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ALGEBRAIC FOUNDATIONS OF MANY-VALUED REASONING TRENDS IN LOGIC Studia Logica Library VOLUME7 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 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 ofTechnology, Germany Assistant Editor Jacek Malinowski, Box 61, UPT 00-953, Warszawa 37, Poland 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 rela tions to other disciplines. These include artificial intelligence, informatics, cogni tive science, philosophy of science, and the philosophy of language. However, this list is not exhaustive, moreover, the range of applications, comparisons and SOUf ces of inspiration is open and evolves over time. The titles published in this series are listed at the end afthis valurne. ROBERTO L.O. CIGNOLI Department of Mathematics. University of Buenos Aires. Argentina ITALA M.L. D'OTfAVIANO Department of Philosophy and The Centre for Logic. Epistemology and the History of Science. State University of Campinas. Brazil and DANIELE MUNDICI Department of Computer Science. University of Milan. Italy ALGEBRAIC FOUNDATIONS OF MANY-VALUED REASONING Springer-Science+Business Media, B.V. A c.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-5336-7 ISBN 978-94-015-9480-6 (eBook) DOI 10.1007/978-94-015-9480-6 Printed on acid-free paper All Rights Reserved © 2000 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2000. Softcover reprint of the hardcover 1s t edition 2000 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, incIuding photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. To the memory 0/ ROLANDO CHUAQUI ENNIO DE GlORGI ANTONIO MONTEIRO great Scientists and Teachers Contents Introduction 1 1 Basic not ions 7 1.1 MV-algebras ........... . 7 1.2 Homomorphisms and ideals '" 12 1.3 Subdirect representation theorem 19 1.4 MV -equations . . 20 1.5 Boolean algebras . . . . 24 1.6 MV-chains ....... . 27 1.7 Bibliographical remarks 29 2 Chang completeness theorem 31 r . . . . . . . . 2.1 The functor 31 2.2 Good sequences ...... . 34 2.3 The partially ordered monoid M 37 A 2.4 Chang's f-group GA' . . . . . 40 2.5 Chang completeness theorem. 43 2.6 Bibliographical remarks 49 3 Free MV- algebras 51 3.1 McN aughton functions . . . . . . . . 51 3.2 The one-dimensional case ...... . 56 3.3 Decomposing McNaughton functions 62 3.4 Ideals in free MV -algebras 64 3.5 Simple MV-algebras .. 70 3.6 Semisimple MV-algebras 72 3.7 Bibliographical remarks 75 vii viii CONTENTS 4 Lukasiewicz oo-valued calculus 77 4.1 Many-valued propositional calculi 78 4.2 Wajsberg algebras. . 82 4.3 Provability.......... 87 4.4 Lindenbaum algebra . . . . 92 4.5 All tautologies are provable 94 4.6 Syntactic and semantic consequence . 97 4.7 Bibliographical remarks ...... . · 101 5 Ulam's game 103 5.1 Questions and answers ...... . · 103 5.2 Dynamics of states of knowledge .. .104 5.3 Operations on states of knowledge . · 107 5.4 Bibliographical remarks .. · 109 6 Lattice-theoretical properties 111 6.1 Minimal prime ideals . . . . . . . . . . . . .112 6.2 Stonean ideals and archimedean elements . .115 6.3 Hyperarchimedean algebras ... .116 6.4 Direct products . . . . . . . . . . · 121 6.5 Boolean products of MV-algebras · 124 6.6 Completeness ......... . · 129 6.7 Atoms and Pseudocomplements · 132 6.8 Complete distributivity . · 134 6.9 Bibliographical remarks · 137 7 MV- algebras and f-groups 139 r 7.1 Inverting the functor · 139 7.2 Applications .... . · 146 7.3 The radical ..... . · 150 7.4 Perfeet MV-algebras . · 151 7.5 Bibliographical remarks · 156 8 Varieties of MV- algebras 157 8.1 Basic definitions. . . . . · 157 8.2 Varieties from simple algebras · 160 8.3 MV-chains of finite rank ... · 161 CONTENTS ix 8.4 Komori's c1assification ....... . · 167 8.5 Varieties generated by a finite chain . · 171 8.6 The cardinality of Free~ .173 8.7 Bibliographical remarks ...... . · 177 9 Advanced topics 179 9.1 McNaughton's theorem . . . . . . . . . 180 9.2 Nonsingular fans and normal forms . . 185 9.3 Complexity of the tautology problem . 187 9.4 MV-algebras and AF C*-algebras . 191 9.5 Di Nola's representation theorem . 193 9.6 Bibliographical remarks ..... . 194 10 Further Readings 197 10.1 More than two truth values . 197 10.2 Current Research Topics . . . 199 10.2.1 Product . . . . . . . . 199 o 10.2.2 States, bservables , Probability, Partitions . . 200 10.2.3 Deduction . . . . . . . . 201 10.2.4 Further constructions . . . . . . . . . . . .. . 201 Bibliography 203 Index 225 Introduction The aim of this book is to give self-contained proofs of all basic results concerning the infinite-valued proposition al calculus of Lukasiewicz and its algebras, Chang's MV -algebras. This book is for self-study: with the possible exception of Chapter 9 on advanced topics, the only prere quisite for the reader is some acquaintance with classical propositional logic, and elementary algebra and topology. In this book it is not our aim to give an account of Lukasiewicz's motivations for adding new truth values: readers interested in this topic will find appropriate references in Chapter 10. Also, we shall not explain why Lukasiewicz infinite-valued propositionallogic is a ba sic ingredient of any logical treatment of imprecise notions: Hajek's book in this series on Trends in Logic contains the most authorita tive explanations. However, in order to show that MV-algebras stand to infinite-valued logic as boolean algebras stand to two-valued logic, we shall devote Chapter 5 to Ulam's game of Twenty Questions with lies/errors, as a natural context where infinite-valued propositions, con nectives and inferences are used. While several other semantics for infinite-valued logic are known in the literature-notably Giles' game theoretic semantics based on subjective probabilities-still the transi tion from two-valued to many-valued propositonallogic can hardly be modelled by anything simpler than the transformation of the familiar game of Twenty Questions into Ulam game with lies/errors. This book is mainly addressed to computer scientists and mathe maticians wishing to get acquainted with a compact body of beautiful results and methodologies-that have found applications in the treat ment of uncertain information, (e.g., adaptive error-correcting codes) as weIl as in various mathematical areas, such as toric varieties, lattice- 1 R. L. O. Cignoli et al., Algebraic Foundations of Many-Valued Reasoning © Springer Science+Business Media Dordrecht 2000 2 INTRODUCTION ordered groups and C"-algebras. As the title indicates, the main em phasis is on algebraic methods. Thus, reversing the historical order, we shall make the reader familiar with MV-algebras before introducing Lukasiewicz's propositional calculus in Chapter 4. This will allow us to get neat and elementary proofs of several deep results, using much less symbolism and detail than in traditional syntax-oriented approaches. The definition-theorem-proof style adopted throughout this book will hopefully result in time saving for the reader who wishes to get the proofs of all main theorems on the infinite-valued calculus as quickly as possible, without embarking on a potentially unbounded search through a scattered literature on ordered groups, lattices, algebraic logic, poly hedra, geometry of numbers, model theory, linear inequalities, et cetera. By definition, an MV-algebra A is a set equipped with an associat ive-commutative operation EB, with a neutral element 0, and with an operation -, such that -,-,x = x, x EB -,0 = -,0, and, characteristically, These six equations are intended to capture some properties of the real unit interval [0, 1] equipped with negation -,x = 1 - x and truncated addition x EB y = min(l, x + y). For instance, once interpreted in [0,1], the left hand term in the last equation coincides with the maximum of x and y; thus the equation states that the max operation over [0,1] is commutative. The fundamental theorem on MV-algebras is Chang's completeness theorem, stating that every valid equation in [0, 1] is auto matically valid in all MV-algebras. A new proof of this theorem is given in Chapter 2. As a preliminary step, in Chapter 1 we prove Chang's subdirect representation theorem, stating that an equation is valid in every MY-algebra iff it is valid in every totally ordered MV-algebra. As in the classical case, one may ask for an effective procedure to decide when an equation is valid. Rather than working in "MY algebraic equationallogic", it is more convenient to give the Lukasiewicz infinite-valued calculus the same role that the classical propositional calculus has for the boolean decision problem. Accordingly, one may write -,x ~ y instead of x EB y