Philosophy and Sociology, Polish Academy of Sciences, Warsaw, Poland Editors Daniele Mundici, Department of Computer Sciences, University of Milan, Italy Ewa Odowska, National Institute of Telecommunications, Warsaw, Poland Graham Priest, Department of Philosophy, University of Queensland, Brisbane, Australia Krister Segerberg, Department of Philosophy, Uppsala University, Sweden Alasdair Urquhart, Department of Philosophy, University of Toronto, Canada Heinrich Wansing, Institute of Philosophy, Dresden University of Technology, Germany SCOPE OF THE SERIES Trends in Logic is a bookseries covering essentially the same area as the journal Studia Logica - that is, contemporary formal logic and its applications and relations to other disciplines. These include artificial intelligence, informatics, cognitive science, philosophy of science, and the philosophy of language. However, this list is not exhaustive, moreover, the range of applications, com parisons and sources of inspiration is open and evolves over time. Volume Editor Heinrich Wansing The titles published in this series are listed at the end of this volume. FRANCESCO PAOLI Universita di Cagliari, Italy SUB STRUCTURAL LOGICS: A PRIMER SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. A C.I.p. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-6014-3 ISBN 978-94-017-3179-9 (eBook) DOI 10.1007/978-94-017-3179-9 Printed on acid-free paper All Rights Reserved © 2002 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2002 Softcover reprint of the hardcover 1st edition 2002 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise. without written permission from the Publisher. with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system. for exclusive use by the purchaser of the work. CONTENTS Preface ....................................................................................................... ix Part I: The philosophy of substructurallogics 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 o/reading a sequent ...................................................................... 30 3.1 The truth-based reading ............................................................ 31 3.2 The proof-based reading ........................................................... 31 3.3 The infonnational reading ......................................................... 32 3.4 The "Hobbesian" reading .......................................................... 34 Part 11: The proof theory of sub structural logics Chapter 2. Basic proof systems for substructurallogics .......................... 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 affme 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 VI 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 formalislDS .................................................................. 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 Ill: The algebra of substructurallogics s. Chapter Algebraic structures. ............................................................. 159 1. *-autonomous lattices ........................................................................... 161 1.1 Definitions and elementary properties ...................................... 161 1.2 Notable *-autonomous lattices ................................................ 165 1.3 Homomorphisms, i-filters, i-ideals, congruences ..................... 171 1.4 Principal, prime and regular i-ideals ....................................... 181 1.5 Representation theory ............................................................. 186 2. Classical residuated lattices ................................................................. 187 2.1 Maximal, prime, and primary f-ideals ..................................... 188 2.2 Subdirectly irreducible c.r.lattices .......................................... 190 2.3 Weakly simple, simple and semisimple c.r.lattices .................. 192 Vl1 Part IV: The semantics ofs ubstructurallogics 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. RelatioDBl sellUU1tics .............................................................. 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 sub structural logics ..................................................... 271 1. Lambek calculus ...................................................................... 271 2. Ono's subintuitionistic logics ..................................................... 277 3. Basic logic ............................................................................... 281 Bibliography ............................................................................................. 289 Index of subjects ....................................................................................... 301 PREFACE 1. AN INTRIGUING CHALLENGE Whoever undertakes the task of compiling a textbook on a relatively new, but already vastly ramified and quickly growing area of logic - and substructurallogics are such, at least to some extent - is faced with a baffling dilemma: he can either presuppose a high degree of logical and mathematical expertise on the reader's part, or else require no background at all except for a "working knowledge" of elementary logic. In our specific case, each one of these policies had its own allure. The former strategy promised to speed up the presentation of some advanced topics and to allow a more refmed expository style; the latter one, on the other side, would have permitted to reach a wider audience, some members of which might have had the opportunity to study for the fIrst time some elementary, but fundamental results - such as Gentzen's Hauptsatz - directly in the perspective of substructurallogics. Teaching logic from this point of view to unexperienced, and presumably still unbiased, students seemed to us an irresistibly intriguing challenge - therefore, we opted for the second alternative. Thus, we assume that the reader of this book has attended an undergraduate course in logic and has a good mastery of the rudiments of propositionallogic (Hilbert-style and natural deduction calculi, truth table semantics) and naive set theory. As for the rest, the volume is self-contained and gradually accompanies the reader up to some of the most recent and specialistic research developments in this area. Some prior acquaintance with either predicate logic or algebra is useful, but not indispensable; in particular, the algebraic notions used throughout the book are surveyed in a special glossary (Appendix A). Of course, this book is not meant only for students. The researcher in the fIeld of sub structural logics will fmd plenty of material she can directly exploit and draw from in her research practice.