Sheaves, Games, and Model Completions A Categorial Approach to Noncl
This book is an example of fruitful interaction between (non-classical) propo sitionallogics and (classical) model theory which was made possible due to categorical logic. Its main aim consists in investigating the existence of model completions for equ
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TRENDS IN LOGIC Studia Logica Library VOLUME 14 Managing Editor Ryszard Wojcicki, Institute of Philosophy and Sociology, Polish Academy of Sciences, Warsaw, Poland Editors Daniele Mundici, Department of Computer Sciences, University of Milan, Italy Ewa Orlowska, 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, comparisons and sources of inspiration is open and evolves over time.
Volume Editor Ryszard Wojcicki
The titles published in this series are listed at the end of this volume.
SILVIO GHILARDI Universita degli Studi di Milano, Milan, Italy
MAREK ZAWADOWSKI Uniwersytet Warszawski, Warsaw, Poland
SHEAVES, GAMES, AND MODEL COMPLETIONS A Categorial Approach to Nonclassical Propositional Logics
1Ir...
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Springer-Science+Business Media, B.Y.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-6036-5 ISBN 978-94-015-9936-8 (eBook) DOI 10.1007/978-94-015-9936-8
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.
To jas, Lavinia, Mis, and Virgilio.
Contents
1 1 4 6 12
1. INTRODUCTION
1 2 3 4
Motivating example An overview of the book How to read the book Historical remarks
2. PRELIMINARY NOTIONS 1 Basic algebraic structures 2 Model theoretic background Subobjects and regular subobjects 3 Finitely presented algebras 4 Principal congruences 5 6 The amalgamation property Exercises 7
15 15 25 30 32 36 40
3. MODEL COMPLETIONS 1 r-Heyting categories Model completions and fp algebras 2 Exercises 3
49 49 60 71
4. HEYTING ALGEBRAS 1 Basic definitions Finitely presented Heyting algebras 2 Duality for Heyting algebras 3 4 A combinatorial result Properties of MH and WH 5
73 73 78 82 88 91
Vll
44
SHEAVES, GAMES, AND MODEL COMPLETIONS
V III
6 7 8 9 10
Projective Heyting al
