The Alternative Mathematical Model of

Linguistic Semantics and Pragmatics

International Federation for Systems Research International Series on Systems Science and Engineering

Series Editor: George J. Klir State University of New York at Binghamton

Editorial Board Gerrit Broekstra Ivan M. Havel Erasmus University, Rotterdam, Charles University, Prague,

The Netherlands John L. Casti

Czechoslovakia Manfred Peschel

Santa Fe Institute, New Mexico Brian Gaines

Academy of Sciences, Berlin, Germany Franz Pichler

University of Calgary, Canada University of Linz, Austria

Volume 1

Volume 2

Volume 3

Volume 4

Volume 5

Volume 6

Volume 7

Volume 8

ANTICIPATORY SYSTEMS: Philosophical, Mathematical, and Methodological Foundations Robert Rosen FOUNDATIONS OF MATHEMATICAL SYSTEM DYNAMICS: The Fundamental Theory of Causal Recursion and Its Application to Social Science and Economics Arvid Aulin MErASYSTEMS MErHODOWGY: A New Synthesis and Unification Arthur D. Hall, III PREDICTIVE SIMPliCITY: Induction Exhumed Kenneth S. Friedman DYNAMICS AND THERMODYNAMICS IN HIERARCHICAlLY ORGANIZED SYSTEMS Pierre Auger SELF-MODIFYING SYSTEMS: A New Frameworkfor Dynamics, Information, and Complexity Gyorgy Kampis FACETS OF SYSTEMS SCIENCE George 1. Klir THE ALTERNATNE MATHEMATICAL MODEL OF liNGUISTIC SEMANTICS AND PRAGMATICS Vilem Novak

IFSR was established "to stimulate all activities associated with the scientific study of systems and to coordinate such activities at international level. " The aim of this series is to stimulate publication of high-quality monographs and textbooks on various topics of systems science and engineering. This series complements the Federation's other publications.

A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further information please contact the publisher. Volumes 1-6 were published by Pergamon Press.

The Alternative Mathematical Model of

Linguistic Semantics and Pragmatics

Vih~m Novak Czechoslovak Academy of Sciences

Mining Institute Ostrava-Poruba, Czechoslovakia

Springer Science+Business Media, LLC

Library of Congress Catalog1ng-1n-Pub11 cation Data

Novak, Vi lern. The a l t e r n a t i v e mathematical model of l i n g u i s t i c semantics and

pragmatics / VHém Novak. p. cm. — ( I n t e r n a t i o n a l F e d e r a t i o n f o r Systems Research

I n t e r n a t i o n a l s e r i e s on systems s c i e n c e and engineering ; v. 8) Includ e s b i b l i o g r a p h i c a l r e f e r e n c e s and Index.

1. Semantics—Mathematical models. 2. Pragmatics. I. T i t l e . I I . S e r i e s : IFSR i n t e r n a t i o n a l s e r i e s on systems s c i e n c e and engineering ; v. 8. P325.5.M36N68 1992 40V .43*015118—dc20 92-28913

CIP

ISBN 978-1-4899-2319-6 ISBN 978-1-4899-2317-2 (eBook) DOI 10.1007/978-1-4899-2317-2

© Springer Science+Business Media New York 1992 Originally published by Plenum Press, New York in 1992

Softcover reprint of the hardcover 1st edition 1992

All rights reserved

No part of this book 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

To my sons, David and Martin

Foreword

The vast domain of meaning includes the semantic-pragmatic patterning of the system of language and also issues of language's logical analysis, and of its relationships to cognitive patterns, to knowledge organization, and so on. Up to now, various component parts of this domain have been analyzed, either by linguistic approaches, which connect them with the layers both of syntax and of the lexicon, or by formal methods of logic and inquiries into knowledge representation and other aspects of cognitive science. However, not much has been achieved in the area that concerns one of the most characteristic features of linguistic meaning: the fuzziness of its articulation, its vagueness.

In the present book, Vilem Novak attacks the problem of vagueness directly, overcoming disadvantages inherent in most of the theoreticiu approaches to linguistics, which perhaps only state and vaguely discuss the fuzziness of meaning-usually without presenting an explicit framework for its description. Thus this book fills a gap in our knowledge of semantics in that it formulates the main points of a new approach to meaning, and this approach deserves to be further elaborated vis-a-vis such detailed questions as those concerning quantifiers and their scopes, comparatives and their complementations, the relations of control and binding, and many other issues the author could not handle in this book.

Vilem Novak's understanding of natural language meaning is based on mathe­matical and linguistic approaches deserving of more attention than has been paid to them; these approaches concern the dependency-based conceptualization of sentence syntax as well as Petr Vopenka's alternative set theory. The latter theory, which represents a new view of the fundamentals of mathematics, is applied here in close connection with Lotfi Zadeh's theory of fuzzy sets, and the level of linguistic (literal) meaning (or of underlying sentence structure) is systematically discussed here, its description substantially enriched through a treatment of the vagueness of meaning.

The present monograph constitutes an important step in the development of our understanding of the nature of meaning because it certainly will evoke further

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viii Foreword

discussions on how the oppositions inherent in the linguistic patterning of semantics and pragmatics are shaped within and captured by the human mind, and how we can adequately describe them.

Petr Sga11

Preface

This book is a result of discussions in the Prague group of theoretical linguists and logicians concerning the semantics of natural language and the possibilities of modeling it using mathematical tools. Within this group, two approaches exist. The first one, represented mainly by Professor Pavel Materna and his colleagues (T. Vlk, T. Chrz, 1. Peregrin, and others), is based on intensional logic. The second is based on alternative set theory and is adhered to in the present volume. Despite the different mathematical backgrounds of the two approaches, they have many common points through which they influence each other. Most important, the linguistic basis of both mathematical models of the semantics of natural language is the functional generative description of natural language initiated by Professor Petr Sgall and developed for many years by him and his group (E. Hajicova, 1. Panevova, P. Pit'ha, and others).

In this book, we present a theory stemming from principles that in many important points differ from those well known in classical mathematics. They provide us with different views of the world, and I am convinced the new approach is more natural and closer to the human way of regarding the world than the classical one. We challenge the reader to judge whether this is true and whether the theory is worthy of further development.

I would like to express my thanks to many people who took part in producing this book: first, to my wife, Daria, for her support; to many of my colleagues who encouraged me, discussed the topic with me, or participated in any way in developing this book; to the leadership of the Mining Institute of the Czechoslovak Academy of Sciences in Ostrava, for allowing me the time and opportunity to write this book; to Professor Ellis Cumberbatch for careful reading of parts of the manuscript and improving its language; and to Professor George 1. Klir for his encouragement in publishing this book with Plenum Publishing Corporation.

My special thanks belong to Professor Petr Sgall, who first suggested that I write this book, and subsequently read the whole text and helped me to improve it

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x Preface

in many places. I am, however, the only person fully responsible for the whole text, and any possible mistakes it contains.

Vilem Novak Ostrava, Czechoslovakia

Contents

Chapter 1. Claims and Assumptions ........................... 1 1.1. The Origins of Our Approach .................. 1 1.2. Why the Alternative Set Theory ................ 2 1.3. Objects and Properties ........................ 7 1.4. Fundamental Assumptions ..................... 9 1.5. Notation and the Main Notions ................. 11

Chapter 2. The Alternative Set Theory .............. 0. . . . . . . . . . . 15 2.1. The Main Features of AST ......... . . . . . . . . . . . . 15 2.2. The Principles of AST ........................ 17

2.2.1. Sets and Their Universe ................. 17 2.2.2. Classes ............................... 21 2.2.3. The Language of Class Formulas ......... 28

2.3. Projective Hierarchy of Classes ................. 30 2.4. Real Classes and Topology in AST .............. 34

2.4.1. Indiscernibility Relations ................ 34 2.4.2. Figures and Elements of Topology. . . . . . . . . 36

2.5. Rational and Real Numbers .................... 40 2.6. Extended Universe and Codable Classes. . . . .. .. . . 42 2.7. Automorphisms of the Extended Universe ........ 46 2.8. Arithmetic Cuts of Classes ..................... 47

Chapter 3. Functional Generative Description of Natural Language. . 49 3.1. Levels of Linguistic Description ................ 49 3.2. The Units of the Tectogrammatical Level ......... 50 3.3. Topic-Focus Articulation ...................... 53 3.4. A Dependency Grammar for the Tectogrammatical

Level ...................................... 55

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xii Contents

3.5. The Representation of Meaning on the Tectogrammatical Level ....................... 58

Chapter 4. Fuzzy Logic and Fuzzy Sets ........................ 61 4.1. The Structure of 'fruth Values .................. 61 4.2. Fuzzy Sets .................................. 63 4.3. First-Order Fuzzy Logic ....................... 66

4.3.1. Language ............................. 66 4.3.2. Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3.3. Deduction . . . .. . . . . . . . .. . . . . . .. . . .. . . . 69 4.3.4. Theories of First-Order Fuzzy Logic ....... 71 4.3.5. Completeness Theorems ................. 74

4.4. Fuzzy Approach in AST ....................... 75 4.4.1. Fuzzy Classes in AST .................. 75 4.4.2. Operations with Fuzzy Classes. . . . . . . . . . . . 84

Chapter 5. The Alternative Mathematical Model of Linguistic Semantics and Pragmatics .......................... 87 5.1. General Approach to the Meaning of Lexical Units 87 5.2. Semantic Universe ........................... 99 5.3. The Meaning of Basic Lexical Units and of Their

Forms ...................................... 109 5.3.1. Nouns ................................ 109 5.3.2. Adjectives .... . . . . . . . . . . . . . . . . . . . . . . . . 116 5.3.3. Verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.3.4. Adverbs .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.3.5. Quantifiers ............................ 136

5.4. The Semantics of Syntagms and Close Coordination 139 5.5. Verb Complementation and the Semantics of Simple

Clauses ..................................... 145 5.6. Free Coordination and the Semantics of Complex

and Compound Sentences ...................... 153 5.6.1. The Semantics of Complex Sentences ...... 153 5.6.2. The Semantics of Compound Sentences .... 159 5.6.3. The Semantics of Conditional Sentences. . .. 160

5.7. Pragmatics and Anaphora ...................... 162 5.8. Cognitive Layer Representation, Interpretation, and

Truth Conditions ............................. 167 5.8.1. The Structure of the Stock of Knowledge ... 167 5.8.2. Interpretation of the Utterance ............ 171

Contents xiii

5.8.3. Truth Conditions and the Semantics of Assertions ............................ 172

5.9. Examples ................................... 174

Chapter 6. Comments on Possible Practical Application and Further Development . . . . . . . . . . .. . . . . . . . .. .. . . . . . . . . . . . . . 185 6.1. Fuzzy Set Representation of the Meaning of

Lexical Units ................................ 185 6.2. Possibilities of Practical Applications ............ 191 6.3. Further Development and Its Limits ............. 193

Bibliography ................................................ 195

List of Symbols 199

Subject Index ............................................... 201

The Alternative Mathematical Model of

Linguistic Semantics and Pragmatics