Table Of ContentSUB 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
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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.
