Probabilistic Lattices: With Applications to Psychology

206 Pages·2014·2.363 MB·English

by  Louis Narens

Probabilistic Lattices
With Applications to Psychology
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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
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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
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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
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is not required from the publisher.
Printed in Singapore
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For Kimberly and Beau
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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.
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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