Non-commutative Multiple-Valued Logic Algebras
This monograph provides a self-contained and easy-to-read introduction to non-commutative multiple-valued logic algebras; a subject which has attracted much interest in the past few years because of its impact on information science, artificial intelligen
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Lavinia Corina Ciungu
Noncommutative Multiple-Valued Logic Algebras
Lavinia Corina Ciungu Department of Mathematics University of Iowa Iowa City, IA, USA
ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-3-319-01588-0 ISBN 978-3-319-01589-7 (eBook) DOI 10.1007/978-3-319-01589-7 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2013947576 Mathematics Subject Classification: 03G10, 06F35, 06D35, 03G25, 03B50, 06F20, 06D50, 08A30, 08B26, 08A72, 03B52, 03G12, 28E10 © Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Dedicated to my precious son David Edward
Introduction
In 1920 Łukasiewicz introduced his three valued logic ([223]), the first model of multiple-valued logic. The n-valued propositional logic for n > 3 was constructed in 1922 and the ℵ0 -valued Łukasiewicz-Tarski logic in 1930 ([224]). The first completeness theorem for ℵ0 -valued Łukasiewicz-Tarski logic was given by Wajsberg in 1935. As a direct generalization of two-valued calculus, Post introduced in 1921 an n-valued propositional calculus distinct from that of Łukasiewicz ([239]). In the early 1940s Gr.C. Moisil was the f
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