Protoalgebraic Logics
The main aim of this book is to present recent ideas in logic centered around the notion of a consequence operation. We wish to show these ideas in a factually and materially connected way, i.e., in the form of a consistent theory derived from several sim
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TRENDS IN LOGIC Studia Logica Library VOLUME 10 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 of Technology, 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 fonnal logic and its applications and relations to other disciplines. These include artificial intelligence, infonnatics, cognitive science, philosophy of science, and the philosophy of language. However, this list is not exhaustive, moreover, the range of applications, comparisons and sources of inspiration is open and evolves over time.
JANUSZ CZELAKOWSKl Institute of Mathematics, Opole University, Opole, Poland
PROTOALGEBRAIC LOGICS
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SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-017-2807-2 (eBook) ISBN 978-90-481-5693-1 DOI 10.1007/978-94-017-2807-2
Printed on acid-Iree paper THIS BOOK WAS TYPESET BY AMS-TEX, the TEX macro system of the American Mathematical Society.
All Rights Reserved © 200 I Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2001 No par! of the material protected by this copyright notice may be reproduced or utiIized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.
CONTENTS
Preface
ix
INTRODUCTION 1
On the book Mathematical prerequisites Exercises Notes for Mathematical Prerequisites
11 17 20
Chapter O. Basic definitions and facts § 0.1. Sentential languages. Consequence operations. Sentential logics § 0.2. Logics as deductive systems . § 0.3. Consequence operations and matrix models. Operations on matrices § OA. Model-theoretic aspects of the theory of sentential logics § 0.5. The Leibniz congruence. Reduced matrices § 0.6. The Characterization Theorem for Mod(C) § 0.7. The structure of the class Mod*(C) § 0.8. Deductive filters Exercises Notes for Chapter 0
21 26 29 36 41 45 49 54 59 64
PART I.
PROTOALGEBRAIC LOGICS. THE LEIBNIZ OPERATOR
Chapter 1. Protoalgebraic logics
§ 1.1. Preliminary remarks 69 § 1.2. Parameterized systems of equivalence sentences 78 § 1.3. Protoalgebraicity and subdirect products 87 § 1A. The characterization problem for the class Mod*(C). Lyndon classes 93 § 1.5. The Suszko operator. Free matrices 97 § 1.6. Injectivity of the Leibniz operator 100 § 1.7. The Transfer Principle 106 Exercises 111 Notes
