Probabilistic Lattices: With Applications to Psychology PDF
Preview Probabilistic Lattices: With Applications to Psychology
Probabilistic Lattices With Applications to Psychology 9345_9789814630412_tp.indd 1 3/12/14 9:25 am ADVANCED SERIES ON MATHEMATICAL PSYCHOLOGY Series Editors: H. Colonius (University of Oldenburg, Germany) E. N. Dzhafarov (Purdue University, USA) Published Vol. 1: The Global Structure of Visual Space by T. Indow Vol. 2: Theories of Probability: An Examination of Logical and Qualitative Foundations by L. Narens Vol. 3: Descriptive and Normative Approaches to Human Behavior edited by E. Dzhafarov & L. Perry Vol. 4: Discovering Cognitive Architecture by Selectively Influencing Mental Processes by R. Schweickert, D. L. Fisher & K. Sung Vol. 5: Probabilistic Lattices: With Applications to Psychology by L. Narens EH - Probabilistic Lattices.indd 1 2/12/2014 9:55:04 AM Advanced Series on Mathematical Psychology Vol. 5 Probabilistic Lattices With Applications to Psychology Louis Narens University of California, Irvine, USA World Scientifi c NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI 9345_9789814630412_tp.indd 2 3/12/14 9:25 am Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Narens, Louis. Probabilistic lattices : with applications to psychology / by Louis Narens. pages cm -- (Advanced series on mathematical psychology ; vol. 5) Includes bibliographical references and index. ISBN 978-981-4630-41-2 (hardcover : alk. paper) 1. Probabilities. 2. Probabilities--Psychological aspects. 3. Lattice theory. 4. Decision making-- Mathematical models. 5. Psychology--Mathematical models. I. Title. QA273.4.N372 2015 519.2--dc23 2014038123 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2015 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore EH - Probabilistic Lattices.indd 2 2/12/2014 9:55:04 AM December1,2014 13:45 WorldScientificBook-9inx6in Book2014a pagev For Kimberly and Beau v May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk December1,2014 13:45 WorldScientificBook-9inx6in Book2014a pagevii Acknowledgments I want to thank Benjamin Feintzeig, Lisa Guo, and Sarita Rosenstock for many suggestions and corrections. They greatly helped to improve the quality of the text and the correctness of the mathematics. The research for this book was supported by grants FA9550-08-1-0389 and FA9550-13-1-0012fromtheAirForceOfficeofScientificResearchandgrant SMA-1416907 from National Science Foundation. Figure6.2wasreprintedwithpermissionfromtheJournal of Mathematical Physics. Copyright 1972, AIP Publishing LLC. vii May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk December1,2014 13:45 WorldScientificBook-9inx6in Book2014a pageix Contents Acknowledgments vii 1. Introduction 1 1.1 What This Book is About . . . . . . . . . . . . . . . . . . 1 1.2 Level and Kind of Mathematics Used . . . . . . . . . . . . 2 1.3 How I Came to Write This Book . . . . . . . . . . . . . . 3 1.4 Differentiation from Fuzziness . . . . . . . . . . . . . . . . 8 1.5 Toward a Generalized Theory of Probability . . . . . . . . 9 1.6 Extended Probability Functions . . . . . . . . . . . . . . . 10 1.7 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.8 Forms of Complementation . . . . . . . . . . . . . . . . . 18 1.9 Rationality . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.10 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2. Basic Lattice Theory 27 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 General Notation, Conventions, and Definitions . . . . . . 28 2.3 Lattices: Definitions and Duality . . . . . . . . . . . . . . 31 2.4 Distributive and Modular Lattices . . . . . . . . . . . . . 37 2.5 Lattices with Negative Operators . . . . . . . . . . . . . . 43 2.6 Representation Theorems . . . . . . . . . . . . . . . . . . 47 2.7 Miscellaneous Results . . . . . . . . . . . . . . . . . . . . 50 2.8 Unique Complementation and Distributivity . . . . . . . . 52 2.9 Orthomodular Lattices . . . . . . . . . . . . . . . . . . . . 61 2.10 Finite Boolean Lattices. . . . . . . . . . . . . . . . . . . . 67 3. Pseudo Complemented Distributive Lattices 71 ix
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