Soft Computing in Humanities and Social Sciences

Rudolf Seising and Veronica Sanz (Eds.) Soft Computing in Humanities and Social Sciences

Studies in Fuzziness and Soft Computing, Volume 273 Editor-in-Chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw Poland E-mail: [email protected] Further volumes of this series can be found on our homepage: springer.com Vol. 258. Jorge Casillas and Francisco José Martínez López Marketing Intelligent Systems Using Soft Computing, 2010 ISBN 978-3-642-15605-2 Vol. 259. Alexander Gegov Fuzzy Networks for Complex Systems, 2010 ISBN 978-3-642-15599-4 Vol. 260. Jordi Recasens Indistinguishability Operators, 2010 ISBN 978-3-642-16221-3 Vol. 261. Chris Cornelis, Glad Deschrijver, Mike Nachtegael, Steven Schockaert, and Yun Shi (Eds.) 35 Years of Fuzzy Set Theory, 2010 ISBN 978-3-642-16628-0 Vol. 262. Zsófia Lendek, Thierry Marie Guerra, Robert Babuška, and Bart De Schutter Stability Analysis and Nonlinear Observer Design Using Takagi-Sugeno Fuzzy Models, 2010 ISBN 978-3-642-16775-1 Vol. 263. Jiuping Xu and Xiaoyang Zhou Fuzzy-Like Multiple Objective Decision Making, 2010 ISBN 978-3-642-16894-9 Vol. 264. Hak-Keung Lam and Frank Hung-Fat Leung Stability Analysis of Fuzzy-Model-Based Control Systems, 2011 ISBN 978-3-642-17843-6 Vol. 265. Ronald R. Yager, Janusz Kacprzyk, and Prof. Gleb Beliakov (eds.) Recent Developments in the Ordered Weighted Averaging Operators: Theory and Practice, 2011 ISBN 978-3-642-17909-9

Vol. 266. Edwin Lughofer Evolving Fuzzy Systems – Methodologies, Advanced Concepts and Applications, 2011 ISBN 978-3-642-18086-6 Vol. 267. Enrique Herrera-Viedma, José Luis García-Lapresta, Janusz Kacprzyk, Mario Fedrizzi, Hannu Nurmi, and Sławomir Zadro˙zny Consensual Processes, 2011 ISBN 978-3-642-20532-3 Vol. 268. Olga Poleshchuk and Evgeniy Komarov Expert Fuzzy Information Processing, 2011 ISBN 978-3-642-20124-0 Vol. 269. Kasthurirangan Gopalakrishnan, Siddhartha Kumar Khaitan, and Soteris Kalogirou (Eds.) Soft Computing in Green and Renewable Energy Systems, 2011 ISBN 978-3-642-22175-0 Vol. 270. Christer Carlsson and Robert Fullér Possibility for Decision, 2011 ISBN 978-3-642-22641-0 Vol. 271. Enric Trillas, P. Bonissone, Luis Magdalena and Janusz Kacprzyk Combining Experimentation and Theory, 2011 ISBN 978-3-642-24665-4 Vol. 272. Oscar Castillo Type-2 Fuzzy Logic in Intelligent Control Applications, 2012 ISBN 978-3-642-24662-3 Vol. 273. Rudolf Seising and Veronica Sanz (Eds.) Soft Computing in Humanities and Social Sciences, 2012 ISBN 978-3-642-24671-5

Rudolf Seising and Veronica Sanz (Eds.)

Soft Computing in Humanities and Social Sciences

ABC

Editors Dr. Rudolf Seising European Centre for Soft Computing Edificio Investigación 3a Planta. C Gonzalo Gutiérrez Quirós S/N 33600 Mieres, Asturias Spain E-mail: [email protected] Dr. Veronica Sanz Science, Technology and Society Center University of California at Berkeley 470 Stephens Hall Berkeley, CA 94720-2350 USA E-mail: [email protected]

ISBN 978-3-642-24671-5

e-ISBN 978-3-642-24672-2

DOI 10.1007/978-3-642-24672-2 Studies in Fuzziness and Soft Computing

ISSN 1434-9922

Library of Congress Control Number: 2011940007 c 2012 Springer-Verlag Berlin Heidelberg  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Typeset & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed on acid-free paper 987654321 springer.com

Preface

The field of Soft Computing in Humanities and Social Sciences is at a turning point. Not very long ago, the very label seemed a little bit odd. Soft Computing is a technological field while Humanities and Social Sciences fall under the other pole of the “two cultures” defined by C.P. Snow in 1959. In the recent years, however, this has changed. The strong distinction between “science” and “humanities” has been criticized from many fronts and, at the same time, an increasing cooperation between the so-called “hard sciences” and “soft-sciences” is taking place in a wide range of scientific projects dealing with very complex and interdisciplinary topics. In the last fifteen years the area of Soft Computing has also experienced a gradual rapprochement to disciplines in the Humanities and Social Sciences, and also in the field of Medicine, Biology and even the Arts, a phenomenon that did not occur much in the previous years (to the surprise of the very founder of the field, Lotfi Zadeh). The collection of this book presents a generous sampling of the new and burgeoning field of Soft Computing in Humanities and Social Sciences, bringing together a wide array of authors and subject matters from different disciplines. Some of the contributors of the book belong to the scientific and technical areas of Soft Computing while others come from various fields in the humanities and social sciences such as Philosophy, History, Sociology or Economics. The six sections in which the volume is divided represent some of the most relevant topics that have result from fruitful exchanges taken place on this topic in the last years in several workshops, seminars and special sessions. These are only an example of what the interesting encounter and conversations between Soft Computing and the Humanities and Social Sciences can yield in the future. As this book will appear in 2011, it feels well to address some special dates and events that happened during the time we worked with the manuscripts that have been collected here. In the year 2009 Abe Mamdani left us unexpectedly and so Ladislaus Kohout passed away. In July 2010 Jaume Casasnovas also left us just a few weeks after he agreed to write a contribution to this book. In memory of him we include a dedication written by his colleague and friend Gaspar Mayor.

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On the other hand, the publication of this book coincides with some happy events as well. In February 2011 Lotfi Zadeh became 90 years old and half a year before his theory of Fuzzy Sets and Systems became 45 years old. In addition, the general field of Soft Computing is approaching its age of majority as it has been alive for 20 years now. To all of these milestones we want to dedicate this book. We want to thank the Foundation for the Advancement of Soft Computing, the Scientific Committee of the European Centre for Soft Computing (ECSC) in Mieres, Asturias (Spain) and, especially, to the General Director of the ECSC, Luis Magdalena, and the two emeritus researchers Claudio Moraga and Enric Trillas of the unit of “Fundamentals of Soft Computing” for their help in the development of this project. We are also very grateful to Springer Verlag (Heidelberg) and in particular to Dr. Thomas Ditzinger for helping this edition find its way onto the publisher’s list, and likewise to Janusz Kacprzyk (Warsaw), who accepted the book into the series Studies in Fuzziness and Soft Computing. Finally, we would like to thank all the contributors for their enthusiastic participation in this book and for creating with us the path for the development of the promising field of Soft Computing in Humanities and Social Sciences.

June, 2011

Rudolf Seising Veronica Sanz González Mieres (Asturias), Spain and Berkeley, California, USA

Jaume Casasnovas In Memoriam

Jaume Casasnovas was born in Palma (Mallorca, Spain) in 1951. He past away too soon, at the age of 59. In 1973, he finished his B.S. degree in Mathematics at the University of Barcelona. He later received his doctorate in Computer Science, in 1989, from the University of the Balearic Islands (UIB). In 1975, after two years working as an Assistant Professor at the University of Barcelona, he became a Secondary School teacher. He remained in this position until September 1980, when he took over as a Secondary School technical supervisor. In 1994, he won a position as Associate Professor at the University of the Balearic Islands. He remained in this position at the department of Mathematics and Computer Science at the UIB until he passed away on July 14th, 2010. His interest in the teaching of Mathematics was what fuelled his participation in designing various projects such as Mathematics for Education, which is still part of the curriculum of the Master of Mathematics, the post-graduate studies of Mathematics and the teaching of Mathematics in Secondary Education in the UIB.

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Jaume Casasnovas In Memoriam

Jaume Casasnovas was without a doubt a role model in the world of Mathematics education in our community and his influence will be felt for a long time, especially among his university students to whom he was able to instil a big dose of enthusiasm and devotion for this subject. Jaume was not only a great teacher but also a good researcher. In 1989, under the leadership of Professor Josep Miró, he finished his Ph.D. in Computer Science, with the thesis “Contribution to a formalization of the direct inference”, in which he made a study of inference of approximate knowledge. This work manages concepts and results of areas such as approximate knowledge, multiple-valued logic, theories of possibility and need, etc. Without a doubt, his relationship with Professor Miró was what turned Jaume’s interest towards research in approximate knowledge. During the early eighties, thanks to the significant support of Nadal Batle, Llorenç Valverde, and Enric Trillas, who was responsible for the introduction of Fuzzy Logic in Spain, there was a growing interest in the study of this budding theory in Palma. Years later Gaspar Mayor was able to create the research group LOBFI at the UIB, whose main interest is the study of the mathematical aspects of Fuzzy Logic. Jaume was part of this group for several years. He also offered his knowledge to the research group BIOCOM (UIB) in the fields of Computational Biology and Bioinformatics. His contribution to these research groups has left a scientific and human imprint that will be remembered for a long time. I would not wish to end without mentioning Jaume’s personal side. The dictionary defines the word bonhomie as affability, plainness, kindness, honesty. I believe that Jaume Casasnovas has been an excellent example of this.

September 2010

Gaspar Mayor Head of LOBFI research group

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

V

Jaume Casasnovas In Memoriam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII Gaspar Mayor

Part I: Introduction 1

From Hard Science and Computing to Soft Science and Computing – An Introductory Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rudolf Seising, Veronica Sanz 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Hard Science and Hard Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 From Hard Science to Artificial Intelligence . . . . . . . . . . . . . . . . . . . . 1.4 Softening Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 From Artificial Intelligence to Soft Computing . . . . . . . . . . . . . . . . . . 1.6 Soft Computing and Soft Sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 The Contributions in This Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 6 13 17 22 26 27 30 34

Part II: General Overviews of Soft Computing in Humanities and Social Sciences 2

On Some “family resemblances” of Fuzzy Set Theory and Human Sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Settimo Termini 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.2 Zadeh’s Question and a Tentative Answer . . . . . . . . . . . . . . . . . . . . . . 42

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2.3 A Few Conceptual Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 A Tentative Analysis of Another Example: Trust Theory . . . . . . . . . . 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 47 52 53

Warren Weaver’s “Science and complexity” Revisited . . . . . . . . . . . . . . Rudolf Seising 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Warren Weaver – A Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . 3.3 Science and Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Lotfi A. Zadeh, Fuzzy Sets and Systems – The Work of a Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 57 59 63 70 84 85

How Philosophy, Science and Technologies Studies, and Feminist Studies of Technology Can Be of Use for Soft Computing . . . . . . . . . . . 89 Veronica Sanz 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.2 Philosophical Approaches to AI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.3 STS Studies of AI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.4 Feminist Analysis of AI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.5 Soft Computing Confronting Philosophical Critiques to AI . . . . . . . . 99 4.6 Soft Computing Confronting STS Critiques to AI . . . . . . . . . . . . . . . . 101 4.7 Soft Computing Confronting Feminist Critiques to AI . . . . . . . . . . . . 103 4.8 Conclusion Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Part III: Philosophy, Logic and Fuzzy Logic 5

On Explicandum versus Explicatum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Settimo Termini 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.2 Carnap’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.3 A Unifying Framework for Looking at Typical Case Studies . . . . . . . 116 5.4 As a Sort of Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6

Axiomatic Investigation of Fuzzy Probabilities . . . . . . . . . . . . . . . . . . . . . 125 Takehiko Nakama, Enric Trillas, Itziar García-Honrado 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.3 Probability Functions as Membership Functions . . . . . . . . . . . . . . . . . 127

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6.4 Fuzzy-Crisp Probability: Probability Measures for Fuzzy Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.5 Conditional Probability and Independence for Fuzzy Events . . . . . . . 132 6.6 Fuzzy-Fuzzy Probability: Fuzzy Numbers as Probabilities . . . . . . . . 135 6.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7

Fuzzy Deontics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Kazem Sadegh-Zadeh 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.2 Deontic Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.3 Deontic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.4 Fuzzy Deontics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

8

Soft Deontic Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Txetxu Ausín, Lorenzo Peña 8.1 Deontic Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.2 Theoretical Assumptions of Standard Deontic Logic . . . . . . . . . . . . . 158 8.3 Shortcomings of Standard Deontic Logic . . . . . . . . . . . . . . . . . . . . . . . 160 8.4 Elements for a Soft Deontic Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 8.5 The Underlying Quantificational Calculus . . . . . . . . . . . . . . . . . . . . . . 164 8.6 DV System of Soft Deontic Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 8.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Part IV: Soft Computing, Natural Language and Perceptions 9

Retrieving Crisp and Imperfect Causal Sentences in Texts: From Single Causal Sentences to Mechanisms . . . . . . . . . . . . . . . . . . . . . 175 Cristina Puente, Alejandro Sobrino, José Ángel Olivas 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 9.2 Causality and Information Retrieval Systems . . . . . . . . . . . . . . . . . . . 176 9.3 Retrieving Conditional Sentences in Texts . . . . . . . . . . . . . . . . . . . . . . 178 9.4 Retrieving Crisp and Imperfect Causal Sentences in Texts . . . . . . . . . 183 9.5 Generating an Example of Imperfect Causal Graph . . . . . . . . . . . . . . 189 9.6 Conclusions and Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

10 Facing Uncertainty in Digitisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Lukas Gander, Ulrich Reffle, Christoph Ringlstetter, Sven Schlarb, Klaus Schulz, Raphael Unterweger 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

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10.2 Language Technology for OCR and Information Retrieval on Historical Document Collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 10.3 The Functional Extension Parser (FEP) . . . . . . . . . . . . . . . . . . . . . . . . 200 10.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 11 The Role of Synonymy and Antonymy in ‘Natural’ Fuzzy Prolog . . . . 209 Alejandro Sobrino 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 11.2 Fuzzy Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 11.3 Towards a Linguistic Fuzzy Prolog: Linguistic Relations of Synonymy and Antonymy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 11.4 Resolution with Synonymy and Antonymy . . . . . . . . . . . . . . . . . . . . . 224 11.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 12 On an Attempt to Formalize Guessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Itziar García-Honrado, Enric Trillas 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 12.2 Towards the Problem: Where Can Knowledge Be Represented? . . . . 239 12.3 Towards the Concept of a Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 240 12.4 The Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 12.5 The Properties of the Operators of Conjectures . . . . . . . . . . . . . . . . . . 246 12.6 Kinds of Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 12.7 On Refutation and Falsification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 12.8 The Relevance of Speculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 12.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 13 Syntactic Ambiguity Amidst Contextual Clarity . . . . . . . . . . . . . . . . . . . 257 Jeremy Bradley 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 13.2 Linguistic Ambiguity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 13.3 Human Indifference to Unclear Language . . . . . . . . . . . . . . . . . . . . . . 258 13.4 Vagueness in the German Language . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 13.5 Aphasia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 13.6 Other Applications of Text Simplification . . . . . . . . . . . . . . . . . . . . . . 265 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 14 Can We Learn Algorithms from People Who Compute Fast: An Indirect Analysis in the Presence of Fuzzy Descriptions . . . . . . . . . 267 Olga Kosheleva, Vladik Kreinovich 14.1 People Who Computed Fast: A Historical Phenomenon . . . . . . . . . 267 14.2 People Who Computed Fast: How They Computed? . . . . . . . . . . . . 268

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14.3

People Who Computed Fast: Their Self-explanations Were Fuzzy and Unclear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 14.4 With the Appearance of Computers, the Interest in Fast Human Calculators Waned . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 14.5 Why Interest Waned: Implicit Assumptions . . . . . . . . . . . . . . . . . . . 269 14.6 A Surprising 1960s Discovery of Fast Multiplication Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 14.7 Fast Multiplication: Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 271 14.8 Interest in Fast Human Calculators Revived . . . . . . . . . . . . . . . . . . . 271 14.9 Direct Analysis Is Impossible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 14.10 Indirect Analysis: Main Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 14.11 Data That We Can Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 14.12 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 14.13 Possible Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 15 Perceptions: A Psychobiological and Cultural Approach . . . . . . . . . . . . 277 Clara Barroso 15.1 The Brain: An Organ That Communicates with Other Organs . . . . . . 277 15.2 The Functional Perspective on Perception . . . . . . . . . . . . . . . . . . . . . . 278 15.3 The Importance of Psychology in the Construction of Perceptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 15.4 Perception and the Construction of Meaning . . . . . . . . . . . . . . . . . . . . 280 15.5 Perception and Artificial Intelligence . . . . . . . . . . . . . . . . . . . . . . . . . . 281 15.6 Concept Maps and Contextual Meaning . . . . . . . . . . . . . . . . . . . . . . . . 283 15.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Part V: Soft Models in Social Sciences and Economics 16 Rule Based Fuzzy Cognitive Maps in Humanities, Social Sciences and Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 João Paulo Carvalho 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 16.2 Dynamic Cognitive Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 16.3 A Qualitative Macro Economic Model as an Example of DCM Modeling in Socio-Economic Systems . . . . . . . . . . . . . . . . . . . . . . . . . 291 16.4 Conclusions, Applications, and Future Developments . . . . . . . . . . . . 298 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 17 Voting on How to Vote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 José Luis García–Lapresta, Ashley Piggins 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 17.2 Majorities Based on the Difference in Support . . . . . . . . . . . . . . . . . . 303

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k Majorities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 17.3 Self-selective M 17.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 18 Weighted Means of Subjective Evaluations . . . . . . . . . . . . . . . . . . . . . . . . 323 Jaume Casasnovas, J. Vicente Riera 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 18.2 Discrete Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 18.3 Subjective Evaluations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 18.4 n-Dimensional Aggregation Functions . . . . . . . . . . . . . . . . . . . . . . . . . 333 18.5 Group Consensus Opinion Based on Discrete Fuzzy Weighted Normed Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 18.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 19 Some Experiences Applying Fuzzy Logic to Economics . . . . . . . . . . . . . 347 Bárbara Díaz, Antonio Morillas 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 19.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 19.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Part VI: Soft Computing and Life Sciences 20 Fuzzy Formal Ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Kazem Sadegh-Zadeh 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 20.2 Ordinary Formal Ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 20.3 Fuzzy Formal Ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 20.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 21 Computational Representation of Medical Concepts: A Semiotic and Fuzzy Logic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 Mila Kwiatkowska, Krzysztof Michalik, Krzysztof Kielan 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 21.2 Representation of Medical Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 21.3 A Framework for Modeling Changeability and Imprecision of Medical Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 21.4 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

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Part VII: Soft Computing and Arts 22 Invariance and Variance of Motives: A Model of Musical Logic and/as Fuzzy Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Hanns-Werner Heister 22.1 Introduction: Congruities, Similarities, Analogies between Fuzzy Logic and Musical Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 22.2 Models: The B-A-C-H-Motif . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 22.3 Outlook: Generalization and Connection of Fuzzy Logic with Further Musical and Artistic Subject Fields . . . . . . . . . . . . . . . . . . . . . 441 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 23 Mathematics and Soft Computing in Music . . . . . . . . . . . . . . . . . . . . . . . 451 Teresa León, Vicente Liern 23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 23.2 Some Concepts and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 23.3 Notes as Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 23.4 A Numerical Experiment and Sequential Uncertainty . . . . . . . . . . . . . 459 23.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 24 Music and Similarity Based Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 Josep LLuís Arcos 24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 24.2 SaxEx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 24.3 TempoExpress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 24.4 Identifying Violin Performers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 25 T-Norms, T-Conorms, Aggregation Operators and Gaudí’s Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 Amadeo Monreal 25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 25.2 Gaudí’s Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 25.3 Generalizing Gaudí’s Columns Using T-Norms and T-Conorms . . . . 485 25.4 Generalizing Gaudí’s Column Using Aggregation Operators . . . . . . . 490 25.5 A Park Güell’s Tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 25.6 The Limit Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 25.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 Abstracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

Part I

1 From Hard Science and Computing to Soft Science and Computing – An Introductory Survey

Rudolf Seising and Veronica Sanz

1.1

Introduction

Humanities and Sciences conform the “two cultures” of modern society. “Literary intellectuals” on the one hand and scientists on the other established a deep gulf or rift in between. That is the point of the Rede’s-Lecture The Two Cultures and the Scientific Revolution that was given by Sir Charles Percy Snow (1905-1980) at May 7, 1959 in the Senate House in Cambridge [40]. In this lecture he made the case of a breakdown of communication between these “two cultures” and he claimed that this breakdown is a major hindrance to solve the problems of the world. Without looking beyond one’s own nose, he claimed, both cultures will become poorer. Offspring of these two cultures is our classification of Hard and Soft Science. “Hardness” of science means stringency, precision and unexceptional validity of the natural laws through mathematical formulation, whereas “soft” sciences do not have these properties. It is said that hard sciences use empirical, experimental and quantifiable data; that hard scientists operate with precise scientific methods and that they are targeted on exactness and objectivity. Soft sciences, on the contrary, are said to be based on conjectures and rely on qualitative data and analysis. Examples of “hard sciences” are physics and chemistry. As examples of “soft” scientific disciplines are included psychology, anthropology, and the social sciences. Although a great deal of humanities and social sciences operates nowadays with quantitative data and “hard” methodologies, the dichotomy of hard/soft sciences is still dominant. Thus, physics, chemistry, engineering are among the so-called hard sciences, while humanities and social sciences are called soft scientific disciplines1 – however superficial this classification can be. 1

There are two different classifications of academic disciplines in the different traditions. The English language distinguishes between the “natural sciences”, which study natural phenomena, and “social sciences”, which study human behavior and societies. These two groups of scientific fields have empirical approaches. Contrariwise there are the “humanities” (often grouped together wiht the “Arts”) and this concept embraces all studies of the human condition using analytical, critical, or speculative methods suchnas philosophy, or literature. In contrast to this, the German language area distinguishes between “Naturwissenschaften” (natural sciences) and “Geisteswissenschaften” (first used as a translation of John Stuart Mill’s (1806-1873) term “moral sciences”) that covers Philosophy,

R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 3–36. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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1 From Hard Science and Computing to Soft Science and Computing

Four years later, in The Two Cultures: A Second Look. An Expanded Version of The Two Cultures and the Scientific Revolution, Snow augured much more optimistically: A new culture, a “third culture” would emerge and close the communication gap between the two old cultures and, because of this, in this third culture the literary intellectuals would be on speaking terms with the scientists [41]. Another 30 years later, John Brockman published The Third Culture: Beyond the Scientific Revolution. In his introduction to this book he disagreed with Snow’s thesis of a third culture: “In Snow’s third culture, the literary intellectuals would be on speaking terms with the scientists. Although I borrow Snow’s phrase, it does not describe the third culture he predicted. Literary intellectuals are not communicating with scientists. Scientists are communicating directly with the general public. Traditional intellectual media played a vertical game: journalists wrote up and professors wrote down. Today, third culture thinkers tend to avoid the middleman and endeavor to express their deepest thoughts in a manner accessible to the intelligent reading public.” [5] For Brockman, the third-culture thinkers are new public intellectuals, “they represent the frontiers of knowledge in the areas of evolutionary biology, genetics, computer science, neurophysiology, psychology, and physics. Some of the fundamental questions posed are: Where did the universe come from? Where did life come from? Where did the mind come from? Emerging out of the third culture is a new natural philosophy, founded on the realization of the import of complexity, of evolution. Very complex systems – whether organisms, brains, the biosphere, or the universe itself - were not constructed by design; all have evolved.” [5] However, in the second half of the 20th century some more open-minded observers of the social, cultural, and scientific systems turned to more complex views than the dichotomic hard/soft-view. In 2007 the historian and journalist of science Michael Shermer wrote in his column in the Scientific American: “I have always thought that if there must be a rank order (which there mustn’t), the current one is precisely reversed. The physical sciences are hard, in the sense that calculating differential equations is difficult, for example. The variables within the causal net of the subject matter, however, are comparatively simple to constrain and test when contrasted with, say, computing the actions of organisms in an ecosystem or predicting the consequences of global climate change. Even the difficulty of constructing comprehensive models in the biological sciences pales in comparison to that of modeling the workings of human brains and societies. By these measures, the social sciences are the hard disciplines, because the subject matter is orders of magnitude more complex and multi-faceted.” [39] Shermer turned the order of hard and soft sciences but, beyond that, he praised those scientists who belong also to the new public intellectuals, the scientist who History, Philology, Social Sciences, and also Theology and Jurisprudence; thus, the majority of these disciplines would come under the anglo-saxonian “Humanities”, but it does not contain any arts. The German historian, philosopher and sociologist Wilhelm Dilthey (1833-1911) considered the “Naturwissenschaften” on the one hand and the “Geisteswissenschaften” on the other, but he did not consider the status of mathematics and of philosophy in this classification.

1.1 Introduction

5

know how to write on their scientific results not only in peer-reviewed journals but in publications for the big audience: “We are storytellers. If you cannot tell a good story about your data and theory that is, if you cannot explain your observations, what view they are for or against and what service your efforts provide then your science is incomplete.” In his view, the distinction between technical and narrative writing on science and to brush off that narrative writing is unsophisticated and shallow-brained. “Between technical and popular science writing is what I call “integrative science”, a process that blends data, theory and narrative. Without the three of these metaphorical legs, the seat on which the enterprise of science rests would collapse”, [39] he wrote, and followed by arguing that in many of these “wellcrafted narratives” we get new and “higher-order works of science” that we can use to find the answers of the last big questions! Here, Shermer argues close to Brockman and his list of “integrative” working scientists which has some members in common with Brockman’s list of scientific writers searching to answer the “last questions” in popular works, such as Stephen Jay Gould2 , Richard Dawkins3 and Steven Arthur Pinker4 . However, Brockman does not believe in a “third culture” that will activate the communication between the two former cultures, but in the popularization of science as a third culture. In contrast, Shermer sees no point in sharping partitions of different “cultures” of scientific work. He argues that there are scientists of various talents to teach, to research, and to tell the stories of scientific developments. If we can integrate these areas, then science is complete, that is to say: “Integrative Science is hard science”! In this introduction we will illuminate the relations between hard and soft sciences and hard and soft computing with each other. In section 1.2 we tackle a route from Hard science to the research program of Artificial Intelligence (AI) in the second half of the 20th century and in section 1.3 we show the merging of hard science and computing in modern sciences. In section 1.4 we will see that around 1900 some “softening methodologies” appeared in the area of modern science that arose from the spot of the important relationships between logic and language. In section 1.5 we show why the new field of Soft Computing broke in this research. In section 1.6 we consider the initiation of Soft Computing in the last decade of the 20th century and in our outlook-section we plea for the use of soft computing methodologies in the soft sciences. The last section gives a short view on the collection of contributions to this volume5 . 2 3 4 5

See e.g. [17], [18]. See e.g. [12], [13]. See e.g. [30], [31]. However, a list of the author’s abstracts of these contributions is given at the end of this book.

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1.2 1.2.1

1 From Hard Science and Computing to Soft Science and Computing

Hard Science and Hard Computing Hard Science

In science we have a traditional division of work: on the one hand we have fundamental, logical and theoretical investigations; on the other, we have experimental examinations and applications. The theoretical work in science uses logics and mathematics to formulate axioms and laws and it is linked with the philosophical view of rationalism. The other aspects of science use experiments to find, prove or refute natural laws and have their roots in philosophical empiricism. In both directions – from experimental results to theoretical laws or from theoretical laws to experimental proves or refutations – scientists have to bridge the gap that separates theory and practice in science. Beginning as early as the 17th century, a primary quality factor in all scientific work has been a maximal level of exactness. Galileo Galilei (1564-1642) and René Descartes (1596-1650) started the process of giving modern science its exactness through the use of the tools of logic and mathematics. The language of mathematics has served as a basis for the definition of theorems, axioms, definitions, and proofs. The works of Sir Isaac Newton (1642-1727), Gottfried Wilhelm Leibniz (1646-1716), Pierre Simon de Laplace (1749-1827) and many others led to the ascendancy of modern science, fostering the impression that scientists were able to represent all the facts and processes that people observe in the world, completely and exactly. But this optimism has gradually begun to seem somewhat naïve in view of the discrepancies between the exactness of theories and what scientists observe in the real world. From the empiricist point of view, on the other hand, the source of our knowledge is sense experience. The English philosopher John Locke (1632-1704) used the analogy of the mind of a newborn as a “tabula rasa” that will be written by the sensual perceptions the baby has later. In Locke’s opinion, this perceptions provide information about the physical world. Locke’s view is called “material empiricism” whereas the so called “idealistic empiricism” was hold by George Berkeley (1684-1753) and David Hume (1711-1776) an Irish and a Scottish philosopher, respectively: their idea is that there exists no material world, only perceptions are real. The basis of science, their assumptions and implications, their methods and results, their theories and experiments, all are reflected by philosophers of science. Here, we can distinguish between the philosophy of astronomy, physics, chemistry, and other empirical sciences, and we can also be interested in philosophy of social sciences and the humanities. These different philosophies of scientific disciplines arose in different historical periods but the earliest philosophical reflections on modern science started with theories and experiments in mechanics in the 17th century. The two main views in philosophy of science by that time, rationalism and empiricism, arose in about the same period and in both directions – from experimental results to theoretical laws or from theoretical laws to experimental proves or refutations –: scientists had to bridge the gap that separates theory and practice in science. However, this tradition is deeply interconnected with the concepts of determinism and prediction as the following very famous paragraph in the Essai Philosophique sur les Probabilités written by Laplace shows:

1.2 Hard Science and Hard Computing

7

“We ought to regard the present state of the universe as the effect of its antecedent state and as the cause of the state that is to follow. An intelligence knowing all the forces acting in nature at a given instant, as well as the momentary positions of all things in the universe, would be able to comprehend in one single formula the motions of the largest bodies as well as the lightest atoms in the world, provided that its intellect were sufficiently powerful to subject all data to analysis; to it nothing would be uncertain, the future as well as the past would be present to its eyes. The perfection that the human mind has been able to give to astronomy affords but a feeble outline of such an intelligence.” [9] Beyond that, Laplace’s expression of determinism shows the connection between the traditions of hard science and the performance of mathematical calculations, i.e. “computing”. Before the 20th century these were “mathematical calculations by hand”, but, since, then it became what an electronic computer does, or simply “computing”. The historical path that linked hard sciences and computing started with astronomy and physics, continued in electrodynamics and electrical engineering, and it opened out after World War II into computing, information and communication technologies. This development has been told as an enormous success story, and in the early part of the 20th century the time was ripe for this prosperity to become a subject of philosophy in two respects: • in Philosophy of Mathematics • in Philosophy of Sciences In the early stages of the 20th century there were several groups of European scientists and philosophers who concerned themselves with the interrelationships between logics, mathematics, science, and the real world, e.g. in Berlin, Cambridge, Göttingen, Jena, Warsaw, and in Vienna. The so-called Vienna Circle was supposed to carry on the analytical tradition of the Austrian physicist and philosopher Ernst Mach (1838-1916) and these scholars regularly debated these issues over a period of years until the annexation of Austria by Nazi Germany in 1938 marked the end of the group. The scientific-philosophical direction of Logical Empiricism (or Neo-Positivism) was originated in this “Vienna Circle”. Here, the conceptual analysis of theories – it was believed – would bring with it advancements in the understanding of nature. The important questions were: • What is the status of our knowledge? • What is the status of statements about this knowledge? • How are science and the world connected by logic? The Vienna Circle was thus concerned with the three-way relationship of world – language – science. Its goal was to found a scientific philosophy with a new perspective and, in particular, to destroy metaphysics. To this end, metaphysical sentences first had to be demarcated from scientific sentences. Moritz Schlick (1882-1936), the leader of the group, had also introduced the criterion of verifiability, which his

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1 From Hard Science and Computing to Soft Science and Computing

student Friedrich Waismann (1896-1956) described as follows: “If one cannot indicate any ways in which a sentence is true, then the sentence has no meaning whatsoever, for the meaning of a sentence is the method by which it is verified.” In his Tractatus Logico-Philosophicus [44], the Viennese philosopher Ludwig Wittgenstein (1889-1951) had espoused a logistical philosophy of language according to which the world can be clearly represented in the form of sentences. This book had been “talked through sentence by sentence” in the Vienna Circle, as Rudolf Carnap (1891-1970) wrote in his autobiography ([7], p. 39.). “Whereof one cannot speak, thereof one must be silent.” (“Wovon man nicht sprechen kann, darüber muss man schweigen.”) is the last proposition in the Tractatus that appeared in German in 1921 and one year later in a bilingual edition (German and English). Wittgenstein and his work were supported by Bertrand Russell (1872-1970), who wrote an introduction to it where he tried to explain Wittgenstein’s thinking: “A picture”, he wrote, “is a model of the reality, and to the objects in the reality correspond the elements of the picture: the picture itself is the fact. The fact that things have a certain relation to each other is represented by the fact that in the picture its elements have a certain relation to one another. “In the picture and the pictured there must be something identical in order that the one can be a picture of the other at all. What the picture must have in common with reality in order to be able to represent it after its manner – rightly or falsely – is its form of representation.” (2.161, 2.17) ([33], p. 10) Wittgenstein’s Tractatus says that the world consists of facts. Facts may or may not contain smaller parts. If a fact has no smaller parts, he calls it an “atom atomic fact.” If we know all atomic facts, we can describe the world completely by corresponding “atomic propositions.” Here it is proposed that “the logical picture of the facts is the thought” and that “the thought is the significant proposition”. Finally: “The totality of propositions is language.” ([44], prop. 4.001) Of course, Wittgenstein argued that sentences in colloquial language are very complex. He conceded that there is a “silent adjustment to understand colloquial language” but it is “enormously complicated.” Therefore it is “humanly impossible to gather immediately the logic of language.” ([44], prop. 4.002) This is the task of philosophy: “All philosophy is ’Critique of language.’ ” ([44], prop. 4.0031) However, Wittgenstein knew that common linguistic usage is vague, but at the time when he wrote Tractatus, he tried to solve this problem by constructing a precise language – an exact logical language that gives a unique picture of the real world. He influenced many philosophers in the era before the Second World War, e.g. most of the members in the Vienna Circle and, first of all, Carnap, the author of Der logische Aufbau der Welt (The Logical Structure of the World) [8]. Therefore, in the years before the Second World War, Wittgenstein and many philosophers thought that this “linguistic turn” solved all philosophical problems to the point that Wittgenstein himself left philosophy and returned to Austria to become a primary school teacher. Contemporaneously, philosophy of mathematics became an spectacular research area. After the establishment of formal logic and set theory, philosophical thinking had to clarify the relationships of mathematics to logic and reality, to science

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and humanities, and especially to psychology. Philosophers and mathematicians asked for the foundations of mathematics and logics and they divided into various schools of thought because of the different pictures of mathematical epistemology and ontology that was given by their different philosophical views, e.g.: • Logicism, the thesis that mathematics is reducible to logic, e.g. represented by Russell and Carnap, • Psychologism, the position that mathematical truths are reducible to psychological facts, e.g. represented by Gustave Le Bon (1841-1931), • Intuitionism, a program that denies non-experienced mathematical truths, represented first by Luitzen E. Brouwer (1881-1966), • Conventionalism, a position that chooses mathematical axioms for the results they produce, not for their apparent coherence with our intuitions about the physical world, e.g. represented by Henri Poincaré (1854-1912), • Formalism, the view that mathematical statements may be thought of as statements about the consequences of certain string manipulation rules, represented first by David Hilbert (1862-1943). In 1900 on the Second International Congress of Mathematicians held in Paris, Hilbert presented “The Problems of Mathematics”. Among these 23 unsolved problems in 1900 - not all of the problems in mathematics but problems on mathematics – some became solved along the 20th century, but others are still unsolved and, for some of them, it is now mathematically proved that they are, as a matter of principle, unsolvable. Hilbert’s second problem was the most famous case: to prove that our arithmetic is a non-contradictory formal system. Since that time, and especially in the 1920s, it was Hilbert’s intention to motivate mathematicians to establish mathematics on a fix ground i.e. a complete and consistent set of axioms, from which researchers could deduce all mathematical truths by the usual methods of first order logic or “predicate” logic. Eight years later, in 1928, he lectured for the International Congress of Mathematicians held in Bologna. In continuation of his list of problems in 1900 he asked three questions: 1. Is mathematics complete? – Can we prove or refute every mathematical statement given a finite set of axioms? 2. Is mathematics consistent? – Can we be sure that we can prove only the true mathematical statements? 3. Is every statement in mathematics decidable? – Can we find an algorithm (with a description of a formal language as input) that takes a mathematical statement in this language and produces “true” or “false” as output either according to whether the statement is true or false. The second question is again Hilbert’s second problem from 1900, and the third question became known as “Hilbert’s Entscheidungsproblem” (English: decision problem). These three questions remained unsolved for two more years and as late as 1930 Hilbert believed that there would be no such thing as an unsolvable problem, as we can read in his conclusion words of his retirement address to the Gesellschaft

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der deutschen Naturforscher und Ärzte (Society of German Scientists and Physicians) in the fall of 1930 in Königsberg: Wir müssen wissen. ... Wir werden wissen. (English: We must know. ... We will know.). But just one day before and almost at the same place, young Viennese mathematician Kurt Gödel (1906-1978) announced in a roundtable discussion during the Conference on Epistemology (held jointly with the Society meetings) that “one can even give examples of propositions (and in fact of those of the type of Goldbach or Fermat) that, while contentually true, are unprovable in the formal system of classical mathematics” [14]. That was the first expression of his incompleteness theorem. Six and seven years later, respectively, the American mathematician Alonzo Church (1903-1995) and Alan Turing published independent papers showing that there exists no general solution to the Entscheidungsproblem [10], [22]: it is impossible to decide algorithmically whether all statements in arithmetic are true or false, which in fact means that everything computable is computable algorithmically. This, in short, is called the “Church-Turing Thesis” – a thesis that cannot be formally proven but that characterizes what is called “(hard) computing”.6 1.2.2

Hard Computing

Traditional histories of computation consider Alan Mathison Turing (1912-1954) as the “father” of both computing and Artificial Intelligence (AI). However, we have to make an excursus here about some precedents of Turing’s ideas that are sometimes not so much recognized. Almost a century before, the English mathematician and mechanical engineer Charles Babagge (1791-1871) tried to construct a physical machine, the “Analytical Engine”, that can be considered the first model of the current computer. With regard to the very idea of applying an algorithm to that machine (that is, the idea of “programming”) it was Ada Lovelace (1815-1852) who, in her extended notes to the translation of a lecture given by Babagge in 1842, recognized for the first time that Babagge’s differential machine could be an excellent tool for mathematical calculations. Although this project was finally never constructed, she defined for the first time what could be a “thought machine”, a computational machine that could be programmed for different purposes7. However, despite these important attempts in the 19th century, the mathematical basis for the theory of computation were set up by Alan Turing, who, while still being a graduate student at Princeton University (in 1936), he developed the concepts that now are considered as the basic elements of computation. His main contribution was to apply the idea of a “methodical process” (what people perform when pursuing any kind of organized action) to something that can be done “mechanically” by 6

7

Of course, in the 1950s it didn’t exist a difference between “hard computing” and “soft computing” but just “computing”, which corresponds with our definition of “hard computing” in this book. It can therefore be said that we owe to Augusta Ada King, Countess of Lovelace the very idea of “computer programming”, an idea that she continued developing in later writings where she also advanced terms such as “looping” and “subroutine” that are nowadays central in software programming.

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a machine. Though he didn’t construct such a device, he mathematically demonstrated that this could be possible by proposing a hypothetical machine known since then as the “Turing machine”. Turing gave for the first time the formal definition of what should count as a “definite method” (or, in modern language, simply “an algorithm”). That machine would be able to perform certain elementary operations by using a series of instructions, which have to be written in symbols of formal language (that is, in a precise form). His idea was that these symbols could be translated into a physical medium (which in Turing’s example consisted on a paper tape). An “effective algorithm” was defined by Turing as a series of instructions that, applied to a set of data, allow us to achieve correct results. Turing’s argument goes on by telling that, if each particular algorithm can be written out as a set of instructions in the same standard form, there could be a universal machine that can do what any other particular Turing machine would do. The “Universal Turing Machine” embodies the essential principle of the computer: a single machine for all possible tasks. He proved that this abstract machine that represented the process of computing on a paper band subdivided into fields could solve every conceivable mathematical problem as long as there was an algorithm for it. In his paper “On Computable Numbers, with an Application to the Entscheidungsproblem” [22], Turing reformulated Gödel’s incompleteness-findings and he replaced Gödel’s universal, arithmetic based, formal language with simple, formal “automata”. On the other hand, Turing also introduced another idea that would have a great influence for the epistemology and ontology of computation and AI: that an “effective algorithm” connects in a bijective way language symbols with actions in the machine (that is: for every symbol of the language, there is one – and only one – action associated to it in the machine, and vice versa). The process of writing the instructions on a physical device and retrieving the output on the same medium implies that the device has to “keep” the information and the instructions for some time. Because of that, Turing supposed that the machine should have some “internal states”, which he assumed to be equivalent to states of the mind in the human brain. In this way Turing draw a bridge between the mental (identified as logical) and the physical worlds, between thought and action, in so crossing previous conventional boundaries. This is not by any means a trivial supposition, and its relevance for the epistemology and ontology of computation and AI was enormous, as we will see. Turing’s machine was a purely theoretical model, a kind of universal computer. However, this abstract idea of an automatic calculating machine was to be realized ten years later in the so-called era of computers that started in the 1940s. The first one was the electro-mechanical Z3 computer that was designed in 1941 by Konrad Zuse (1919-1995) in Germany; the second one was the “the first electronic computer ABC” (Atanasoff-Berry-Computer) created by John V. Atanasoff (1903-1995) and Clifford E. Berry (1918-1963), but it was not general-purpose computer; the third one (also a not general-purpose) were the digital and electronic “Colossus” computers in England designed by Tommy Flowers (1905-1998) with the help of Alan Turing. These were used to decrypt the Germans’ Enigma codes in the II. World

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War in 1943. And, finally, we have the electro-mechanical computer Harvard Mark I, designed by Howard Aiken (1900-1973) in 1944. (For details see [32]). Based on Turing’s achievements, the idea of “computing machine” changed in the late 1940s from the earlier conception of “computers” (or sometimes “computors”) as humans that performed computations (mainly women), to apply that name to the machine that, based on digital equipment, was able to perform anything that could be described as “purely logical”. Many thousands of women “computers” were employed in the 1930s and 40s in commerce, government, and research establishments in the USA. In fact, the relationship of women with math and their work as “computers” is prior to the creation of computers. Since the eighteenth century many women were employed to perform calculations in the field of astronomy, aviation, navigation and military research. Their math abilities and their patience, persistence and organizational skills were appreciated when performing complex tasks (all of which was related to the female gender.) This association remained when earlier mechanical calculators were introduced. Because of Turing’s demonstration that computation could be used for more than just mathematical calculations, the study of computability began to be a ’science’ and the expression “computing machine” started to refer to this new machines such as the first general-purpose electronic computers that were the war products such as ENIAC (Electronic Numerical Integrator and Computer) and EDVAC (Electronic Discrete Variable Computer), designed by John Presper Eckert (1919-1995) and John William Mauchly (1907-1980). ENIAC was announced in the year 1946 but already in the spring of 1945 the mathematician John von Neumann (1903-1957) was asked to prepare a report on the logical principles of its successor, the EDVAC (since the ENIAC had not had any such description and it had been sorely missed). The ENIAC had an electronic working memory, so the individual processing operations of the entered data were exceptionally fast. However, each program that was going to be run had to be hard wired, and so reprogramming required several hours of work.8 Von Neumann recognized very quickly, though, that this was a major drawback to the huge computer, and he was soon looking for ways to modify it. Today, the novel concept of a central programming unit in which programs are stored in a coded form is attributed to John von Neumann. Instead of creating the program by means of the internal wiring of the machine, the program is installed directly in the machine. Basic operations like addition and subtraction remain permanently wired in the machine, but the order and combinations of these basic functions could be 8

By 1946 almost all the people working as programmers in the ENIAC were women, including supervisory positions. These women were commonly known as “the ENIAC girls” (see [20], [16] and [23]). As happens with all gender divisions, this one was also related to a hierarchy of values in which the work of women “computers” was considered routinary and not too creative. Partly because of this, historians of computing have focused their attention on the development of hardware (performed mainly by electrical engineers) which was considered the real groundbreaking “innovation”, in the early era of computers, in so devaluing the initial development of programming (or software) more connected with the feminine.

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varied by means of instructions that were entered into the computer just like the data. The EDVAC was not supposed to suffer from the “childhood diseases” that had afflicted the ENIAC. To this end, Neumann’s principle of store programming was used, and the principle that went down in the history of the computer as “Von Neumann architecture” was realized for the first time. [28] Today, computers are used in all scientific disciplines and as well in humanities and social sciences, and (Hard) computing is used to find exact solutions. However, not all problems can be resolved with these methods. As Lotfi A. Zadeh (born 1921), the founder of the theory of Fuzzy Sets, mentioned many times over the last decades, humans are able to resolve tasks of high complexity without measurements or computations. In conclusion, he stated that “thinking machines” — i.e. computers as they were named in their starting period -– do not “think” as humans do. From the mid-1980s he focused on “Making Computers Think like People” [54]. For this purpose, the machine’s ability “to compute with numbers” — hard computing -– has to be supplemented by an additional ability more similar to human thinking: “soft computing”. He explained: “what might be referred to as soft computing -– and, in particular, fuzzy logic — to mimic the ability of the human mind to effectively employ modes of reasoning that are approximate rather than exact.” He claimed: “In traditional — hard — computing, the prime desiderata are precision, certainty, and rigor. By contrast, the point of departure in soft computing is the thesis that precision and certainty carry a cost and that computation, reasoning, and decision making should exploit – wherever possible – the tolerance for imprecision and uncertainty.” [54]. – But before we will consider this Soft Computing approach, let’s have a look at the development of the field of Artificial Intelligence and, then, how “hard science” became soften!

1.3

From Hard Science to Artificial Intelligence

A decade after the constructionof the first computers, a new discipline was going to flourish also out of the ideas of Alan Turing, a discipline of computer science that was going to be named as Artificial Intelligence. The term “Artificial Intelligence” was used for the first time in 1956. In the summer of that year the American mathematician John McCarthy (born 1927) organized “The Dartmouth Summer Research Project on Artificial Intelligence” held in Dartmouth, New Hampshire (USA), which is considered the key milestone in the development of AI9 . This conference brought together four of the researchers that were to become the developers of the new field in the following decades: Marvin Minsky (born 1927), John McCarthy, Herbert A. Simon (1916-2001) and Allen Newell (1927-1992). There, McCarthy coined for the first time the term “Artificial Intelligence”, which was defined later by Minsky 9

The Dartmouth Conference of 1956 was organized together with Marvin Minsky and two scientists from the MIT and the Bell Labs and IBM, resp.: the mathematician and electrical engineer Claude E. Shannon (1916-2001) and the electrical engineer Nathaniel Rochester (1919-2001).

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as “the science of making machines do things that would require intelligence if done by men” (Minsky, [27]).10 . At that seminar, Simon and Newell presented their first AI system named LT (Logical Theorist) that they had being developing during the previous year. The program followed a symbolic-reasoning way, where a set of symbols represent concepts and a set of rules let the system manage these symbols to arrive deductively to conclusions. Simon and Newell claimed that LT “reasons logically”, being one of the achievements of the systems to demonstrate some theorems of Russell and Whitehead’s Principia Mathematica. Following Turing’s ideas, Symbolic AI (also known as Symbol Processing Theory) is the tradition in AI that emphasizes representation in symbolic form, and became the main paradigm in AI for the following three decades. As Trillas pointed out ([42]), Symbolic AI implies not only the use of symbols but also the use of first-order binary-logic: “The machine could only be able to compute if it is previously programmed with complete detail and precision, that is, with no ambiguity (the algorithm would be effective only if it can be traduced trough precise rules to actions in the machine)” ([42], p. 23) The proposal for the Darmouth conference included this assertion: “[We believe that] every aspect of learning or any other feature of intelligence can be so precisely described that a machine can be made to simulate it”.11 Though Symbolic AI was the main approach from the 1960s to 1980s, it was not the only approach, not even in the early days. In fact, another approach was developed already in the 1940s: the Artificial Neural Networks approach, also called the “Connectionism”. In 1943, Warren McCulloch (1898-1969) and Walter Pitts (19231969), a neurophysiologist and a mathematician respectively, proposed the idea of building a physical machine that simulates neural connections in our brain. Their computational approach to mind was not based on the symbolic aspect of reasoning but on the human brain’s neurological model, which they try to simulate through artificial neural networks. McCulloch and Pitts’ early development was constructed in an electromechanical way. The basic idea was that training an artificial neural network involves the reinforcement of the right associations between inputs and outputs by altering electromechanical (or electronic) weights, until a stable state is achieved, and then tested against new input data to achieve new correct activations. McCulloch and Pitts’ idea was later extended by Donald Hebb (1904-1985), who, based on Santiago Ramón y Cajal’s (1852-1934) development that the repeated activation of one neuron through synapses increments the conductivity of the network, proposed that artificial neural networks could also “learn”. In this model each neuron is seen as an elemental processor of information and the brain as a “Parallel Distributed Processing System” (PDP). A main influence for early connectionism was the Behaviourist paradigm prevalent in psychology by that time. Behaviourism (with Burrhus Frederic Skinner (1904-1990) as his main representative) didn’t believe in the existence of any mental states (as the model of cognition of symbolic paradigm affirmed) but described 10 11

It is interesting to point out that, though unconciously, the definition recalls a traditional masculinist way of talking as it is said “done by men” instead of “humans”. Original in Spanish, translated by the authors.

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cognition as a process of association between stimuli and reinforcements called “learning” or “training (something that, on the other hand, scientists can observe). By the end of 1960s, behaviourism was substituted by the Cognitive approach as the main paradigm in psychology, which affected the development of this branch in AI. Much of the problem of the abandonment of connectionism was precisely its close relation to the decline of behaviourism. Another factor for this abandonment was the fierce fight for funding in those years, being the representatives of Symbolic AI who “won the battle” – mainly because of their strong criticisms of some (on the other hand rather practical) failures of the Artificial Neural Networks approach.12 Going back to Symbolic AI as it was the main paradigm during many decades and the most influential in terms of foundations of the discipline (also called Classical AI, Traditional AI or “GOFAI”13) it can be defined as the branch of artificial intelligence research that attempts to explicitly represent human knowledge in a declarative form (i.e. facts and rules). The main assumptions of Symbolic AI are: a) Human reasoning is equivalent to “symbol processing”. In symbolic-type reasoning a set of symbols represents concepts, and a well-defined set of rules can handle these concepts to infer conclusions in a deductive way14 . Reasoning is, then, equivalent to formal logic but, moreover, it also rests in the possibility that this can be described in terms of physical symbols (the “physical symbol system hypothesis (PSSH)” proposed by Allen Newell and Herbert A. Simon). This is precisely what goes on into a computer, a principle that Turing had already demonstrated theoretically. b) The way reason is applied to solve problems of the world (“problem solving”) is through serial decision-making process: a process of listing alternative choices (understood as discrete states), determining the consequences of each of them and evaluating them in terms of ’means-ends’ analysis, which allows a rationaldecision making. It is implicit in this conception that there is always a “best” (rational) choice. The claim of Symbolic AI is that this procedural and sequential knowledge (which can be implicit in some human performances) can be made explicit and formalized by means of symbols and operations for their manipulation. Although Simon himself recognized that this idealized model was not always followed by humans (in part because they don’t always know all the choices and their consequences), this was not seen as an objection to his rationalistic account of a 12

13 14

Minsky and Papert wrote in 1969 a book with the only goal of criticizing connectionism, which is considered to be the main cause of the shortage of funding to connectionist projects in the following years. The return of artificial neural networks in the late 1980s (although in combination with other developments) was in turn due to the failures of some traditional Symbolic AI projects to cope with their early predictions. A detailed discussion of the fight between Symbolic AI and Connectionist AI can be found in [29]. GOFAI is the acronym for “Good-Old-Fashioned-AI”, termed by John Haugeland [21] 1985, see [1], p. 38). Although in logic there are three kind of reasoning methods (induction, abduction, and deduction) the most relevant and widely used was deduction.

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machine that can deduce all consequences from a set of data and rules. Based on this idea and after presenting LT in Dartmouth, Simon and Newell developed in 1957 another program called GPS (General Problem Solving) which, in their terms, was devoted to mimic “general problem solving abilities”. This system was, again, a rule-based symbolic reasoning system. They based their data and rules on examples of people’s accounts of the ways they solved logical problems (which was called “protocols”). The subjacent assumption was that any problem can be always characterized and formally described by a number of discrete operating states. The interesting conclusion of these authors from an epistemic point of view is that, though their examples consisted on problem solving strategies within a very concrete bounded realm of logic problems (i.e. numerical cryptograms), they extrapolated it to “the very nature of general problem solving” ([1], p. 37)15. However, after some years, GPS failed to fulfil most of its promises and, around 10 years after they have launched it, Newell announced that they abandoned the GPS program. Nevertheless, this failure was seen by Simon and Newel as a failure of implementation – that is, a technical problem caused by the scarce capacity of that time computers to cope with the complexity of real-life problems –, but not a problem of the approach itself. Symbolic AI achieved its major success in the 1970s with the sub-discipline known as “Expert Systems” or “Knowledge-Based Systems (KBS)”. This success was in part due to the more modest expectations of these systems compared to previous symbolic systems, focusing instead on more concrete domains of human knowledge performances. Seeing the difficulty of finding general principles of problem solving, AI researchers reduced the scope to well-known problems that human experts are able to solve in concrete domains. Expert systems are based on the knowledge of one or more human experts in a particular domain (i.e. medicine, chemistry or engineering) assuming that these experts are able to explain their knowledge in an explicit and precise way. The experts are asked about the knowledge and strategies they use to solve problems in their work, and, then, this knowledge is explicitly represented in a knowledge database mostly composed by logical statements and “if. . . then” rules. Then, an inference (deductive) engine is added to handle the stored data and to infer new conclusions. The optimist claims about knowledge-based systems was due to the success of some programs that were applied to real-world problems. The two most cited examples are MYCIN and PROSPECTOR; MYCIN was an interactive program that simulates a specialist in infectious blood diseases that was able to diagnose and prescribe the appropriate medication based on the medical data available and on the answers provided. (The system also added an explanation of how it came to that diagnosis). This system was in fact used in several hospitals16 . On the other hand, the system known as PROSPECTOR was used to advice mineral exploration, and acquired a great recognition because it was able to predict the drilling at a site 15 16

This, of course, implies a deterministic view that is also shown in the use of the term “search” that assumes the existence of one “best solution” to any (real) problem. However, because of its experimental character this system was used as an “adviser” to physicians and not as an isolate “diagnoser”.

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that contained a large deposit of molybdenum which was previously dismissed by geologists, see [42]. Despite these successes and some other industrial applications of KBS, not all the expert systems were so successful. AI researchers recognized that the “bottle-neck” of expert systems was the so-called “knowledge-acquisition” phase (where they try to “extract” the knowledge from the experts) and its formalization processes. To cope with that challenge was created a new sub-discipline within AI called “Knowledge Engineering” where the design stage of an expert system involved a great deal of interaction with the human experts not only during the “acquisition” phase but also during the multiple feedback and improvement cycles. Because of this, scholars from other disciplines such as Philosophy and Science and Technology Studies have been interested in founding different explanations to the failures of Symbolic AI systems [1], [11], [15]. Critiques to the traditional approaches to AI have not been in vain and some authors within AI started to challenge some of the claims of the traditional paradigm, presenting themselves as alternatives approaches. That is the case of Lotfi A. Zadeh, the theory of Fuzzy Sets and his subsequent development of the whole area of Soft Computing.

1.4

Softening Science

History of science shows also some roots in Philosophy of Science and Mathematics that lead us to a process that we name “softening science”. Philosophical reflections of science and mathematics in the 20th century followed not just one linear path but during the same time there were various deviant developments as we will demonstrate in three examples: 1.4.1

Logical Tolerance

Our first road starts in the late Vienna Circle and it leads to a sort of conventionalism in the area of logics. Here, it was the young Austrian mathematician Karl Menger (1902-1985) who tried to “soften” the group’s philosophical view on science. This was not easy and it started with little success, but at the end the “principle of logical tolerance” was established and even emphasized by Carnap in his famous book of 1934: Logical Syntax of language [6]. While the prevailing opinion in the Vienna Circle was that there could be just one logic and one language in science, Menger – conditioned by his involvement with Brouwer’s intuitionism during the year he spent in Amsterdam to receive his qualification for a professorship (Habilitation) – expoused the view that it was perfectly reasonable to introduce and use different languages and logics in science. When he came back from Amsterdam as a professor of geometry, he was invited to give a lecture on Brouwer’s intuitionism to the Vienna Circle. In this talk he rejected the view held by almost all members of this group that there is one unique logic. He claimed that we are free to choose axioms and rules in mathematics, and thus we

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are free to consider different systems of logic. He realized the philosophical consequences of this assumption, which he shared with his student Kurt Gödel: “the plurality of logics and language entail some kind of logical conventionalism” ([25], p. 88). Later in the 1930s, when the Vienna Circle became acquainted with different systems of logics – e.g. the three and multi-valued logic founded by the Polish logician Jan Łukasiewicz (1878-1956), (which was also discussed by Alfred Tarski (1902-1983) in one of his Vienna lectures on “Metamathematics”,) and Brouwer’s intuitionistic logic – Carnap also defended this tolerant view of logic. In his lecture The New Logic: A 1932 Lecture, Menger wrote. “What interests the mathematician and all that he does is to derive propositions by methods which can be chosen in various ways but must be listed. And to my mind all that mathematics and logic can say about this activity of mathematician (which neither needs justification nor can be justified) lies in this simple statement of fact.” ([26], p. 35.) This “logical tolerance” proposed by Menger later became well-known in Carnap’s famous book Logische Syntax der Sprache (Logical Syntax of Language) in 1934 [6]. This was a basic principle for the contemplation of “deviant logics”, that is, systems of logic that differ from usual bivalent logic, for fuzzy logic! 1.4.2

Vagueness

The second route we consider as “softening science” is the concept of “Vagueness”. It was Bertrand Russell who published the first logico-philosophical article on “Vagueness” in 1923 [34]: “Let us consider the various ways in which common words are vague, and let us begin with such a word as ‘red’. It is perfectly obvious, since colours form a continuum, that there are shades of colour concerning which we shall be in doubt whether to call them red or not, not because we are ignorant of the meaning of the word ‘red’, but because it is a word the extent of whose application is essentially doubtful. This, of course, is the answer to the old puzzle about the man who went bald. It is supposed that at first he was not bald, that he lost his hairs one by one, and that in the end he was bald; therefore, it is argued, there must have been one hair the loss of which converted him into a bald man. This, of course, is absurd. Baldness is a vague conception; some men are certainly bald, some are certainly not bald, while between them there are men of whom it is not true to say they must either be bald or not bald.” ([34], p. 85). Russell showed that concepts are vague even though there have been and continue to be many attempts to define them precisely: “The metre, for example, is defined as the distance between two marks on a certain rod in Paris, when that rod is at a certain temperature. Now, the marks are not points, but patches of a finite size, so that the distance between them is not a precise conception. Moreover, temperature cannot be measured with more than a certain degree of accuracy, and the temperature of a rod is never quite uniform. For all these reasons the conception of a metre is lacking in precision.” ([34], p. 86) Russell also argued that a proper name cannot be considered to be an unambiguous symbol even if we believe that there is only one person with this name. Every person “was born, and being born is a gradual process. It would seem natural to suppose that the name was not attributable before birth; if so, there was doubt, while

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birth was taking place, whether the name was attributable or not. If it be said that the name was attributable before birth, the ambiguity is even more obvious, since no one can decide how long before birth the name become attributable.” ([34], p. 86) He reasoned “that all words are attributable without doubt over a certain area, but become questionable within a penumbra, outside which they are again certainly not attributable.” ([34], p. 86f) Then he generalized that words of pure logic such as “or”, “not” and “false” have no precise meanings either (e.g. in classical logic the composed proposition “p or q” is false only when p and q are false and true elsewhere). He went on to claim that the truth values “‘true’ and ‘false’ can only have a precise meaning when the symbols employed – words, perceptions, images . . . — are themselves precise”. As we have seen above, this is not possible in practice, so he concludes: “every proposition that can be framed in practice has a certain degree of vagueness; that is to say, there is not one definite fact necessary and sufficient for its truth, but certain region of possible facts, any one of which would make it true. And this region is itself ill-defined: we cannot assign to it a definite boundary.” Russell emphasized that there is a difference between what we can imagine in theory and what we can observe with our senses in reality: “All traditional logic habitually assumes that precise symbols are being employed. It is therefore not applicable to this terrestrial life, but only to an imagined celestial existence.” ([34], p. 88f). He proposed the following definition of accurate representations: “One system of terms related in various ways is an accurate representation of another system of terms related in various other ways if there is a one-one relation of the terms of the one to the terms of the other, and likewise a one-one relation of the relations of the one to the relations of the other, such that, when two or more terms in the one system have a relation belonging to that system, the corresponding terms of the other system have the corresponding relation belonging to the other system.” And in contrast to this, he stated that “a representation is vague when the relation of the representing system to the represented system is not one-one, but one-many.” ([34], p. 89) He concluded that “Vagueness, clearly, is a matter of degree, depending upon the extent of the possible differences between different systems represented by the same representation. Accuracy, on the contrary, is an ideal limit.” ([34], p. 90). “Vagueness. An exercise in logical analysis”, published in 1937 [2] was an answer to Russell’s article by the Cambridge philosopher and mathematician Max Black (1909-1988). Influenced by Russell, Wittgenstein and other analytical philosophers, Black differentiated vagueness from ambiguity, generality, and indeterminacy. He emphasized that “the most highly developed and useful scientific theories are ostensibly expressed in terms of objects never encountered in experience. The line traced by a draughtsman, no matter how accurate, is seen beneath the microscope as a kind of corrugated trench, far removed from the ideal line of pure geometry. And the ’point-planet’ of astronomy, the ’perfect gas’ of thermodynamics, and the ’pure species’ of genetics are equally remote from exact realization.” ([2], p. 427). Black proposed a new method to symbolize vagueness: “a quantitative differentiation, admitting of degrees, and correlated with the indeterminacy in the divisions made by a group of observers.” ([2], p. 441) He assumed that the

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vagueness of a word involves variations in its application by different users of a language and that these variations fulfil systematic and statistical rules when one symbol has to be discriminated from another. He defined this discrimination of a symbol x with respect to a language L by DxL(= Dx¬L).

Fig. 1.1. Consistency of application of a typical vague symbol ([2], p. 443).

Most speakers of a language and the same observer in most situations will determine that either L or ¬L is used. In both cases, among competent observers there is a certain unanimity, a preponderance of correct decisions. For all DxL with the same x but not necessarily the same observer, m is the number of L uses and n the number of ¬L uses. On this basis, Black stated the following definition: “We define the consistency of application of L to x as the limit to which the ratio m/n tends when the number of DxL and the number of observers increase indefinitely. [. . . ] Since the consistency of the application, C, is clearly a function of both L and x, it can be written in the form C(L, x).” ([2], p. 442; see figure 1.1) More than a quarter century later Black published the article “Reasoning with loose concepts”[3]. In this article he labelled concepts without precise boundaries as “loose concepts” rather than “vague” ones, in order to avoid misleading and pejorative implications . Once again, he expressly rejected Russell’s assertion that traditional logic is “not applicable” as a method of conclusion for vague concepts: “Now, if all empirical concepts are loose, as I think they are, the policy becomes one of abstention from any reasoning from empirical premises. If this is a cure, it is one that kills the patient. If it is always wrong to reason with loose concepts, it will, of course, be wrong to derive any conclusion, paradoxical or not, from premises in which such concepts are used. A policy of prohibiting reasoning with loose concepts would destroy ordinary language – and, for that matter, any improvement upon ordinary language that we can imagine.” ([3], p. 7). 1.4.3

Family Resemblances

Our last example of “softening science” is the late philosophy of Wittgenstein. When he returned to Cambridge after the World War II, he resigned his position at

1.4 Softening Science

21

Cambridge in 1947 to concentrate on a totally new philosophical system. He turned away from his “Tractatus philosophy” and its ideal mapping between the things in reality and a logical precise language. If we are not able to find such an exact logical language, then we have to accept the fact that for all languages there is a vague lingual usage. Then, the images, models, and theories that we build with words and propositions of our languages to communicate on them are, and will always be, vague. Already in his so-called Blue Book (a collection of Wittgenstein’s lecture manuscripts in 1933/34), we find the following paragraph: “This is a very one-sided way of looking at language. In practice we very rarely use language as such a calculus. For not only do we not think of the rules of usage – of definitions, etc. – while using language, but when we are asked to give such rules, in most cases we aren’t able to do so. We are unable clearly to circumscribe the concepts we use; not because we don’t know their real definition, but because there is no real ’definition’ to them. To suppose that there must be would be like supposing that whenever children play with a ball they play a game according to strict rules.” ([45], p. 49) Later, in the Philosophical Investigations, he wrote: “And this is true. – Instead of producing something common to all that we call language, I am saying that these phenomena have no one thing in common which makes us use the same word for all, – but that they are related to one another in many different ways. And it is because of this relationship, or these relationships, that we call them all “language”. I will try to explain this.” ([46], § 65). And continues: “Consider for example the proceedings that we call “games”’. I mean board-games, card-games, ball-games, Olympic games, and so on. What is common to them all? – Don’t say: “There must be something common, or they would not be called “games” “– but look and see whether there is anything common to all. – For if you look at them you will not see something that is common to all, but simi-larities, relationships, and a whole series of them at that. To repeat: don’t think, but look! – Look for example at boardgames, with their multifarious relationships. Now pass to card-games; here you find many correspondences with the first group, but many common features drop out, and others appear. When we pass next to ball-games, much that is common is retained, but much is lost. – Are they all “amusing”? Compare chess with noughts and crosses. Or is there always winning and losing, or competition between players? Think of patience. In ball games there is winning and losing; but when a child throws his ball at the wall and catches it again, this feature has disappeared. Look at the parts played by skill and luck; and at the difference between skill in chess and skill in tennis. Think now of games like ring-a-ring-a-roses; here is the element of amusement, but how many other characteristic features have disappeared! sometimes similarities of detail. And we can go through the many, many other groups of games in the same way; can see how similarities crop up and disappear. And the result of this examination is: we see a complicated network of similarities overlapping and criss-crossing: sometimes overall similarities.” ([46], § 66) Furthermore, Wittgenstein created a new concept to describe this new epistemological system: “I can think of no better expression to characterize these similarities than “family resemblances”; for the various resemblances between members of a

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family: build, features, colour of eyes, gait, temperament, etc. etc. overlap and criss-cross in the same way. And I shall say: “games” form a family.” ([46], § 67) Concepts and their families have no sharp boundaries as he also wrote in paragraph 119 of the Philosophical Investigations: “One might say that the concept “game” is a concept with blurred edges. – “But is a blurred concept a concept at all?” Is an indistinct photograph a picture of a person at all? Is it even always an advantage to replace an indistinct picture by a sharp one? Isn’t the indistinct one often exactly what we need? Frege compares a concept to an area and says that an area with vague boundaries cannot be called an area at all. This presumably means that we cannot do anything with it. – But is it senseless to say: “Stand roughly there”?” ([46], § 71) What do we want to argue, then, regarding these three approaches to “softening science” in the first half of the 20th century? That there existed possible ways that hard science would have became more “soft”. But, then, the computer came into play, hard computing and hard sciences got so strong stimulations that the fiction that hard science and hard computing will manage all problems of the world was almost generally accepted. With this mainstream in science and technology in the middle of the 20th century, the “softening science approaches” had to trickle away – for the time being. A renaissance of “Softening science” started in the 1990’s whith the growth of a group of new scientific methodologies that rubbed salt into hard science and hard computing’s wounds that were linked to non-exact knowledge, vague concepts, non-optimal solutions etc.: the methodologies of Fuzzy sets and systems, Neural networks and Evolutionary strategies, that is, Soft Computing.

1.5

From Artificial Intelligence to Soft Computing

In April 1965 the Berkeley researcher Lotfi Zadeh (born 1921) gave a speech on “A New View on System Theory” in Brooklyn. It was one of his first talks on Fuzzy sets and systems and his seminal article “Fuzzy Sets” [48] was already in press. In this talk he anticipated its substance, i.e. a new “way of dealing with classes in which there may be intermediate grades of membership”. He maintained that this new theory provided a “convenient way of defining abstraction – a process which plays a basic role in human thinking and communication.” [49] In 1969, Zadeh proposed his new theory of Fuzzy Sets to the life science community: “The great complexity of biological systems may well prove to be an insuperable block to the achievement of a significant measure of success in the application of conventional mathematical techniques to the analysis of systems.” [50] “By ‘conventional mathematical techniques’ in this statement, we mean mathematical approaches for which we expect that precise answers to well-chosen precise questions concerning a biological system should have a high degree of relevance to its observed behaviour. Indeed, the complexity of biological systems may force us to alter in radical ways our traditional approaches to the analysis of such systems. Thus, we may have to accept as unavoidable a substantial degree of fuzziness in the description of the behaviour of biological systems as well as in their characterization.” [50]

1.5 From Artificial Intelligence to Soft Computing

23

In the previous decade, methods of Artificial Intelligence (AI) had become methods to compute with numbers to find exact solutions. However, not all problems could (and can) be resolved with these methods. On the other hand, humans are able to solve such problems very well without any measurements or computations, as Zadeh has mentioned over and over again in the last decades. Therefore, to “make computers think like people”, the machine’s ability “to compute with numbers” needs to be supplemented by an additional ability more similar to human thinking. In 1990, Zadeh coined the label ’Soft Computing’ to name an interdisciplinary field that covers different approaches to Artificial Intelligence that had been developing during the last decades but weren’t part of the mainstream of AI: “The concept of soft computing crystallized in my mind during the waning months of 1990”, Zadeh wrote about 20 years ago. He formulated this new scientific concept when he wrote that “what might be referred to as soft computing – and, in particular, fuzzy logic – to mimic the ability of the human mind to effectively employ modes of reasoning that are approximate rather than exact. In traditional – hard – computing, the prime desiderata are precision, certainty, and rigor. By contrast, the point of departure in soft computing is the thesis that precision and certainty carry a cost and that computation, reasoning, and decision making should exploit – wherever possible – the tolerance for imprecision and uncertainty. [...] Somewhat later, neural network techniques combined with fuzzy logic began to be employed in a wide variety of consumer products, endowing such products with the capability to adapt and learn from experience. Such neurofuzzy products are likely to become ubiquitous in the years ahead. The same is likely to happen in the realms of robotics, industrial systems, and process control. It is from this perspective that the year 1990 may be viewed as a turning point in the evolution of high MIQ-product17 and systems. Underlying this evolution was an acceleration in the employment of soft computing – and especially fuzzy logic – in the conception and design of intelligent systems that can exploit the tolerance for imprecision and uncertainty, learn from experience, and adapt to changes in the operation conditions.” [54] What motivated Zadeh to define a new approach for AI was his recognition that traditional AI couldn’t cope with the challenges that AI had to face (in which were implicit the critiques to Symbolic AI we have gathered in previous sections). However, Zadeh directed his critique not only to Symbolic AI but to the general approach he thought was characteristic of Computer Science and Engineering, which he calls “hard computing”, against which he offers a new approach named “soft computing”. As we have seen in previous sections, the characteristic of traditional (hard) computing is to use explicit models that have to be precise in order to allow the systems to function. These explicit models involve the values of precision and certainty. Zadeh thought that “precision and certainty carry a cost” ([54], p. 77) which is, precisely, that those systems are not able to solve many problems that humans do very well. If AI was the field devoted to build computer systems that act “intelligently” it appeared very surprising that AI traditional methods (methods that 17

MIQ – Machine Intelligence Quotient.

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compute with numbers, use explicit models and whose aim is to find exact solutions) can not design systems that simulate many human abilities that apparently require “less intelligence” like, as Zadeh suggests, recognizing and classifying images, summarizing a text, understanding distorted speech and many other everyday activities like parking a car. The opposition of soft computing to hard computing is expressed not only by its goals but also by its methodology. Zadeh considers that traditional AI is trapped in “the hammer principle” which he describes as “when the only tool you have is a hammer, everything looks like a nail”. This mentality underlies “a commitment to a particular methodology” but, more dangerously: “the proclaim that it is superior to all others” ([60], pp. 1-2). Zadeh adjudicates this reductionist approach to mainstream AI, even referring at some point to classical authors like Guha and Lenat or Minsky and MacCarthy [63] where he refers to very recent texts of these authors to emphasize that mainstream AI has not changed very much since the early days despite its problems. Soft Computing (from now SC) is characterized by replacing the “hammer metaphor” by the “toolbox metaphor” ([24], p. 48). In 1994, Zadeh defined Soft Computing as “a collection of methodologies (. . . ). Its principal constituents are fuzzy logic, neurocomputing, and probabilistic reasoning (. . . ) with the latter subsuming genetic algorithms, belief networks, chaotic systems, and parts of learning theory” ([56], p. 48). Luis Magdalena in [24] offers diagram 1.2 to describe the different approaches that constitute SC and how the hybridization and symbiotic approaches of these techniques have been developed till now:

Fig. 1.2. Diagram to describe the different approaches that constitute Soft Computing ([24], p. 151).

1.5 From Artificial Intelligence to Soft Computing

25

The common characteristics of the three methodologies are that, to certain extent, try to approximate some “natural processes”, relying all of them on the concept of “approximation”. Soft Computing solutions can not be considered as “the best ones” from a scientific point of view but sub-optimal ones that are “good enough” from a technological/engineering point of view. Because of that, they allow a great scope of applicability, and their successes lie in their high performance and low cost. Zadeh presented his approach as a radically new way of thinking and practising computing, which he himself characterizes as “a paradigm shift” ([63], p. 12): “To make significant progress toward achievement of human level machine intelligence, extension of existing techniques will not be sufficient. Basically, what it is needed is a paradigm shift”. As a whole, SC represents a very strong challenge to the whole “programme” of mainstream AI of previous decades. However, with “paradigm shift” Zadeh refers not only to Symbolic AI but to something more profound for Western though: mathematics and formalization, which rely on Aristotelian classical logic. In this context, it is possible to interpret the extreme reactions of some computer scientists to fuzzy logic as a sign of their commitment with the traditional paradigm and the “danger” that such a new approach could include for their “safe environment”. The meaning of “soft” is constructed by Zadeh by opposition of what he himself characterizes as “hard computing”. We can see this opposition in table 1.1: Table 1.1. Hard versus Soft as presented in Soft Computing. Hard Computing Soft Computing R IGID /C RISP /P RECISE Flexible/Approximate B I - VALUED Fuzzy-valued T OTAL ORDER Partial order A BSTRACT BASED Empiricaly (Contextually) based U NIQUE Hybrid/Plural N UMBERS Words

SC considers that humans perform many more activities requiring intelligence that the ones AI have considered before. The activities reclaimed by SC are examples of “ordinary” activities that all humans can do (such as walking, seeing or talking), activities that, traditionally, were not considered as requiring much intelligence, maybe because everybody can do it. On the contrary, quoting Trillas et. al., [43]: “Intelligence has been defined as scarce quality that only a few group of people have in high degree. [. . . ] which is in turn defined on not so common cognitive skills, such us, play chess, mathematics, spatial reasoning, novel writing, etc(. . . .) The key role of Soft Computing is to help to define and develop systems with high Machine Intelligence (. . . ). What we expect and want from computers can serve to settle these measurements, since they can serve as a guide for researchers. For example, if we want computers able to communicate and interact with humans, then we should include these problems in the pool”.

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The surprising observation that, though computers can do very well and fast some tasks, they weren’t able to do the majority of things that are easy for humans, lead some people to think that maybe these activities required more “intelligence” (or, as more generally stated, “cognitive functions”) than what was thought before. SC turned precisely to these kind of human abilities above mentioned. To differentiate themselves from traditional AI, SC people don’t use the term “artificial intelligence” but the broader term of “machine intelligence” or “computational intelligence”. When Zadeh says that “the role model for Soft Computing is the human mind”, he observes that humans reason and handle knowledge in a very different way: “Soft Computing is inspired by the remarkable human capability to perform a wide variety of physical and mental tasks without any measurements and any computations (for example common-sense knowledge). What these activities have in common is that make rational decisions in an environment of uncertainty and imprecision” ([63], p. 11). Imprecision, uncertainty, partial truth and approximation are features not much “appreciated” by traditional AI (neither by science and engineering in general). However, most of the human reasoning and abilities rely on these features. Because of this, these are the types of knowledge that SC is interested in.

1.6

Soft Computing and Soft Sciences

Hard computing (or computing with numbers) is used in hard sciences to find exact solutions. However, not all problems can be resolved with these methods. But, as it was also emphasized by Zadeh in 2001, we have methods of “non-hard”computing that can do this job. “As we move further into the age of intelligent systems, the problems that we are faced with become more complex and harder to solve. To address these problems, we have an array of methodologies – principally fuzzy logic, neurocomputing, evolutionary computing and probabilistic computing. In large measure, the methodologies are complementary; and yet, there is an element of competition among them. In this setting, what makes sense is formation of a coalition. It is this perception that motivated the genesis of soft computing – a coalition of fuzzy logic, neurocomputing, evolutionary computing, probabilistic computing and other methodologies.”[61] The “great complexity” we find it not only in biological systems but also in social sciences and humanities. At the end of the 1960s, and for a greater audience two years later, Zadeh wrote more generally: “What we still lack, and lack rather acutely, are methods for dealing with systems which are too complex or too illdefined to admit of precise analysis. Such systems pervade life sciences, social sciences, philosophy, economics, psychology and many other “soft” fields.” [51], [52] He distinguished between mechanic (or inanimate) systems on the one hand, and humanistic systems on the other, and he saw the following state of the art in computer technology: “Unquestionably, computers have proved to be highly effective in dealing with mechanistic systems, that is, with inanimate systems whose behavior

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27

is governed by the laws of mechanics, physics, chemistry and electromagnetism. Unfortunately, the same cannot be said about humanistic systems, which – so far at least – have proved to be rather impervious to mathematical analysis and computer simulation.” He defined a “humanistic system” as “a system whose behaviour is strongly influenced by human judgement, perceptions or emotions. Examples of humanistic systems are: economic systems, political systems, legal systems, educational systems, etc. A single individual and his thought processes may also be viewed as a humanistic system.” [53], p. 200) Zadeh summarized that “the use of computers has not shed much light on the basic issues arising in philosophy, literature, law, politics, sociology and other human-oriented fields. Nor have computers added significantly to our understanding of human thought processes – excepting, perhaps, some examples to the contrary that can be drawn from artificial intelligence and related fields.” ([53], p. 200) Thus, hard computing has been very successful in hard sciences but it could not be that successful in humanistic systems in the field of the “soft sciences”. Therefore we should open the field of applications of soft computing to the soft sciences. This was what Zadeh had in mind already in the commencement of soft computing, the starting time of Fuzzy Sets: “I expected people in the social scienceseconomics, psychology, philosophy, linguistics, politics, sociology, religion and numerous other areas to pick up on it. It’s been somewhat of a mystery to me why even to this day, so few social scientists have discovered how useful it could be. Instead, Fuzzy Logic was first embraced by engineers and used in industrial process controls and in “smart” consumer products such as hand-held camcorders that cancel out jittering and microwaves that cook your food perfectly at the touch of a single button. I didn’t expect it to play out this way back in 1965.” [54]

1.7

Outlook

Since those words by Zadeh, in the last times more and more scientists started to realize Zadeh’s vision and enlarge the usage of his theory of Fuzzy Sets and Systems and – more generally – Soft Computing in social sciences and humanities. We review here some of these developments: • About 10 years ago, a group of researchers built the BISC special interest group in Philosophy of Soft Computing, founded by Vesa Niskanen18 • In 2007 the EUSFLAT Working Group on Philosophical Foundations was founded to motivate philosophers, educators, and scientists, to approach the roots of Soft Computing, as well as its results in real applications 19 . This working group consists in: – A forum for the analysis of the grounds and methodologies of SC and, in particular, of Fuzzy Logic and Computing with Words/Meanings, where 18 19

See: http://www.helsinki.fi/˜niskanen/ bisc.html See: http://www.eusflat.org/research/phil.htm

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• •

•

•

new points of view could be openly and quickly discussed, and from which some scientific work/publications could follow up, – An effort to create an adequate environment to interest educators, and specially High School teachers, to approach Fuzzy Logic, with the general goal to lengthen their horizons of knowledge. Associated with this working group, the I. Workshop on Soft Computing in Humanities and Social Sciences was organized at the European Centre for Soft Computing (ECSC) in March 5-6, 2009.20 Also in the ECSC in Mieres the I. International Seminar on Soft Computing in Humanities and Social Sciences (SCHSS) took place at September 10-11, 2009. There, about 30 European scientists (from Austria, Finland, Germany, Italy, Portugal and Spain) met for three days to discuss topics, surveys, applications, case studies and visions on the field of SC in history, philosophy, education science, economics, linguistics, political and life sciences, and in the arts (especially in the fields of music and architecture).21 In 2009 the first volume on the topic of Soft Computing in Humanities and Social Sciences appeared in the series “Studies in Fuzziness and Soft Computing”: Views on Fuzzy Sets and Systems from Different Perspectives. Philosophy and Logic, Criticisms and Applications [36]. The second volume is the book at hand. In 2011 took place the First International Symposium on Fuzzy Logic, Philosophy and Medicine and the First International Open Workshop on Fuzziness and Medicine took place in March 23-25 at the ECSC in colaboration with the International Fuzzy Systems Association (IFSA), European Society for Fuzzy Logic and Technology (EUSFLAT) and the Hospital Universitario Central de Asturias. It is planned that there will appear a book of extended contributions of these presentations in the same Springer-series “Studies in Fuzziness and Soft Computing” in 2012 under the title Fuzziness and Medicine: Philosophical Reflections and Application Systems in Health Care[38]. This book will act as a “Companion Volume” to Kazem Sadegh-Zadeh’s Handbook on Analytical Philosophy of Medicinecit:KSadegh-Zadeh2011.

• In addition there have been Special Topic Sessions on this subject in various conferences during the last years: – “The Meaning of Information, Cognition, and Fuzziness” at the North American Fuzzy Information Society Annual Conference (NAFIPS-09), University of Cincinnati, Ohio, USA, June 14-17, 2009.22 20 21 22

See: http://www.softcomputing.es/schss2009/en/program.php See: http://www.softcomputing.es/schss2009/en/home.php See: http://nafips2009.ewu.edu/wiki/index.php/The_Meaning_of_Information%2C_Cognition %2C_and_Fuzziness.

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29

– “Philosophical, Sociological and Economical Thinking” at the 2009 IFSA World Congress und 2009 EUSFLAT Conference July 19-23, Lisbon, Portugal, 2009.23 – “Uncertainty, Vagueness and Fuzziness in Humanities and Social Sciences” at the 13th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU-2010), Conference Center Westfalenhallen, Dortmund, Germany, June 28 - July 2, 2010.24 – “Fuzzy Sets and Systems and the “Soft Sciences” ” at the FUZZ-IEEE part of this 2010 IEEE World Congress on Computational Intelligence (WCCI 2010), IEEE World Congress on Computational Intelligence, Centre de Convencions Internacional de Barcelona, Spain, July 18-23, 2010.25 – “Soft Computing in Soft Sciences” at the 2011 World Conference on Soft Computing – A Better Future Using Information and Technology, (WConSC 2011), May 23-26, 2011, San Francisco, California, USA.26 – “Thinking – Language – Meaning” at the conference EUSFLAT-LFA 2011, July 18-22, 2011, Aix-Les-Bains, France.27 Our view on the relations of Soft Computing disciplines to Humanities and Social Sciences led also to the formation of new “areas” for paper submissions in relevant conferences: • The new “area” on “Fuzzy Logic in Social Sciences” was established to the 2009 IFSA World Congress und 2009 EUSFLAT Conference July 19-23, in Lisbon, Portugal.28 • Also the new “area” on “Uncertainty and Social Sciences” was established to the 13th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU-2010), Conference Center Westfalenhallen, Dortmund, Germany, June 28 - July 2, 2010.29 Furthermore, in the year 2010 it started a new format of scientific conversation meetings called “Saturday Scientific Conversations” (SSC): • The SSC 2010 took place under the heading “Philosophy, Science, Technology and Fuzzy Logic” in Gijon, in the Asturias region of Spain, in May 15, 2010. It was organized by the ECSC under the sponsorship of the Government of Asturias and CajAstur. This first edition received the participation of 15 young researchers coming from many European countries (plus the conversants).30 23 24 25 26 27 28 29 30

See: http://ifsa2009.ist.utl.pt/index.php?option=com_content&view=article&id=59 &Itemid=63 See: http://www.mathematik.uni-marburg.de/˜ipmu2010/spse.html See: http://www.wcci2010.org/special-sessions See: http://www.ece.ualberta.ca/ reform/wconsc/special-sessions.html See: http://www.polytech.univ-savoie.fr/index.php?id=1638 See: http://ifsa2009.ist.utl.pt/index.php?option=com_content&view=article&id=59 &Itemid=63 See: http://www.mathematik.uni-marburg.de/˜ipmu2010/area Chairs. html. See: http://ssc.unipa.it/SSC_2011/Past_Edition.html

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• For the SSC 2011 edition Palermo in Sicily, Italy has been chosen as the guest city. The conversations have been held in Palazzo Steri in May 14, 2011.31 Finally, we should note that two years ago Alejandro Sobrino and Martin Pereira (University of Santiago de Compostela) launched the series of the Newsletter Philosophy & Soft Computing with articles, book reviews and interviews concerning the area of Soft Computing to Humanities and Social Sciences, especially Philosophy. The first volume appeared in 2009 and was published in the ECSC web. 32 The next volumes did appear in 2010 and 2011.33 The prosperity of these activities shows that Soft Computing is getting the attention of the non-technical disciplines and, in our view, there will be more and more lively interest. In the future we hope to take notice that Fuzzy Sets and Systems, Artificial Neural Networks, Genetic and Evolutionary Algorithms and other methods of Soft Computing will be part of research projects in philosophy, education science, economics, linguistics, political sciences, life sciences, law and arts. It is our aim to spread off the knowledge of SC to the areas of the “soft sciences”!

1.8

The Contributions in This Volume

The 24 contributions of this book offer a selected and great example of this burgeoning field on Soft Computing in Humanities and Social Sciences bringing together a wide array of authors and subject matters from different disciplines. Many of these contributions are extended works presented in the I. Workshop and the I. International Seminar held in the ECSC in Mieres in 2009, and also the follow-up work of special sessions related to our topic in different Soft Computing conferences in the last three years. While some of the contributors belong to the scientific and technological areas of Soft Computing, others come from various fields in the Humanities and Social Sciences such as Philosophy, History, Sociology or Economics. The six sections in which the volume is divided represent the most relevant topics resulted the fruitful exchanges taken place in the last years in several workshops, seminars and special sessions on the topic. The first section of the book titled General overviews of Soft Computing in Humanities and Social Sciences, includes three contributions. Settimo Termini’s article “On some ’family resemblances’ of Fuzzy Set Theory and Human Sciences” offers a philosophical approach on the topic. He identifies some “family resemblances” between fuzzy logic and the humanities and social sciences that, in his view, can help to create a bridge between them. Rudolf Seising’s contribution “Warren Weaver’s ’Science and Complexity’ revisited” presents an historical approach by 31 32 33

See: http://ssc.unipa.it/SSC_2011/2011.html See: http://docs.softcomputing.es/public/NewsletterPhilosophyAndSoftComputing Number_1.pdf See: http://docs.softcomputing.es/public/NewsletterPhilosophyAndSoftComputing Number_2.pdf, http://docs.softcomputing.es/public/NewsletterPhilosophyAndSoft ComputingNumber_3.pdf, and http://docs.softcomputing.es/public/NewsletterPhilosophy AndSoftComputingNumber_4.pdf

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showing how Warren Weaver’s concept of “complexity” developed in 1949 represents an important precedent to Zadeh’s concept of “fuzzy sets” of 1965. The author argues, as well, that Weaver’s idea of collaboration of “mixed teams” of scientists from different disciplines is a great example for the future collaboration between Soft Computing researchers and those in the Humanities and Social Sciences. Finally, Verónica Sanz’s paper “How Philosophy, Science and Technologies Studies, and Feminist Studies of Technology can be of use for Soft Computing” offers an approach to the topic using three different disciplinary areas: Philosophy of Artificial Intelligence, Science and Technology Studies, and Feminist Studies of Technology. She argues that analyzing the challenges that these three disciplines have presented to classical approaches in AI can be very useful for the future development of Soft Computing that will have to deal with the old problems in a different manner. The second section is titled Philosophy, Logic and Fuzzy Logic and includes four contributions. Settimo Termini’s second article “Explicatum and explicandum and Soft Computing” uses the distinction between “explicatum” and “explicandum” (a distinction proposed by the analytical philosopher of science Rudolf Carnap to explain the development of scientific concepts) to analyze some conceptual and epistemological questions affecting Fuzzy Set Theory. The article by Takehiko Nakama, Enric Trillas, and Itziar García-Honrado “Axiomatic Investigation of Fuzzy Probabilities” focuses on the interesting area of Fuzzy Probabilities, arguing that the probabilistic concepts used in Fuzzy Logic (and their mathematical foundations) have not been fully established to date. In order to clarify this topic, they offer a theoretical review of some concepts such as probability functions, probability measures and fuzzy-valued probabilities. The other two contributions in this section deal with the topic of Deontic Logic, that is, the logic used to represent moral and legal forms. Kazem Sadegh-Zadeh’s article “Fuzzy Deontics” proposes a “fuzzy version” of deontic logic by using both numerical and qualitative fuzzy rules that may aid fuzzy-deontic decision making. Along the same line, Txetxu Ausin and Lorenzo Peña’s contribution also propose an “alternative deontic logic” that they call “Soft Deontic Logic” that uses the tools of fuzzy logic to “soften” the traditional assumptions of standard deontic-logic, for which they indeed develop a formalized axiomatic system. The third section, called Soft Computing, Natural Language and Perception, addresses one of the most important topics in Soft Computing: its relation with human natural language. Cristina Puente, Alejandro Sobrino and José Angel Olivas’ article “Retrieving conditional and causal sentences in texts” is especially relevant since it is an extended version of the paper they presented at the IEEE World Congress on Computational Intelligence (Barcelona, 2010) that received the Best Paper Award. This award represents in some way the recognition from part of the general community of Soft Computing to the topic of Soft Computing in Humanities and Social Sciences. In this contribution, the authors deal with the concept of causality and its relation with conditionality, a topic that is at the core of the very scientific method. In order to study that relationship, the authors develop a method to extract causal and conditional sentences from texts of different scientific disciplines and use them as a database to explore the role of causality both in the

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so-called “hard sciences” and in the social sciences. Klaus Schulz, Ulrich Reffle, Lukas Gander, Raphael Unterweger and Sven Schlarb’s article “Facing Uncertainty in Digitisation” also deals with texts, though in a different way. The authors present a very interesting application of fuzzy logic to the area of digitization of written language using texts from different disciplines. They present a fuzzy-rule based system to be used in the post-processing steps of text digitization and recognition. For his part, in his individual contribution “The Role of Synonymy and Antonymy in a ’Natural’ Fuzzy Prolog” Alejandro Sobrino addresses two important features of natural language: synonymy and antonymy. He applies these concepts to the programming language Prolog (a system that traditionally uses crisp logic). Sobrino extend the “fuzzy version” of Prolog (Fuzzy Prolog) by introducing the possibility of developing a “natural Fuzzy Prolog”. Enric Trillas and Itziar García-Honrado’s second contribution “On an Attempt to Formalize Guessing” approaches the topic of conjectures, which are considered one of the first steps in the scientific method. The authors present a way to formalize conjectures by analyzing new operators of consequences that “free” them from previous methods of deduction. Jeremy Bradley’s paper “Syntactic Ambiguity Amidst Contextual Clarity – Reproducing Human Indifference to Linguistic Vagueness” focuses on the topic of ambiguity and imprecision in human language by studying a software system used to help people affected by “aphasia” (a language disorder that involves difficulty in producing or comprehending spoken or written language). For their part, Olga Kosheleva and Vladik Kreinovitch’s contribution “Can we learn Algorithms from People Who Compute Fast: An Indirect Analysis in the Presence of Fuzzy Descriptions” focus on the topic of performing fast calculations and whether people who do that use “faster” algorithms or just standard ones in a very fast way. They turn to the self-descriptions of human “calculators” which, as expressed in natural language, have a fuzzy nature. Finally, Clara Barroso’s article “Perceptions: a psychobiological and cultural approach” is the one dealing with the topic of perception. She argues that a review of different approaches to the study of human perception can be very useful to inform models to represent the meanings of perceptions to be used in artificial intelligence systems. The fourth section of the book: Soft Models in Social Sciences and Economics, turns now to the area of applications of Soft Computing to different fields in the social sciences. For example, Joao Paulo Carvalho’s article “Rule Based Fuzzy Cognitive Maps in Humanities, Social Sciences and Economics” offers an idea to apply fuzzy models to a social science’s tool known as “Cognitive Maps”. Originally developed as a tool to model real-world dynamic systems, Carvalho shows how traditional versions of cognitive maps have several shortcomings that could be solved by using fuzzy rules, for which he proposes the idea of “rule-based fuzzy cognitive maps”. José Luis García-Lapresta and Ashley Piggins’ “Voting on how to vote” present a fuzzy model to be used in voting systems. The model represents voters’ preferences as unit intervals and the thresholds as fuzzy numbers. Introducing the notion of “self-selective threshold”, the authors offer a procedure that can be used in voting decision-making. On their part, Juan Vicente Riera and Jaume Casasnovas’ contribution “Weighted Means of Subjective Evaluations” present an

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example of how to use fuzzy logic models in the area of Education and, in particular, in the methods for the evaluation of students. By focusing on the problem of aggregation of fuzzy information (for example in the grades awarded by different evaluators to the same students), the authors propose a theoretical method to build n-dimensional aggregation functions on discrete fuzzy numbers. Finally, Bárbara Díaz Diez and Antonio Morillas turn to the field of Economy in their article “Some experiences applying fuzzy logic to Economics” where they present several of their previous works on the application of fuzzy logic to economic systems. In concrete, they explain their use of fuzzy inference systems to model and predict different economic aspects such as wage-earning employment levels, the profit value of Andalusian agrarian industry and Input-Output economic analyses. Section five, Soft Computing and Life Sciences, offer two examples of how to apply Soft Computing to medicine. Kazem Sadegh-Zadeh’s second contribution tittled “Fuzzy Formal Ontology” proposes a formal ontological framework based on fuzzy logic to be applied in medical knowledge-based systems, which he denominates as a “Fuzzy Formal Ontology”. On their part, Mila Kwiatkowska, Krzysztof Michalik and Krzysztof Kielan’s article “Computational Representation of Medical Concepts: a Semiotic and Fuzzy Logic Approach” proposes a meta-modeling framework that uses fuzzy logic and semiotic approaches for computational representation of medical concepts to be used for computer-based systems in Medicine. Their system is intended to overcome the shortcomings of previous systems that cannot deal with the malleability and imprecision characteristic of medical concepts. The last section of this volume, Soft Computing and Arts, collects several examples of the interesting encounter of Soft Computing with the arts, being Music the most addressed. Hanns-Werner Heister’s article “Invariance and variance of motives: A model of Musical logic and/as Fuzzy Logic” introduces the concept of “musical logic” (the idea that music follows a logical rationale, widely recognized in music theory). Interpreting this concept as the logical manner in which the motive (“core”) of a piece relates to the “contours”, Heister affirms that this relation is a dynamic process that is indeed a matter of fuzzy logic. Teresa León and Vicente Liern’s paper “Mathematics and Soft Computing in Music” also follows the idea of “musical logic” and, more specifically, the mathematical character of music that can be described at the very physical processes of frequencies. Drawing on the fact that human musicians allow small changes in frequency values for the same notes, the authors affirm that music involves a relative level of imprecision that is better explained in terms of fuzzy logic, for which they offer an experiment with the help of a saxophonist. Along the same line, Josep Lluis Arcos’ article “Music and Similarity-Based Reasoning” focus on the fact that musicians, when performing, intentionally deviate from the original score in order to offer expressiveness. In order to design a computer model of musical expressiveness, the author uses the notion of “local similarity” and presents a case-based reasoning methodology that has been successfully developed. Finally, we find an example of the application of Soft Computing to an area different from music: architecture. Amadeo Monreal’s article “T-norms, t-conorms, aggregation operators and Gaudí’s columns” uses the fuzzy logic concepts or t-norms, t-conorms and aggregation operators to provide a

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mathematical model of the characteristic “Gaudí columns” (also known as “double twist” columns) that can be used to generate new types of Gaudí-like columns in future architecture projects. As we have already stated, the twenty-four contributions of this book are only an example of what the interesting encounter and conversations between Soft Computing and the Humanities and Social Sciences can yield in the future.

References [1] Adam, A.: Artificial Knowing. Gender and the Thinking Machine. Routledge, London (1998) [2] Black, M.: Vagueness. An exercise in logical analysis. Philosophy of Science 4, 427– 455 (1937) [3] Black, M.: Reasoning with loose concepts. Dialogue 2, 1–12 (1963) [4] Blair, B.: Interview with Lotfi Zadeh, Creator of Fuzzy Logic by Azerbaijada International (2.4) (winter 1994), http://www.azer.com/aiweb/categories/magazine/24_folder/24_articles/ 24_fuzzylogic.html [5] Brockman, J.: The Third Culture: Beyond the Scientific Revolution. Simon & Schuster (1995) [6] Carnap, R.: Logische Syntax der Sprache (1934); English translation, The Logical Syntax of Language. Kegan Paul (1937) [7] Carnap, R.: Mein Weg in die Philosophie. Philipp Reclam jun, Stuttgart (1993); Original English edition: Carnap, R.: Intellectual Autobiography. In: Schilpp, P. A. (ed.), The Philosophy of Rudolf Carnap = The Library of Living Philosophers. 11, pp. 1-84. Open Court, London (1963) [8] Carnap, R.: Der logische Aufbau der Welt. Weltkreis Verlag, Berlin-Schlachtensee (1928); English translation by George, R.A.: The Logical Structure of the World. Pseudoproblems in Philosophy. University of California Press (1967) [9] Laplace, P.S.: Essai Philosophique sur les Probabilités forming the introduction to his Théorie Analytique des Probabilités. V Courcier (1820); repr. In: Truscott, F.W., Emory, F.L. (trans.) A Philosophical Essay on Probabilities. Dover, New York (1951) [10] Church, A.: An unsolvable problem of elementary number theory. The American Journal of Mathematics 58, 345–363 (1935) [11] Collins, H.M.: Artificial Experts: Social Knowledge and Intelligent Machines. MIT Press, Cambridge (1990) [12] Dawkins, R.: The Selfish Gene. Oxford University Press, Oxford (1976) [13] Dawkins, R.: The Blind Watchmaker. W. W. Norton & Company, New York (1986) [14] Dawson, J.W.: Logical Dilemmas: The Life and Work of Kurt Gödel, p. 71 (1997) [15] Forsythe, D.: Engineering Knowledge: The Construction of Knowledge in Artificial Intelligence. Social Studies of Science 23(3), 445–477 (1993) [16] Fritz, W.B.: The Women of ENIAC. IEEE Annals of the History of Computing 18(3), 13–28 (1996) [17] Gould, S.J.: The Mismeasure of Man. Norton, New York (1996) [18] Gould, S.J.: The Structure of Evolutionary Theory. The Belknap Press of Harvard University Press (2002) [19] Lenat, D.B., Guha, R.V.: Building Large Knowledge-Bused Systems Representation and Inference in the Cyc. Project. Addison Wesley, Reading (1990)

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[20] Gürer, D.: Women in Computing History. SIGSCE Bulletin 34(2), 116–120 (2002) [21] Haugeland, J.: Artificial Intelligence: The Very Idea. MIT Press, Cambridge (1985) [22] Turing, A.M.: On Computable Numbers, with an Application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, Series 2 42, 230–265 (1936-1937); with corrections from Proceedings of the London Mathematical Society, Series 2 43, 544–546 (1937) [23] Light, J.S.: When Computers Were Women. Technology and Culture 40(3), 455–483 (1999) [24] Magdalena, L.: What is Soft Computing? Revisiting Possible Answers. International Journal of Computational Intelligence Systems 3(2), 148–159 (2010) [25] Menger, K.: Memories of Moritz Schlick. In: Gadol, E.T. (ed.) Rationality and Science. A Memorial Volume for Moritz Schlick in Celebration of the Centennial of his Birth, pp. 83–103. Springer, Vienna (1982) [26] Menger, K.: The New Logic: A 1932 Lecture. In: Selected Papers in Logics and Foundations, Didactics, Economics, Vienna Circle Collection, vol. 10. D. Reidel, Dordrecht (1979) [27] Minsky, M., Papert, S.: Perceptrons. MIT Press, Cambridge (1969) [28] von Neumann, J.: First Draft of a Report on the EDVAC, http://www.alt.ldv.ei.tum.de/lehre/pent/skript/VonNeumann.pdf [29] Papert, S.: One AI or many? In: Graubard, S.R. (ed.) The Artificial Intelligence Debate: False Starts, Real Foundations. MIT Press, Cambridge (1988) [30] Pinker, S.A.: How the Mind Works. Norton & Company, New York (1999) [31] Pinker, S.A.: The Stuff of Thought: Language as a Window into Human Nature. Penguin, New York (2008) [32] Rojas, R., Hashagen, U. (eds.): The First Computers: History and Architectures. MIT Press (2000) [33] Russell, B.: Introduction. In: [44] [34] Russell, B.V.: Vagueness. The Australasian Journal of Psychology and Philosophy 1, 84–92 (1923) [35] Sadegh-Zadeh, K.: Handbook of Analytic Philosophy of Medicine. Philosophy and Medicine, Springer, Berlin (to appear) (2012) [36] Seising, R. (ed.): Views on Fuzzy Sets and Systems from Different Perspectives. Philosophy and Logic, Criticisms and Applications. Studies in Fuzziness and Soft Computing, vol. 243, pp. 1–35. Springer, Berlin (2009) [37] Seising, R.: What is Soft Computing? – Bridging Gaps for the 21st Century Science. International Journal of Computational Intelligent Systems 3(2), 160–175 (2010) [38] Seising, R., Tabacchi, M. (eds.): Fuzziness and Medicine: Philosophical Reflections and Application Systems in Health Care. Studies in Fuzziness and Soft Computing. Springer, Berlin (to appear) (2012) [39] Shermer, M.: The Really Hard Science. Scientific American, September 16 (2007) [40] Snow, C.P.: The Two Cultures and the Scientific Revolution. Cambridge University Press (1960) [41] Snow, C.P.: Two Cultures: And a Second Look. An Expanded Version of the Two Cultures and the Scientific Revolution. The University Press, Cambridge (1964) [42] Trillas, E.: La inteligencia artificial. Máquinas y personas, (ed.) Debate, Madrid (1998) [43] Trillas, E., Moraga, C., Guadarrama, S.: A (naïve) glance at Soft Computing. International Journal of Computational Intelligence and Systems (in Press 2011) [44] Wittgenstein, L.: Tractatus logico-philosophicus. Routledge & Kegan Paul, London (1922)

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[45] Wittgenstein, L.: Das Blaue Buch. In: Werkausgabe in acht Bänden. Frankfurt am Main: Suhrkamp, Bd. 5 (1984) [46] Wittgenstein, L.: Philosophical Investigations (1953) [47] Zadeh, L.A.: From Circuit Theory to System Theory. Proceedings of the IRE 50, 856– 865 (1962) [48] Zadeh, L.A.: Fuzzy Sets. Information and Control 8, 338–353 (1965) [49] Zadeh, L.A.: Fuzzy Sets and Systems. In: Fox, J. (ed.) System Theory. Microwave Research Institute Symp., vol. XV, pp. 29–37. Polytechnic Press, Brooklyn (1965) [50] Zadeh, L.A.: Biological Application of the Theory of Fuzzy Sets and Systems. In: Proctor, L.D. (ed.) The Proceedings of an International Symposium on Biocybernetics of the Central Nervous System, pp. 199–206. Little, Brown and Comp., London (1969) [51] Zadeh, L.A.: Toward a Theory of Fuzzy Systems, Electronic Research Laboratory, University of California, Berkeley 94720, Report No. ERL-69-2 (June 1969) [52] Zadeh, L.A.: Towards a theory of fuzzy systems. In: Kalman, R.E., DeClaris, N. (eds.) Aspects of Network and System Theory, pp. 469–490. Holt, Rinehart and Winston, New York (1971) [53] Zadeh, L.A.: The Concept of a Linguistic Variable and its Application to Approximate Reasoning I. Information Science 8, S199–S249 (1975) [54] Zadeh, L.A.: Making Computers Think like People. IEEE Spectrum 8, 26–32 (1984) [55] Zadeh, L.A.: Fuzzy Logic, Neural Networks and Soft Computing. Communications of the ACM 37(3), 77–84 [56] Zadeh, L.A.: Soft computing and fuzzy logic. IEEE Software 11(6), 48–56 (1994) [57] Zadeh, L.A.: Fuzzy Logic = Computing with Words. IEEE Transactions on Fuzzy Systems 4(2), 103–111 (1996) [58] Zadeh, L.A.: From Computing with Numbers to Computing with Words - From Manipulation of Measurements to Manipulation of Perceptions. IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications 45(1), 105–119 (1999) [59] Zadeh, L.A.: The Birth and Evolution of Fuzzy Logic – A Personal Perspective. Journal of Japan Society for Fuzzy Theory and Systems 11(6), 891–905 (1999) [60] Zadeh, L.A.: Applied Soft Computing - Foreword. Applied Soft Computing 1(1) (June 2001) [61] Zadeh, L.A.: A New Direction in AI. Toward a Computational Theory of Perceptions. AI-Magazine 22(1), 73–84 (2001) [62] Zadeh, L.A.: Foreword. Applied Soft Computing 1(1), 1–2 (2001) [63] Zadeh, L.A.: Toward Human Level Machine Intelligence – Is It Achievable? The Need for a Paradigm Shift. IEEE Computational Intelligence Magazine, 11–22 (August 2008)

Part II

2 On Some “family resemblances” of Fuzzy Set Theory and Human Sciences Settimo Termini

2.1

Introduction

The aim of this paper is to underline the importance of detecting similarities or at least, ’family resemblances’ among different fields of investigation. As a matter of fact, the attention will be focused mainly on fuzzy sets and a few features of human sciences; however, I hope that the arguments provided and the general context outlined will show that the problem of picking up (dis)similarities among different disciplines is of a more general interest. Usually strong dichotomies guide out attempts to understand the paths along which scientific research proceed; i.e., soft versus hard sciences, humanities versus the sciences of nature, Naturwissenschaften versus Geisteswissenschaften, Kultur versus Zivilization, applied sciences and technology versus fundamental, basic (or, as has become recently fashionable to denote it, “curiosity driven”) research. However, an open minded look at the problem under scrutiny shows that the similarity or dissimilarity of different fields of investigation is – to quote Lotfi Zadeh – “a matter of degree”. This is particularly evident in the huge, composite, rich and chaotic field of the investigations having to do with the treatment of information, uncertainty, partial and revisable knowledge (and their application to different problems). The present paper is partly based on my [4] from which it borrows Section 2.2 and 2.3. However, the general attitude is different. Here, in fact, an attempt is done of extending and enlarging the general context in which must be seen the specific analyses and considerations already done. Section 2.4, finally, is completely new and introduces the attempt – done along the same line of thought – of comparing a few epistemological questions raised by Trust Theory with those of Fuzzy Set Theory. This change of perspective induces also to consider the specific points treated here as case studies of a more general crucial question: a question which could be important in treating in a non superficial way the problems posed by interdisciplinarity. In the remaining of this introduction, I shall, first, briefly focus the setting or general context in which these considerations should be done; secondly, I will state in a synthetic way a related difficult problem; and, finally, some reference to a few related questions will be done. In Section 2.2, starting from a well known question asked by Lotfi Zadeh, I shall argue in favor of the thesis that what impeded a rapid R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 39–54. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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and widely diffused use of fuzzy sets theory results and methodologies in Human Sciences was not only a lack of correct communication but resides in deep and difficult questions connected with the fact that the valuation of precision and rigor is different in different disciplines. Let me add that – although no single formula is presented in the paper – all the considerations and remarks done here spring out from a rethinking of technical results, in the light of the problems discussed. A clarification of conceptual questions – besides being of great intrinsic interest – plays a crucial role also in the development of purely technical results in new directions. I am, in fact, firmly convinced that conceptual clarifications subsequent to, and emerging from, epistemological analyses help to focus innovative paths of investigation. Also from this point of view – besides their intrinsic value and interest – both the scholarly investigation of Rudolf Seising [10] and his edited volume [11] are very important references which have been very useful in clarifying a few aspects along the writing of the present paper. The interaction between fuzzy sets and human sciences must be seen as an episode of a larger question. There is a long history, in fact, regarding the mutual relationship existing between the (so-called) humanities and the (so-called) hard sciences. More than fifty years have elapsed from the publication of a precious booklet on “The Two Cultures” by the English physicist and novelist Charles Percy Snow [12], dealing with the problem of the breakdown of communication between the two main forms of culture of our times, the sciences and the humanities, and whose appearance provoked – 50 years ago – intense debates and strong discussions. It is not clear whether this debate can ever reach a definite conclusion at this very general level whilst it is relatively clear that the problems present themselves with different nuances and specificities according to the particular fields of inquiry and disciplines considered and the way in which it is presumed or asked they should interact. It seems that also the general and specific tradition of a given country can play a no negligible role. For instance, in Italy, where – in the past century – there has been a strong contraposition between the “two cultures” the situation (the context) should be considered different from the one described by Snow in which more than a strong contraposition there was a very profound lack of communication. It is also interesting to observe that remaining always in Italy – in contrast to the just mentioned contraposition that dominated many decades of the twentieth century, it seems possible also to trace an old and long lasting tradition in Italian Literature that stresses a dialogue between scientists and humanities [2]. This tradition has the same roots of the scientific revolution of the Seventeenth Century and is also connected to the founding father of Italian literature. According to this school of thought, there is, in fact, a line connecting Dante, Galileo, Leopardi, Italo Calvino or Primo Levi, all authors in whose writings there not only is no opposition between the “two cultures” but an innovative literary language is a tool a powerful tool for expressing new scientific results or a new (scientific) Weltanschaung. All this, in turn, contribute to establishing new clear relationships between Science and Society. In this setting also the

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way in which science is communicated and the audience which is explicitly chosen (or seen) as the privileged target plays a central role [3]. This is the general context in which the questions asked in the paper should be seen, although to reach not too vague conclusions it is important to examine specific questions and problems. Many of the questions and proposals done along the years by Zadeh are utmost original and innovative so that they have required also to look anew at some epistemological aspects for a satisfactory assessment of the considered problems before trying to tackle (and solve) them. This applies also to his question of the limited use in Human Sciences of results and techniques of Fuzzy Sets and Soft Computing (from now on, FS&SC), a question which, then, must not be considered as an occasional comment as it could – prima facie – appear. Let me add that I am firmly convinced that a revisitation of the so-called problem of the two cultures in the light of the typical epistemological questions of information sciences and, in particular, of the intriguing presence of something not numerical in typically scientific problems as the ones proposed by soft computing, from one side, could provide new points of view to eventually show, that it is nothing more than a very complex episode of the interdisciplinary interaction of (very) distant fields.1 On the other side, it could provide very useful indications for treating the problems asked by interdisciplinarity whithin a very general and adequate setting. The general motivations previously outlined indicate that, to approach in concrete the question posed by the title of this paper, we must look carefully at the meaningful features of every specific discipline and, also, to the context in which a certain question is asked or a given concept is used. Therefore, here we cross and are obliged to face the old (and also very difficult besides under many aspects, still mysterious2 ) problem of interdisciplinarity (see, for instance, [20], [17]). Everyone who has crossed interdisciplinary problems knows that the best way to obtain good results is to be very cautious and prudent in using concepts (and tools) outside the domain in which they were initially conceived and developed. This fact generates also some “family resemblance” between different disciplines at least from the epistemological point of view (see [13]). In the case considered in the present paper we have to be particularly careful since we are trying to establish bridges between very distant domains. Some related questions which will not be treated in these preliminary attempts to focus the problem, such the following ones. First, one must remember that many problems (technical, epistemological, psychological, etc) posed by the use of 1 2

To consider it under this light, of course, does not mean to transform it into a simple problem! Of course, those aspects of interdisciplinarity still remaining mysterious are not mysterious at all but (simply!) those emerging from the extreme complexity of forcing different disciplines to interact while asking them to preserve their proper specificities, methodologies, levels of rigor, etc., (and, then , aspects very difficult to treat). In our case we must also play a specific tribute to the constraints imposed by the novelty of the theory (FS&SC) on one side, and the refractoriness to be imbedded into (more or less) formal methods on the other side (Human Sciences).

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formal methods in human sciences are still crucial, unsettled and strongly debated. Secondly, the process leading from an informal notion as used in everyday language to its regimentation inside scientific theories (but also in specialized languages) has to be carefully taken into account. This second problem has been superbly discussed by Rudolf Carnap who introduced the two notions of explicandum and explicatum for explaining aspects of this process. The help that Carnap’s analysis can provide also for the general study of the relationships between FS&SC and Human Sciences has been briefly indicated in [21]. In the present paper, for reason of space, these questions – however crucial they are – will not be taken into account.

2.2

Zadeh’s Question and a Tentative Answer

One of the problems addressed in this paper is the way in which uncertainty, and in particular its specific facet represented by fuzziness, presents itself in different disciplines, looking – in particular – for possible meaningful differences when human sciences or hard sciences are, respectively, taken into account. As a matter of fact, I have been induced to consider this general problem starting from one observation done by Lotfi Zadeh who, rightly, judged strange the fact that Fuzzy Sets Theory has been so less used in Human Sciences. An attempt will be done, in these pages, to propose some hypotheses to explain this fact. If a phenomenon is persistent for many decades, in fact, it cannot be simply ascribed to casualty but it is reasonable to think that unusual, deep (or anyway, unknown) reasons have worked – in the underground – to produce it. In the present section, I start from Zadeh’s question to focus a few general points aiming at a preliminary chartering of the territory. More complete analyses of the topics briefly surveyed here will involve – as already stressed – a careful consideration of many other aspects (at least, the ones mentioned above). 2.2.1

Zadeh’s Question

Rudolf Seising, proposing and presenting a special session on “Uncertainty, Vagueness and Fuzziness in Humanities and Social Sciences” rightly remembered that Lotfi Zadeh has been struck by the fact that the theory of fuzzy sets has not been very diffused and strongly used in human sciences and reports a quotation from a 1994 interview: “I expected people in the social sciences, economics, psychology, philosophy, linguistics, politics, sociology, religion and numerous other areas to pick up on it. It’s been somewhat of a mystery to me why even to this day, so few social scientists have discovered how useful it could be.”3 I want to address this question now, trying to find out some reasons of this fact and so contributing to reduce the mystery to which Zadeh refers. In the following 3

“Lotfi A. Zadeh. Creator of Fuzzy Logic”, Interview by Betty Blair, 1994 Azerbaijan International Winter 1994 (2.4).

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pages I shall, then, present a tentative answer to Zadeh’s question or, better, an hypothesis regarding the reasons why – forty five years after the appearance of the founding paper of the theory – there has not been a wide use of Zadeh’s theories in Human and Social Sciences. I shall not provide an answer, however tentative, to the question asked, but I shall formulate a few hypotheses regarding what can be involved in the situation. I am, in fact, convinced that Zadeh has not simply asked a question but has touched a crucial point of the interaction between human sciences and hard sciences which cannot be solved simply by asking a question and providing an answer. So, if these hypotheses grasp some truths we shall remain with a lot of additional work to be done (as always happens in science, a (possible) solution of a specific point opens a lot of new questions). 2.2.2

A Remark and a Few Tentative Hypotheses

The remark has to do with the epistemological (and ontological) features of both. Although many people are probably convinced that there exist strong epistemological similarities between FS&SC and Human Sciences, it is more difficult to find analyses and descriptions of these similarities. (In this paper I leave completely out of the considerations done about the problem of possible similarities of ontological type, but see [15] and [14]). Let us consider the problem of rigour. One of the crucial points in my view is that in both fields we are looking for rigour in the same way and this way is in many aspects different from the one in which the rigour is looked for in Hard Sciences. The kind of rigour that is meaningful and useful in both FS&SC and Human Sciences, in fact, is different from the one that displays his benefic effects in hard sciences. In all the fields of investigation one is looking – among many things – also for rigour. Let us consider the case of Hard Sciences. Roughly, what we require from a good theory is that it – at least – should model the chosen piece of reality and be able to forecast the output of experiments. For this second aspect, one is looking then for a stronger correspondence between the numerical output computed by the theory and the numerical output measured by the experimental apparatus. This is not certainly, in general, the case in Human Sciences. Let us make a digression to clarify this point. The exacteness of a writer or of a resercher in philology is not based on the number of decimal ciphers after the dot that a given theory produces for some meaningful parameters (and which allows a comparison with the measurements done in corresponding experiments). Human Sciences are also looking for exactness but a sort of different one, which is difficult even to define if we have in mind the numerical-previsional model typical of Hard Sciences. We could, perhaps, say that is the exactness of having grasped in a meaningful way some of the central aspect of a given problem. For instance, in a literary text of very good quality we see that we cannot change the words without loosing something, we cannot change in some cases even the simple order of the words. The text produced by an outstanding writer is exactly what was needed to

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express something. This is certainly true for poetry, but also for any literary text of very high quality.4 In a certain sense the same is true also in FS&SC. It suffices to remember the very frequent remarks done by Zadeh on the fact that humans very efficiently act without doing any “measurement” (and numerical computations). Also in the case of FS&SC, then, we are moving in a universe in which we have to do with an exactness which is different from the one of measurements and numerical precision. Let us now go back to the problem of a possible dialogue between FS&SC and Human Sciences (and the fact that it has not been very strong in the past decades). The (tentative) hypothesis I want to propose is that the causes reside just on the fact that both FS&SC and Human Sciences share the same methodological and epistemological attitude towards the problem exactness and rigour: both look – as already observed – for a sort of exactness not based on numerical precision. But why an epistemological similarity could be the cause of a difficulty in the interaction of the two fields? My answer is twofold. On the one hand, I think that this very similarity is difficult to master. We are accustomed to think that the interaction with formal, hard, sciences is a way for introducing their kind of precision (a quantitative one) into a (still) imprecise field. On the other hand, I think that the very fact that the kind of precision can be different from “numerical precision” obliges to reflect on the kind of operation one is trying to do. Of course, the problems proper to FS&SC and Human Sciences respectively are different (and different also from those appearing in the traditional approaches of hard sciences) notwithstanding the epistemological similarities mentioned above. To pick up the differences while acknowledging the epistemological similarities is, in my view, a crucial passage. Regarding their methodological similarity, let us limit, here, to take into account only their not considering numerical precision as crucial. The challege is to evaluate what kind of advantage can emerge for these two worlds if the traditional passage through the caudin forks of numerical measurements and computation is not the crucial point. FS&SC provide a very flexible language in which we can creatively pick up locally meaningful tools which can efficiently clarify some specific aspects of the considered problems. But there is no formal machinery that automatically solves the problems. If we do not have this point clear in mind it is easy to have difficulty in contrasting the objections affirming that the formalism and language of fuzzy set theory can complicate instead of simplify the description and understanding of some pieces of reality. Just, to give an example: the full machinery of many valued logic is more complex than the one of classical logic. It is difficult to have a non occasional interaction, if we are not able to convince people working in human sciences that: a) we can have advantages even 4

And this is in fact one of the central points and difficulties of the translation of a text in different languages. Commenting the difficulties faced in early Cybernetics by the mechanical, cybernetic, translation from one natural language to another one, it was observed that some of the difficulties are caused by the lack of a formal (mechanical) theory of meaning. But for having a good mechanical translation of literary tests there are other things still missing, among which a mechanical theory of “exactness”, which is difficult also to envisage.

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without numerical evalutions, and b) by introducing “degrees” we are not opening the Pandora’s box of the full machinery of a more complex formalism. Considerations regarding methodological and epistemological similarities between FS&SC in Human Sciences can help, then, to understand aspects of the relationships between these fields of inquiry. The natural question that arises at this point is whether an examination also of their respective ontological basic assumptions could provide relevant information. But, as was said above, I leave this as a completely untouched problem here.

2.3

A Few Conceptual Corollaries

Let us list in a very rough and rudimentary way some of the consequences that in my view immediately emerge from the previous attempt at analyzing the question. A. Precision is not an exclusive feature of hard sciences. However the way in which this concept is needed and is used in Human Sciences and in Hard Sciences is different. B. We must observe that the same is true also when we consider the problem in different scientific disciplines. Also when mathematics and physics are taken, for instance, into account, we can easily observe differences. Dirac’s delta function, for instance, was considered in no way acceptable by mathematicians until Schwartz developed the Theory of Distributions, while physicists – still being completely aware and sharing all the mathematical objections to the contradictions conveyed by this notion – used it by strictly controlling the way in which it was used. The same could be argued for other notions which have been used in Theoretical Physics, i.e., Renormalization, Feynman’s diagrams. C. The notion of precision that can be fruitfully used in Fuzzy Sets and Soft Computing has features belonging partly to Human Sciences and partly to Hard Sciences. This fact makes particularly interesting and flexible the tools provided by FS&SC but, at the same time, makes more complex their use in new domains since it requires their intelligent adaptation to the considered problems and not a “mechanical application”. D. The “mechanical application”, however, is exactly what each of the two partners in an interdisciplinary collaboration, usually, expect each from the other. Let us consider an imaginary example. Let A and B two disciplines involved, Ascientist sees a difficulty for the solution of a certain problem arising from features and questions typical of discipline B and asks to his colleague B-scientist for their solution, presuming that he can provide on the spot a (routine) answer (what above, was called a mechanical solution). However, for non-trivial problems, usually B-scientist is unable to provide a mechanical solution, since, for instance, the posed problem either can be particularly complex or outside the mainstream of B discipline. The expected mechanical interaction does not happen, since it is not feasible. However, an interaction is possible. What A and B – representing with these letters both the disciplines and the scientists – can do is

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to cooperate for a creative solution and not for a routine mechanical application of already existing tools. E. The problem appears particularly delicate when Human Sciences and formal techniques are involved for reasons which I am tempted to define of sociological nature. I am, in fact, convinced that the epistemological problems are indeed strong, but not crucial in creating difficulties. These can emerge more from both the attitudes and the expectations of the people involved. We could, for instance, meet categories of the following types: people that so strongly believe in the autonomy of their discipline that they do not think other disciplines can provide any help; people so strongly trusting to the power of the tools coming from hard sciences that they expect these techniques may work very well without any creative adaptation to the considered problem. F. The difficulties of point E) above are magnified if the tools to be used cannot, by their same nature, be mechanically applied and – in the light of point C above – this is often the case for tools borrowed from FS&SC. One is tempted to ask if one could use Carnap’s procedure to see whether it is used – in different disciplines – different explicata of the intuitive notions of rigor and precision. This question, besides being very slippery, presents the additional difficulty of not forgetting the fact that rigor and precision are also – in some sense – metatheoretic notions and, in the case of interdisciplinary investigations, intertheoretic notions. Another type of corollaries has to do with the fact that all these considerations impinge (also) strongly with technical work and specific investigations. This is the right moment to look at typical concepts and problems that could benefit from such an analysis. Among these ones I should certainly mention vagueness (in itself and in connection to the notion of fuzziness), see [23], [16], [18], [5]; the revisiting of logical principles [24], [9]; the problem of controlling booleanity, i. e., how much a fuzzy approach departs from a standard boolean situation (which, looked the other way around, can be seen as the problem of measuring the fuzziness of a certain description), see [6], [7], [8]. A preliminary analysis of these problems done by applying the approach outlined here (and taking into account – as an epistemological guide – Carnap’s analysis of the notions of explicandum and explicatum) can be found in [21], present in this same volume. There, the reader will find also the attempt at analyzing in a general context the notion of information and a warning about the problems one meet when trying to develop an “information dynamics” [19]. I shall conclude this Section by listing some hot topics that, in my view, could also strongly benefit from a preliminary conceptual analysis based on the approach outlined here and in [21]. They all share some features that can be characterized by the common property (which could be used as a slogan) of “using the notion of fuzziness as a key for approaching and modelling problems in new ways”. Among them one should, first, once again, recall the problems of the relationship existing between words and numbers as well as the one of “manipulating perceptions” instead of “manipulating measurements”. These two recent ideas of Lotfi Zadeh are really disruptive [29], also with respect to [28] His idea of “computing with words”,

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for instance, really breaks with the basic notion of computation and the idea of starting from perceptions instead that from measurements goes nearer to Husserl’s ideas than to the ones usual to the Galilean tradition. Another very interesting topic is the one of understanding the role and presence of “conjectures” in ordinary reasoning (see, e.g., [25], [26]). On one hand, the questions that can be asked have their roots in very old philosophical (and scientific) problems, but it seems that new flexible tools allow today to approach them in a new way. On the other hand, these problems allow to construct useful bridges among purely logical problems and the way in which we humans behave in every day reasoning. This is a crucial and urgent problem both for A.I. and for the deep problems of “reasoning” (the informal idea, the explicandum) in the same way of the most sophisticated results of mathematical logic. As already observed all these topics and, specifically, the way in which they are actually approached – or it is proposed to approach them – pose the following question: Are we going away from Galilean tradition? Told in another way, the question has to do with the fact whether we are strictly following the classical scientific methodology or are departing from it in an essential way. For a preliminary analysis of this question, the reader is referred to [22]. In the next Section, I shall ask the question whether these problems are an exclusive feature of soft computing.

2.4

A Tentative Analysis of Another Example: Trust Theory

I think that one should stress again and again the fact that the features analyzed above would acquire a larger significance should we be able to show that they apply not only to arguments and concepts sprung out from fields of investigation belonging to the realm of ideas of the community of fuzzy sets. To obtain this evidence it is instrumental to show that such features are also present (in very similar although not identical modalities) in other new research fields. One such example could be provided by Trust theory. I shall consider here a recent monograph by Castelfranchi and Falcone [1] which presents a very general model of the notion of Trust. The book (more than 350 pages) tries, along his 13 chapters, “to systematize a general theory of ’trust’, to provide an organic model of this very complex and dynamic phenomenon on cognitive, affective, social (interactive and collective) levels” (page 1). Here, I shall not try to review the book, stress its merits or synthesize its main concerns. Not only for the obvious reason that a paper is not the right place for such a thing as a review, but, mainly, since this is not what is of interest for the conceptual reflections developed here. What is of interest for our analysis, now, – as I have already stressed above – is to see whether some of the points picked up in the previous pages (as well as in the small constellation of a few of my recent contributions to the reflection on the foundations of fuzziness) have to do also with crucial problems of other, different (although not unrelated)

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fields of investigation.5 So from this point of view, a (recent) book that aims at providing/systematizing a general theory of a new topic is the best thing a researcher could hope in order to test new ideas and topics along the lines outlined above. For all these reasons (but not only because of them), my report here of Castelfranchi and Falcone’ approach is very partial and incomplete and, as such, certainly unfaithful. I only hope that it is not so unfaithful as to distort and misrepresent their point of view. My aim here – to extract some elements from their approach to support my thesis that some features of the developments of fuzzy set theory are not a unique peculiarity –, hopefully, will be unfaithful only in the sense of not representing the richness and variety of the topics and aspects described in the book, but not in the sense of misrepresenting their aims. Let me add that if the reconnaissance technique in this new land will – eventually – be successful one could arrive at understanding better (and perhaps be able to classify) the features common to new approaches that try to model fields of investigation of unusual type. In these new fields, in fact, we should on the one hand move according to the methodology of scientific tradition; on the other hand, we must implicitly accept some modifications (and in some case also drastic changes); and finally we could also accept strong departures from classical scientific methodology seeing them as necessary phases to be crossed in order to arrive to new types of models. At the end of such a reconnaissance flight we should be able to see and examine many problems and questions from an enlarged and more general perspective. I shall then limit mainly to consider the way in which the authors present their effort and aim in writing the book and the path followed to construct an adequate model of such a new innovative notion. In a subsection of the introductory chapter with the significant title “Respecting and Analyzing Concepts”, the authors – after reporting that in disciplines “still in search of their paradigmatic status and recognition” very frequently it follows an attitude, called, using a Piagetian terminology, “assimilation” – declare that their approach will try to follow a different attitude which stresses, instead, the importance of accomodating the formal tools (concepts and schemes) “to the complexity and richness of the phenomenon in object”. The three reasons they advance for this choice are: 5

Interestingly enough, the work done in [1] crosses some topics of fuzzy sets and this helps in showing possible interactions that could be beneficial for both fields. The connection with fuzzy set theory is, in fact, stronger than it can appear at first sight. Besides having an application of the model obtained by using Fuzzy Cognitive Maps, there is a second reason. This has to do with the fact that the importance of taking into account uncertainty in forms that do not easily match standard mathematical treatments and techniques emerges in many places in the book. Moreover, by reading the book, that many other connections and applications come immediately to mind (just one example: the problem of “how to measure trust” versus “how to measure fuzziness”). However, this is not the central point of the present discussion. At the moment, the focus is on “non casual” conceptual similarities in the approaches followed.

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““First, because the current trust ’ontology’ is really a recognized mess, not only with a lot of domain-specific definitions and models, but with a lot of strongly contradictory notions and claims. Second, because the separation from the current (and useful) notion of trust (in common sense and languages) is too strong, and loses too many interesting aspects and properties of the social/psychological phenomenon. Third, because we try to show that all those ill-treated aspects not only deserve some attention, but are much more coherent than supposed, and can unified and grounded in a principled way.” I quoted in extenso this passage, since its complete reading can be useful for at least two reasons. The first has to do with the fact that many of the problems involved are very similar to questions frequently debated and discussed at the beginning of the development of fuzzy set theory. The second reason is that the situation so clearly described in the quoted passage, impinges more generalon the problem of the methodological similarities and differences of soft sciences and hard sciences (and, indirectly, on the dialogue between humanities and natural sciences, or the “Two cultures’ questions”). The assumed attitude seems to be perfectly tuned with many remarks done in this paper, moreover, it adds new arguments (and a different viewpoint) to approach many of the open problems encountered along the present intellectual “excursion”. But let us go back to Trust Theory. As a consequence of their preliminary analysis and the epistemological choice just reported, the authors present their model of trust, which appears to be very articulated and complex. They list, in fact, seven main features of the model. The reader is referred to pages 3-4 of [1] for a detailed analyses of them, but it is important here to notice, at least, that after characterizing their model as “non-prescriptive” (last listed feature), the Section in question ends by stressing the following point: “our aim is not to abuse the concept, but at the same time to be able to situate in a precise and justified way a given special condition or property” (in a suitable way according to the context) – the italic is mine. As I have already written, the authors of the book want “to systematize a general theory of thrust”. But, for doing this it is necessary to start from a general and widely accepted definition of the notion at the base of the theory. However, they find that “there is not yet a shared or prevailing, clear and convincing notion of trust”. This situation can produce “an unconscious alibi, a justification for abusing this notion, applying it in any ad hoc way, without trying to understand if, beyond the various specific uses and limited definitions, there is some common deep meaning, a conceptual core to be enlightened”. After a brief description of the situation fixed in the literature, they continue affirming that: “the consequence is that there is very little overlapping among the numerous definitions of trust, while a strong common conceptual kernel for characterizing the general notion has yet to emerge. So far the literature offers only partial convergences and ’family resemblances’ among different definitions, i.e. some features and terms may be common to a subset of definitions but not to other subsets. This

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book aims to counteract such a pernicious tendency, and tries to provide a general, abstract, and domain-independent notion and model of trust.” Let me first observe that, curiously enough, Castelfranchi and Falcone use the same Wittgesteinian image of ’family resemblance’ used in the title of the present paper. But – besides the obvious fact that both want to indicate a situation in which there is something in common but in an unusual (and hardly quantifiable) way – , the respective uses of this same image is very different. In this paper it has been used – positively – to stress the fact that the two fields of “fuzzy sets”, on one side, and “human sciences”, on the other side, though apparently completely disjoint, share nonetheless a sort of (patchwork) similarity. In their book, the authors use this image to complain the (negative) fact that – in the same research field – the same name stands for different definitions not sharing a solid common core, but only “locally” connected. The fact of using – by chance – the same image, however, is – perhaps – the indication that we are facing very similar problems, although they look at it from different angles or at different stages of evolution.6 I was led to use the image of ’family resemblance’ – positively – in trying to indicate similarities between fuzzy sets and human sciences that could explain some questions arising in the attempt of applying fuzzy techniques to problems in human sciences. Thus, the conceptual tool of ’family resemblance’ is used to evidence a certain kind of similarity that, at the same time, is loose and presents local variations. It is, then, a tool adequate to show that the fields are not so separate as one could first think, but they are neither so similar as to allow a standard, routine, mechanical application of the techniques of one field to solve problems of the other. In the case of Thrust Theory, on the contrary, the problem is that we must start from a basic concept and cannot be satisfied with something that – in the best case – presents only a ’family resemblance’ (in the technical sense that “some features and terms may be common to a subset of definitions but not to other subsets”). Read the other way around, this means that the features and terms common to all the definitions conform the empty set. (That is, their common core is empty. Let me add that, also with reference to my proposal of using Carnap’s dialectics between explicandum and explicatum as an epistemological guide (heralded in the companion paper present in this volume and briefly referred here in the introductory Section), Castelfranchi and Falcone’s proposed that ther systematization of a general theory of thrust presents interesting questions to be asked (before being answered). First, although the corpus of analyses, techniques and results presented in the book appears really impressive, it seems that (the authors explicitly affirm that “they try to develop a model”) we are still looking for the “final” focusing of 6

Let me stress that the problem of facing a plurality of different definitions is something that certainly sounds familiar to fuzzy theorists, at least to those that remember the old foundational papers. On the one hand, it was the presence of a proliferation of different basic definitions of crucial concepts in some parts of System Theory which definitely convinced Zadeh to look for a way of escaping this situation. He thought that the basic root for the proliferation could be an unnecessary “exactness” connected to the standard modeling in this field. On the other hand, the problem of picking up a satisfactory definition can be also related to the debates connected to the very concept of ’fuzzy set’.

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the explicandum, and from this point of view, it is not strange that the viable paths to propose an adequate explicatum are still numerous and open to future work. So there is both a similarity and a difference with the situation of the notion of fuzziness. The difference is that fuzziness presented itself in the first paper of Zadeh – in the terminology I propose to use now – as the explicatum of something. This something (the explicandum) was not completely clear, but it was sufficiently delimited regarding the kernel of the new notion. All the discussions – at the time – had to do with the ambitions of the new theory. The similarity between the two cases has to do at least with the fact that they both start from a situation in which the present status induces to propose a very huge numbers of definitions of notions of interest. When thinking about the best way to present this comparison, I felt that it could be seen as an imposed, undue and unnatural forcing, and was, first, tempted, to cancel these remarks. However, it may be useful to stress some new features of innovative notions that are springing out of these and many other interdisciplinary fields, however preliminary and hazy the first comments could seem and, often, actually are. It is clear from comparing and noticing the emergence of unforeseen similarities and specific differences, that we can obtain an empirical knowledge and an epistemological understanding of the way in which new emerging fields of investigation are slowly structuring themselves. I have not touched on many problems connected to “quantitative evaluations”7 as well as to the types of formal models suggested, used and developed in the book. Also in this case there are many interesting suggestions valid in themselves and good for stressing similarities and differences. I refer only to two points: the problems posed by the definition of “degree of trust” (page 96 of [1], and the ones arising by the “very diffused reduction of trust to subjective probability” (pages 235 and 247 and ff.). All these – crucial and important – problems would deserve an accurate analysis, which is not possible to provide here. Let me limit to observe that many points in common with (and slightly dissimilar to) fuzzy sets would emerge also in relation to the search of the required accuracy and precision. A precision that, in many cases, can be unrelated to numerical accuracy.8 But this would really deserve another paper. 7 8

Let me limit here to recall the effective and forceful definition of “quantificatio precox” coined by the authors to indicate a dangerous use of quantitative evaluations. A numerical evaluation seems to be – at the same time – an important goal to obtain and something very complex to do since it is crucial not to flatten all the nuances of a multifaceted complex concept, projecting all richness of the informal notion into only (let me say, a Marcusian) “one dimension”. Castelfranchi and Falcone pose this problem with respect to the “reduction of trust to subjective probability”. But one could observe that every numerical evaluation, eventually, flattens – however complex the strategy to arrive to this final number is – all the possible used procedures. So the alternatives present at the moment seem to be (but this is only my interpretation) either to arrive at a single numerical evaluation (as a result of a complex strategy followed and not as a numerical assignment to a single notion (a more or less modified and corrected belief function or a subjective probability assignment) but preserving a trace of the procedure followed or to define such a notion by means of more innovative tools (but still in an embryonic stage of development) as “linguistic labels”, as they discuss in the chapter related to fuzzy techniques. In both cases, it is paid the price of working with heavier formal machineries.

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2.5

Conclusions

Dichotomies can be useful. By looking at the large dicothomy between Hard and Soft Sciences, we have been led to see that the Theory of Fuzzy Sets interestingly manifests selected aspects and features of both. This has induced us to propose a partial answer to Zadeh’s question: the interaction between Fuzzy sets and Human sciences has been very limited not only due to a generic lack of communication (and of mutual knowledge), but also (or mainly) since a true, profound and stable interaction requires the clarification of what each of the two actors is asking (and – reciprocally – is able to give) to the other one. The interaction while appearing, in principle, very natural and, apparently, straightforward, in practice, poses – in fact – very subtle conceptual problems (different from the ones arising in a “more traditional” interdisciplinary interaction between near fields or very distant fields). It is precisely for this reason that it is not a trivial thing to realize. But we have learned more than that, I presume. Dichotomies can be, often, very helpful: very helpful to start with the study of a new problem as well as to focus the main questions to be addressed. And also for looking at the implications of the choices done at the preliminary stages of a new investigation.9 However, in the subsequent stages of the research work, it can be useful (and, sometimes, necessary) to take into account also some nuances, some more subtle differences. This implies that clear-cut dichotomies become blurred. What I have described is common experience of every investigator, and so it seems that there is no justification for all these comments which, being well known, appear as nothing else than sheer banalities. However, what is not equally trivial is a widespread acknowledgement that this apply not only inside well established fields of investigation but also when different disciplines interact. Along this interaction, something happens so that the original individuality of the single disciplines seems to disappear. Information sciences and, in a very specific way, a few central topics of soft computing interestingly suggest that one can find out something which is correct to call “family resemblances” among different fields of investigations. (It is interesting to recall that a way of describing the notion of ’family resemblance’ with fuzzy sets has been recently proposed.) These “resemblances” can vary, appear and disappear in, apparently, unpredictable ways. Thus, a new, interesting, epistemological challenge issued by these new fields of investigation (and, perhaps, also a new point of view for better understanding interdisciplinarity).

Acknowledgements I want to thank Enric Trillas for many thought-provoking questions and discussions along too many years and Rudi Seising for equally interesting discussions along the last few years. 9

That distinction is, obviously, related to problems of ’identity’ versus ’difference’, well known to anthropologists. Problems unknown – or, at least, certainly not central – in the epistemology of natural sciences. Perhaps, they could help in assessing better the features of new emerging fields and the way in which they could (and should) interact.

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References [1] Castelfranchi, C., Falcone, R.: Trust Theory – A Socio-cognitive and Computational Model. John Wiley, Chichester (2010) [2] Greco, P.: L’astro narrante. Springer, Milan (2009) [3] Greco, P.: Storia della comunicazione della scienza nel Seicento. In: L’idea pericolosa di Galileo. UTET, Turin (2009) [4] Termini, S.: Do Uncertainty and Fuzziness Present Themselves (and Behave) in the Same Way in Hard and Human Sciences? In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds.) IPMU 2010. CCIS, vol. 81, pp. 334–343. Springer, Heidelberg (2010) [5] Kluck, N.: Some Notes on the Value of Vagueness in Everyday Communication. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds.) IPMU 2010. CCIS, vol. 81, pp. 344– 349. Springer, Heidelberg (2010) [6] De Luca, A., Termini, S.: A definition of a non probabilistic entropy in the setting of fuzzy sets theory. Information and Control 20, 301–312 (1972) [7] De Luca, A., Termini, S.: Entropy and energy measures of a fuzzy set. In: Gupta, M.M., Ragade, R.K., Yager, R.R. (eds.) Advances in Fuzzy Set Theory and Applications, pp. 321–338. North-Holland, Amsterdam (1979) [8] De Luca, A., Termini, S.: Entropy Measures in the Theory of Fuzzy Sets. In: Singh, M.G. (ed.) Encyclopedia of Systems and Control, pp. 1467–1473. Pergamon Press, Oxford (1988) [9] Lukasiewicz, J.: Philosophical remarks on many-valued systems of propositional logic. In: Borkowski, L. (ed.) Jan Lukasiewicz: Selected Works. Studies in Logic and the Foundations of Mathematics, pp. 153–178. North-Holland Publ. Comp./Pol. Scientif. Publ., Amsterdam, Warszawa (1970) [10] Seising, R.: The Fuzzification of Systems. The Genesis of Fuzzy Set Theory and Its Initial Applications – Its Development to the 1970s. STUDFUZZ, vol. 216. Springer, Berlin (2007) [11] Seising, R. (ed.): Views on Fuzzy Sets and Systems from Different Perspectives. Philosophy and Logic, Criticisms and Applications. STUDFUZZ, vol. 243. Springer, Berlin (2009) [12] Snow, C.P.: The Two Cultures and the Scientific Revolution. Cambridge University Press, Cambridge (1959) [13] Tamburrini, G., Termini, S.: Do Cybernetics, System Science and Fuzzy Sets share some epistemological problems? I. An analysis of Cybernetics. In: Proceedings of the 26th Annual Meeting Society for General Systems Research, Washington, D.C, January 5-9, pp. 460–464 (1982) [14] Tamburrini, G., Termini, S.: Some Foundational Problems in the Formalization of Vagueness. In: Gupta, M.M., Sanchez, E. (eds.) Fuzzy Information and Decision Processes, pp. 161–166. North-Holland (1982) [15] Termini, S.: The formalization of vague concepts and the traditional conceptual framework of mathematics. In: Proceedings of the VII International Congress of Logic, Methodology and Philosophy of Science, Salzburg, vol. 3, Section 6, pp. 258–261 (1983) [16] Termini, S.: Aspects of vagueness and some epistemological problems related to their formalization. In: Skala Heinz, J., Termini, S., Trillas, E. (eds.) Aspects of Vagueness, pp. 205–230. Reidel, Dordrecht (1984) [17] Termini, S.: Remarks on the development of Cybernetics. Scientiae Matematicae Japonicae 64(2), 461–468 (2006)

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[18] Termini, S.: Vagueness in Scientific Theories. In: Singh, M.G. (ed.) Encyclopedia of Systems and Control, pp. 4993–4996. Pergamon Press (1988) [19] Termini, S.: On some vagaries of vagueness and information. Annals of Mathematics and Artificial Intelligence 35, 343–355 (2002) [20] Termini, S.: Imagination and Rigor: their interaction along the way to measuring fuzziness and doing other strange things. In: Termini, S. (ed.) Imagination and Rigor, pp. 157–176. Springer, Milan (2006) [21] Termini, S.: On Explicandum versus Explicatum: A few elementary remarks on the birth of innovative notions in Fuzzy Set Theory (and Soft Computing). In: Seising, R., Sanz, V. (eds.) Soft Computing in Humanities and Social Sciences. Studies in Fuzziness and Soft Computing, vol. I, Springer, Heidelberg (2011) [22] Termini, S.: Concepts, Theories, and Applications: the role of “experimentation” for formalizing new ideas along innovative avenues. In: Trillas, E., Bonissone, P., Magdalena, L., Kacprycz, J. (eds.) Experimentation and Theory: Hommage to Abe Mamdani. STUDFUZZ, Physica-Verlag (to appear, 2011) [23] Terricabras, J.-M., Trillas, E.: Some remarks on vague predicates. Theoria 10, 1–12 (1988) [24] Trillas, E.: Non Contradiction, Excluded Middle, and Fuzzy Sets. In: Di Gesù, V., Pal, S.K., Petrosino, A. (eds.) WILF 2009. LNCS (LNAI), vol. 5571, pp. 1–11. Springer, Heidelberg (2009) [25] Trillas, E., Mas, M., Miquel, M., Torrens, J.: Conjecturing from consequences. International Journal of General Systems 38(5), 567–578 (2009) [26] Trillas, E., Pradera, A., Alvarez, A.: On the reducibility of hypotheses and consequences. Information Sciences 179(23), 3957–3963 (2009) [27] Trillas, E., Moraga, C., Sobrino, A.: On ’family resemblance’ with fuzzy sets. In: Proceedings of the 13th IFSA World Congress and 6th EUSFLAT Conference - IFSAEUSFLAT 2009, pp. 306–311 (2009) [28] Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965) [29] Zadeh, L.A.: From Computing with Numbers to Computing with Words-from Manipulation of Measurements to Manipulation of Perceptions. International Journal of Applied Mathemtics and Computer Science 12, 307–324 (2002)

3 Warren Weaver’s “Science and complexity” Revisited

Rudolf Seising

3.1

Introduction

This historical paper deals with two important changes in science and technology in the 20th century which are associated with the concepts of complexity and fuzziness. Here “complexity” refers to the big change in science – roughly spoken – from physics to the life sciences that started in the thirties of the 20th century. The initiation of this change was deeply connected with the work of the American mathematician and science administrator Warren Weaver (1894-1978). Weaver was a very active and exciting man of science in a long period of the 20th century. In the late 1940s he did not only publish the most well-known introductory and popularizing paper “The Mathematics of Communication” [33] on Shannon’s “Mathematical Theory of Communication” [25], [26] and the very influential memorandum “Translation” [35] on the possible use of early computers to translating natural languages. But he also wrote the article “Science and Complexity” [32] where he identified a class of scientific problems “which science has as yet little explored or conquered”. ([32], p. 539) Weaver argued that these problems can neither be reduced to a simple formula nor they be solved with methods of probability theory, and to solve such problems he pinned his hope on the power of digital computers and on interdisciplinary collaborating “mixed teams”. “Fuzziness” refers to another big change in science that Lotfi A. Zadeh (born 1921) initiated when he established the theory of Fuzzy sets in the mid-1960s. As one of the leading scientists in System theory in that time he started looking for new mathematics: “. . . I began to feel that complex systems cannot be dealt with effectivity by the use of conventional approaches largely because the description languages based on classical mathematics are not sufficiently expressive to serve as a means of characterization of input-output relations in an environment of imprecision, uncertainty and incompleteness of information.” [58] As Weaver did already more than a decade before, Zadeh denied in 1962 that probability theory is an adequate mathematical tool to manage the analysis of high complex systems. However, afterborn, Zadeh became like Weaver a contemporary witness for the beginning “computer age”. In the year 1949 at Columbia University in New York he served as a moderator at a debate on digital computers between Claude Elwood Shannon (1916-2001), Edmund C. Berkeley (1909-1988), the author of the book Giant Brains or Machines That Think published in 1949 [2], and Francis J. Murray (1911-1996), R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 55–87. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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a mathematician and consultant to IBM. In the following year he published the article “Thinking Machines – A New Field in Electrical Engineering”, which appeared in the student’s journal The Columbia Engineering Quarterly [40] (see figure 3.11 ). Since then he stated very often over the last decades that “thinking machines” do not think as humans do and from the mid-1980s he focused on “Making Computers Think like People” (see figure 3.1) using his theory of Fuzzy Sets as an asset for these machines. [60]

Fig. 3.1. Left: Frontpage of the Columbia Engineering Quarterly, 1950; right: Zadeh’s article “Making Computers Think like People, [60].

In this contribution I bring forward the argument that the scientific problems that Weaver tried to catch and the scientific problems that found “good enough”solutions with the methods of the theory of fuzzy set and systems are almost the same! In the sections 3.2 and 3.4 I give short surveys of Weaver’s and Zadeh’s vitas and in the sections 3.3 and 3.4.1 I describe their scientific changes in some details, respectively. Section 3.5 gives some reflections on similarities and differences of the two approaches and section 3.6 gives an outlook to other papers of Weaver and Zadeh that also concern similar topics: communication of information and translation of natural languages by computers. 1

The author is very sorry about the non-optimal quality of some of the figures in this chapter but the quality of the originals is alomost the same.

3.2 Warren Weaver – A Biographical Sketch

3.2

57

Warren Weaver – A Biographical Sketch

Warren Weaver (see figure 3.2) was born on July 17, 1894 in Reedsburg, Wisconsin. He studied mathematics and engineering at te University of WisconsinMadison and he became friends with his professor in mathematics, Charles Sumner Slichter (1864 - 1946).2 Another influencial teacher and friend of the young student Weaver in Madison was the mathematician and physicist Max Mason (1877-1961) who recommended Weaver to the physicist Robert Andrews Millikan (1868-1953). When Millikan commuted from Chicago to Throop College in Pasadena, California,3 Weaver became assistant professor in Pasadena after he finished his studies in Madison in the year 1917. He spent one year with the US Air Force, in the Washington Bureau of Standards and after that time he changed between Madison and Pasadena. Weaver and Mason started writing a book on electromagnetic field theory4 and in 1921 Weaver earned the grade of a Ph D and he became professor in the mathematical department at Madison University (1928-1932). Since 1925 Max Mason was president of the University of Chicago and in 1928 he became Natural Science Director of the Rockefeller Foundation (1929-1932). In 1931, when Mason became president of the Rockefeller Foundation, he offered his former position to Weaver who accepted after some time to think it over. Weaver held this position for a quarter century from 1932 until 1957! Already as a professor of mathematics in Wisconsin Weaver has been interested in biology. It was expected that this field would show interesting and exciting results in the close future: “On the campuses there was talk that the century of biology was upon us. At Wisconsin, for example, there was a lively program in biology at the School of Agriculture as well as in the College of Arts and Sciences.” ([16], p. 505 f) When he was asked to accept the position as the Natural Science Director he demanded to change the Rockerfeller Foundation’s program: “I explained that, satisfied as I was with being immersed in the physical sciences, I was convinced that the great wave of the future in science, a wave not yet gathering its strength, was to occur in the biological science. The startling visions that were just then beginning to open up in genetics, in cellular physiology, in biochemistry, in developmental mechanics, these were due for tremendously significant advances.” He asked for a long-range establishment to support quantitiative biology: “The idea that the time was ripe for a great new change in biology was substantiated by the fact that the physical sciences had by then elaborated a whole battery of analytical and experimental procedures capable of probing into nature with a fineness and with a quantitative precision that would tremendously supplement the previous tools of biology – one almost say “the previous tool of biology”, since the optical microscope had furnished so large a proportion of the detailed evidence.” ([36], p. 60) In his Annual Report for the year 1938 Weaver discussed a new research area that he named “Molecular Biology”: “Among the studies to which the Foundation 2 3 4

In the year 1925 Weaver edited the third edition of Slichter’s textbook Elementary Mathematical Analysis [17]. Since 1929 this college is called California Institute of Technology. This textbook The Electromagnetic Field [7] was published in 1929.

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is giving support is a series in a relatively new field, which may be called molecular biology, in which delicate modern techniques are being used to investigate ever more minute details of certain life processes.” (quoted by [16], p. 503) Retrospective, in his Autobiography “Scene of Change” that was published in 1970, Weaver wrote: “My conviction that physics and chemistry were ripe for a fruitful union with biology, necessarily somewhat tentative and amateurish when I first accepted the post as a director of the Rockefeller Foundation, steadily became more firm and more enthusiastic as my European visits brought me into contact with scientists after scientists who expressed a desire to participate in our program.” [36] He improved his skills in genetics, cellular physiology, organic chemistry, biochemistry, and he learned techniques to analyze molecular structures. His division finanzed research in life sciences that became very well-known in the later years.5

Fig. 3.2. Warren Weaver.

Retrospective, Weaver wrote: “I believe that the support which the Rockefeller Foundation poured into experimental biology over the quarter century after 1932 was vital in encouraging and accelerating and even in initiating the development of molecular biology. Indeed, I think that the most important thing I have ever been able to do was to reorient the Rockefeller Foundation science program in 1932 and direct the strategy of development of the large sums which that courageous and 5

Weaver mentioned among others in his “Scene of Change” James Dewey Watson (born 1928) and Francis Harry Compton Crick (1916-2004), who discovered the structure of DNA in 1953, George Wells Beadle (1903-1989), the American biologist and later president of the University of Chicago and Max Ludwig Henning Delbrück (1906-1981), the German-American biophysicist and Nobel laureate (1969 together with Alfred D. Hershey (1908-1997) and Salvador E. Luria (1912-1991) “for their discoveries concerning the replication mechanism and the genetic structure of viruses”). Beadle was Nobel Prize laureate in Physiology or Medicine (1958 together with Edward Lawrie Tatum (19091975)) “for their discovery that genes act by regulating definite chemical events.” ([36], p. 72f) In the late 1960s Beadle reported that in the period between 1954 and 1965 15 of 18 Nobel laureates who had been involved in molecular biology had received assistance from the Rockefeller Foundation. ([36], p. 73), [16], p. 504).

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imaginative institution made available. It was indeed a large sum, for between 1932 and my retirement from the Rockefeller Foundation in 1959 the total of the grants made in the experimental biology program which I directed was roughly ninety million dollars.”([36], p. 72) In the Second World War Weaver became Head of the Applied Mathematics Panel of the US Office of Scientific Research and Development (1943-46). In the wartimes he also wrote his papers on Information theory and language translation that we will discuss in the outlook section 3.6 of this paper. After the war he accepted among his position as Rockefeller’s Natural Science Director being Science Consultant at the Sloan-Kettering Institute for Cancer Research (1947-1951), Appointed Director of the Division of Science and Agriculture of the Rockefeller Foundation(1951-1955) and vice president “Natural and Medical Sciences” of the Rockefeller Foundation(1955-1959). Furthermore he was vice president of the Alfred P. Sloan Foundation (1958-1964). Warren Weaver died on November 24, 1978 in New Milford in Connecticut.

3.3

Science and Complexity

Very short after the Second World War Warren Weaver wrote the important article “Science and Complexity” [32] based upon material for his introductory contribution to a series of radio talks, presenting aspects of modern science by 81 scientists, given as intermission programs during broadcasts of the New York PhilharmonicSymphonies. Weaver edited the written contributions in the book The Scientists Speak [31] and one year later “Science and Complexity”, which arose from the book’s first chapter, was published in the American Scientist (see figure 3.4). In this article Weaver gave an overview on “three and a half centuries” of modern science and he took “a broad view that tries to see the main features, and omits minor details.” [32] Regarding the history of sciences he wrote “that the seventeenth, eighteenth, and nineteenth centuries formed the period in which physical sciences learned variables, which brought us the telephone and the radio, the automobile and the airplane, the phonograph and the moving pictures, the turbine and the Diesel engine, and the modern hydroelectric power plant.” ([32], p. 536) Compared to this historical path of modern (hard) sciences, he assessed the development of life sciences elsewise: “The concurrent progress in biology and medicine was also impressive, but that was of a different character. The significant problems of living organisms are seldom those in which one can rigidly maintain constant all but two variables. Living things are more likely to present situations in which a halfdozen or even several dozen quantities are all varying simultaneously, and in subtly interconnected ways. Often they present situations in which the essentially important quantities are either non-quantitative, or have at any rate eluded identification or measurement up to the moment. Thus biological and medical problems often involve the consideration of a most complexly organized whole.” ([32], p. 536)

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Fig. 3.3. Left: Frontpage of Weaver’s autobiography “Scene of Change [36]; right: Frontpage of the collections of radio talks The Scientists Speak [31].

3.3.1

Disorganized Complexity

To revisit Weaver’s reasoning: he distinguished between “problems of simplicity” that “physical science before 1900 was largely concerned with”, and another type of problems that “life sciences, in which these problems of simplicity are not so often significant”, are concerned with. The life sciences “had not yet become highly quantitative or analytical in character", he wrote. Then, he enlarged on the new developed approach of probability and statistics in the area of exact sciences at around 1900: “Rather than study problems which involved two variables or at most three or four, some imaginative minds went to the other extreme, and said. “Let us develop analytical methods which can deal with two billion variables.” That is to say, the physical scientists, with the mathematician often in the vanguard, developed powerful techniques of probability theory and statistical mechanics to deal with what may be problems of disorganized complexity”, a phrase that “calls for explanation” as he wrote, and he entertained this as follows: A problem of disorganized complexity “is a problem in which the number of variables is very large, and one in which each of the many variables has a behavior which is individually erratic, or perhaps totally unknown. However, in spite of this helter-skelter, or unknown, behavior of all the individual variables, the system as a whole possesses certain orderly and analyzable average properties.”[32] Weaver emphasized that probability theory and statistical techniques “are not restricted to situations where the scientific theory of the individual events is very well

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known” but he also attached importance to the fact that they can also “be applied to situations [. . . ] where the individual event is as shrouded in mystery as is the chain of complicated and unpredictable events associated with the accidental death of a healthy man.” He stressed “the more fundamental use which science makes of these new techniques. The motions of the atoms which form all matter, as well as the motions of the stars which form the universe, come under the range of these new techniques. The fundamental laws of heredity are analyzed by them. The laws of thermodynamics, which describe basic and inevitable tendencies of all physical systems, are derived from statistical considerations. The entire structure of modern physics, our present concept of the nature of the physical universe, and of the accessible experimental facts concerning it, rest on these statistical concepts. Indeed, the whole question of evidence and the way in which knowledge can be inferred from evidence are now recognized to depend on these same statistical ideas, so that probability notions are essential to any theory of knowledge itself.” [32]

Fig. 3.4. Left: Frontpages of “Science and Complexity” as the first chapter in The Scientists Speak [31]; right: title page “Science and Complexity” in the American Scientist [32].

3.3.2

Organized Complexity

In addition to, and in-between “problems of simplicity” that are solvable with hard sciences and “problems of disorganized complexity” to deal with probability theory and statistics, Weaver identified a third kind and therefore a trichotomy of scientific problems – may be for the first time at all: “One is tempted to oversimplify, and say that scientific methodology went from one extreme to the other – from two variables to an astronomical number – and left untouched a great middle region. [See figure 3.5.] The importance of this middle region, moreover, does not depend primarily on the fact that the number of variables involved is moderate – large compared to two, but small compared to the number of atoms in a pinch of salt. The problems in this middle region, in fact, will often involve a considerable number of variables. The really important characteristic problems of this middle region, which science has as yet little explored or conquered, lies in the fact that these problems, as contrasted with the disorganized situations which statistics can cope, show the

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essential feature of organization. In fact, one can refer to this group of problems as those of organized complexity.” ([32], p. 539) He listed examples of such problems and some of them are: • What makes an evening primrose open when it does? • Why does salt water fail to satisfy thirst? • Why does the amount of manganese in the diet affect the maternal instinct of an animal? • What is the description of aging in biochemical terms? • What meaning is to be assigned to the question: Is a virus a living organism? • What is a gene, and how does the original genetic constitution of a living organism express itself in the developed characteristics of the adult? Although these problems are complex, they are not problems “to which statistical methods hold the key” but they are “problems which involve dealing simultaneously with a sizable number of factors which are interrelated into an organic whole”. All these are not problems of disorganized complexity but, “in the language here proposed, problems of organized complexity.” [32] Weaver specified some more of these questions: • On what does the prize of wheat depend? • How can currency be wisely and effectively stabilized? • To what extend is it safe to depend on the free interplay of such economic forces as supply and demand? • To what extend must systems of economic control be employed to prevent the wide swings from prosperity to depression? Weaver stressed that the involved variables are “all interrelated in a complicated, but nevertheless not in helter-skelter, fashion” that these complex systems have “parts in close interrelations”, and that “something more is needed than the mathematics of averages.” ([32], p. 69)

Fig. 3.5. Weavers three regions of scientific problems.

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Here Weaver identified the problems of organized complexity “which science has as yet little explored or conquered” to be problems that can neither be reduced to a simple formula nor can they be solved with methods of probability theory: “These problems – and a wide range of similar problems in the biological, medical, psychological, economic, and political sciences – are just too complicated to yield to the old nineteenth-century techniques . . . ” [and] . . . “these new problems, moreover, cannot be handled with the statistical techniques so effective in describing average behaviour in problems of disorganized complexity.” ([32], p. 70) Finally, Weaver wrote: “These new problems – and the future of the world depends of many of them, requires science to make a third great advance, an advantage that must be even greater than the nineteenth-century conquest of problems of simplicity or the twentieth-century victory over problems of disorganized complexity. Science must, over the next 50 years, learn to deal with these problems of organized complexity.” ([32], p. 70) In my judgment science performed this task in fact with some new concepts and theories, which have – of course – their roots in earlier decades or centuries, but have got developed in the second half of the 20th century, e.g. self-organization, synergetic, chaos theory, and fractals. Yet another approach to find solutions for problems of organized complexity was given by Lotfi Zadeh’s theory of Fuzzy Sets and Systems. In the following sections I will present some remarks on this scientist and his new theory.

3.4

Lotfi A. Zadeh, Fuzzy Sets and Systems – The Work of a Lifetime

In the late 1940s and early 1950s “Information theory”, “The Mathematical Theory of Communication” [25], and Cybernetics [37] developed during the Second World War by Shannon, the American mathematician Norbert Wiener (1894–1964) (see figure 3.6), the Russian mathematician Andrej Nikolaevich Kolmogorov (1903–1987), the English statistician and biologist Ronald Aylmer Fisher (1890–1962) (see figure 3.7) and others became well-known. When Shannon and Wiener went to New York to give lectures on their new theories at Columbia University in 1946, they introduced these new milestones in science and technology to the electrical engineer Lotfi A. Zadeh who just have moved from Tehran, Iran, to the USA in 1944 with a BS degree in Electrical Engineering from Tehran in 1942. At MIT he continued his studies and he received an MS degree in 1946. He then removed to New York, where he joined the faculty of Columbia University as an instructor. In the early 1950s, the engineering-oriented so-called System Theory was a rising scientific discipline “to the study of systems per se, regardless of their physical structure”. Engineers in that time were, in general, inadequately trained to think in abstract terms, but nevertheless, Zadeh, who was then assistant professor at Columbia University, believed that it was only a matter of time before system theory would attain acceptance. 1958, when he was already professor of electrical engineering at Berkeley, he could describe problems and applications of system theory and its

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relations to network theory, control theory, and information theory. Furthermore, he pointed out “that the same ‘abstract systems’ notions are operating in various guises in many unrelated fields of science is a relatively recent development. It has been brought about, largely within the past two decades, by the great progress in our understanding of the behavior of both inanimate and animate systems—progress which resulted on the one hand from a vast expansion in the scientific and technological activities directed toward the development of highly complex systems for such purposes as automatic control, pattern recognition, data-processing, communication, and machine computation, and, on the other hand, by attempts at quantitative analyses of the extremely complex animate and man-machine systems which are encountered in biology, neurophysiology, econometrics, operations research and other fields” ([43], pp. 856f).

Fig. 3.6. Left: Claude Elwood Shannon; right: Norbert Wiener,

Fig. 3.7. Left: Anddrej N. Kolmogorov; right: Ronald A. Fischer.

3.4.1

Fuzzy Sets and Systems

In his first journal article on this subject he introduced the new mathematical entities – “Fuzzy Sets” – as classes or sets that “are not classes or sets in the usual sense of these terms, since they do not dichotomize all objects into those that belong to the class and those that do not”. In fuzzy sets “there may be a continuous infinity of

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grades of membership, with the grade of membership of an object x in a fuzzy set A represented by a number fA (x) in the interval [0, 1].” [44]

Fig. 3.8. Left: Lotfi A. Zadeh, 1950; right: Title page of his article “System Theory” [42].

In the same year he also defined Fuzzy Systems and a deeper view to this historical development shows that it was his aim to generalize System theory for all systems – including those that were too complex or poorly defined to be accessible to a precise analysis. Alongside the systems of the “soft” fields, the “non-soft” fields were replete with systems that were only “unsharply” defined, namely “when the complexity of a system rules out the possibility of analyzing it by conventional mathematical means, whether with or without the computers”. ([49], p. 469f). He started this generalization program by moving from systems to fuzzy systems.6

Fig. 3.9. Zadeh’s seminal article “Fuzzy Sets” [44] and his conference contribution “Fuzzy Sets and Systems” [45].

Already in the last third of April 1965, when the “Symposium on System Theory” took place at the Polytechnic Institute in Brooklyn, Zadeh presented “A New View on System Theory”. A shortened version of the paper delivered at this symposium appeared in the proceedings under the title “Fuzzy Sets and Systems”. Here, Zadeh 6

For more details to this aspect of the genesis of “Fuzzy Sets and Systems” see in [19] mainly the chapters 4 and 5.

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defined for the first time the concept of “fuzzy systems”: A system S is a fuzzy system if (input) u(t), output y(t), or state s(t) of S or any combination of them ranges over fuzzy sets ([45], p. 33). With the turn of the decade of the 1960s to the 1970, Zadeh was already pointing out the usefulness of Fuzzy Sets and Systems in computer science, in the life sciences, and also in the social sciences and humanities:7 • In describing their fields of application, he enumerated the problems that would be solved by future computers. Alongside pattern recognition, these included traffic control systems, machine translation, information processing, neuronal networks and games like chess and checkers. We had lost sight of the fact that the class of non-trivial problems for which one could find a precise solution algorithm was very limited, he wrote. Most real problems were much too complex and thus either completely unsolvable algorithmically or – if they could be solved in principle – not arithmetically feasible. In chess, for instance, there was in principle an optimal playing strategy for each stage of the game; in reality, however, no computer was capable of shifting through the entire tree of decisions for all of the possible moves with forward and backward repetitions in order to then decide what move would be the best in each phase of the game. The set of good strategies for playing chess had fuzzy limits similar to the set of tall men – these were fuzzy sets. ([49], p. 469f, [47], p. 199f) • Zadeh proposed his new theory of Fuzzy Sets to the life science community. In the second paragraph of his 1969-paper on a “Biological Application of the Theory of Fuzzy Sets and Systems” he wrote: “On scanning the literature of mathematical biosciences, a system theorist like myself cannot but be impressed with the sophistication of some of the recent work in this field [...]. Yet, as pointed out so cuccinctly by Richard Bellman8 , one cannot help feeling that, on the whole, the degree of success achieved by the use of mathematical techniques in biosciences has been quite limited. What is more disturbing, however, is the possibility that classical mathematics – with its insistence on rigor and precision – may never be able to provide totally satisfying answers to the basic questions relating to the behavior of animate systems.” [47], p. 199) In the already cited paper “Towards a Theory of Fuzzy Systems” he wrote: “The great complexity of biological systems may well prove to be an insuperable block to the achievement of a significant measure of success in the application of conventional mathematical techniques to the analysis of systems.” . . . “By ‘conventional mathematical techniques’ in this statement, we mean mathematical approaches for which we expect that precise answers to well-chosen precise questions concerning a biological system should have a high degree of 7 8

See also the argumentation in the introduction to this book. Richard Ernest Bellman (1920-1984) and Zadeh have been very close friends. Bellman was employed at RAND Corporation in Santa Monica until 1965, then he was appointed Professor of Mathematics, Electrical Engineering and Medicine at the University of California, Los Angeles (UCLA). For more biographical information on Bellman, see the IEEE History Center website: http : //www.ieee.org/organizations/history/center/

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relevance to its observed behaviour. Indeed, the complexity of biological systems may force us to alter in radical ways our traditional approaches to the analysis of such systems. Thus, we may have to accept as unavoidable a substantial degree of fuzziness in the description of the behaviour of biological systems as well as in their characterization.” ([49], p. 471) • About one decade later, in the 2nd volume of Kluwer’s series “Frontiers in System Research”, titled Systems Methodology in Social Science Research: Recent Developments, Zadeh wrote: “The systems theory of the future – the systems theory that will be applicable to the analysis of humanistic systems – is certain to be quite different in spirit as well as in substance from systems theory as we know it today.9 I will take the liberty of referring to it as fuzzy systems theory because I believe that its distinguishing characteristics will be a conceptual framework for dealing with a key aspect of humanistic systems – namely the pervasive fuzziness of almost all phenomena that are associated with their external as well as internal behavior.” ([59], p. 26) • Zadeh said also already in 1969 and again in 1971: “What we still lack, and lack rather acutely, are methods for dealing with systems which are too complex or too ill-defined to admit of precise analysis. Such systems pervade life sciences, social sciences, philosophy, economics, psychology and many other “soft” fields.” ([46], p. 1), [49], p. 469) He was intended to open the field of its applications to humanities and social sciences. Also reading an interview that was printed in the Azerbaijan International, in 1994, we can improve this view: when Zadeh was asked, “How did you think Fuzzy Logic would be used at first?” his retrospective answer was: “In many, many fields. I expected people in the social sciences-economics, psychology, philosophy, linguistics, politics, sociology, religion and numerous other areas to pick up on it. It’s been somewhat of a mystery to me, why even to this day, so few social scientists have discovered how useful it could be.” He was very surprised when Fuzzy Logic was first in the 1970s “embraced by engineers” and later in that decennium Fuzzy Sets and Systems have been successful “used in industrial process controls and in ’smart’ consumer products such as hand-held camcorders that cancel out jittering and microwaves that cook your food perfectly at the touch of a single button. I didn’t expect it to play out this way back in 1965.” [3] He concluded the 1982-paper as follows: “Fuzzy systems theory is not yet an existing theory. What we have at present are merely parts of its foundations. Nevertheless, even at this very early stage of its development, fuzzy systems theory casts some light on the process of approximate reasoning in human decision making, planning, and control. Furthermore, in the years ahead, it is likely to develop into an 9

Already in Part I of his seminal paper “The Concept of a Linguistic Variable and its Application to Approximate Reasoning” in 1975 he explained that a “humanistic system” is “a system whose behaviour is strongly influenced by human judgement, perception or emotions. Examples of humanistic systems are: economic systems, political systems, legal systems, educational systems, etc. A single individual and his thought processes may also be viewed as a humanistic system.” ([53], p. 200)

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effective body of concepts and techniques for the analysis of large-scale humanistic as well as mechanistic systems.” ([59], p. 39) After some of these “years ahead” have passed he presented perception-based system modeling: “A system, S, is assumed to be associated with temporal sequences of input X1 , X2 , . . . ; output Y1 , Y2 , . . . ; and states S1 , S2 , . . . S2 is defined by state-transition function f with St+1 = f (St , Xt ), and output function g with Yt = g(St , Xt ), for t = 0, 1, 2, . . . . In perception-based system modelling, inputs, outputs and states are assumed to be perceptions, as state-transition function, f , and output function, g.” ([61], p. 77.) This view on future artificial perception-based systems (see figure 3.10) – “Computing with Words”-systems and therefore systems for reasoning with perceptions – is the goal of Zadeh’s “Computational Theory of Perceptions” (CTP). It is closely linked by regarding the human brain as a fundamentally fuzzy system. Only in very few situations do people reason in binary terms, as machines classically do. This human characteristic is reflected in all natural languages, in which very few terms are absolute. The use of language is dependent on specific situations and is very seldom 100% certain. For example, the word “thin” cannot be defined in terms of numbers and there is no measurement at which this term suddenly stops being applicable. Human thinking, language and reasoning can thus indeed be called fuzzy. The theory of Fuzzy Sets and Systems has created a logical system far closer to the functionality of the human mind than any previous logical system. Fuzzy Sets and Systems and Fuzzy Logic enable them to express uncertainty regarding measurements, diagnostics, evaluations, etc. In theory, this should put the methods of communication used by machines and human beings on levels that are much closer to each other.

Fig. 3.10. Left: Lotfi A. Zadeh in his office, University of California at Berkeley, Evans Hall, 1989; right: Zadeh’s Perception-based system modelling [61].

Zadeh’s addition of scientific methods and techniques that are different from usual (classical) mathematics are rooting in his call for a non-probabilistic and nonstatistical mathematical theory already in 1962. It is understood that Zadeh kept

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sets of problems at the back of his mind, that are very similar to Weaver’s newlydiscovered scientific problems, when he described problems and applications of System theory and its relations to network theory, control theory, and information theory in the paper “From Circuit Theory to System Theory” [43]. He pointed out that “largely within the past two decades, by the great progress in our understanding of the behaviour of both inanimate and animate systems-progress which resulted on the one hand from a vast expansion in the scientific and technological activities directed toward the development of highly complex systems for such purposes as automatic control, pattern recognition, data-processing, communication, and machine computation, and, on the other hand, by attempts at quantitative analyses of the extremely complex animate and man-machine systems which are encountered in biology, neurophysiology, econometrics, operations research and other fields” [43]. Then he wrote the famous paragraph where he used for the first time the word “fuzzy” to name unsharp entities in mathematics: “In fact, there is a fairly wide gap between what might be regarded as “animate” system theorists and “inanimate” system theorists at the present time, and it is not at all certain that this gap will be narrowed, much less closed, in the near future. There are some who feel that this gap reflects the fundamental inadequacy of the conventional mathematics – the mathematics of precisely-defined points, functions, sets, probability measures, etc. – for coping with the analysis of biological systems, and that to deal effectively with such systems, which are generally orders of magnitude more complex than man-made systems, we need a radically different kind of mathematics, the mathematics of fuzzy or cloudy quantities which are not describable in terms of probability distributions. Indeed, the need for such mathematics is becoming increasingly apparent even in the realm of inanimate systems, for in most practical cases the a priori data as well as the criteria by which the performance of a man-made system is judged are far from being precisely specified or having accurately-known probability distributions” [43]. In this 1962-paper Zadeh had called for “fuzzy mathematics” without exact knowing, what kind of theory he would create about three years later when he established the theory of Fuzzy Sets and Systems.10 The potential of the new methods of the theory of Fuzzy sets and Fuzzy systems urged Ebrahim H. Mamdani (1942–2010), a professor of electrical engineering at Queen Mary College in London, to attempt the implementation of a fuzzy system under laboratory conditions. He expressed the intention to his doctor student Sedrak Assilian, who designed a fuzzy algorithm to control a small steam engine within a few days. The concepts of so-called linguistic variables and Zadeh’s max–min composition were suitable to establish fuzzy control rules because input, output and state of the steam engine system range over fuzzy sets. Thus, Assilian and Mamdani designed the first real fuzzy application when they controlled the system by a fuzzy rule base system [6].11 In 1974, Assilian completed his Ph.D. thesis on this first fuzzy control system [1]. The steam engine heralded the Fuzzy boom that started 10 11

For details of the prehistory of the theory of Fuzzy Sets and Systems see:[19], chapters 1-4. Fore more details on Ebrahim Mamdani and the history of Fuzzy Control systems see chapter 6 in [19] and the appearing volume [29] “in memoriam Ebrahim Mamdani”, especially my contribution [22].

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in the 1980s in Japan and later pervaded the Western hemisphere. Many fuzzy applications, such as domestic appliances, cameras and other devices appeared in the last two decades of the 20th century. Of greater significance, however, was the development of fuzzy process controllers and fuzzy expert systems that served as trailblazers for scientific and technological advancements of fuzzy sets and systems.

3.5

Reflections

In this chapter we collect some similarities in the scientific works of Warren Weaver and Lotfi Zadeh – both became mathematical orientated engineers. Weaver started his studies of engineering and in his autobiography he described this and what followed: “Early in my sophomore year I began to sense the semantic error that had previously equated the words “engineering” and “science”. I don not mean to imply that I did not appreciate and enjoy the strictly engineering subjects, for I did. In those days one could enroll at Wisonsin for a five-year course that earned a professional degree – in my case that of C.E. – and I stuck out the whole five years and have never for a moment regretted that I did so, even though I have never explicitely made use of this civil engineering training. The awakening occured during the first semester of my second year when, according to the now outdated system then in use, my sophomore mathematics subject was differential calculus. Not only did I then for the first time meet a really poetic branch of mathematics, all alive with excitement and power and logical beauty – I had the great good fortune to have a truly great teacher.”[36]12 In an unpublished interview that Zadeh gave me in 1999 he said “I was always interested in mathematics, even when I was in Iran, in Teheran, but I was not sufficiently interested to become a pure mathematician. In other words, I never felt that I should pursuit pure mathematics or even applied mathematics. So, this mixture of an engineer was perfectly suited for me. So, essentially, I’m sort of mathematical engineer, that’s the way I would characterize myself. But I’m not a mathematician. I was somewhat critical of the fact that mathematics has gone away from the real world.. . . I criticized the fact that mathematics has gone too far away from the real world.” [63] Weaver and Zadeh worked in the starting phase of Information Theory in the last 1940s and the beginning 1950s. However, they both saw a great field of scientific progress in the non-technical sciences, especially in the life sciences. Both of them had a deep interest in languages and especially in the subject of meaning and they also have been very intersted in research programs to computerize the translation of natural languages in each other. Last but not least they both recommended interdisciplinarity and team work – and in anti-nationalism. 12

This teacher was the applied mathematician Charles Sumner Slichter, see section 3.2.

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Life Sciences

In section 3.2 on Weaver’s biography I described his initiative for a “great change” in the world’s scientific system. With Rockefeller’s new program in the 1930s to support research in the life sciences he established “a fruitful union” of physics, chemistry, and biosciences. Zadeh presented his new theory of fuzzy sets in the already in section 3.4.1 quoted contribution “Biological Application of the Theory of Fuzzy Sets and Systms” at the International Symposium on Biocybernetics of the Central Nervous System, held at the Sheraton Park Hotel, Washington, D.C. in the year 1969. In my view, Zadeh approached with this call for Fuzzy applications in the life sciences to Weaver’s reasoning about 20 years before. In this paper Zadeh proposed already one possible application of his fuzzy sets in the medical sciences; when he wrote that “a human disease, e.g., diabetes, may be regarded as a fuzzy set in the following sense. Let X = x denote the collection of human beings. Then diabetes is a fuzzy set, say D, in X, characterized by a membership function μD (x) which associates with each human being x his grade of membership in the fuzzy set of diabetes” ([47], p. 205). Zadeh formulated his thoughts on medical applications of the theory of fuzzy sets very accurately and pointed: “In some cases, it may be more convenient to characterize a fuzzy set representing a disease not by its membership function but by its relation to various symptoms which in themselves are fuzzy in nature. For example, in the case of diabetes a fuzzy symptom may be, say, a hardening of the arteries. If this fuzzy set in X is denoted by A, then we can speak of the fuzzy inclusion relation between D and A and assign a number in the interval [0, 1] to represent the “degree of containment” of A in D. In this way, we can provide a partial characterization of D by specifying the degrees of containment of various fuzzy symptoms A1 , . . . , Ak in D. When arranged in a tabular form, the degrees of containment constitute what might be called a containment table.” ([47], p. 205) In the 1980s the Iranian-German physician and philosopher of medicine Kazem Sadegh-Zadeh (born 1942, see figure 3.11) was aware of the vagueness of medical thinking and saw a solution in Zadeh’s theory of fuzzy sets: “The main source of misunderstandings and disagreements both in nosology and in philosophy of health and disease is basically this inherent fuzziness of the subject that has not yet been sufficiently recognized to evoke the need for an appropriate conceptual apparatus and methodology”, he said already in the year 1982 ([13], p. 629). He discussed the meaning of the notions “health”, “illness”, and “diseases”. To this end he took a fuzzy theoretic approach towards a novel theory of these concepts: “health is a matter of degree, illness is a matter of degree, and disease is a matter of degree.” [13] He rejected the conceptual opposition that an individual could be either healthy or ill. In his philosophy of medicine, health and illness are particular fuzzy states of health. He defined the state of health to be a linguistic variable called state-of-health, “whose term set may be conceived of as something like:

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Tstate-of-health = { well, not well, very well, very very well, extremely well, ill, not ill, more or less ill, very ill, very very ill, extremely ill, not well and not ill, . . . etc. . . . }” This state-of-health linguistic variable operates over the fuzzy set H (health), the set of healthy people, and it assigns to degrees of health, H(x) values, elements of the term set Tstate-of-health [14] (see figure 3.11).

Fig. 3.11. Left: Kazem Sadegh-Zadeh, 2010; right: the linguistic variable State-of-health [14].

Sadegh-Zadeh introduced the term “disease”not with a linguistic but with a social definition: There are complex human conditions “that in a human society are termed diseases”, and he specified potential candidates: “like heart attack, stroke, breast cancer, etc. Such complex human conditions are not, and should not be, merely confined to biological states of an organism. They may be viewed and represented as large fuzzy sets which also contain parts that refer to the subjective, religious, transcendental and social world of the ill, such as for example, pain, distress, feelings of loneliness, beliefs, behavioral disorders, etc.” [14] In Sadegh-Zadeh’s philosophy of medicine, disease entities “may be conceptualized as fuzzy sets’ and “symptoms, and signs would then belong to an individual disease to particular extents. Thus, an individual disease would appear as a multidimensional cloud rather than as a clear-cut phenomenon.” [13]13 3.5.2

Information Theory

Claude E. Shannon’s article “A Mathematical Theory of Communication” appeared in two parts in the July and October 1948 editions of the Bell System Technical 13

For more details on Fuzzy Sets and Systems in medicine see [18] and also chapter 6 in [19]. Sadegh-Zadeh’s recent Handbook on Analytical Philosophy of Medicine [15] and a companion volume to this work [24] will appear in 2011 or 2012. See also Sadegh-Zadeh’s contributions to this volume.

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Journal [25]. Shannon was a Bell employee and at the Bell Telephone Laboratories there existed already a tradition in “communication technology” with Harry Nyquist (1889-1976) and Ralph Vinton Lyon Hartley (1888-1970). However, in the 1920s, when these two engineers published their seminal articles [10], [11], [4], the term “communication” was still not used.14 These authors used the terms “transmission of intelligence (by telegraph)” whereas the concept of “information” arose with Hartleys article “Transmission of Information” in 1928 [4]. In 1948, Shannon included “new factors” to the theory of information transmission, “in particular the effect of noise in the channel, and the savings possible due to the statistical structure of the original message and due to the nature of the final destination of the information.” ([25], p. 379).

Fig. 3.12. Left: Heading of Weaver’s article “The Mathematics of Communication” in the Scientific American [33].; right: Editor’s note on the forthcomming book The Mathematical Theory of Communication [26] in Weaver’s article [33].

However, it is probable that Shannon’s article wouldn’t have become famous without the help of Weaver’s popular text “The Mathematics of Communication” [33] that re-interpreted Shannon’s work for broader scientific audiences. This article came into being when Chester I. Barnard (1948-1952) was president of the Rockefeller Foundation, as Weaver told in his autobiography: “On day at lunch he [Barnard] asked me if I had read, in the Bell Technical Journal, an article by Claude E. Shannon on a mathematical theory of communication. I had. Did I understand the article? Not realizing the risk I was running, I answered yes. Could these ideas be explained in less formidably mathematical terms? Again, yes. “All right,” he said, “do so.” The article appeared in July 1949 in the Scientific American, and the editor added to this article already: “The University of Illinois Press will shortly publish a memoir on communication theory. This will contain the original articles on communication by Claude E. Shannon of the Bell Telephone Laboratories, together with an expanded and slightly more technical version of Dr. Weaver’s article.” [33] (see figure 3.12) One year later, Weaver had modified this text a little and accentuated it with the new title “Recent Contributions to the Mathematical Theory of Communication” [34] as a kind of “introduction” into Shannon’s article and both manuscripts 14

For more detail on the history of Information Theory and especially its connection to the history of Fuzzy Sets and Systems see [21] and chapter 3 in [19].

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appeared in the book The Mathematical Theory of Communication [26] that represents the beginning of the then so-called “Information Theory”. To illustrate this theory, Shannon had drawn a diagram of a general communication model and the very similar schema was also printed in Weaver’s article (see figure for Weaver’s version of the schema in figure 3.13). Shannon’s communication systems have the following parts: • An information source produces a series of messages that are to be delivered to the receiver side. Transmission can occur via a telegraph or teletypewriter system, in which case it is a series of letters. Transmission can also occur via a telephone or radio system, in which case it is a function of time f (t), or else it is a function f (x, y,t), such as in a black and white television system, or it consists of complicated functions. • The transmitter transforms the message in some way so that it can produce signals that it can transmit via the channel. In telegraphy, these are dotdash codes; in telephony, acoustic pressure is converted into an electric current. • The channel is the medium used for transmission. Channels can be wires, light beams and other options. • The receiver must perform the opposite operation to that of the transmitter and in this way reconstructs the original message from the transmitted signal. • The destination is the person or entity that should receive the message.

Fig. 3.13. Shannon’s diagram of communication in the version of Weaver in [33], p. 12 f.

According to this, Shannon formulated the essential difficulty of the communication process: “The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point.” ([25], p. 379) This problem was considered by the young electrical engineer Lotfi Zadeh as a “basic problem in communication of information”. After he had published his Ph.D. thesis on “Frequency Analysis of Variable Networks” in 1950 in the journal Proceedings of the I.R.E. [39] he was appointed assistant professor at Columbia Unversity in New York in the same year. He was enhanced by Shannon’s work (and also by Wiener’s Cybernetics) and in February 1952, when he presented “Some Basic Problems in Communication of Information” at the meeting of the Section of Mathematics and Engineering of the New York Academy of Sciences, he sketched the so-called recovery process of transmitted signals: “Let X = x(t) be a set of signals. An arbitrarily selected member of this set, say x(t), is transmitted through a noisy channel C and is received as y(t). As a result of the noise and distortion introduced by C, the received signal y(t) is, in general, quite different from

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x(t). Nevertheless, under certain conditions it is possible to recover x(t) — or rather a time-delayed replica of it -– from the received signal y(t).” ([41], p. 201). In this paper, he restricted his view to the problem to recover x(t) from y(t) “irrespective of the statistical character of x(t)”. Corresponding to the relation y = Cx between x(t) and y(t) he represented the recovery process of y(t) from x(t) by x = Γ −1 y, where Γ −1 is the inverse of Γ , if it exists, over y(t). Zadeh represented signals as ordered pairs of points in a signal space S, which is imbedded in a function space. To measure the disparity between two signals x(t) and y(t) he attached a distance function d(x, y) with the usual properties of a metric. Then he considered the special case in which it is possible to achieve a perfect recovery of the transmitted signal x(t) from the received signal y(t). He supposed that “X = x(t) consist of a finite number of discrete signals x1 (t), x2 (t), ..., xn (t), which play the roles of symbols or sequences of symbols. The replicas of all these signals are assumed to be available at the receiving end of the system. Suppose that a transmitted signal xk is received as y. To recover the transmitted signal from y, the receiver evaluates the ‘distance’ between y and all possible transmitted signals x1 , x2 , ..., xn , by the use of a suitable distance function d(x, y), and then selects that signal which is ‘nearest’ to y in terms of this distance function (see figure 3.14). In other words, the transmitted signal is taken to be the one that results in the smallest value of d(x, y). This in brief, is the basis of the reception process.” ([41], p. 201).

Fig. 3.14. Recovery of the input signal by comparing the distances between the obtained signals and all possible transmitted signals

By this process the received signal xk is always ‘nearer’ – in terms of the distance functional d(x, y) – to the transmitted signal y(t) than to any other possible signal xi , i.e. d(xk , y) < d(xi , y), i = k, for all k and i. But at the end of his reflection of this problem Zadeh conceded that “in many practical situations it is inconvenient, or even impossible, to define a quantitative measure, such as a distance function, of the disparity between two signals. In such cases we may use instead the concept of neighborhood, which is basic to the theory of topological spaces” ([41], p. 202).

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Spaces such as these, Zadeh surmised, could be very interesting with respect to applications in communication engineering. Therefore, this problem of the recovery process of transmitted signals which is a special case of Shannon’s fundamental problem of communication, the problem “of reproducing at one point either exactly or approximately a message selected at another point”, was one of the problems that initiated Zadeh’s thoughts about not precisely specified quantitative measures, i. e. of cloudy or fuzzy quantities. 3.5.3

Meaning

When Weaver considered Shannon’s “Mathematical Theory of Communication” in a philosophical way, he was already familiar with the Foundations of the Theory of Signs that have been published in 1938 by the American engineer and philosopher Charles William Morris (1901-1979) [9]. This monograph contains a general theory of signs. It concerns not only sign processing of human beings but sign processing of all organisms. It seems that Weaver adopted Morris’ classification into syntactic, semantic, and pragmatic problems, when he wrote: “In communication there seem to be problems at three levels: 1) technical, 2) semantic, and 3) influential. The technical problems are concerned with the accuracy of transference of information from sender to receiver. They are inherent in all forms of communication, whether by sets of discrete symbols (written speech), or by a varying two-dimensional pattern (television). The semantic problems are concerned with the interpretation of meaning by the receiver, as compared with the intended meaning of the sender. This is a very deep and involved situation, even when one deals only with the relatively simple problems of communicating through speech. [. . . ] The problems of influence or effectiveness are concerned with the success with which the meaning conveyed to the receiver leads to the desired conduct on his part. It may seem at the first glance undesirable narrow to imply that the purpose of all communication is to influence the conduct of the receiver. But with any reasonably broad definition of conduct, it is clear that communication either affects conduct or is without any discernible and provable effect at all.” ([9], p. 11) In the revised version of his article [34] that was published in [26], Weaver explained this trichotomy in connection with the communication problem and he divided it into three levels: • Level A contains the purely technical problem involving the exactness with which the symbols can be transmitted, • Level B contains the semantic problem that inquires as to the precision with which the transmitted signal transports the desired meaning, • Level C contains the pragmatic problem pertaining to the effect of the symbol on the destination side: What influence does it exert? Weaver underscored very clearly the fact that Shannon’s theory did not even touch upon any of the problems contained in levels B and C, that the concept of information therefore must not be identified with the “meaning” of the symbols: “In fact, two messages, one of which is heavily loaded with meaning and the other of

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which is pure nonsense, can be exactly equivalent, from the present viewpoint, as regards information.” [34] However, there is plenty of room for fuzziness in the levels B and C. It is quite plain: Weaver went over and above Shannon’s theory: “The theory goes further. Though ostensibly applicable only to problems at the technical level, it is helpful and suggestive at the levels of semantics and effectiveness as well.” Weaver stated, that Shannon’s formal diagram of a communication system (see figure 3.13) “can, in all likelihood, be extended to include the central issues of meaning and effectiveness. [. . . ] One can imagine, as an addition to the diagram, another box labelled “Semantic Receiver” interposed between the engineering receiver (which changes signals to messages) and the destination. This semantic receiver subjects the message to a second decoding the demand on this one being that it must match the statistical semantic characteristics of the message to the statistical semantic capacities of the totality of receivers, or of that subset of receivers which constitutes the audience one wishes to affect. Similarly one can imagine another box in the diagram which inserted between the information source and the transmitter, would be labelled “Semantic Noise” (not to be confused with “engineering noise”. This would represent distortions of meaning introduced by the information source, such as a speaker, which are not intentional but nevertheless affect the destination, or listener. And the problem of semantic decoding must take this semantic noise into account. It is also possible to think of a treatment or adjustment of the original message that would make the sum of message meaning plus semantic noise equal to the desired total message meaning at the destination.” ([34], p. 13)

Fig. 3.15. Shannon’s diagram with Weaver’s addition of the two boxes “Semantic Receiver” and “Semantic Noise” in my version.

Figure 3.15 shows my version of Shannon’s diagram of a communication system with Weaver’s two additional “fuzzy” boxes. I interpret the “first coding” between the information source and the “Semantic Noise” as a fuzzification and the “second decoding” between the “Semantic Receiver” and the destination as a defuzzification.15 In the late 1960s, when Zadeh became interested in combining Fuzzy Sets and Systems and linguistics, his occupation with natural and artificial languages gave 15

For more details in this subject see [22].

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rise to his studies in humanities and social sciences, especially on semantics. “Can the fuzziness of meaning be treated quantitatively, at least in principle?” he asked in his 1971 article “Quantitative Fuzzy Semantics” ([50], p. 160) that started with a hint to this studies: “Few concepts are as basic to human thinking and yet as elusive of precise definition as the concept of “meaning”. Innumerable papers and books in the fields of philosophy, psychology, and linguistics have dealt at length with the question of what is the meaning of "meaning" without coming up with any definitive answers.” ([50], p. 159) Now, Zadeh was intended “to point to the possibility of treating the fuzziness of meaning in a quantitative way and suggest a basis for what might be called quantitative fuzzy semantics.” In the section entitled “Meaning” he set up the basics: “Consider two spaces: (a) a universe of discourse, U, and (b) a set of terms, T , which play the roles of names of subsets of U . Let the generic elements of T and U be denoted by x and y, respectively. Then he started to define the meaning M(x) of a term x as a fuzzy subset of U characterized by a membership function μ (y|x) which is conditioned on x. One of his examples was: “Let U be the universe of objects which we can see. Let T be the set of terms white, grey, green, blue, yellow, red, black. Then each of these terms, e.g., red, may be regarded as a name for a fuzzy subset of elements of U which are red in color. Thus, the meaning of red, M(red), is a specified fuzzy subset of U.” In the following section of this paper, that is named “Language”, Zadeh regarded a language L as a “fuzzy correspondence”, more explicit, a fuzzy binary relation, from the term set T = x to the universe of discourse U = y that is characterized by the membership function μL : T × U → [0, 1]. If a term x of T is given, then the membership function μL (x, y) defines a set M(x) in U with the following membership function: μM(x) (y) = μL (x, y). Zadeh called the fuzzy set M(x) the meaning of the term x; x is thus the name of M(x). In the 1970s Zadeh published important papers summarizing and developing these concepts of language and meaning: Already at the International Conference Man and Computer in Bordeaux, 1970, he established the basic aspects of a theory of fuzzy languages that is “much broader and more general than that of a formal language in its conventional sense”, as he wrote in his contribution to the proceedings that appeared in 1972. ([51], p. 134) One year later he published his famous paper “Outline of a new approach to the analysis of complex systems and decision processes” in the IEEE Transaction on Systems, Man, and Cybernetics [52] and in 1975 the three-part article “The concept of a Lingustic Variable and its Application to Approximate Reasoning” appeared in the journal Information Sciences ([53], [54], [55]). In the same year Zadeh published “Fuzzy Logic and Approximate Reasoning” in the philosophical journal Synthese [56] and in 1978 it followed “PRUF a meaning representation language for natural languages” in the International Journal of Man-Machine Studies [57].16 The subjects of languages and meaning are deeply connected with the problem of the language translation and therefore it is not surprising that Weaver and Zadeh both considered this topic as well. This will be shown in the next section. 16

For more details on Fuzzy Sets int he areas of Languages and Semantics see [23].

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3.5.4

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Language and Translation

One world of many different languages spoken by many different people, especially by scientists – this fact became well-known to Weaver when he started travelling very often due to his job at the Rockefeller Foundation. In his autobiography he wrote: “I had fair German but only rudimentary oral French, so that language was at once a problem.” ([36], p. 64) In the 1930s, when he came in touch with different cultures on various continents, he was confronted with the problem of language translation. During the Second World War, he was engaged in decipherment or decoding of enemy messages and he saw that machines could give support to do this job. When the war was over he brooded whether it is unthinkable to design digital computers which would translate documents between natural human languages: “Early in 1947, having pondered the matter for nearly two years, I started to formulate some ideas about using computers to translate from one language to another. I first wrote this suggestion to the American mathematician Norbert Wiener (18941964), who was then teaching at MIT. I chose him because I knew him as a gifted linguist and brilliant logician. To my surprise and disappointment he was almost completely skeptical and discouraging about the idea. But I continued to turn the subject over in my mind, and in July 1949 I wrote a thirteen-page memorandum explaining why there was some real hope that translation could be done with computers. Obviously, an efficient procedure for translating would be of great social service to the world, even if it did not produce elegant prose. It was by that time clear that computers could carry out any logically planned sequence of steps, however complex and extensive, at lightning speed. Memory organs would be available for storing vast data such as are found, for example in Russian-English, TurkishFrench, and other dictionaries.” ([36], p. 107) Weaver was looking for “invariant properties” – that “again not precisely but to some statistically useful degree” – are equal for all languages. In “Translation” he wrote: “All languages – at least all the ones under consideration here – were invented and developed by men, whether Bantu or Greek, Islandic or Peruvian, have essentially the same equipment to bring to bear on this problem. They have vocal organs capable of producing about the same set of sounds (with minor exceptions, such as the glottal click of the African native). Their brains are of the same general order of potential complexity. The elementary demands for language must have emerged in closely similar ways in different places and perhaps at different times. One would expect wide superficial differences; but it seems very reasonable to expect that certain basic, and probably very nonobvious, aspects be common to all the developments.” He conjectured “that the way to translate from Chinese to Arabic, or from Russian to Portuguese, is not to attempt the direct route [. . . ]. Perhaps the way is to descend, from each language, down to the common base of human communication – the real but as yet undiscovered universal language – and – then re-emerge by whatever particular route is convenient.” ([35], p. 23) In “Translation” Weaver quoted from Wiener’s answer to his letter in 1947: “Second – as to the problem of mechanical translation, I frankly am afraid the boundaries of words in different languages are too vague and the emotional and international

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Fig. 3.16. First page of Machine translation of Languages: fourteen essays [12]; right: Heading of Weaver’s memorandum “Translation” in this collection.

connotations are too extensive to make any quasimechanical translation scheme very hopeful. I will admit that basic English seems to indicate that we can go further than we have generally done in the mechanization of speech, but you must remember that in certain respects basic English is the reverse of mechanical and throws upon such words as get a burden which is much grater than most words carry in conventional English. At the present tune, the mechanization of language, beyond such a stage as the design of photoelectric reading opportunities for the blind, seems very premature ...” ([35], p. 18) In Scene of Change Weaver wrote: “There appeared to be two really serious difficulties: the ambiguity of meaning of many words (“fire” means to shoot, to set ablaze, to discharge from a job); and the apparently distinct and complicated ways in which the grammar (and more especially syntax) of various languages accomodate the expression of ideas. For both of these difficulties I had suggestions of ways in which they might be handled. Being no student of linguistics, I was conscious that my ideas be naïve or unworkable. But the possibilities so intrigued me that I had the memorandum mimeographed. I sent it to twenty or thirty persons – students of linguistics, logicians, and mathematicians. The first reaction was almost universally negative.” ([36], p. 107) However, already in his memorandum Weaver also wrote that “The idea has, however, been seriously considered elsewhere. The first instance known to W. W. [Warren Weaver], subsequent to his own notion about it, was described in a memorandum dated February 12, 1948, written by Dr. Andrew D. Booth who, in Professor J. D. Bernal’s department in Birbeck College, University of London, had been active in computer design and construction. Dr. Booth said: A concluding example, of possible application of the electronic computer, is that of translating from one language into another. We have considered this problem in some detail, and it

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transpires that a machine of the type envisaged could perform this function without any modification in its desgn.” ([35], p. 19)17 Also the linguist and information scientist William John Hutchins (born 1939), an expert in the history of machine translation, wrote in the year 2000: “The memorandum met with scepticism among some recipients, and enthusiasm among others, including Erwin Reifler (University of Wahington) and Abraham Kaplan (Rand), who began investigations. It was a major stimulus to research activity, thanks in large part to Weaver’s personal standing and influence. It led to the appointment at MIT of Bar-Hillel and the convening of the first MT18 conference. The outcome of the conference was the first book-length treatment of MT, to which Weaver contributed the foreword.19 In this, his last contribution to the new field of MT, he states his optimism for the “new Tower of Anti-Babel” under construction “not to charm or delight, not to contribute to elegance or beauty; but to be of wide service in the work-a-day task of making available the essential content of documents in languages which are foreign to the reader.” And so it has proved.” ([5], p. 20)20 Hutchins emphazised in this brief note that one of Weaver’s proposals in “Translation” “was founded on the logical bases of language. He drew attention to the work of McCulloch and Pitts (1943)21 on the analogies between the neural structure of the human brain and ’logical machines’, wich suggested that “insofar as written language is an expression of logical character,” the problem of translation is formally solvable.” ([5], p. 19) Nowadays we know much more on natural neural networks and against the whole background of scince-history it seems interesting to read what the the mathematician Enric Trillas said in his invited lecture “Fuzzy Logic. Looking at the Future” for the AGOP 2009 conference: “As one of the brain’s activity, language appeared after brain is actually commanded through some brain functions and, in this sense, perhaps language is no less complex than brain functioning is. In addition, there is a relevant aspect that makes language less known than brain is. This lies in the amount of specific knowledge expressed in scientific terms neurobiologists do have on the functioning of the brain, something that is not the case with language since it is not currently treated as a natural experimental discipline concerning a special type of living being. Such a new discipline, provided it was created, could result in an upmost interest for the advancement of Computing with Words and Perceptions 17

18 19 20

21

Andrew Donald Booth (1918-2009) was a British electrical engineer, physicist and computer scientist; John Desmond Bernal (1901-1971) was a British scientist and historian of science. MT means Machine Translation. This is the volume Machine translation of Languages: fourteen essays that was edited by William. N. Locke and A. Donald Booth [12]. Erwin Reifler (1903-1965) was an Austrian-american sinologist and linguist, Abraham Kaplan (1918-1993) was an American philosopher, Yehoshua Bar-Hillel (1915-1975) was an Austrian-Israeli philosopher, mathematician, and linguist. The mathematician, psychologist and neurologist Warren McCulloch (1898-1969) and his co-worker Walter Pitts (1923-1969) published in 1943 the first ideas on artificial neural networks: “A Logical Calculus of the Ideas Immanent in Nervous Activity” [8].

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(CWP). At the end, brain functioning is closely related with the electro-chemical processing of perceptions, and language with the representation and communication, or spreading of such processing. [. . . ] Concerning the mathematical modelling of language, and the possible benefits that can follow from some results in fuzzy logic, it could be interesting to recall, for instance, the presence of symmetry in the way of obtaining the membership function on an antonym, or opposite, of a linguistic term P from a membership function of P. Antonymy is an important feature of language, and since symmetry is a pervasive concept in the world, and also in the brain, possibly it should play some relevant role in language, and hence it could deserve to be studied for a deeper characterization of language. This is a challenging subject for mathematicians and computer scientists interested in CWP.” [28] Weaver’s midcentury expectations on the progress in science and technology seem to be anticipating important topics of vague, fuzzy or approximate reasoning, the meaning of concepts, and “to descend from each language, down to the common base of human communication the real but as yet undiscovered universal language” [35] that seems similar to Zadeh’s concept of “Precisiated Natural language” (PNL) that he introduced already in his 2001 AI-Magazine article. [64] and obviously he perceived that there will be a big change in science and technology in the 21st century. In 2004 Zadeh described the conceptual structure of PNL as a basis for CTP in greater detail. [66] However, there is no direct relation between the work of Weaver and Zadeh22 but it seems to me that it is worth to study Weavers writings in this context. 3.5.5

Interdisciplinarity and Anti-nationalism

Weaver was familiar with the computer development during the Second World War and to solve problems of organized complexity he pinned his hope on the power of the “wartime development of new types of electronic computers” on the one hand and on the other hand a second wartime advance, the “mixed-team” approach of operational analysis: “Although mathematicians, physicists, and engineers were essential, the best of the groups also contained physiologists, biochemists, psychologists, and a variety of representatives of other fields of the biochemical and social sciences. Among the outstanding members of English mixed teams, for example, were an endocrinologist and an X-ray crystallographer. Under the pressure of war, these mixed teams pooled their resources and focused all their different insights on the common problems. It is found, in spite of the modern tendencies toward intense scientific specialization, that members of such diverse groups could work together and could form a unit which was much greater than the mere sum of its parts. It was shown that these groups could tackle certain problems of organized complexity, and 22

In a personal message Zadeh answered to the author’s question whether he was familiar with Weaver’s papers in the 1940s and 1950s that he did not read the papers [32] and [32]. He also wrote: “It may well be the case that most people near the center [of the “world of information theory and communication” in that time] did not appreciate what he had to say. In a sense, he may have been ahead of his time.”(L. A. Zadeh, e-mail to the author, May 23, 2009.)

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get useful answers” ([32], p. 541) and some paragraphs later Weaver “suggested that some scientists will seek and develop for themselves new kinds of collaborative arrangements; that these groups will have members drawn from essentially all fields of science; and that these new ways of working, effectively instrumented by huge computers, will contribute greatly to the advantage which the next half century will surely achieve in handling the complex, but essentially organic, problems of the biological and social sciences.” ([32], p. 542)23 In the 1990s, Zadeh pleaded for the establishment of the research field of Soft Computing (SC). In his foreword to the new journal Applied Soft Computing he recommended that instead of “an element of competition” between the complementary methodologies of SC “the coalition that has to be formed has to be much wider: it has to bridge the gap between the different communities in various fields of science and technology and it has to bridge the gap between science and humanities and social sciences! SC is a suitable candidate to meet these demands because it opens the fields to the humanities. [...] Initially, acceptance of the concept of soft computing was slow in coming. Within the past few years, however, soft computing began to grow rapidly in visibility and importance, especially in the realm of applications which related to the conception, design and utilization of information/intelligent systems. This is the backdrop against which the publication of Applied Soft Computing should be viewed. By design, soft computing is pluralistic in nature in the sense that it is a coalition of methodologies which are drawn together by a quest for accommodation with the pervasive imprecision of the real world. At this juncture, the principal members of the coalition are fuzzy logic, neuro-computing, evolutionary computing, probabilistic computing, chaotic computing and machine learning.” ([65], p. 1-2) However, this means also that researchers of these fileds have to bridge the borders of their diciplines. Weaver wrote already at the end of “Science and Complexity”: The great gap, which lies so forebodingly between our power and our capacity to use power wisely, can only be bridged by a vast combination of efforts. Knowledge of individual and group behavior must be improved. Communication must be improved between peoples of different languages and cultures, as well as between all the varied interests which use the same language, but often with such dangerously differing connotations. A revolutionary advance must be made in our understanding of economic and political factors. Willingness to sacrify selfish shortterm interests, either personal or national, in order to bring about longterm improvement for all must be developed.” ([32], p. 544) Zadeh took the same line when he wrote in the year 2001, : “The concept of soft computing crystallized in my mind during the waning months of 1990. Its genesis reflected the fact that in science, as in other realms of human activity, there is a tendency to be nationalistic . . . .” And in the same text he said that the “launching of Berkeley Initiative in Soft Computing (BISC) at UC, Berkeley in 1991, represented a rejection of this mentality.” [65] 23

For more details see also [20].

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3.6

Outlook

Almost eight years ago Warren Weaver’s “Science and Complexity” was reproduced as a “classical paper” in Emergence - Journal of Complexity Issues (E:CO - Emergency: Complexity and Organization). In the inroduction to this issue Peter M. Allen, the editor in chief, wrote that this “classic paper is classic classic” because Weaver anticipated many of our today deeper insights. Another new but short introduction to Weaver’s paper was given by the management-professor Ross Wirth, and in his last section he said: “When reading this, there is a bit of déjà vu in what we sometimes hear today of our study of complexity. So too in the statement that “science has, to date, succeeded in solving a bewildering number of relatively easy problems, whereas the hard problems, and the ones which perhaps promise most for man’s future lie ahead” (Weaver, 1948). In the end the reader is left with conflicting feelings of surprise that we are not further along in our understandings of complexity given Weaver’s ideas nearly 60 years ago, while also still being optimistic in our success for the same reasons Weaver was optimistic.” [38] Weaver’s optimism concerning the two ways of computerization and interdisciplinarity is understandable because science in the USA realized these ways successfully in the time after the Second World War. 60 years ago this view was an important opinion of a science manager like Weaver. Are there reasons to be optimistic for our world society today too? In the year 2007 the science journalist Michael Shermer24 pointed to this circumstances in his Scientific American column when he wrote: “Over the past three decades I have noted two disturbing tendencies in both science and society; first, to rank the sciences from “hard” (physical sciences) to “medium” (biological sciences) to soft (social sciences). . . ”, second, to divide science writing into two forms, technical and popular. And, as such rankings and divisions are wont to do, they include an assessment of worth, with the hard sciences and technical writing respected the most, and the soft sciences and popular writing esteemed the least. Both these prejudices are so far off the mark that they are not even wrong. [27]. As Warren Weaver and Lotfi Zadeh wrote very often, the solution seems to be a bridging of the gab between hard sciences and soft sciences and it seems that Soft Computing and the theory of Fuzzy Sets and Systems in the centre of this coalition of modern scientific methodologies are suitable tools to do so.

Acknowledgments Work leading to this paper was partially supported by the Foundation for the Advancement of Soft Computing Mieres, Asturias (Spain). I would like to thank the members of the Scientific Committee of the European Centre for Soft Computing in Mieres and, most notably, I thank Claudio Moraga for his important help to write this chapter. 24

See the Introdcution to this book.

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References [1] Assilian, S.: Artificial Intelligence in the Control of Real Dynamic Systems. Ph. D. Thesis DX193553, University London (August 1974) [2] Berkeley, E.C.: Giant Brains or Machines that Think. John Wiley & Sons, Chapman & Hall, New York, London (1949) [3] Blair, B.: Interview with Lotfi Zadeh, Creator of Fuzzy Logic by Azerbaijada International, winter (2.4): (1994), http://www.azer.com/aiweb/categories/magazine/24_folder/24_articles/ 24_fuzzylogic.html [4] Hartley, R.V.L.: Transmission of Information. The Bell System Technical Journal VII(3), 535–563 (1928) [5] Hutchins, J.: Warren Weaver and the Launching of MT. Brief Biographical Note. In: John Hutchins, W. (ed.) Early Years in Machine Translation, pp. 17–20. John Benjamins, Amsterdam (2000) [6] Mamdani, E.H., Assilian, S.: An experiment in linguistic synthesis with a fuzzy logic controller. International Journal of Man-Machine Studies 7(1), 1–13 (1975) [7] Mason, M., Weaver, W.: The Electromagnetic Field. The University of Chicago Press (1929) [8] McCulloch, W.S., Pitts, W.: A Logical Calculus of the Ideas Immanent in Nervous Activity. Bulletin of Mathematical Biophysics 5, 115–133 (1943) [9] Morris, Charles W.: Foundations of the Theory of Signs, (International Encyclopedia of Unified Science, ed. Otto Neurath, vol. 1 no. 2.) Chicago: University of Chicago Press, 1938. Rpt. University of Chicago Press,Chicago 1970-71. Reprinted in Charles Morris, Writings on the General Theory of Signs. The Hague, Mouton, pp. 13-71 (1971) [10] Nyquist, H.: Certain Factors Affecting Telegraph Speed. Journal of the AIEE 43, 124 (1924); Reprint in The Bell System Technical Journal III, 324-346 (1924) [11] Nyquist, H.: Certain Topics in Telegraph Transmission Theory. Transactions of the AIEE, 617–644 (1928) [12] Locke, W.N., Booth, A.D. (eds.): Machine translation of Languages: fourteen essays. Technology Press of the MIT. John Wiley & Sons, Inc., Cambridge, Mass, New York (1955) [13] Sadegh-Zadeh, K.: Organism and disease as fuzzy categories. Presented at the Conference on Medicine and Philosophy. Humboldt University of Berlin, July 2 (1982) [14] Sadegh-Zadeh, K.: Fuzzy Health, Illness, and Disease. Journal of Medicine and Philosophy 25(5), 605–638 (2000) [15] Sadegh-Zadeh, K.: Handbook on Analytical Philosophy of Medicine. STUDFUZZ. Springer, Berlin (2012) [16] Rees, M., Weaver, W.: A Biographical Memoir. National Academy of Sciences, Washington D. C (1894-1978, 1987) [17] Slichter, C.S.: A Textbook for First-year College Students. In: Weaver, W. (ed.) Elementary Mathematical Analysis, 3rd edn. McGraw-Hill, New York (1925) [18] Seising, R.: From Vagueness in Medical Thought to the Foundations of Fuzzy Reasoning in Medical Diagnosis. Artificial Intelligernce in Medicine 38, 237–256 (2006) [19] Seising, R.: The Fuzzification of Systems, The Genesis of Fuzzy Set Theory and Its Initial Applications - Developments up to the 1970s. STUDFUZZ, vol. 216. Springer, Berlin (2007) [20] Seising, R.: What is Soft Computing? – Bridging Gaps for the 21st Century Science! International Journal of Computational Intelligent Systems 3(2), 160–175 (2010)

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[21] Seising, R.: Cybernetics, System(s) Theory, Information Theory and Fuzzy Sets and Systems in the 1950s and 1960. Information Sciences 180, 459–476 (2010) [22] Seising, R.: The Experimenter and the Theoretician – Linguistic Synthesis to tell Machines what to do. In: [29] (to appear) [23] Seising, R.: From Electrical Engineering and Computer Science to Fuzzy Languages and the Linguistic Approach of Meaning: – The non-technical Episode: 1950-1975. International Journal of Computers, Communications & Control (IJCCC), Special Issue – Celebrating the 90th birthday of Lotfi A. Zadeh, (to appear in 2012) [24] Seising, R., Tabacchi, M. (eds.): Fuzziness and Medicine: Philosophical Reflections and Application Systems in Health Care. A Companion Volume to Sadegh-Zadeh’s Handbook on Analytical Philosophy of Medicine. STUDFUZZ. Springer, Berlin (2012) [25] Shannon, C.E.: A Mathematical Theory of Communication. The Bell System Technical Journal 27, 379–423, 623–656 (1948) [26] Shannon, C.E., Weaver, W.: The Mathematical Theory of Communication. Univ. of Illinois Press, Urbana (1949) [27] Shermer, M.: The Really Hard Science. Scientific American, September 16 (2007) [28] Trillas, E.: Fuzzy Logic. Looking to the Future, Manuscript. Invited Lecture for AGOP 2009 (Fifth International Summer School of Aggregation Operators), Palma de Mallorca, Spain, July 6-10 (2009) [29] Trillas, E., Bonissone, P., Magdalena, L., Kacprycz, J.: Experimentation and Theory – Homage to Abe Mamdani. STUDFUZZ. Physica-Verlag (to appear, 2011) [30] Turing, A.M.: Computing Machinery and Intelligence. Mind 49(236), 433–460 (1950) [31] Weaver, W.: The Scientists Speak. Boni & Gaer Inc. (1947) [32] Weaver, W.: Science and Complexity. American Scientist 36, 536–544 (1948) [33] Weaver, W.: The Mathematics of Communication. Scientific American 181, 11–15 (1948) [34] Weaver, W.: Recent Contributions to the Mathematical Theory of Communication. In: [26] [35] Weaver, W.: Translation. In: [12], pp. 15–23 [36] Weaver, W.: Scene of Change. A Lifetime in Americans Science. Charles Scribner’s Sons, New York (1970) [37] Wiener, N.: Cybernetics or Control and Communications in the Animal and the Machine. MIT Press, Cambridge (1948) [38] Wirth, R.: Introduction to: Weaver, Warren: Science and Complexity, classical paper. Emergence - Journal of Complexity Issues (E:CO - Emergency: Complexity and Organization) 6(3), 65 (2004) [39] Zadeh, L.A.: Frequency analysis of variable networks. Proceedings of the IRE 38, 291– 299 (1950) [40] Zadeh, L.A.: Thinking machines - a new field in electrical engineering. Columbia Engineering Quarterly,12–13, 30–31 (1950) [41] Zadeh, L.A.: Some basic problems in communication of information. The New York Academy of Sciences, Series II 14(5), 201–204 (1952) [42] Zadeh, L.A.: System Theory. Columbia Engineering Quarterly 8, 16–19, 34 (1954) [43] Zadeh, L.A.: From Circuit Theory to System Theory. Proceedings of the IRE 50, 856– 865 (1962) [44] Zadeh, L.A.: Fuzzy Sets. Information and Control 8, 338–353 (1965) [45] Zadeh, L.A.: Fuzzy Sets and Systems. In: Fox, J. (ed.) System Theory. Microwave Research Institute Symp. Series XV, pp. 29–37. Polytechnic Press, New York (1965) [46] Zadeh, L.A.: Towards a Theory of Fuzzy Systems, Electronic Research Laboratory, University of California, Berkeley 94720, Report No. ERL-69-2 (June 1969)

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[47] Zadeh, L.A.: Biological Application of the Theory of Fuzzy Sets and Systems. In: Proctor, L.D. (ed.) The Proceedings of an International Symposium on Biocybernetics of the Central Nervous System, pp. 199–206. Little, Brown and Comp., London (1969) [48] Zadeh, L.A.: Fuzzy Languages and their Relation to Human and Machine Intelligence. In: Man and Computer. Proc. Int. Conf. Bordeaux 1970, pp. 130–165. Karger, Basel (1970) [49] Zadeh, L.A.: Towards a Theory of Fuzzy Systems. In: Kalman, R.E., DeClaris, N. (eds.) Aspects of Network and System Theory, pp. 469–490. Holt, Rinehart and Winston, New York (1971) [50] Zadeh, L.A.: Quantitative Fuzzy Semantics. Information Sciences 3, 159–176 (1971) [51] Zadeh, L.A.: Fuzzy Languages and their Relation to Human and Machine Intelligence. In: Man and Computer. Proceedings of the International Conference, Bordeaux 1970, pp. 130–165. Karger, Basel (1972) [52] Zadeh, L.A.: Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans. SMC SMC-3(1), 28–44 (1973) [53] Zadeh, L.A.: The Concept of a Linguistic Variable and its Application to Approximate Reasoning. Information Science 8, 199–249 (1975) [54] Zadeh, L.A.: The Concept of a Linguistic Variable and its Application to Approximate Reasoning II. Information Science 8, 301–357 (1975) [55] Zadeh, L.A.: The Concept of a Linguistic Variable and its Application to Approximate Reasoning III. Information Science 9, 43–80 (1975) [56] Zadeh, L.A.: Fuzzy Logic and Approximate Reasoning. Synthese 30, 407–428 (1975) [57] Zadeh, L.A.: PRUF - a meaning representation language for natural languages. International Journal of Man-Machine Studies 10, 395–460 (1978) [58] Zadeh, L.A.: Autobiographical Note 1, Undated two-pages type-written manuscript, written after (1978) [59] Zadeh, L.A.: Fuzzy Systems Theory: A Framework for the Analysis of Humanistic Systems. In: Cavallo, R.E. (ed.) Systems Methodology in Social Science Research: Recent Developments, pp. 25–41. Kluwer Nijhoff Publishing, Boston (1982) [60] Zadeh, L.A.: Making Computers Think like People. IEEE Spectrum 8, 26–32 (1984) [61] Zadeh, L.A.: Fuzzy Logic, Neural Networks, and Soft Computing. Communications of the ACM 37(3), 77–84 (1994) [62] Zadeh, L.A.: The Birth and Evolution of Fuzzy Logic - A Personal Perspective. Journal of Japan Society for Fuzzy Theory and Systems 11(6), 891–905 (1999) [63] R.S. Interview with Lotfi A. Zadeh on September 8, in Zittau, at the margin of the 7th Zittau Fuzzy Colloquium at the University Zittau/Görlitz (1999) [64] Zadeh, L.A.: A New Direction in AI. Toward a Computational Theory of Perceptions. AI-Magazine 22(1), 73–84 (2001) [65] Zadeh, L.A.: Foreword. Applied Soft Computing 1(1), 1–2 (2001) [66] Zadeh, L.A.: Precisiated Natural Language (PNL). AI Magazine 25(3), 74–92 (2004)

4 How Philosophy, Science and Technologies Studies, and Feminist Studies of Technology Can Be of Use for Soft Computing Veronica Sanz

4.1

Introduction

Since its origins, Artificial Intelligence (from now AI) has been one of the fields within Computer Science that has generated more interest among humanities and social scientists. On the one hand, philosophy has been the humanist discipline devoting the more interest in it Artificial Intelligence since it deals with some of the most traditionally topics of philosophy such as knowledge and reasoning. The philosophical debate on AI was mainly focused in the question of whether computers can or cannot think. Authors like John Searle, Daniel Dennett, Hubert Dreyfus or Roger Penrose were among the first ones to cope with these issues [20, 9, 10, 11, 19]. Later on, the most recent area of Science, Technology and Society Studies (or STS), also has paid some attention to Artificial Intelligence. As an interdisciplinary field containing history, sociology, anthropology and many other approaches to science and technology, STS takes a different stand with regard to AI than philosophy. They do not take part in abstract discussions about the ultimate goal of AI such as the possibility (or not) of creating “an artificial mind”. STS methodology looks instead at the actual practices of AI practitioners, trying to analyse the social and cultural contexts where scientific and technological developments take part. The main authors in this approach have been Lucy Suchman [21, 23], Harry Collins [8] and Diana Forsythe [12]. Finally, in addition to philosophy and STS there also has been some interesting and innovative analysis of AI from the field of Feminist Studies of Science and Technology. The British computer scientist and feminist Alison Adam is the one who has pursued the most extended study about gender and Artificial Intelligence. She has accurately applied the insights of feminist epistemology to the main traditions in AI such as Symbolic AI and Expert Systems. In the following sections (4.2 to 4.4) I will review these three different traditions and the way they have analyzed and criticized traditional approaches to AI. In the remaining sections I will turn the focus to Soft Computing as presented by its practitioners: as an “alternative paradigm” to traditional AI. As a different paradigm, my main question in this paper will be: how does Soft Computing (SC) confront the previous critiques made from philosophers, social scientists and feminist scholars R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 89–109. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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to AI? Is Soft Computing overcome these critiques or still challenged by some of them?

4.2

Philosophical Approaches to AI

As I mentioned before, philosophy has been the humanist discipline devoting more interest in AI, mainly because AI deals with some of the most important topics in philosophy such as “knowledge” or “reasoning”. The philosophical debate on Artificial Intelligence during the 1970s and 80s was mostly focused on Symbolic AI (the branch of artificial intelligence that attempts to explicitly represent human knowledge in a declarative form – i.e. facts and rules –), since this was the main approach in the field for many decades. In concrete, some of the ambitious statements of early AI researchers such as Minsky, Newell and Simon about the possibility of creating “an artificial mind” touched very sensitive “fibers” of philosophers. The philosophical debate, then, revolved around the question of whether computers can or cannot “think” (either in the present or in the future). With the exception of very few authors, the philosophical debate exhibited a significant resistance to the very possibility of the existence of “thinking machines”. Trying to assure the specificity of the human nature with regard to computational artifacts, philosophers often appealed to non-cognitive aspects of the mind such as intentionality, emotion or perception. But let’s review with some more detail these critiques and their main proponents. 4.2.1

Alan Turing: The Origins of the Debate

In addition to being the one who set the basic elements of computation, the British mathematician Alan Turing was also the one who posed the first stone of the debate about the possibility of creating “thinking machines”, proposing a mental experiment known since then as the “Turing Test” [26]. The experiment consisted on an interrogator who, through a separating barrier, asks questions to a machine and a human, trying to guess who (or which) is answering. If the interrogator cannot tell whether the responder is the human or the machine then the intelligence of the machine would be proved.1 The main feature of Turing’s idea regarding machine intelligence lies on the practical way he is proposing the test as an “imitation game”. He is not defining a priori what intelligence is but establishing a procedure were the performance of human intelligence is showed. Whether the machine thinks “in the way humans do” or not 1

We should note here an aspect that has not been not pointed out neither by philosophers of AI nor by Turing’s biographers. If we carefully read Turing’s article we realize that the first step of the “imitation game” consists on distinguishing between a man and a woman, affirming the inability to differentiate the gender of the participants in terms of their intelligence. From my point of view, this represents in some sense a de-gendering approach to intelligence which, (“curiously”) has been overlooked by most authors along the years.

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is not addressed by the Turing’s test (in fact, it is precisely this kind of philosophical question what he tried to avoid.) 4.2.2

John Searle: The Critique to “Strong AI”

On the contrary, the American philosopher John Searle in his famous paper from 1980 [20] argued against the strong claim that computers can be intelligent in the same way humans are (which, in his view, implies having cognitive or psychological states, i.e. consciousness and intentionality). To develop his argument Searle distinguishes two strands of AI. One is called “Strong AI”, which refers to a kind of AI that tries to create intelligent machines that replicate not only intelligent products, but the intelligent processes. By contrast, “Weak AI” is the perspective that understands computers as useful tools that mimic some human abilities that are considered to be products of human intelligence. In contrast to Strong AI, Weak AI does not make the claim that computers actually understand or have mental states. Following this argument, Searle’s critique is directed to Strong AI. To do that, Searle follows Turing’s tradition of proposing a mental experiment and he presented one called ‘The Chinese room”. In this case, a native English speaker who doesn’t speak Chinese is set in a room and is provided with a book of instructions to manipulate the symbols of Chinese language. These instructions make him/her able to give correct answers to questions asked by an interrogator, which enable him/her to pass the Turing Test. But, as Searle argues, that doesn’t mean that he/she understands a word of Chinese. The axis of Searle’s argument relies on the idea that syntactic manipulation of symbols (that is, “computation”) does not imply understanding the semantics (or meanings). From Searle’s point of view, we cannot attribute intelligence to such a system since “real intelligence” is related with the understanding of meanings, an activity that involves the existence of mental states such as consciousness and intentionality, something that computers – by definition – do not (and cannot) have. 4.2.3

Daniel Dennet: A Positive Approach

Though the majority of philosophers have been quite critical with AI, Daniel Dennett is an exception [9]. He shares with Symbolic AI researchers what in philosophy of mind and cognitive sciences is called the “Computational Theory of Mind”: the idea that the human mind works by manipulating formal symbols. Dennett has even been involved in some AI projects. He disagrees with Searle in two important points: 1. He criticizes the way Searle understands intentionality as being by definition limited to humans. Against that, Dennett affirms that human consciousness and intentionality are not as specific of humans as we use to think, which is showed in the way we use a “mentalist language” when describing the actions of animals, machines and so on.

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2. He also criticizes the Chinese Room thought experiment for being very far from the actual practices of everyday AI research. However ingenious it is, says Dennet, it is impossible to reproduce it in reality, so it is very doubtful that can be used as a proof of anything2. 4.2.4

Hubert Dreyfus: The Phenomenological Critique

On the other side of the debate, the philosopher who has devoted most energy and time to the critique of AI during the past 30 years has been the phenomenologist Hubert Dreyfus [10, 11]. His arguments are also directed against the “Computational Theory of Mind” underlying Symbolic AI, but, unlike Searle and other philosophers of mind, he is not concerned with issues of intentionality but, drawing upon Heidegger and existential phenomenology, with the problem of representing the human ways of “being in the world”. Dreyfus criticizes the epistemological assumption of Symbolic AI that all activity can be formalized in the form of propositional knowledge (which would allow its computation in the form of rule-based symbol manipulation), as it is done in many Symbolic AI-based expert systems. For Dreyfus, many parts of human knowledge and competences cannot be reduced to an algorithmic procedure, like, for example, the kind of knowledge one might employ when learning a skill like walking, riding a bike or ice skating. This learning is done through practicing with our bodies: to learn how to do it one must try it, it’s not enough to “follow” written rules3 . And this requires having a body and experimenting with it in a changing environment. Although Dreyfus concedes that certain well-circumscribed AI experts systems have succeeded in particular domains, he predicted that many others will never fulfill the expectations. From Dreyfus’ point of view, this is not because of technical shortages (which could be overcome in the future) but because symbolic type of artificial systems are incapable of representing bodily-skills knowledge, a kind of knowledge that cannot be captured in an algorithm.

4.3

STS Studies of AI

Among the different non-technical disciplines that have been interested in the study of AI, philosophy has not been the only one. Though certainly not so much as philosophy, other areas such the recent one known as Science, Technology and Society studies (STS) have also paid some attention to AI. 2 3

Actually Dennett elaborated this critique in general against all kind of “thought experiments”. Dreyfus’s argument also relies on Wittgenstein (1953) analysis of rule-following. Wittgenstein thesis is that a rule-like activity can not be completely spell out without a regress to other rules on which they depend on, because this process takes to an infinite regress. This suggests that there is always a background of practices that form the conditions of possibility of all rule-like activity. Because of that, it is not possible to translate all skill knowledge (or “know-how”) into propositional knowledge (or “know-that”).

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STS is an interdisciplinary field of studies about science and technology that emerged during the 1970s and 1980s from the confluence of a variety of disciplines and disciplinary subfields in social sciences and humanities, all of which shared an understanding of science and technology as socially embedded enterprises. STS scholars resist classical accounts that science and technology are totally objective and neutral with respect to the context in which they are made 4 . An STS methodology of research implies the analysis of the social and cultural contexts where all scientific and technological developments take part. In tune with this methodological approach, STS takes a different stand than philosophy regarding the study of AI. They do not take part in abstract discussions about the ultimate goal of AI or about the success or failure of the enterprise of creating “an artificial mind”. For STS researchers, the problem with these philosophical discussions is that they are not close enough to the everyday practices in AI laboratories. An STS methodology consists on looking at the real practices of AI practitioners (including, for example, ethnographic observation and interviews). From the STS view, the important thing to investigate is the way these engineers understand, model and use concepts such as “knowledge” and “reasoning” in their daily practice and how they operate to create devices with them. Within AI, the subfield of Expert Systems has been the one that has attracted more attention from part of STS scholars, as showed in the works I review in the following section. 4.3.1

Lucy Suchamn: The Critique to the “Planning Approach”

Probably, the most influential contribution from part of social sciences to the field of AI has been Lucy Suchman’s Plans and Situated Actions [21] published in 19875 . In the early 1980s, Suchman was appointed by Xerox PARC Co. to carry out an ethnographical research on a prototype of a so-called “intelligent” photocopier that the company was testing by that time6 . The work of Suchman consisted on observing and commenting the testing phase of the copier where a group of people interacted with it, in order to verify if the system worked as expected or needed improvements. The new photocopier included an electronic display screen which incorporated an AI-based expert system. The system was designed to analyse what the users intended to do and help them through the process by offering a set of instructions. The intention of the company was that Suchman and her collaborators might serve as mediators between designers and users. 4

5 6

Typical issues treated in STS are the study of how social, political, and cultural values affect scientific research and technological innovation, and how these, in turn, affect society, politics, and culture. Her work has been in fact very much quoted in the field of expert systems design in the 1990s. The case of Suchman is especially interesting since she was hired by the very company to do her ethnographical work with the intention to get advantage of her findings in order to design better “interactive systems”.

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Contrary to their expectations, Suchman’s work turned to be a strong critique to the “planning-approach” used by the expert systems community of the time. By observing the practice of users’ interactions with the copier, Suchman found that the traditional AI way of understanding actions thorough the so-called “planning approach” used a very simplistic model of human action described in terms of “plans”. As the machine was designed with the assumption that users always have a clear plan in mind, this leaded to much confusion when they were using the copier. At the same time, as users were trying to make sense of the “machine’s actions” it became even more difficult to use the copier properly7 . Suchman showed that “plans” are super-imposed on actions in a retrospective way, while in real-life situations, actions are much more fluid and contingent, responding to the interactions among the actors (be those between humans, between humans and their physical environment or between humans and machines). 4.3.2

Harry Collins: Tacit Knowledge and “Social Embededness”

Harry Collins’ study of AI [8] is another important contribution from part of STS to the study of AI and, in concrete, to the field of Experts Systems. As we said before about STS in general, Collins’ interest is not whether future systems will or will not be more intelligent than humans. On the contrary, Collins is interested in those systems which are already being used in practice and seem to be functioning. Contrary to some philosophers such as Dreyfus or Searle, he is not interested in explaining or foreseeing the “failures” of AI but in explaining its “successes”. Collins explains the functioning of AI systems by interpreting them not as “artificial brains” but as “social prostheses” that have a role in our society (a kind of “artificial members of society”). He claims that machines can perform this role due to the ability of humans to interpret their actions and repair their deficiencies. This human ability normally remains invisible, but, for Collins, it is the place where, to a great extent, the intelligence attributed to machines rests on. However its success, AI is also criticized by Collins in a way quite similar to Dreyfus’ with regard to the impossibility of formalizing certain types of knowledge. His work is based on recent Sociology of Scientific Knowledge developed by David Bloor [4] and on the so-called Laboratory Studies (Latour [17]), where it is shown how scientists need to share a common culture in order to produce science. Collins maintains that know-how knowledge is principally acquired through enculturation. As this way of transmitting skills is mostly implicit, it cannot, therefore, be fully spelt out in formal algorithms. This kind of knowledge is commonly known in STS as “tacit knowledge”, which refers to the “common knowledge” that, within a particular community, is taken for granted8. However, Collins understanding of “tacit knowledge” is wider than Dreyfus’ “embodied knowledge”. The acquisition of skills-type knowledge is, in Collins’s 7 8

As many times happens when users are faced with a new device, they reported they very much preferred the old copier. Very much related with a very important concept in STS known as “epistemic cultures”.

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view, related to socialization rather than embodiment, stressing the importance of natural language. In this way, Collins brings the discussion to the cultural arena. 4.3.3

Diana Forsythe: An Etnogranphic Approach to Knowledge Engineering

Finally, the work of the STS scholar Diana Forsythe [12] also involved (as in the case of Suchman) an ethnographic study of an expert systems laboratory in the US9 . She was interested in “the culture of Knowledge Engineering”, what she defines as the values and assumptions (some explicit, but mostly implicit) that constitute what they take for granted in their discipline and work practice. Forsythe also refers to this as “common sense truths” or those things that “everybody knows” within a given setting (in this case, the expert systems community). Investigating the culture of knowledge engineers Forsythe found that they share a very restricted notion of knowledge that has important implications for the products they make. She focused on the first step of the process of building an expert system, known as the ‘knowledge acquisition’ phase (also ‘knowledge elicitation’) where designers have to gather the knowledge from the human experts. She shows how the use of terms such “extraction” or “store” are metaphors that imply a very simple understanding of knowledge as an “object” that can be susceptible of easy manipulation and being formally encoded to be transferred into a machine. The elicitation process is considered by knowledge engineers themselves a very difficult phase which causes many problems and some failures in the systems when encountering ‘real-world situations’ (mostly situations that the systems builders did not anticipate). However, knowledge engineers blame the human experts for not being able to clearly specify their expert knowledge -that is: in the way they can encode it-10 . As Forsythe’s interviews show, these engineers do not think about the possibility that their implicit assumptions about knowledge can be the cause of their problems. On the contrary, Forsythe affirms that it is precisely the epistemological stance about the simplicity of knowledge assumed by knowledge engineers what causes the ‘brittleness’ of their systems11 . In general terms, the kind of expert systems studied by Forsythe are included in the more general approach of Symbolic AI, since, in the end, knowledge “extracted” from human experts has to be written-down in rules-type propositional knowledge. With respect to this, the same critique from Dreyfus and Collins about the inability to model other types of knowledge (particularly social knowledge) is applicable to Forsythe’s work. However, Forsythe points out an issue largely unexamined by 9

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The importance of the ethnographic method relies on the main thesis of cultural anthropology about the complex relation between beliefs and practices: usually what people do, what they think they do, and what they report they do, are someway different. It is not casual that the kinds of expert knowledge they prefer to work with are already those very formalized in well-defined narrow domains. As an anthropologist and social scientist, Forsythe’s understanding of knowledge is precisely the opposite: knowledge is a highly problematic issue, mostly tacit and unconscious and whose “locus” is the social group or community.

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philosophers of AI and only slightly suggested by Collins: the politics of knowledge involved in AI projects. In the terms used by Forsythe, knowledge engineers have the power to impose their concept of knowledge, since their selections, translations and deletions are built into their systems. The relevance of STS and anthropological approaches to technical practices (and so of gender analysis, as we will see soon) is to “uncover” the implicit assumptions about knowledge and scientific practice, and the political consequences of these assumptions.

4.4

Feminist Analysis of AI

Partly rooted on classical philosophy and epistemology, partly on the STS tradition, feminism and gender studies have developed in the last twenty years an important and extensive scholarship regarding issues of science, technology. We should clarify here that the concept of “gender” in gender studies this does not refer (only) to the gender of the particular individuals. That is to say that gender studies of science and technology are not (only) focused in pointing to the scarce number of women in computing technology or recovering the contributions of individual women in the history of AI (which both are important issues), but also in analyzing how the gender system (defined as the structural organization of society, of individuals’ identities and cultural imaginary) is intertwined with science and technology. The main idea in contemporary feminist studies is that the category of gender is constructed in relation to other social factors such us class, race and culture, and also in relation to science and technology. For example, it is an important topic to investigate how in our Wester culture science and technology have been so highly associated with masculinity. One of the most important developments within Science, Technology and Gender studies has been done in the area of epistemology, developed by the so-called area of Feminist Epistemology. Starting from a concern about how women and women-related issues have been traditionally excluded or neglected from the practices of science, feminist epistemology has ended developing a strong challenge to the very possibility of an objective method that can assure the value-neutrality of knowledge. In this regard, feminist epistemologists have developed a critique to the unconditioned knowledge subject of traditional epistemology. As Alcoff and Potter posed it [3]. “[Feminist epistemologies] share the scepticism about the possibility of a general or universal account of the nature and limits of knowledge, an account that ignores the social context and status of knowers”. For feminist epistemologies, the gender of the knower (interrelated with other factors such as race, class, ethnicity, etc) is relevant for the production and results of knowledge, or, in words of the philosopher Lorrain Code [6], “the sex of the knower is epistemologically significant”. The program of feminist epistemology has been, then, “taking subjectivity into account”.

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As AI is an area where epistemological issues are very important (as most of philosopher’s analyses of AI have shown), feminist epistemological analysis of AI enter the debate in its own right. However, they take a different stand than classical epistemology. Feminist epistemologists are going to criticize the way in which the philosophical critiques of AI that we have reviewed in section 4.2 leave unexplored the social and cultural dimensions of epistemology (i.e. its social contingence and the historical roots of certain assumptions). Therefore, the critical point of view characteristic of feminism (and of STS, for this matter) states that every part of science and technology includes unavoidable implicit epistemological, methodological and ontological assumptions that are part of the “culture” of each epistemic community, which configures their practice and their techno-scientific products. But let’s proceed now to review the most important contributions from part of feminist studies to the field of AI. 4.4.1

Alison Adam: Applying Feminist Epistemology to Symbolic AI

Alison Adam is a British computer scientist and feminist who has pursued the most extended study about gender and Artificial Intelligence in a book titled Gender and the Thinking Machine [1]. Adam starts her analysis by stating that the traditional philosophical critique to AI is “epistemologically conservative”, since neither questions traditional rationalistic epistemology nor takes into account critical versions to it. On the contrary, Adam makes use of the insights of feminists epistemology and applies it to AI (mainly to Symbolic AI and Expert Systems, but she also makes some reference to Connectionism and to more recent approaches such as Artificial Life and Situated Robotics). Adam’s work is devoted look at “the gendered models of knowledge represented and inscribed in AI” ([1]: 4), which she rightly presumed to be implicit. One of Adam’s theses is that traditional AI systems are based on the Cartesian ideal of a disembodied mind and the over-valuation of mental (abstract) knowledge over corporeal (concrete) knowledge, being the former historically associated with masculine realm and the latter with the feminine. Following the work of the philosopher Lorrain Code [6, 7] Adam shows that traditional AI is based on the rationalistic epistemological model of“S knows that p” where ‘S’ refers to the knowledge subject (who knows), and ‘p’ the object of that knowledge (what can be known). Regarding ‘S’, traditional epistemology has defined it as the ideal knower of Modern science: an anonymous, universal and disinterested subject who “knows from nowhere”. Due to that ‘S’ is considered to be universal, there was no concern about who is the concrete ‘Ss’ pursuing the observations or developing theories. On the contrary, feminist epistemologies emphasize the “situatedness” of the knower who is always a subject situated within a concrete social, economical, racial and gendered context. Borrowing Sandra Harding’s words (1991), Adam’s feminist inquiry of AI asks ‘whose knowledge’ is inscribed in AI systems. For example, Adam asks who are the “knowledge subjects” of systems like Cyc. The Cyc project, leaded by Lenat and Guha [18], tries to build a vast knowledge base which includes “most of human common-sense knowledge”. Since the “human knower” is considered to be universal, Lenat and Guha assume that there is

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one and only one consensus reality available to all humans. The developments of STS, anthropology and feminist studies in the last decades have shown that it is quite problematic to assume that hypothesis. We “see” the world according to our theories, and these theories depend on our cultural location. Adam points out that, in the end, what Cyc knowledge base would consist on is what Adam’s calls “TheWorld-As-The-Builders-Of-Cyc-Believe-It-To-Be”, who happened to be group of predominantly middle-class male US university professors, that, without realizing it or intending it, are privileging their “consensus reality” over other groups’. Regarding ‘p’ (the object of knowledge), traditional epistemology has considered it to be an “objective knowledge” which takes the form of propositions in 2nd order logic. As Dreyfus and Collins argued before, propositional knowledge (or know-that) is considered to be the superior type knowledge, relegating and leaving out other types such as bodily-knowledge and skill-type knowledge (or know-how). Drawing on feminist epistemology, Adam claims that this distinction is not innocent of gender values but, rather, has a clear gendered character (a point that is absent in Dreyfus and Collins’ critiques). In our highly dichotomized and hierarchically structured culture, different types of knowledge have been related to particular social and gendered groups. As many other dualisms such as mind/body, culture/nature or technical/social, this has been connected to gender dualism. Abstract or propositional knowledge has been historically associated with the masculine realm, while issues related with the body (concrete and practical thinking but also emotional and interpersonal skills) have been largely related to the feminine gender. In accordance with the alleged lower value of the feminine with respect to the masculine in our culture, these types of knowledge have been considered less valued, so they suffer from both an epistemic hierarchization and an epistemic discrimination. Adam rightly applies this argument to the kind of knowledge constructed in Symbolic AI. 4.4.2

Lucy Suchman: The Situatedness of the Designer

We already have referred to Lucy Suchman and her very influential book of 1987 Plans and Situates Actions [21] in the section dedicated to STS analysis of AI. But in later works, she has also introduced a feminist perspective to her analysis of AI [22, 23]. Suchman roots her critique in Donna Haraway’s concept of “situated knowledge” [14], which is the most important concept of feminist epistemology12. Suchman affirms that, the same as traditional scientists claim to observe the world from a “nowhere position”, designers of AI also assume the same un-located stand when designing their systems. The “design from nowhere” assumption involves the ignorance of their own position, which is encouraged by designers’ training in engineering schools and in their later work settings (no matter to this regard if they work for private or public institutions). Closely related to the traditional view of technologists as “lonely creators” detached from the future users of their technologies 12

Donna Haraway is one of the most influential scholars in feminist technoscience studies. The concept of “situated knowledge” refers to the material embodiment and the social embedded-ness of any kind of subject who perform a knowledge action.

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-who, consequently, are assumed to be mere “passive recipients”-, their technological developments are seen as “decontextualized products” developed through the rationalist engineering methods of design and production. In opposition to this traditional unawareness of the designers’ community with respect to their methods and assumptions and to their location within the nets of relations that make their work even possible, Suchman stresses the fact that this position is unreal Traditional epistemology (which is inherited by AI) has not only claimed a position that does not exist, but also has served as a way of domination over those subordinate groups who – for different reasons – do not participate in the creation of “valuable” knowledge13. Donna Haraway has insisted on the fact that unconsciousness about one’s own location doesn’t relieve knowers from the responsibility of their actions in technoscientific production. In the same way, Suchman applies it to the designers of AI systems, claiming for the necessity of them to assume their responsibility in the effects of the products they design.

4.5

Soft Computing Confronting Philosophical Critiques to AI

Being Soft Computing self-considered an “alternative paradigm” to traditional (Symbolic) AI our main question in this paper is: how does Soft Computing perform in relation to the critiques that philosophy, STS and feminist studies have made to traditional AI? Is Soft Computing overcoming these critiques or is still challenged by some of them? In this section I will show how Soft Computing can confront the philosophical critiques to traditional AI, as I will do the proper with STS and feminist critiques in sections 4.6 and 4.7, respectively. 4.5.1

Soft Computing and Searle’s “Strong AI”

One of the main philosophical critiques of AI was centred on the type of AI that Searle named “Strong AI”. This critique (and, in fact, the very distinction between it and “Weak AI”) was motivated by the excessively optimistic predictions of some of the founding fathers of AI. However, most of researchers in AI do not claim they are creating an “artificial mind”. Instead, they rather say that their aim is to construct devices that can perform some tasks that are considered “intelligent” when done by humans (see [24]), which is more in tone with what Searle called “Weak AI”. Surely this is the way Soft Computing practitioners think about their work, as far as I could observe. In this way, Soft Computing folks do not take issue with intentionality or any kind of “mental states” of an alleged “thinking machine”. This debate is in fact 13

Therefore, for feminist epistemology knowledge has a political dimension: the question about who is considered a knowing subject is in close relation the one about who has the power to perform such a designation. Feminist epistemologists show that while the knowing subject was said to be “neutral” in fact happened to be a masculine, white, Western knower.

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a philosophical one (the old debate between Behaviorism and Mentalism14 ) that is far from the interests of Soft Computing’s researchers. Because of that, these kind philosophical critiques do not apply to Soft Computing. However, in relation to this topic it would be interesting to analyze what Zadeh means when he uses the term “human-level machine intelligence” in some of his recent articles [33]. 4.5.2

Soft Computing and the Question of Semantics

The other axis of Searle’s critique to AI relied on the idea that syntax (formal symbol manipulation, or computation) does not imply semantics (understand meanings), and that “real intelligence” involves something more than mastering syntax. Searle’s critique is directed to the “Computational Theory of Mind” which is at the fundaments of Symbolic AI. In the case of Soft Computing, on the contrary, its aim is to cope with the kind of imprecise knowledge and ways of reasoning that humans use in their daily activities. This has lead SC to deal with the problem of natural language, and so with meaning (that is semantics). One of the main themes in Fuzzy Logic is precisely to deal with meaning in natural language which, due to the non-static character of meaning, requires tools which can deal with imprecision and uncertainty. The enormous importance of natural language for SC is shown in the turn Zadeh proposed in 1996 from Fuzzy Logic to “Computing with Words” [29]. Computing with words (CW), Zadeh says, is a methodology in which words are used in place of numbers for computing and reasoning. As fuzzy sets are able to model some natural language’s concepts, it would be possible to use these models to compute directly with them. In this way, we can say that Soft Computing involves a step further that overcome one of the most important critiques made to previous Symbolic AI approaches. 4.5.3

Soft Computing and the “Bodily/Skills-Knowledge” Problem

As we explained in section 4.2.4., Dreyfus’s main critique to the Symbolic type of artificial systems is its incapability to represent bodily-skills knowledge. As Symbolic AI is based exclusively on propositional knowledge, Dreyfus argues, it is not capable of dealing with other types of knowledge such us the so-called “bodily-skill knowledge”, which is the kind of knowledge one employs while driving a car, riding a bike, ice skating, dancing and so on. This “knowing-how” knowledge is acquired through practice, which requires having a body and experimenting with it; to learn how to do it one must get on and try it in a changing environment, so it’s not enough to follow written explicit rules. Dreyfus defines human uniqueness in terms of action so, for him, computers could never carry out many tasks humans do since they do not have “embodied” knowledge and experience. 14

That is, whether or not there are “internal mental states” in the mind or only observable behavior.

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In the decade of the 1980s, new types of approaches in AI have tried to avoid this problem, like the case of Rodney Brook’s work in the so-called Situated Robotics. The work of Brooks and his colleagues [5] focus on the idea that intelligent behavior is an emergent phenomenon resulting from embodiment and the system’s interactions with its environment. He developed "bottom-up" strategies that are more related to action than to representation of intelligence in a formal way. What Soft Computing has to say about this debate? In general, we could say that not much since this is a problem typical in Robotics. However, when we read Zadeh’s appellation to activities that deal with imprecision he sometimes refers to the same ones Dreyfus’s refers (i.e. driving a car, riding a bike, etc.). Though, on the one hand, Soft Computing (via Fuzzy Logic) have dealt much more with language and representation of meaning than with actions, on the other hand one of the most important traditional applications of Fuzzy Logic has been in the area of control systems, where they have to deal with systems-in-action. One clear example was the problem of the “inverted pendulum”, which was solved by Yamakawa [28] using fuzzy rules. The subfield of fuzzy-control systems is based on empirical methods (basically trial-and-error), which is indeed quite similar to the way humans learn bodily-skills knowledge. Therefore, it would be a very interesting topic for future research to look at how fuzzy control theory, in addition to new approaches in SC on “computing with actions and perceptions” (see [13]) can be applied to solve the so-called “bodily-skills knowledge” problem that have presented a great challenge to traditional approaches of AI.

4.6 4.6.1

Soft Computing Confronting STS Critiques to AI Soft Computing, Contextualism and “Social Embededness”

As we said before, researchers in the field of Symbolic AI claim that cognition consists on the manipulation of symbols by means of logical rules, so, therefore, human knowledge is, to a large extent, context free. We have already explained how this idea has been criticized by authors like Dreyfus or Collins. On the contrary, one of the fundaments of Fuzzy Logic when dealing with natural language is Wittgenstein’s pragmatist turn in The Philosophical Investigations [9] and his definition of meaning as “its use in language”, as it is cited by many SC authors [25]. This conception of meaning implies that meaning is contextual, that is: it can only be grasped when used by concrete people in concrete contexts. As Wittgenstein says, one must "look and see" the variety of uses to which a word is put in particular cases to be able to grasp its meaning. To translate this to SC research implies that previous empirical analysis of the context of use it is an obligatory step to construct AI systems. This obligates SC to be an empiricallybased research (that is, they have to take into account the users of the words when designing concrete systems) in opposition to the abstract orientation of Symbolic AI.

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However, though this idea of contextualism may be assumed and implicit in SC practitioners, it is not exactly the same understanding of “contextualism” we use in STS or feminist studies where it is used to stress the “situatedness” of the designer regarding his/her own culture, class, nation, race or gender15. In STS and feminist studies, for example, we would ask for questions such as the followings: how it is contextualized the design of the experiment? Who choose the group of speakers of a language that are going to participate in the experiment about the use of some words? Does this take into account that most of the time the people who participate in the experiments are computer science colleagues or students (who normally share the same race, class, nation and gender of the designer)? It would be very interesting to confront SC researchers with these kinds of questions that, on the other hand, may have a positive outcome for the results of their own projects. 4.6.2

Soft Computing and “Common-Sense Knowledge”

As we have seen throughout the critiques of the philosopher Hubert Dreyfus and the STS scholar Harry Collins, the appellation to the inability of Classical AI to deal with common-sense knowledge is, together with the one about bodily-skills knowledge, the most important challenged presented to Symbolic AI. The so-called “common-sense knowledge” is slightly different than “bodily-skill” knowledge in which refers to the kind of (mostly implicit) knowledge that humans have due to living within a culture16. The argument about the impossibility of representing all common-sense knowledge in a “knowing-that” propositional form is related, then, to the difficulty of knowing what is relevant in each particular situation. Dreyfus and Collin’s argument is that this kind of knowledge is mostly tacit and took for granted, and, because of that, it cannot be formalized in explicit rules. On the contrary, many Soft Computing articles contain the affirmation that fuzzy logic can represent common-sense knowledge. The most explicit article from Zadeh about the topic [30] explains that common-sense knowledge is expressed in dispositions, as opposed to propositions. A disposition in philosophy is a kind of belief that is stored in the mind but is not currently being considered (in this way we can say is a kind of tacit knowledge). For the purpose of representation Zadeh defines a disposition as a proposition with implicit fuzzy quantifiers; therefore fuzzy logic can be used to deal with that kind of knowledge. Although this strategy is only in its first steps, I think is a very promising way of dealing with common-sense knowledge, at least much better than Guha and Leant kind of propositional knowledge data base of the project Cyc. However, there are many issues that are not addressed or solved yet in SC. A very important one has 15 16

About the use of same terms in different disciplinary fields see Termini in this volume. Traditionally, designers of AI systems did not considered this kind of knowledge as important, therefore they did not include rules like "If President Clinton is in Washington, then his left foot is also in Washington," or, as Adam (1998) shows, they construct medical expert systems that may ask a male patient if he was pregnant.

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to do with the very definition of “common-sense knowledge” and its interrelations with concepts such as “tacit knowledge”, “situated knowledge” and so on. Is it all common-sense knowledge tacit and “taken for granted”? What is the difference between common-sense knowledge and skill-knowledge when it comes to formalize them? And, what Zadeh exactly means when he talks about “world knowledge” [33]?

4.7

Soft Computing Confronting Feminist Critiques to AI

As we explained in section 4.4.1, Alison Adam constructs her critique to traditional AI by challenging the rationalistic epistemological model of “S knows that p” where ‘S’ refers to the anonymous, universal and disinterested subject of Modern science and ‘p’ to the kind of propositional knowledge that can be written down in formal first order logic. Adam argues that traditional AI systems are based on the Cartesian ideal of a disembodied mind and the over-valuation of mental, abstract –and masculine related- knowledge over corporeal, concrete – and feminine related – knowledge. For Adam (as for other feminist epistemologists), this is very much related to the “mathematization” of the Western culture which, started in ancient Greece and increased in the Modern Age, claims that the entire universe is governed by numbers. Although already among the Greeks took place a plea for other ways of knowing, it was the Platonic and Aristotelian conception of the world which finally succeeded. As it is well known, Aristotle is considered “the father of Logic”, which he defined in binary terms (there are only true or false statements and valid or invalid inferences). As we know, the rejection of this thesis was the starting point of Zadeh’s Fuzzy Logic. Zadeh’s repeated claim that humans reason without necessity of numbers or measurements is in tune with feminist epistemologists’ critiques to the mathematization of knowledge. But, if Soft Computing involves such a change of paradigm with respect to the elitist conceptions of intelligence of classical AI, which are the ‘p’ and the ‘S’ of Soft Computing? 4.7.1

The ‘S’ of Soft Computing

Following Adam’s argument, I have already explained how Symbolic AI incorporates the claim of traditional philosophy of science about the possibility if a universal, unconditioned and context-free subject. On the contrary, Soft Computing researchers use to point out the contextual character of meaning in natural language, which they try to represent with the tools offered by Fuzzy Logic. For example, assuming that ordinary language is “context-dependent”, the membership function of a linguistic variable (which represents its meaning in fuzzy logic) must be calibrated in the context of its use. As Zadeh says ([31]: 26) “The grade of membership is subjective in nature; it is a matter of definition rather than measurement”. That is why, as we already mentioned, previous empirical analysis of the context of use is an obligatory step for SC developments.

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Let us show an example used in SC to illustrate the dependency of the model with respect to the context. The sentence “John is tall” can mean different things in different contexts: for instance, if the context is a kinder-garden and John is 5 years old child, “tall” can be applied to a child of 1 meter tall; but if the context is a team of professional basketball players, one meter person wouldn’t be considered “tall”. Due to this, the fuzzy set “tall” is represented in fuzzy logic by different membership functions dependent of the context of use. However, if we push further this conception of “contextualism” we can differentiate two aspects of it: 1. On the one hand there is what I would name an “objective context”, which refers to the realm of application of the predicate (that is, its universe of discourse). In the former cases, this “objective context” would be the group of children in the kinder garden or the group of basketball players17. 2. On the second hand, there is a kind of ‘contextualism’ that refers to the subjectivity of the person who predicates the property, that is, a “subjective context”. Even with respect to the group of children or basketball players, different persons can apply the predicates “tall”, “quite tall” or “a little bit tall” to different persons. Each one may interpret the membership function as the degree that fits with his/her perception and experience of the world (this is explicitly recognized by Zadeh in [33]) I think that Soft Computing’s acknowledgement of the contextual character of knowledge is a very important change with relation to previous AI approaches, and that makes SC very appealable to the interest of social scientists and feminist researchers. However, the question of how to deal with the second type of “contextualism” is still very unclear in Soft Computing in the sense that it remains implicit in many cases. SC researches have not made so far further considerations about the epistemological consequences of this important challenge they are deploying. For example -as we already stated in section 6.1.-, it would be very interesting if they interrogate themselves about what influences the subjective contexts of the knowers/speakers they use as examples for their investigations like, for example: which race, class, nation or gender have the people who use this or that word? Should we do always our experiments in English? What kind of education background has the people participating in our experiment? Is the experiment biased by our expectations? 4.7.2

The ‘p’ of Soft Computing

In Symbolic AI, ‘p’ refers to the kind of propositional knowledge that can be written down in a preciseway by means of first order logic. On the contrary, Soft Computing 17

Trillas ([24]: 113) explains this quite clearly: “Because in natural language we use the concept “small” in a very variety of discourse scopes, the affirmation “X is small” is only informative as far as we know the scope of objects in which we predicate it” (Our translation).

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tries to model a wide variety of human physical and mental activities that involve imprecision and uncertainty such as common-sense and approximate reasoning that can not be formalized in binary logic. As feminist epistemologies have shown, these two types of knowledge have been traditionally related to the feminine gender, as it has been the ordinary use of language (in opposition to mathematics and formal languages)18. Contrary to previous AI approaches, SC is interested in these types of “ordinary reasoning” (the kind of reasoning processes humans use in their daily activities). Everyday practices and activities involve a great deal of reasoning -that might be also called “common sense reasoning”- that is made without any measurement or computation. This kind of reasoning can be equivalent to the “practical” or “concrete” type of knowledge not very much valued by traditional science and many times related to the feminine gender19. The fact that these activities are concrete and not abstract doesn’t mean there is no reasoning processes involved in them. However, classical logic was not able to model this type of reasoning. Fuzzy Logic provides the method of Generalized Modus Ponens which uses fuzzy “if-then” rules (or “approximate rules of inference”) in which both the antecedent and the consequents are propositions containing linguistic variables. This method allows inferring “approximate conclusions” (not deductions) by making “conjectures” from imprecise information. For example: If P then Q P* Q*

It is raining a little bit If it is raining then Elisa wear a hat It is likely that Elisa wear a hat

In this way, Soft Computing allows to mode the ability of people to reason and make decisions in absence of precise, complete and secure knowledge, that is, is able to deal with the common-sense knowledge and practical reasoning that we express in natural language. SC does not only presents the technical advantage of being able to do what previous AI approaches weren’t, but, in my point of view, involves an “epistemological step forward” in the sense that re-valuates types of knowledge that have been regarded as less important. Though SC hasn’t explicitly made this point, these types of knowledge and reasoning were the ones traditionally related to the feminine gender. 4.7.3

Soft Computing and Issues of Accountability and Responsibility

In section 4.4.2 we explained that Lucy Suchman’s critique to traditional AI approaches is rooted on the key feminist epistemological concept of “situated knowledge” coined by Donna Haraway. In the same way that traditional scientists claim to observe the world from a “nowhere position”, Suchamn argues, designers of AI also 18 19

For example, common-sense knowledge is normally expressed in natural language, that is, by linguistics terms that are intrinsically imprecise. Since the Aristotelian times, women have been considered not much capable of “abstract” reasoning and mathematical abilities, though were “allowed” to hold some kind of reasoning and common sense knowledge necessary to perform their domestic and care tasks.

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assume the same un-located stand when designing their systems, which she calls the “design from nowhere” assumption. Haraway has been insisting on the fact that unconsciousness about one’s own location doesn’t relieve scientist from the responsibility of their actions in technoscientific production. In the same way Suchman applies this ethical statement to the designers of AI systems, claiming for the necessity of them to assume their responsibility in the effects of the products they design. To put that into practice Suchman proposes what she names located accountability’. This process requires that AI system developers and researchers become aware of their location within the historical, cultural, epistemological and ontological networks that they are inevitable part of, and that, then, become responsible for their performances20. In the same line as Suchman, the computer scientist Philip Agre [2] -himself a former AI designer who became very critical of the traditional paradigm- calls this possible way of doing AI a critical technical practice. In the case of Soft Computing, as far as I could observed, issues of accountability and responsibility had not been introduced explicitly in the practice –though, on the other hand, these issues are not common in any area of computer science except but in some particular approaches such as “participatory design”-. For example, while SC researchers acknowledge that different group of language users will determine different meanings of the words they will use to model a system, they do not question their own role in choosing the “selected group” to fill the inputs of the experiments (which usually turns to be many times the very researchers and students who work with them) and how this choice can laden the experiment’s inputs and the systems designed upon them. When SC researchers use terms such us “situated” or “contextual” knowledge (or meaning), they are not using in it in the same way we understand them in STS or gender studies, because, in the end, they do not question the universality of the designer. However, my suggestion is that, because of the particular characteristics we have been pointed out along the article, Soft Computing is more suitable to include these kinds of issues than other more abstract AI approaches such, for example, Symbolic AI.

4.8

Conclusion Remarks

Along the article I have enumerated various critiques made to traditional approaches in Artificial Intelligence from different areas in the social sciences and humanities, in particular Philosophy, Science and Technology Studies and Feminist Studies of Technology. As Soft Computing distinguishes itself very much from traditional AI approaches it seems to me as a very interesting topic to question how Soft Computing could cope with these critiques and challenges. As we have seen, in some 20

When we talk about the “networks” in which AI researchers are part of, we do not only refer only the local networks of their research institutions or companies, nor even their international scientific communities, but to the extended network that, in our highly globalized world, involves transnational, racialized and gendered divisions of labour.

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cases (particularly in philosophical critiques) SC has overcame the problems that were present in previous AI approaches. However, some STS and feminist critiques are still applicable and should be addressed by SC researchers. Although this research is still in a preliminary state, I think the issues treated in this article can provide a good frame for future work in the new field of Soft Computing in Humanities and Social Sciences. In addition to that, it could be also useful for Soft Computing itself. Soft Computing has to face many challenges in order to improve previous AI methodologies and achieve what they weren’t able to do. Having a clear picture of these challenges will be very useful. Philosophical and sociological studies of science could help to identify problems and challenges, and, although far from engineering practice, can inspire and suggest new ways of facing the most durable challenges. As Suchman ([22], p. 97) says, the development of useful systems must be a boundary-crossing activity, taking place through the deliberate creation of situations that allow for the meeting of different partial knowledges. Therefore, when we humanities adn social sciences’ scholars seek to establish proposals we shouldn’t do it from "outside" claiming for a “total change” of the ways in which the community we are studying works. In the sense of actor-network theory [17], we must "recruit allies” to influence the network in the direction that our responsibility suggests. In my experience working with the Soft Computing community, I have found many engaged researchers who attempt to change some of the traditional practices and are open to new ideas and criticism, though, as far as I could observe, Soft Computing practitioners have not introduced the issues of accountability and responsibility explicitly in their practice. However, within SC I have found recognizably efforts by some of their practitioners to do it, and I feel it would be worthy to join forces with them in engagement and “partial translations”. I strongly believe that interdisciplinary translations and cooperation of us as philosophers, STS scholars and feminist researchers with engineers and scientists themselves is very important. Our critical analyses beg for imaginings of how technological practice might be done differently. In the future of my research my aim will be to look for ways to foster cooperation among disciplines in interdisciplinary translations, which will enable mutual learning and new ways of creating better, more responsible and more social just technologies.

Acknowledgments This work has been carried out within the project “Philosophy of Social Techno-sciences” with reference FFI 2008-03599 funded by Spanish Ministry of Science and Innovation. The author wants to thank Sergio Guadarrama for his comments and fruitful discussions about the topic of the paper. In addition, I want to thank the Foundation for the Advancement of Soft Computing for providing me the best place and environment to proceed with my research in this topic.

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References [1] Adam, A.: Artificial Knowing. Gender and the Thinking Machine. Routledge, London (1998) [2] Agre, P.: Computation and Human Experience. Cambridge University Press, Cambridge (1997) [3] Alcoff, L., Potter, E. (eds.): Feminist Epistemologies. Routledge, New York (1993) [4] Bloor, D.: Knowledge and Social Imagery. Routledge, London (1976) [5] Brooks, R.A.: Intelligence without representation. Artificial Intelligence 47, 139–159 (1991) [6] Code, L.: What Can She Know? Feminist Theory and the Construction of Knowledge. Cornel Univ. Press, Ithaca (1991) [7] Code, L.: Taking Subjectivity into Account. In: Alcoff, L., Potter, E. (ed.) Feminist Epistemologies, pp. 15–48 [8] Collins, H.M. (ed.): Artificial Experts: Social Knowledge and Intelligent Machines. MIT Press, Cambridge (1990) [9] Dennett, D.: The Intentional Stance. MIT Press, Cambridge (1987) [10] Dreyfus, H.L.: What Computers Can’t Do: The Limits of Artificial Intelligence. Harper and Row, New York (1972) [11] Dreyfus, H.L.: What Computers Still Can’t Do: A Critique of Artificial Reason. MIT Press, Cambridge (1992) [12] Forsythe, D.: Engineering Knowledge: The Construction of Knowledge in Artificial Intelligence. Social Studies of Science 23(3), 445–477 (1993) [13] Guadarrama, S.: Computing with Actions: The case of driving a car in a simulated car race. In: International Fuzzy Systems Association-European Society for Fuzzy Logic and technologies (IFSA-EUSFLAT) World Congress, pp. 1410–1415 (2009) [14] Haraway, D.: Simians, Cyborgs, and Women: The Reinvention of Nature. Routledge, New York (1991) [15] Harding, S.: Whose Science? Whose Knowledge? Cornell University Press, Ithaca (1991) [16] Keller, E.F.: Reflections on Gender and Science. Yale University Press, New Haven (1985) [17] Latour, B., Woolgar, S.: Laboratory Life. Sage Publications, Beverly Hills (1979) [18] Lenat, D.B., Guha, R.V.: Building Large Knowledge-Bused Systems: Representation and Inference in the Cyc Project. Addison Wesley, Reading (1990) [19] Penrose, R.: The Emperor’s New Mind. OUP, Oxford (1989) [20] Searle, J.: Minds, brains and programs. Behavioral and Brain Sciences 3, 417–457 (1980) [21] Suchman, L.: Plans and Situated Action: The Problem of Human-Machine Communication. Cambridge University Press, New York (1987) [22] Suchman, L.: Located Accountabilities in Technology Production and Use. Scandinavian Journal of Information Systems 14(9), 1–105 (2002) [23] Suchman, L.: Human-Machine Reconfigurations. Plans and Situated Action, 2nd edn. Cambridge University Press, Cambridge (2007) [24] Trillas, E.: La Inteligencia artificial. Máquinas y personas. Madrid, Ed, Debate (1998) [25] Trillas, E., Moraga, C., Guadarrama, S.: A (naïve) glance at Soft Computing. International Journal of Computational Intelligence and Systems (in Press, 2011) [26] Turing, A.: Computing Machinery and Intelligence. Mind 59, 433–460 (1950) [27] Wittgenstein, L.: Philosophical Investigations (PI). In: Anscombe, G.E.M., Rhees, R. (eds.). Blackwell, Oxford (1953)

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[28] Yamakawa, T.: Stabilization of an inverted pendulum by a high-speed fuzzy logic controller hardware system. Fuzzy Sets and Systems 32, 161–180 (1989) [29] Zadeh, L.A.: Fuzzy Logic = Computing with Words. IEEE Trans. Fuzzy Systems 4(2), 103–111 (1996) [30] Zadeh, L.A.: Commonsense Knowledge Representation based on Fuzzy Logic. IEEE, 61–65 (October 1983) [31] Zadeh, L.A.: A computational theory of dispositions. In: Proceedings of the 10th International Conference on Computational Linguistics, pp. 312–318 (1984) [32] Zadeh, L.A.: Soft computing and fuzzy logic. IEEE Software 11(6), 48–56 (1994) [33] Zadeh, L.A.: Toward Human Level Machine Intelligence- Is It Achievable? The Need for a Paradigm Shift. IEEE Computational Intelligence Magazine, 11–22 (August 2008)

Part III

5 On Explicandum versus Explicatum A Few Elementary Remarks on the Birth of Innovative Notions in Fuzzy Set Theory (and Soft Computing) Settimo Termini

5.1

Introduction

The aim of this paper is twofold. First of all I want to present some old ideas revisited in the light of some of the many interesting new developments occurred in the course of these last ten years in the field of the foundations of fuzziness. Secondly I desire to present a tentative general framework in which it is possible to compare different attitudes and different approaches to the clarification of the conceptual problems arising from fuzziness and soft computing. In what follows, then, I shall use some names as banners to indicate a (crucial) problem (i.e., Carnap’s problem, von Neumann’s problem, Galileian science, Aristotelian science and so on). Although it will be clear by reading the following pages, the use of these reference names (the association of a name to a certain problem) should not be considered as the result of a historically based profound investigation but only as a sort of slogan for a specific position and point of view, an indication which, of course, I hope (and trust) does not, patently, contradict historical evidence regarding the scientific attitudes, approaches and preferences of the named persons. In some cases the problem associated to a certain scientist could be not so central in his scientific interests as it could appear from the connection proposed here and as my slogan could suggest. It should, then, be taken as a sort of working hypothesis which remains independent from the historical accuracy of the etiquette assigned to the problem itself. Above, I mentioned “old” views, ideas and reflections on these themes. They refer maily to discussions I had along the years with old friends; some of them involved also in the 2009 meeting. Finally let me add that some (relatively) recent work (see, [14], [15]) has vigourously indicated additional interesting points of view which open new ways of affording already emerged and new emerging problems.

5.2

Carnap’s Problem

It is well known that Rudolf Carnap in the first pages of his Logical foundations of probability [2] faced the (difficult) problem of the ways and procedures according to which a prescientific concept (which by its very nature is inexact) is trasformed R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 113–124. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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into a (new) exact scientific concept. He called this transformation (the transition from the explicandum, the informal, qualitative, inexact prescientific notion to the explicatum, its scientific, quantitative, exact substitute) the procedure of explication, a procedure which, as Carnap immediately observed, presents a paradoxical aspect. While in ordinary scientific problems, he, in fact, observes, “both the datum and the solution are, under favorable conditions, formulated in exact terms ... in a problem of explication the datum, viz., the explicandum, is not given in exact terms; if it were, no explication would be necessary. Since the datum is inexact, the problem itself is not stated in exact terms; and yet we are asked to give an exact solution. This is one of the puzzling peculiarities of explication”. A comment of general type is necessary at this point, although it may look as a digression. Two immediate corollaries of Carnap’s argumentation are the following ones. First, we shall never able to state that we have a proof that a proposed formal candidate is THE explicatum of a certain explicandum. Secondly, we can have different competing explicata of the same explicandum. Regarding the first point let me recall what Carnap has rightly observed: “It follows that, if a solution for a problem of explication is proposed, we cannot decide in an exact way whether it is right or wrong. Strictly speaking, the question whether the solution is right or wrong makes no good sense because there is no clearcut answer. The question should rather be whether the proposed solution is satisfactory, whether it is more satisfactory than another one, and the like.” The impossibility of having a rigourous proof does not imply, however, that we cannot but proceed in a hazy way. The observation should not encourage sloppiness. In fact, “There is a temptation to think that, since the explicandum cannot be given in exact terms anyway, it does not matter much how we formulate the problem. But this would be quite wrong.”1 Although we cannot reach absolute precision, we can do, and must do our best for using the maximum of exactness or precision that is allowed by the treated problem, under given conditions. All these Carnap’s comments are of extrordinary importance for our issues. There is not a sharp separation 1

He continues affirming: “On the contrary, since even in the best case we cannot reach full exactness, we must, in order to prevent the discussion of the problem from becoming entirely futile, do all we can to make at least practically clear what is meant as the explicandum. What X means by a certain term in contexts of a certain kind is at least practically clear to Y if Y is able to predict correctly X’s interpretation for most of the simple, ordinary cases of the use of the term in those contexts. It seems to me that, in raising problems of analysis or explication, philosophers very frequently violate this requirement. They ask questions like: ’What is causality?’, ’What is life?’, ’What is mind?’, ’What is justice?’, etc. Then they often immediately start to look for an answer without first examining the tacit assumption that the terms of the question are at least practically clear enough to serve as a basis for an investigation, for an analysis or explication. Even though the terms in question are unsystematic, inexact terms, there are means for reaching a relatively good mutual understanding as to their intended meaning. An indication of the meaning with the help of some examples for its intended use and other examples for uses not now intended can help the understanding. An informal explanation in general terms may be added. All explanations of this kind serve only to make clear what is meant as the explicandum; they do not yet supply an explication, say, a definition of the explicatum; they belong still to the formulation of the problem, not yet to the construction of an answer”.

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between the exactness of the proofs or of the empirical verifications and the world of everyday language and use of the words. Also in the case of “unsystematic, inexact terms, there are means for reaching a relatively good mutual understanding as to their intended meaning”. At the very root of “the procedure of explication” there is, then, a puzzling aspect and the indication of a way for not remaining blocked by the puzzle. And the way makes implicit reference to a a different kind of “exactness” which is working in practice, although it would be useless (if not impossible) to try to construct a general theoretical understanding of the phenomenon. Let us briefly dwell on the second corollary indicated above. While looking obviously true, there exists, however, today a notheworthy exception. The informal, intuitive, everyday notion of “computable”, presents a unique explicatum (under the demonstrably equivalence of variuos different formal definitions). So, the so called and well known Church-Turing’s thesis, while remaining “a thesis” (something which cannot be rigourously proved just for the reasons discussed by Carnap – we are confronting an informal idea with a formal definition) is, however, corroborated in a very strong way, stronger than one could think possible by prima facie considering the motivations provided by Carnap: any conceivable proposed new explicatum turns out to be “equivalent” to previously considered proposals. Let me briefly summarize what in my view can be fruitfully “stolen” from Carnap’s observations for our problem. First of all, his “procedure of explication” can be used as a very good starting point for looking in a unified way to the foundational problems of Soft Computing (SC). We have a uniform way for looking at and comparing the different notions used in SC, the ways in which they have been and are “regimented”. Also the ways suggested by Carnap to clarify the aims and qualifications of the explicandum go hand in hand with what has been done in SC. A difference remains in the general approach. The explicatum, in Carnap’s view, cannot but be something exactly defined in the traditional terms of Hard Sciences. All his observations, however, on the ways in which we can transform a very rough idea into something sufficiently clear to be considered reasonable as the explicandum of a certain concept, provide tools for using his ideas also along different paths. In order to proceed a little bit further, let me quote in extenso what Carnap writes regarding the methodology we could follow in order to define a formal explicatum starting from the original informal intuitive notion. “A concept must fulfil the following requirements in order to be an adequate explicatum for a given explicandum: (1) similarity to the explicandum, (2) exactness, (3) fruitfulness, (4) simplicity. Suppose we wish to explicate a certain prescientific concept, which has been sufficiently clarified by examples and explanations as just discussed. What is the explication of this concept intended to achieve? To say that the given prescientific concept is to be transformed into an exact one, means, of course, that an exact concept corresponding to the given concept is to be introduced. What kind of correspondence is required here between the first concept, the explicandum, and the second, the explicatum? Since the explicandum is more or less vague and certainly more so than the explicatum, it is obvious that we cannot require the correspondence between the two concepts to be a complete coincidence. But one might perhaps think that the

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explicatum should be as close to or as similar with the explicandum as the latter’s vagueness permits.” A few comments are needed. Let us preliminary recall the already mentioned unique known conterexample to the very natural general situation described by Carnap (namely “that we cannot require the correspondence between the two concepts to be a complete coincidence”): the concept of computation. My insisting on this fact is motivated both by its intrinsic interest and by a question which will be subsequently asked in the paper. Secondly, let me observe that the second requirement (which is also the unique of a “formal” type) asked for having a good explicatum is exactness. Carnap does not require mathematization or similar things but only (only!) exacteness. Here we see in action the hand of the Master. It is not asked a formalization, axiomatization or the like, but exactness. So its scheme can be used also in fields different from traditional hard sciences paying only the (necessary) price of a preliminary clarification of the form of exactmness we are able to use in the given domain and that the explicata of some central concepts in the theory aspiring at grasping some aspects of the considered domain are such just in virtue of the fact that they satisfy the requirement of the proposed form of exactness. In what follows, I shall call Carnap’s problem, the one of analysing the concepts and notions used to model and describe selected pieces of reality according to the procedure indicated by him and briefly described and commented above.

5.3

A Unifying Framework for Looking at Typical Case Studies

As I have already observed at the end of the previuos Section, Carnap’s problem can be used for (at least) two different scopes. First, as a guide to look at problems of interpretations and of new developments in some classical areas: in these cases it plays the role of providing an important reference point. Secondly, it can be used as a tool for putting order in the analysis of some difficult new problems. In our case a very important example is provided just by the general theme of this Meeting, namely, the relationships existing between SC and Human Sciences. Another important and crucial topic is the one of analyzing the long range methodological innovation provided by Zadeh’s CW and CTP (see, for instance, [37], [38]). In the present Section I shall briefly indicate how some classical problems can be seen in the perspective outlined in the previous Section. What will be presented, then, is nothing more than a very brief outline of a research programme, leaving to future contributions a detailed treatment of the problems and subjects involved. What follows has also the aim of providing a preliminary test of the validity of the epistemological working hypothesis that Carnap’s problem can be a useful tool. In the remaining part of this Section I shall, then, present some scattered observations on a few selected topics. 5.3.1

The Revisitation of Basic Logical Principles

Recently Enric Trillas (see, for instance, [28]) has posed the problem of the validity of logical principles starting from various different technical results of fuzzy

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logic. Of course, that the same acceptance of different truth values besides true and false would pose interpretative problems to the unquestioned logical principles was observed already by the founding fathers of many valued logic, namely, Jan Lukasiewicz and Emil Post (see, for instance, [5]). However, the way in which the problem is posed now is more general since also different aspects, not purely logical are involved. A complete and satisfactory analysis of the problem asks that both the old questions and the new context in which we move today are taken into account. Among these last ones we can remember both the richness and the proliferation of technical tools and results, sometimes associated with a scarce awareness, we could say, of the conceptual implications of the implicit assumptions behind some of the paths followed. And in some cases a sort of (technical and conceptual) lack of significance of some developments. Carnap’s scheme can be fruitfully used for a preliminary classification and clarification of at least some problems and developments. 5.3.2

Vagueness vs. Fuzziness

What is the correct relationship between the two notions of vagueness and fuzziness is an old and debated problem; incidentally, more challenging – in my view – than the one of the relationship between fuzziness and probability. While in the last case, in fact, we are confronting two explicata, (the interesting problem being the “inverse” one of understanding of which explicanda they are good explicata), by comparing vagueness and fuzziness we are faced with a lot of intersting open problems: is vagueness the explicandum and fuzziness the explicatum (or one of its explicata)? If not, which kind of relations can we establish or study of these two different notions (uncomparable for what regards their different level of formalization? For a review of different aspects of these questions see [18], [19], [20] and the whole volume [16]. Carnap’s scheme allows, in my view, to look at this net of interesting questions in a very general way allowing to examine differents facets inside a unified setting. For instance, one interesting comment done by Terricabras and Trillas [17]: “Fuzzy sets can be see as the best approximation of a quantitative, extensional representation of vagueness” acquires a new light in this context. 5.3.3

Measuring Fuzziness (Or Controlling Booleanity?)

If one accepts the existence of fuzzy theories as mathematically meaningful descriptions of concrete systems of a high level of complexity, the question arises whether it is possible to control (and measure) the level of fuzziness present in the considered description. This is the starting point which triggered the development of the the so called “entropy measures” or “measures of fuzziness” of a fuzzy set (see, for instance, [6], [8], [7], [10]). This problem was tackled in a very general way by following the simple idea of proceeding in an two stages axiomatic way by strictly connecting requirements to be imposed and measures satisfying these requirements, keeping in mind, however, the fact that not every requirement one could abstractly envisage could always be imposed in each specific situation. In other words, the

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axioms should not be put all on the same level. One should pick up some basic properties and requirements, necessary to characterize something as “measure of fuzziness” of a fuzzy set. Other requirements could be imposed depending on the particular situation under study. By proceeding in this way, one could have available a wide class of measures from which one could choose the most adequate for the specific problem under study. For what concerns the measures of fuzziness, in my view, the basic kernel of the theory may be considered fairly complete now, after the general classification of the various families of measures provided by Ebanks [3]. However some interesting problems remain. First, at the interpretative level, should we interpret these measures as measures of fuzziness in a very general sense, or should look at them as tools for controlling “booleanity”, as Enric Trillas has often posed the problem? Secondly, and strictly related to the previous question, is the analysis of conceptual relationships existing between the standard (axiomatic) theory and conceptually different approaches. I refer, in particular, to the original way of facing the problem of measuring how far a fuzzy set is from a classic characteristic function due to Ron Yager [31] (for a few remarks, see also [9]). His proposal allows one to look at the problem of intuitive ideas versus formal results from another point of view. His challenging idea is that of measuring the distance or the distinction between a fuzzy set and its negation and the technical tool to do so is provided by the lattice theoretical notion of “betweeness”. It can be shown that in all the cases in which it is possible to define Yager’s measure it is also possible to define a measure of fuzziness in the axiomatic sense, discussed above. The point of view of Yager, then, provides a new very interesting visualization but technically it does not allow one to extend the class of measures, as one would expect, due to the conceptual difference of the starting point. The general problem of the nature and differences of various ways of affording the problem is still in need of a conceptual clarification. Moreover, the discovery of new general requirements to be imposed under the form of new axioms does not appear very likely as long as we move inside the basic scheme of the theory of fuzzy sets with the standard connectives. What is open to future work is the adaptation of the proposed axiomatic scheme to some variants of it. We may, for instance, change the connectives or also the range of the generalized characteristic functions. The problem of measuring vagueness in general remains completely open, i.e., the problem of constructing a general theory of measures of vagueness, a goal which presupposes the existence of a general formal theory of vague predicates. The alternative of finding out paths that can be followed “to measure” in a non quantitative (numerical) way, however interesting and appealing it can appear, seems extremely more difficult also to envisage. 5.3.4

von Neumann’s Problem

All the developments of the theory of fuzzy sets corroborate the conviction of von Neumann that escaping the constraints of all-or-none concepts would allow one to introduce (and use) results and techniques of mathematical analysis in the field of logic, increasing, then, the flexibility of logical tools and their wider application to

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different fields.2 But one can see also a non trivial connection between measures of fuzziness and von Neumann’s remarks on the role of the presence of error in logics seen as as an essential part of the considered process.3 The program of constructing a calculus of thermodynamical type which could be considered a development of von Neumann’s vision of the role of error in “the physical implementation of logic”, to quote his own words, was explicitly mentioned already in 1979 (see [8]). Let me observe that, from my knowledge, the first attempt to proceed technically in the direction indicated by von Neumann is due to Enric Trillas who in his [27] approached the problem of the logical connective of “negation” (in a subsequent paper [1] the other one connective was taken into account). Measures of fuzziness are indeed an element which could contribute, inside the general framework of the theory of fuzzy sets, to the construction of a sort of “thermodynamical logic”. They can, in fact, be viewed as a particular way of studying the levels of precision of a description. From this point of view they can already represent a treatment of error of a “thermodynamical” type in some definite – albeit still vague – sense. They, moreover, can be inserted in logical inference schemes in which approximation, vagueness, partial or revisable information play a role either at the level of the reliability of the premises or of the inference rules (or both). Although a satisfactory and fairly complete integration of all these aspects remains to be done, we can mention, however, a possible development based on the fact that inside the theory of fuzzy sets a lot of different measures has been developed. This will clearly appear, after the description of an attempt of outlining the basis of a possible treatment of the dynamics of information (and in this context it will be clear my use of the expression “thermodynamical logic”. 5.3.5

Towards an “information dynamics”

The informal notion of “information” (the explicandum) is very rich and multifaceted and so it is not strange that the formal theories that have been proposed do not capture all the nuances of the informal notion. One could consider isolating some meaningful and coherent subsets of the properties and features of the explicandum and look for satisfying formalizations of these aspects. Since they are 2

3

“There exists today a very elaborate system of formal logic, and, specifically, of logic as applied to mathematics. This is a discipline with many good sides, but also with certain serious weaknesses. . . . About the inadequacies . . . this may be said: Everybody who has worked in formal logic will confirm that it is one of the most refractory parts of mathematics. The reason for this is that it deals with rigid, all-or-none concepts, and has very little contact with the continuous concept of the real or of the complex number, that is, with mathematical analysis. Yet analysis is the technically most successful and best-elaborated part of mathematics. Thus formal logic is, by the nature of its approach, cut off from the best cultivated portions of mathematics, and forced onto the most difficult part of the mathematical terrain, into combinatorics.” ([11], p. 303) “The subject matter . . . is the role of error in logics, or in the physical implementation of logics – in automata synthesis. Error is viewed, therefore, not as an extraneous and misdirected or misdirecting accident, but as an essential part of the process under consideration . . . ” [12], p. 329)

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different aspects of one unique general concept anyway we must also pick up and study the way in which these subaspects interact. The process suggested above points then not to a very general but static theory of information in which a unique formal quantity is able to take the burden of a multifaceted informal notion, but instead pinpoints an information dynamic in which what the theory controls is a whole process (along with – under the pressure of changes in the boundary conditions – the relative changes of the main central (sub)notions involved in the theory itself and their mutual interactions). In this way we pass from a situation in which there is only one central notion on the stage to another one in which a report of what is happening in a process (in which information is transmitted, exchanged and the like) is provided by many actors on the stage, each of which represents one partial aspect of what the informal use of the word information carries with it. This scenario resembles the one of thermodynamics: no single notion suffices for determining what is happening in the system and the knowledge of the value assumed by one of the thermodynamical quantities can be obtained only as a function of (some of) the others, by knowing (and working out) the quantitative relationships existing among them. That is what an information dynamic must look for: its principles, laws which quantitatively state the connections existing among some of the central quantities of the theory. 5.3.6

Infodynamics of Fuzzy Sets

But let us see what can happen if we try to apply the ideas of this very general scheme to the case of fuzzy sets. In [22], I tried to outline how a program of this type could be looked for in the setting of the theory of fuzzy sets. I shall briefly summarize now the general ideas showing the connection with the remarks done above on “von Neumann’s problem”. It is well known that many quantities have been introduced to provide a global (one could say, “macroscopic”) control of the information conveyed by a fuzzy set; for instance, measures of fuzziness, energy measures (or “fuzzy cardinalities”) (see [6], [8], [7], [10]) or measures of specificity (see [33], [34]). Measures of fuzziness want to provide an indication of how far a certain fuzzy set departs from a classical characteristic function; measures of specificity, instead, want to provide an indication of how much a fuzzy set approaches a singleton. These two classes of measures certainly control different aspects of the information conveyed by a fuzzy set; they are not, however, conceptually unrelated. For instance, if the measure of fuzziness of a certain fuzzy set is maximum (i.e., all the elements have a “degree of belonging” equal to 0.5) then we indirectly know something about specificity. Conversely, if the measure of specificity informs us that we are dealing exactly with a singleton, we can immediately calculate the corresponding measure of fuzziness. But a relationship between these measures also exists in not such extreme cases; in order to refine the way in which this kind of knowledge can be exchanged one could also think to introduce other measures. An important role in this sense can be played by (generalized) cardinality of the fuzzy set and (if not all the elements are on an equal footing) also by a weighted cardinality, for instance, the so-called “energy” of a fuzzy set [8]. It would then be very

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interesting to have some explicit quantitative relations among these measures, since it would provide a way of transforming our knowledge regarding one or two of these quantities in a (more or less approximate) knowledge of the remaining one(s). All this stuff – this was the suggestion given in [22] – should be organized in a way similar to the structure of thermodynamics, listing principles and equations connecting the various central quantities. The final goal of the project is to obtain ways of calculating the values of one of these quantities, once the values of other ones are known, or to reconstruct the fuzzy set given the values of appropriate quantities. So, we see that, by taking Carnap’s problem as a general guide, it is possible to outline a possible development of von Neuman’s problem far beyond the short (although very dense and – considering the time in which they were written – very innovative) comments we have quoted above.

5.4

As a Sort of Conclusion

To examine the relationships existing between SC and human and social sciences is very important and crucial since, through a simple inspection, many forgotten old epistemological problems reemerge, showing a major role that they can play for understanding in a non sectorial way, what we could call the “enterprise of scientific investigation”.4 At the same time a host of new problems interesting completely new questions appear. So there exist different planes along which we can (and should) move. First of all, it is very important to collect all the possible interactions and applications and explore different ways for obtaining other useful applications. Secondly we should explore the reason why the interaction has not been till now more extense. I have briefly indicated why, in my view, this phenomenon is not casual but the true reasons can reside in the fact that to interact in a substantial way and not occasional, can cause the emergence of deep (and, of course, both difficult and interesting) problems. But the question deserves to be investigated more extensively than I have done in the previous pages in a very cursory way. Third, there exist crucial and difficult problems which in my view are a corollary of the reasons of the missed interaction, but which, independently of this, deserve to be studied and analyzed with intellectual passion and scientific care . Before concluding, let me, then, list a few items that – in my view – will be of more and more crucial interest in the future since they stand at the crossroad of conceptual questions and the possibility of scientific (technical) developments along non traditional paths: 4

In this regard, let me stress that I am looking at these problems from the specific (and limited) point of view which can be of interest to the working scientist. For instance, when reflecting on vagueness, I share the very pragmatic point of view of van Heijenoort in [4] and leave out all the interesting and subtle considerations present in the philosophical literature on the subject. Let me stress that all these questions should be read in the general context of the development of Cybernetics and Information Sciences. I refer to [21], [23], [24] for a few (still very preliminary) remarks. Please note that many argumentations in Section 3 are borrowed almost literally from [23].

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a) Galileian science vs Aristotelian science. With these very pompous terms, I want to refer to the two different attitudes which have been and still are present in scientific investigation. The traditional (Galileian) attitude which in its standard fields of investigation proceeds by bold hypotheses and rigourous empirical control of the consequences of these same hypotheses by using sophisticated formal machineries on the theoretical part and equally sophisticated technological machineries for the empirical control. What I called “Aristotelian” science refer to a more descriptive attitude. A first preliminary discussion of my view on this contrapposition can be found in [25] and related comments on “family resemblances” of human sciences and fuzzy set theory in [26] in this same volume. b) It remains also another non trivial problem, namely the one of taking as starting point the perceptions, not to be reduced or recovered, reconstructed from other numerical classical components. But this is another story, which indicates both a strong connection of Zadeh’s new proposals which the questions arised in point a) above and the fact that we should take into account also Husserl’s conception of science. c) Relationships existing between Zadeh’CWW and Church-Turing’s Thesis. To my knowledge, point b) has not been previously discussed, at least with reference to problems of SC. Point c) has been discussed, for instance in [35], [30], along the traditional line of proving that the new model does not violate Church-Turing Thesis; or, told in different terms, c) has been looked for in the sense of proving (under suitable “numerical” translations of CWW procedures) that the model of computation inspired by CWW is equivalent to one of the classical models of computation. Although this is very interesting, however, a major challenge goes exactly in the other direction: to analyze whether by manipulating words we can do “computations” (in a specific sense) to be defined in a more general way, which is not reducible to the classical notion of computation. Let me conclude by saying that all the developments of the last 45 years in what now is known as the huge field of Soft Computing should be periodically confronted with the original setting proposed by Zadeh in [36], looking also to the first developments of the original ideas (see [32]). This could help in appreciating extensions of the general perspective but also – epistemologically relevant – conceptual shifts and programmatic drifts. Finally, let me indicate that Enric Trillas in his recent paper [29], which recollects some scientific exchanges with Italy, indicates in a very sinthetic and challenging way a few crucial problems to approach in our young and intellectually stimulating field of investigation.

Acknowledgements I want to thank Enric Trillas for many thought-provoking questions and discussions along many many years and Rudi Seising for equally interesting discussions along the last few years.

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References [1] Alsina, C., Trillas, E., Valverde, L.: On Some Logical Connectives for Fuzzy Sets Theory. Journal of Mathematical Analysis and Applications 93, 15–26 (1983) [2] Carnap, R.: Logical Foundations of Probability. Chicago University Press (1950) [3] Ebanks, B.R.: On Measures of Fuzziness and their Representations. Journal of Mathematical Analysis and Applications 94, 24–37 (1983) [4] van Heijenoort, J.: Frege and Vagueness. In: van Heijenoort, J. (ed.) Selected Essays, pp. 85–97. Bibliopolis, Naples (1985) [5] Lukasiewicz, J.: Philosophical remarks on many-valued systems of propositional logic. In: Borkowski, L., Lukasiewicz, J. (eds.) Selected Works. Studies in logic and the foundations of mathematics, pp. 153–178. North-Holland Publ. Comp., Pol. Scientif. Publ., Warszawa, Amsterdam (1970) [6] De Luca, A., Termini, S.: A definition of a non probabilistic entropy in the setting of fuzzy sets theory. Information and Control 20, 301–312 (1972); reprinted in: Dubois, D., Prade, H., Yager, R.R. (eds.): Readings in Fuzzy Sets for Intelligent Systems. pp. 197-202, Morgan Kaufmann (1993) [7] De Luca, A., Termini, S.: Entropy of L-fuzzy Sets. Information and Control 24, 55–73 (1974) [8] De Luca, A., Termini, S.: Entropy and energy measures of a fuzzy set. In: Gupta, M.M., Ragade, R.K., Yager, R.R. (eds.) Advances in Fuzzy Set Theory and Applications, pp. 321–338. North-Holland, Amsterdam (1979) [9] De Luca, A., Termini, S.: On Some Algebraic Aspects of the Measures of Fuzziness. In: Gupta, M.M., Sanchez, E. (eds.) Fuzzy Information and Decision Processes, pp. 17–24. North-Holland, Amsterdam (1982) [10] De Luca, A., Termini, S.: Entropy Measures in Fuzzy Set Theory. In: Singh, M.G. (ed.) Systems and Control Encyclopedia, pp. 1467–1473. Pergamon Press, Oxford (1988) [11] von Neumann, J.: The General and Logical Theory of Automata. In: Cerebral Mechanisms in Behaviour – The Hixon Symposium, J. Wiley, New York (1951); (reprinted in [13], pp. 288–328 ) [12] von Neumann, J.: Probabilistic Logics and the Synthesis of Reliable Organisms from Unreliable Components. In: Shannon, C.E., MacCarthy, J. (eds.) Automata Studies. Princeton University Press (1956); (reprinted in [13] pp. 329–378 ) [13] von Neumann, J.: Design of Computers, Theory of Automata and Numerical Analysis. In: Collected Works, vol. V. Pergamon Press, Oxford (1961) [14] Seising, R.: The Fuzzification of Systems. The Genesis of Fuzzy Set Theory and Its Initial Applications – Its Development to the 1970s. STUDFUZZ, vol. 216. Springer, Berlin (2007) [15] Seising, R. (ed.): Views on Fuzzy Sets and Systems from Different Perspectives. Philosophy and Logic, Criticisms and Applications. STUDFUZZ, vol. 243. Springer, Berlin (2009) [16] Skala, H.J., Settimo, T., Trillas, E.: Aspects of Vagueness. Reidel, Dordrecht (1984) [17] Terricabras, J.-M., Trillas, E.: Some remarks on vague predicates. Theoria 10, 1–12 (1988) [18] Termini, S.: Aspects of Vagueness and Some Problems of their Formalization. In: [16], pp. 205–230 [19] Termini, S.: Vagueness in Scientific Theories. In: Singh, M.G. (ed.) Systems and Control Encyclopedia, pp. 4993–4996. Pergamon Press, Oxford (1988)

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[20] Termini, S.: Vague Predicates and the Traditional Foundations of Mathematics. In: International Congress for Logic, Methodology and the Philosophy of Science, Salzburg (1983) [21] Termini, S.: Remarks on the development of Cybernetics. Scientiae Matematicae Japonicae 64(2), 461–468 (2006) [22] Termini, S.: On some vagaries of vagueness and information. Annals of Mathematics and Artificial Intelligence 35, 343–355 (2002) [23] Termini, S.: Imagination and Rigor: their interaction along the way to measuring fuzziness and doing other strange things. In: Termini, S. (ed.) Imagination and Rigor, pp. 157–176. Springer, Milan (2006) [24] Termini, S. (ed.): Imagination and Rigor. Springer, Milan (2006) [25] Termini, S.: Concepts, Theories, and Applications: the role "experimentation” for formalizing new ideas along innovative avenues. In: Trillas, E., Bonissone, P., Magdalena, L., Kacprycz, J. (eds.) Experimentation and Theory: Hommage to Abe Mamdani. STUDFUZZ. Physica-Verlag (to appear, 2011) [26] Termini, S.: On some " family resemblances” of Fuzzy Set Theory and Human Sciences. In: Seising, R., Sanz, V. (eds.) This same volume [27] Trillas, E.: Sobre funciones de negación en la teoría de subconjuntos difusos, Stocastica III, 47-59; an english version appeared in the volume. : Barro, S., Alberto, B., Sobrino, A. (eds.) Advances of Fuzzy Logic, pp. 31–43. Press of the Universidad de Santiago de Compostela, Spain (1998) [28] Trillas, E.: Non Contradiction, Excluded Middle, and Fuzzy Sets. In: Di Gesù, V., Pal, S.K., Petrosino, A. (eds.) WILF 2009. LNCS (LNAI), vol. 5571, pp. 1–11. Springer, Heidelberg (2009) [29] Trillas, E.: Il Laboratorio/Istituto di Cibernetica e la mia vita. In: Greco, P., Termini, S. (eds.) (a cura di) Memoria e progetto, GEM, Bologna, pp. 23–32 (2010) (an english version, not published, is also available) [30] Wang, H., Qiu, D.: Computing with words via Turing machines: a formal approach. IEEE Transactions on Fuzzy Systems 11(6), 742–753 (2003) [31] Yager, R.R.: On the Measures of Fuzziness and Negation. II Lattices. Information and Control 44, 236–260 (1980) [32] Yager, R.R., Ovchinnikov, S., Tong, R.M., Nguyen, H.T. (eds.): Fuzzy Sets and Applications: Selected Papers by L.A. Zadeh. Wiley, New York (1987) [33] Yager, R.R.: On the specificity of a possibility distribution. In: Fuzzy Sets and Systems, vol. 50, pp. 279–292 (1992); Reprinted in: Dubois, D., Prade, H., Yager, R.R. (eds.) Readings in Fuzzy Sets for Intelligent Systems, pp. 203-216. Morgan Kaufmann (1993) [34] Yager, R.R.: Measures of information in generalized constraints. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 6, 519–532 (1998) [35] Ying, M.S.: A formal model of computing with words. IEEE Transactions on Fuzzy Systems 10(5), 640–652 (2002) [36] Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965) [37] Zadeh, L.A.: Foreword to: Dubois, D., Prade, H. (eds.) Fundamentals of Fuzzy Sets, Kluwer Academic Publishers (2000) [38] Zadeh, L.A.: From Computing with Numbers to Computing with Words – From Manipulation of Measurements to Manipulation of Perceptions. International Journal of Applied Mathematics and Computer Science 12, 307–324 (2002)

6 Axiomatic Investigation of Fuzzy Probabilities Takehiko Nakama, Enric Trillas, and Itziar García-Honrado

6.1

Introduction

The mathematics of probability has a long history. The famous correspondence between Pascal and Fermat in the seventeenth century about the “problem of the points” substantially advanced the mathematics of games of chance, and other prominent mathematicians such as James and Jacques Bernoulli, De Moivre, and Laplace established notable limiting results in probability (see, for instance, [11]). The axiomatic foundation of probability theory was established by Kolmogorov in the twentieth century. Today the importance of probability theory is widely recognized in a variety of research fields, and probability continues to fascinate the general public because of its relevance to everyday life. In his book Analytical Theory of Probability, Laplace states, “We see that the theory of probability is at bottom only common sense reduced to calculation; it makes us appreciate with exactitude what reasonable minds feel by a sort of instinct, often without being able to account for it. ... It is remarkable that this science, which originated in the consideration of games of chance, should become the most important object of human knowledge. ... The most important questions of life are, for the most part, really only problems of probability”(see [11]). The rigorously established measure-theoretic probability that we have today deals with probabilities of events, which are “crisp” subsets of the sample space. On the other hand, Zadeh [14, 16] developed a mathematical system of fuzzy sets and fuzzy logic, and he introduced what he called “probability measures” on collections of fuzzy sets [15]. He also invented “fuzzy probabilities”, which are potential functions highly related to probabilities [17]. Evidently probabilistic concepts have been discussed in fuzzy logic since its very inception, and this is hardly surprising as the importance of probability is ubiquitous both in theory and in practice. However, there has not been any systematic effort to satisfactorily extend classical probability theory to fuzzy logic. In this paper we take a first step toward establishing axiomatic foundations for several forms of probability in fuzzy logic, and we discuss various issues regarding probabilistic concepts developed in fuzzy logic. The remainder of this paper is organized as follows. In Section 6.2, we review fundamental concepts in probability theory that are essential for this study. In Section 6.3, we examine fuzzy sets whose membership functions are probability functions; we investigate probability mass functions, probability density functions, R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 125–140. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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and distribution functions as membership functions. We describe some unusual properties of the collection of such fuzzy sets. In Section 6.4, we investigate probability measures for fuzzy events. Conditional probability and independence have not been thoroughly developed in fuzzy logic, and we attempt to extend these important concepts to probabilities of fuzzy sets. In Section 6.6, we examine how to axiomatically treat the fuzzy numbers that have been described as “fuzzy probabilities”. We present natural extensions of the classical probability axioms and show that Zadeh’s model of fuzzy probability satisfies them. Other properties of fuzzy probability are also discussed. A preliminary report of this study was presented at First International Seminar on Philosophy and Soft Computing [6] and at the International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU) [7].

6.2

Preliminaries

First we briefly describe the fundamentals of Kolmogorov’s axiomatic probability theory (see, for instance, [1, 3]). Let Ω denote the sample space, which is often described as the “set of all possible outcomes”. We consider a σ -field F (also called a σ -algebra or a Borel field) of subsets of Ω: F is a nonempty collection of subsets of Ω such that it is closed under complementation and countable union. A probability measure P on F is a numerically valued set function with domain F that satisfies the following three axioms: (i) P(E) ≥ 0 ∀E ∈ F . (ii) P(Ω) = 1. (iii) If {Ei } is a countable collection of (pairwise) disjoint sets in F , then P ( i Ei ) = ∑i P(Ei ). Each set in F is called an event and considered measurable, and P(E) represents the probability of E ∈ F . The triple (Ω, F , P) is called a probability space, and (Ω, F ) is described as a measurable space. Axiom (iii) is called “countable additivity”, and it can be difficult to directly check whether a set function satisfies this property. Instead, we consider the following two axioms: / then P(Ei ) → 0. (iii.a) If {Ei } is a sequence in F such that Ei ↓ 0, (iii.b) If {E1 , E2 , . . . , En } is a finite collection of (pairwise) disjoint sets in F , then  P ( ni=1 Ei ) = ∑ni=1 P(Ei ). Axioms (iii.a) and (iii.b) are called the “axiom of continuity” and “finite additivity”, respectively. It can be proved that these two axioms together are equivalent to (iii), so we will check (iii.a) and (iii.b) to determine whether a set function satisfies the axiom of countable additivity. It is important to note that, by induction on n, P is finitely additive if (iii.b) holds for n = 2—if P(A ∪ B) = P(A) + P(B) for disjoint sets A and B in F .

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Conventional probability theory does not deal with fuzzy sets; each set in F is a crisp set, and with the set-theoretic operations of union, intersection, and complementation, F is a boolean algebra with 0/ its minimum element and Ω its maximum element. Thus probability measures are defined on boolean algebras. For our discussions, it is important to note that in boolean algebras, it follows from the three axioms of probability that E1 ⊂ E2 implies P(E1 ) ≤ P(E2 ). In the quantum theory of probability, measurable elements are those in an orthomodular lattice, which is usually that of the closed vector subspaces of an infinite dimensional Hilbert space [2]. Note that in ortholattices, a ⊂ b implies a ∩ b = 0, but the converse does not hold in general; the two are equivalent only in boolean algebras. In quantum probability theory, if a ⊂ b , then P(a + b) = P(a) + P(b). As described in Section 6.1, we would like to extend axiomatic probability theory to fuzzy sets. If the extension is not possible, then it is desirable to establish an analogous axiomatic theory of probability in fuzzy logic. There exist two main schools of thought in the philosophy of probability: the frequentist school and the Bayesian school. Frequentists view probability as the long-run (limiting) frequency of occurrence, whereas Bayesians view it as the degree of belief. Both positions have advantages and disadvantages, but in either case, we must carefully consider the following two linguistic components of fuzzy sets in order to establish a theory of probability in fuzzy logic. One is the semantic component, which pertains to objects or their information necessary in specifying events that are predicated as probable. Thus this component concerns extending the concept of the measurable space (Ω, F ) to fuzzy logic. The other is the syntactic component, which pertains to a context in which probable events are clearly represented and assigned probabilities. For this, we must establish rules that assign probabilities and specify computations with them. Therefore this component concerns extending the probability measure P to fuzzy logic.

6.3

Probability Functions as Membership Functions

In this section we investigate properties of probability functions as membership functions. Let (X, F , P) be a probability space. In fuzzy logic, X is treated as the universe of discourse. Let F denote the distribution function of P. It is well known that given P on F , there is a unique distribution function (and vice versa) and that every distribution function can be written as the convex combination of a discrete, a singular continuous, and an absolutely continuous distribution function (see, for instance, [3]). For simplicity, we assume that F is either absolutely continuous or discrete. When F is absolutely continuous, then there exists a positive function that equals the derivative of F almost everywhere, and this function is called the density of F. If X is countable, we consider the probability mass function derived from P. Let f represent a probability density or mass function on (X, F ). Any f can be used as a membership function of a probabilistic predicate on X and thus be considered a fuzzy set. We will describe a membership function that is also a probability density function as a probability density membership function. If a

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membership function is a probability mass function, we will describe it as a probability mass membership function. What predicates can be properly represented by f ? Certainly it is reasonable to use such a membership function for the predicate “probable”. Suppose that X = {0, 1, . . . , n}, and consider a binomial distribution on Xwith parameters n and α : The probability mass function f is given by f (x) = nx (α )k (1 − α )n−k for each x ∈ X. Figure 6.1 plots the function with n = 10 and α = 1/2. This probability mass membership function of “probable” indicates that 5 is more probable than the other values in X.

Fig. 6.1. Binomial probability mass function with parameters 10 and 1/2.

The two main schools of thought in probability theory described in Section 6.2 suggest two types of probabilistic predicate. From the frequentist perspective, it is justifiable to use a probability density or mass membership function for a predicate that refers to the frequency of occurrence, such as “frequent”. From the Bayesian perspective, it is appropriate to use a probability density or mass membership function for a predicate that refers to the degree of belief, such as “likely”. When we use a probability function to represent a predicate in fuzzy logic, we must ensure that the predicate pertains to a proper probabilistic concept and that the probability function is appropriate as its membership function. The set of probability density or mass membership functions on (X, F ) forms a rather peculiar family of fuzzy sets. First we consider the ordering of such fuzzy sets. Let f1 and f2 be two probability density or mass functions on (X, F ), and suppose that f1 ≤ f2 . We let P1 and P denote the probability measures of f1 and f2 , respectively. Then f1 (x) ≤ f2 (x) for all x ∈ X, so we have P1 (E) ≤ P2 (E) for any E ∈ F . Also we have P1 (E  ) = 1 − P(E) ≤ P2 (E  ) = 1 − P(E), whence P2 (E) ≤ P1 (E). Thus P1 = P2 , and it follows that f1 = f2 . Therefore two fuzzy sets with these membership functions are either coincidental or uncomparable, and it appears that the pointwise ordering of such fuzzy sets is not the “right” way to compare them; these probability membership functions seem to be of a different type compared to other fuzzy sets. There may be cases where it is appropriate to order probability

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density or mass membership functions in a probabilistic manner. For instance, we can compare their means to determine the relation among them. In this case, the set of probability density or mass functions is linearly ordered, since the mean is a scalar. However, we must ensure that the ordering scheme suites the context in which the probabilistic membership functions are used. Compositions of probability density or mass membership functions by the three typical connectives in any algebra of fuzzy sets ([0, 1]X , ·, +, ) (see [5]) also show some odd properties. Since f1 · f2 ≤ f1 and f1 · f2 ≤ f2 , it follows that f1 · f2 = f1 = f2 . We also have f1 + f2 ≥ f1 and f1 + f2 ≥ f2 . Hence f1 + f2 = f1 = f2 = f1 · f2 . In general, the negation of a probability density or mass function is not a probability function. Thus the collection { f | f is a probability density or mass function} ⊂ [0, 1]L is rather peculiar in any algebra of fuzzy sets. It is also possible to use distribution functions as membership functions. Figure 6.2 plots two normal distribution functions, F1 (shown in blue) and F2 (shown in red). F1 is normal with mean 0 and variance 1, whereas F2 is normal with mean −1 and variance 1. We describe a membership function that is also a distribution function as a distribution membership function. Note that for each x ∈ X and for any predicate p represented by a distribution function, “x is p” should pertain to the likelihood of being less than or equal to x. For distribution membership functions, the canonical definition of order among functions can be used to compare them; for two distribution functions G1 and G2 on X , G1 ≤ G2 if G1 (x) ≤ G2 (x) for all x ∈ X . For example, suppose that G1 is a binomial distribution with parameters n (n ∈ ) and α (0 ≤ α ≤ 1) and that G2 is a binomial distribution with parameters n and β (0 ≤ β ≤ 1). If α ≥ β , then it is easy to see that G1 ≤ G2 . Also, several typical forms of the two connectives · and + can be used for distribution membership functions. If we set · and + to min and max, respectively, then for the normal distribution functions F1 and F2 plotted in Figure 6.2, we have F1 · F2 = F1 and F1 + F2 = F2 . We can also set · to prod and + to its dual t-conorm. Figure 6.3 visualizes their outcomes for F1 and F2 . Notice that the outcomes of these operations are also distribution functions.

Æ

6.4

Fuzzy-Crisp Probability: Probability Measures for Fuzzy Events

In this section, we consider probabilities of fuzzy events, which we call “fuzzycrisp probabilities”. We should keep in mind that for no algebra of fuzzy sets ([0, 1]X , ·, +, ) does there exist L ⊆ [0, 1]X such that (i) L = {0, 1}X , and (ii) (L, ·, +, ) is an ortholattice. A fortiori, (L, ·, +, ) is not a boolean algebra or an orthomodular lattice. Therefore, neither conventional probability theory nor quantum probability theory is immediately applicable to such L. However, this does not imply that, for any L ⊆ [0, 1]X such that L = {0, 1}X , there is no function P : L → [0, 1] that satisfies the three axioms of probability. Indeed, Zadeh [15] found such functions and described them as probability measures of fuzzy events. We examine his concept of fuzzy-crisp probability.

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Fig. 6.2. Distribution membership functions F1 (shown in blue) and F2 (shown in red). The distribution of F1 is normal with mean 0 and variance 1. The distribution of F2 is normal with mean −1 and variance 1.

Fig. 6.3. Operations of · = prod (blue) and + = prod ∗ (red) on F1 and F2 .

Let X := {x1 , x2 , . . . , xn } (thus X is finite). For all μ ∈ [0, 1]X , |μ | := ∑ni=1 μ (xi ) is called the crisp cardinal or sigma-count of μ . Notice that for μ ∈ {0, 1}X satisfying μ −1 (1) = A ⊆ X, we have |μ | = Card(A), since μ (xi ) ∈ {0, 1} for each i. Clearly | μ0 | = 0 and |μ1 | = |X| = n. Let L = [0, 1]X be a standard algebra of fuzzy sets with (T, S, N) such that N := 1 − id, T (a, b) + S(a, b) = a + b.

(6.1) (6.2)

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Condition (6.2) is satisfied by Frank’s family, which includes (a) T = min, S = max; (b) T = prod, S = prod ∗ ; and (c) T = W, S = W ∗ . With such an algebra, the mapping P : L → [0, 1] defined by |μ | ∀μ ∈L (6.3) n satisfies the three axioms of probability. Axioms (i) and (ii) are clearly satisfied. As described in Section 6.2, we check (iii.a) and (iii.b) to determine whether (iii) is satisfied. Clearly (iii.a) holds. For (iii.b), we have P(μ ) =

| μ + σ | | μ · σ | | μ | |σ | + = + = P(μ ) + P(σ ). (6.4) n n n n Hence (iii) holds. Also it is easy to verify the identify P(μ  ) = 1 − P(μ ): P(μ + σ ) + P(μ · σ ) =

|μ | |μ  | ∑ni=1 (1 − μ (xi)) n − ∑ni=1 μ (xi ) = = = 1− = 1 − P(μ ). n n n n For μ , σ ∈ L, suppose that μ ≤ σ . Then μ (xi ) ≤ σ (xi ) for 1 ≤ i ≤ n, so ∑ni=1 μ (xi ) ≤ ∑ni=1 σ (xi ). Hence |μ | ≤ |σ |, and we have P(μ ) ≤ P(σ ), as desired. Due to these properties, we find it agreeable to consider P as a probability measure for the algebra ([0, 1]X , T, S, 1 − id). Suppose that X := [a, b] ⊂ (a ≤ b), and let [0, 1]X be endowed with (T, S, N) satisfying (6.1)–(6.2). Consider P(μ  ) =

Ê

L := { μ ∈ [0, 1]X |μ is Riemann-integrable over [a, b]}. Clearly L contains all the Riemann-integrable functions in {0, 1}X . Notice that L is closed under T , S, and N: μ · σ , μ + σ , and 1 − μ are all in L for any μ , σ ∈ L. Define P : L → [0, 1] by P(μ ) :=

1 b−a

 b a

μ (x) dx ∀ μ ∈ L.

(6.5)

If we have a constant fuzzy set, μr (x) = r ∀ x ∈ X , then clearly P(μr ) = r. We check whether P satisfies the three axioms of probability. Obviously (i) holds, and (ii) is also satisfied since P(μ1 ) = b−a b−a = 1. For (iii), we again check (iii.a) and (iii.b). Clearly (iii.a) holds, and it is easy to show that (6.4) holds for P. Hence it can be considered a probability measure. It is also easy to show that P(μ  ) = 1 − P(μ ): 



b b 1 1 b−a (1 − μ (x)) dx = − μ (x) dx = 1 − P(μ ). b−a a b−a b−a a The supposition of equipossibility is evident in (6.3) and (6.5). However, this supposition is unnecessary, and these measures can be generalized by considering

P(μ  ) =

P(μ ) :=



μ (x) d λ (x),

where λ is a probability measure on the σ -field in X (see [15]). Also notice that, in general, μ ≤ σ  does not imply μ · σ = μ0 , so it does not imply P(μ + σ ) = P(μ ) + P(σ ). However, if T = W , S = W ∗ , and N = 1 − id, then μ ≤ σ  implies μ · σ = μ0

132

6.5

6 Axiomatic Investigation of Fuzzy Probabilities

Conditional Probability and Independence for Fuzzy Events

One of the most important concepts in probability theory is conditional probability, and we will examine this concept for fuzzy-crisp probability. Conditioning is a profound concept in measure-theoretic probability, but for simplicity, we will focus on cases where conditioning events have positive measures. Let (L, ·, +, ) be a boolean algebra. We let 0 and 1 denote its minimum and maximum, respectively. Let P denote a probability measure on L. For each a ∈ L − {0}, aL := {a · x | x ∈ L} is a subalgebra with the restriction of · and + and with a complementation operator ∗ defined by (a · x)∗ = a · x . This subalgebra can represent a conditional boolean algebra. The minimum of aL is 0 · a = 0, and its maximum is 1 · a = a. Provided that P(a) = 0, define P∗ : aL → [0, 1] by P∗ (a · x) =

P(a · x) . P(a)

(6.6)

Then P∗ is a probability measure on aL (but not on L), and we use P(x|a) to denote P∗ (a · x). Two events a, b ∈ L are said to be independent if P(b|a) = P(b) [hence P(a · b) = P(a)P(b)]. If L is a boolean algebra with 2n elements resulting from n atoms, then the simCard(a) for all a ∈ L, where Card(a) plest way of defining P on L is by P(a) = n denotes the number of atoms in a. In this case, for all x ∈ aL, we have P(x|a) = Card(a·x) Card(a) . In general, if {a1 , a2 , . . . , an } is the set of n atoms, then we can define a probability measure P by P(ai ) = αi ≥ 0 for 1 ≤ i ≤ n such that ∑ni=1 αi = 1. In the equipossible case, we have αi = 1/n for all i. By the distributive law, we have a · x + a · y = a · (x + y) ∈ aL for all a, b, x ∈ L. Thus this law is important for aL to be a boolean algebra, and P(x|a) := P∗ (a · x) is a probability when L is a boolean algebra. Unfortunately, if L is an orthomodular lattice, then aL is not an orthomodular lattice, so conditional probability cannot be defined in this manner. We investigate whether the concept of conditional probability for crisp sets can be properly extended to fuzzy sets. We will consider (6.3) and (6.5), which satisfy the axioms of probability. First we suppose that X is finite. Thus we let X := {x1 , x2 , . . . , xn } and examine (6.3). Consider a fuzzy set σ ∈ [0, 1]X as a conditioning event. We suppose that P(σ ) = 0. If we apply (6.6) to this case, then we obtain P(μ |σ ) =

P(μ · σ ) 1 n = ∑ T (μ (xi ), σ (xi )). P(σ ) |σ | i=1

(6.7)

In order for this conditional probability to satisfy P(σ |σ ) = 1,

(6.8)

6.5 Conditional Probability and Independence for Fuzzy Events

133

we must have n

|σ | = ∑ T (σ (xi ), σ (xi )). i=1

This equality is satisfied only by T = min: n

n

i=1

i=1

∑ min{σ (xi ), σ (xi )} = ∑ σ (xi ) = |σ |.

Note that the conditional probability defined by Zadeh [15] is problematic because it does not satisfy (6.8). (Zadeh uses prod for T .) If (T, S) is in Frank’s family, then we have S = max for T = min. In this case, we have P(μ + λ |σ ) + P(μ · λ |σ ) = P(μ |σ ) + P(λ |σ ), as desired (with crisp sets, P(· |σ ) satisfies the three axioms of probability and is thus a probability measure). Thus by imposing (6.8), which should be considered rather natural or desirable, we obtain a different definition of conditional probability compared to that of Zadeh [15], who did not provide any rationale for using T = prod instead of T = min in defining it. We show that P(·|σ ) satisfies the three axioms of probability. For any μ ∈ [0, 1]X , we have P(μ |σ ) ≥ 0 since P(σ ) > 0 and P(μ · σ ) ≥ 0. Hence (i) holds. We also have P(μ1 |σ ) =

P(μ1 · σ ) P(σ ) = = 1, P(σ ) P(σ )

so (ii) holds. To show (iii), we check (iii.a) and (iii.b). Regarding (iii.a), consider a sequence of fuzzy sets { μ j } j∈N decreasing to μ0 : For all x ∈ X , μ j (x) ↓ μ0 (x) = 0. Clearly

∑ μ j (x) ↓ ∑ μ0 (x) = 0.

x∈X

x∈X

Therefore P(μ j |σ ) ↓ 0, and (iii.a) is satisfied. For all μ , λ ∈ [0, 1]X such that μ ·λ = μ0 , it is P(μ + λ |σ ) = P(μ |σ ) + P(λ |σ ), because P(μ0 |σ ) = 0. Hence (iii.b) holds. Therefore, the three axioms are satisfied, and P(·|σ ) is a probability measure. Next we analyze the independence of two fuzzy events. We have 1 n ∑ min{μ (xi ), σ (xi )}, n i=1    1 n 1 n P(μ )P(σ ) = ∑ μ (xi ) n ∑ σ (xi ) . n i=1 i=1 P(μ · σ ) =

Thus, for T = min, two fuzzy events are said to be independent if    1 n 1 n 1 n ∑ min{μ (xi ), σ (xi )} = n ∑ μ (xi ) n ∑ σ (xi ) . n i=1 i=1 i=1

(6.9)

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This definition of independence is different from Zadeh’s [15], because we have chosen T = min so that the conditional probability (6.7) satisfies (6.8). Notice that (6.9) holds for two crisp independent events and that it is also valid when one of the sets is μ1 . (Recall that in classical probability theory, Ω is independent of any event.) Example 1. For X = {1, 2, 3, 4}, consider the following two fuzzy sets:

μ = .75/1 + .75/2 + .25/3 + .25/4. σ = .25/1 + .25/2 + .75/3 + .75/4. By (6.9), these two sets are independent since 1 (min{0.75, 0.25} + min{0.75, 0.25} + min{0.75, 0.25} + min{0.75, 0.25}) 4



0.75 + 0.75 + 0.25 + 0.25 0.25 + 0.25 + 0.75 + 0.75 1 = = . 4 4 4 

Ê

We analyze the case that X = [a, b] ⊂ . For σ ∈ [0, 1]X such that P(σ ) = 0, we use the measure P defined at (6.5) and consider P(μ · σ ) P(μ |σ ) = = P(σ )

b a

T (μ (x), σ (x)) dx . (b − a)P(σ )

(6.10)

In order for (6.10) to satisfy P(σ |σ ) = 1, we again set T to min. In this case, two fuzzy events are said to be independent if



 b  b  b 1 1 1 min{ μ (x), σ (x)} dx = μ (x) dx σ (x) dx . b−a a b−a a b−a a (6.11) Example 2. For X = [0, 1], consider the following two fuzzy sets:

φ (x) = x. ρ (x) = 1 − x. Figure 6.4 plots the two sets. By (6.11), these two sets are independent since  1

 1

 1 1 1 1 1 min{φ (x), ρ (x)} dx = φ (x)dx ρ (x) dx = . 1 0 1 0 1 0 4  There actually exist many pairs of strictly fuzzy (non-crisp) events that satisfy our definition of independence [(6.9) for the discrete case and (6.11) for the continuous case]. However, we must thoroughly examine whether our definition (or any other definition of fuzzy independence) makes sense theoretically or practically, and we will do so in our future studies.

6.6 Fuzzy-Fuzzy Probability: Fuzzy Numbers as Probabilities

135

Fig. 6.4. Fuzzy sets φ (blue) and ρ (red).

6.6

Fuzzy-Fuzzy Probability: Fuzzy Numbers as Probabilities

In this section we examine how “probabilistic” fuzzy numbers can be axiomatically treated as probabilities. Zadeh [17] introduced the idea of using fuzzy numbers to quantify probabilities that are derived from imprecise, incomplete, or unreliable sources. We will call these fuzzy numbers “fuzzy-fuzzy probabilities”. Jauin and Agogino [9], Dunyak and Wunsch [4], and Halliwell and Shen [8] also presented models of fuzzy-fuzzy probability. We describe Zadeh’s concept of fuzzy-fuzzy probability using a simple example. Suppose that there are 20 balls in an urn and that “about 10” of them are white whereas the other balls are black. Let X = {0, 1, . . ., 20}. We represent the fuzzy set “about 10” on X by the membership function μ plotted in Figure 6.5. Suppose that we select a ball from the urn uniformly at random. What is the probability that the selected ball is white? Zadeh uses a potential function to quantify this probability. Figure 6.6 plots the potential function, call it FP(μ ), for the probability. In this figure, the horizontal axis represents the probability, and the vertical axis represents the possibility. It is intuitive that, given the membership function μ , FP(μ ) peaks at .5; the possibility of probability .5 is maximum, whereas the possibility of probability less than .25 or larger than .75 is minimum. Notice that Figure 6.6 can be obtained from Figure 6.5 by dividing values on the horizontal axis of Figure 6.5 by 20. Zadeh describes FP(μ ) as a fuzzy probability and expresses it as μ /20. Notice that in order for this scheme to make sense, the values in X must be able to represent probabilities when divided by a normalizing constant. In the example, the normalizing constant is 20. Zadeh also invented distribution functions for fuzzy-fuzzy probabilities; see [17]. To our knowledge, no study has extended the concept of fuzzy-fuzzy probability to membership functions defined on uncountable X, but it is possible. We demonstrate this extension with a simple example. Consider a floor whose area is 20m2 and

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Fig. 6.5. Membership function μ for “about 10”.

Fig. 6.6. Fuzzy-fuzzy probability FP(μ ).

a carpet whose area is “about 10m2 ”. Suppose that we spread the carpet on the floor. If we randomly drop a pin on the floor (uniformly at random), what is the probability that the pin hits the carpet? In this case, we set X to the closed interval [0, 20]. As the membership function for “about 10”, we consider the function φ plotted in the top panel of Figure 6.7. Then we obtain the possibility distribution FP(φ ) of the probability by dividing the values on the horizontal axis of the top panel by 20. The bottom panel plots the resulting potential function. Clearly FP(φ ) is analogous to FP(μ ), and we express this fuzzy-fuzzy probability as φ /20. Again, note that this scheme is valid when probabilities can be obtained by dividing each value in X by an appropriate normalizing constant. It can be said that the probability that the pin hits the carpet is the fuzzy number “about .5”.

6.6 Fuzzy-Fuzzy Probability: Fuzzy Numbers as Probabilities

137

Fig. 6.7. Extension of fuzzy-fuzzy probability to uncountable X. The top panel plots the membership function φ for “about 10m2 ”, and the bottom panel plots the fuzzy-fuzzy probability FP(φ ).

Can fuzzy-fuzzy probability be considered a form of probability? The answer is clearly no in classical probability theory, where probabilities must be real numbers between 0 and 1 by definition. For this reason, it is probably inappropriate to describe FP as a probability measure, which in classical probability theory is defined as a real-valued set function (see Section 6.2). Therefore, it seems appropriate to distinguish FP from classical probability measures and to call it, for example, “Zadeh’s measure”. In order to treat fuzzy-fuzzy probability axiomatically, we must extend the classical axioms of probability described in Section 6.2 to include “fuzzy-valued” probabilities. Let L := [0, 1]X and L := [0, 1]X/c , where c denotes the normalizing con We consider two algebras stant for FP. Then for each μ ∈ L, we have FP(μ ) ∈ L.  of fuzzy sets: (L, ·, +, ) for the domain of FP and (L, , ⊕,∗ ) for the range of FP with the unit element 1 := μ1 /c and the neutral element 0 := μ0 /c. The following are natural extensions of the classical probabiity axioms in this case: (a) FP(μ ) ≥ 0 ∀ μ ∈ L. (b) FP(μ1 ) = 1. (c) If {μi | μi ∈ L} is a countable set so that μi · μ j = μ0 for i = j, then FP(μ1 + μ2 + μ3 + · · ·) = FP(μ1 ) ⊕ FP(μ2 ) ⊕ FP(μ3 ) ⊕ · · ·. It is informative to carefully compare (a)–(c) with the axioms (i)–(iii) in Section 6.2. If we set , ⊕, and ∗ to ·, +, and  , respectively, then notice that FP clearly satisfies (a)–(c):

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(a) FP(μ ) = μ /c ≥ 0 ∀ μ ∈ L. (b) FP(μ1 ) = μ1 /c = 1. (c) If { μi | μi ∈ L} is a countable set so that μi · μ j = μ0 for i = j, then FP(μ1 + μ2 + μ3 + · · · ) = (μ1 + μ2 + μ3 + · · · )/c = μ1 /c + μ2 /c + μ3/c + · · · = FP(μ1 ) + FP(μ2 ) + FP(μ3 ) + · · · . Note that FP also satisfies the following desirable property: (6.6.1) For μ , σ ∈ L, if μ ≤ σ , then FP(μ ) ≤ FP(σ ). It is also desirable to define operations  and  so that the following properties are satisfied: (6.6.2) FP(μ  ) = 1  FP(μ ). (6.6.3) For any σ ∈ L satisfying FP(σ ) = 0, we have a conditional probability FP(μ |σ ) := FP(μ · σ )  FP(σ ) with which the independence of μ and σ can be defined by FP(μ |σ ) = FP(μ ) or by FP(μ · σ ) = FP(μ )  FP(σ ). Unfortunately, the four existing models of fuzzy-fuzzy probability proposed by [4], [8], [9], and [17] fail to satisfy some of the axioms (a)–(b) and of the properties (6.6.1)–(6.6.3). For instance, using an axiom, Halliwel and Shen [8] ensure only finite subadditivity: If μ · σ = μ0 , then FP(μ + σ ) ≤ FP(μ ) ⊕ FP(σ ). The property (6.6.2) does not always hold for Zadeh’s model [17]. However, consider (μ ⊕ σ )(t) := sup min{ μ (x), σ (y)}. t=x+y

In words, we add fuzzy numbers according to the extension principle in fuzzy arithmetic. Then Zadeh’s model achieves the identity FP(μ + σ ) ⊕ FP(μ · σ ) = FP(μ ) ⊕ FP(σ ). To date, no studies have thoroughly examined the concept of conditioning or independence for fuzzy-fuzzy probability. It is important to carefully establish the operations , ⊕, , and . They may not be universal (different cases may require different sets of these operations), and we should be able to provide justification for using a particular set of them. The effectiveness of these operations will also depend on what type of fuzzy sets are used in deriving fuzzy-fuzzy probabilities. Therefore, to fully establish a mathematically rigorous foundation for fuzzy-fuzzy probabilities, we must properly design various components of fuzzy systems.

6.7

Discussion

We have rigorously examined various forms of probability and probabilistic concepts in fuzzy logic and discussed several issues associated with them. To axiomatically treat fuzzy-fuzzy probability, we have extended the classical axioms of probability. Kolmogorov’s axiomatic, measure-theoretic approach revolutionized

6.7 Discussion

139

probability theory and led to numerous important results, such as those on conditional expectation, sequences of random variables, infinite-dimensional distributions, and random functions—these form the foundation of the modern theory of random processes, which are essential in various fields such as control theory and ergodic theory (see, for instance, [13]). We believe that the axiomatic approach must be extended to fuzzy probabilities in order to fully establish their theoretical foundation and to obtain rigorous theoretical results, which will have significant practical implications. In our future research, we intend to properly extend fundamental principles or properties of classical probability theory to fuzzy-fuzzy probability. For instance, the concepts of conditioning and independence in fuzzy-fuzzy probability must be examined. It is also conceivable to consider random variables whose distributions are characterized in terms of fuzzy-fuzzy probability. It will be interesting to investigate how the four modes of convergence (almost sure convergence, convergence in L p , convergence in probability, and convergence in distribution) in classical probability theory extend to such random variables. Regarding fuzzy-crisp probability, we should keep in mind that there are other measures of uncertainty for fuzzy events. For instance, Sugeno’s λ -measures [12] hold for the algebra ([0, 1]X ,W,W ∗ , 1−id), although they were originally introduced in order to deal with boolean algebras. The value of λ ∈ (−1, ∞) determines the additivity of the measures; they can be sub-additive, super-additive, or additive, and they may not have any of these properties. They have been useful in certain applications (see [10]). Many theoretical issues remain to be addressed in order to develop a rigorous axiomatic theory of probability in fuzzy logic. We hope that our paper can serve to stimulate axiomatic studies of fuzzy probability theory.

Acknowledgments This work has been supported by the Foundation for the Advancement of Soft Computing (ECSC, Asturias, Spain) and by the Spanish Department of Science and Innovation (MICINN) under project TIN2008-06890-C02-01.

References [1] Billingsley, P.: Probability and Measure, 3rd edn. Wiley Interscience, New York (1995) [2] Bodiou, G.: Théorie Dialectique des Probabilités (englobant leurs calculs classique et quantique). Gauthier-Villars, Paris (1965) [3] Chung, K.L.: A Course in Probability Theory, 3rd edn. Academic Press, London (2001) [4] Dunyak, J.P., Wunsch, D.: Fuzzy probability for system reliability. In: Proceedings of the 37th IEEE Conference on Decision & Control, vol. 3, pp. 2934–2935 (1998) [5] Alsina, C., Trillas, E., Pradera, A.: On a class of fuzzy set theories. In: Proceedings of FUZZ-IEEE, pp. 1–5 (2007)

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[6] Moraga, C., Trillas, E., Guadarrama, S.: Some doubts on coupling the words ‘fuzzy’ and ‘probability’. Talk given at the First International Seminar on Philosophy and Soft Computing (2009) [7] Nakama, T., Trillas, E., García-Honrado, I.: Fuzzy probabilities: Tentative discussions on the mathematical concepts. In: Proceedings of the International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (2010) [8] Halliwell, J., Shen, Q.: Linguistic probabilities: Theory and applications. Soft Computing 13, 169–183 (2009) [9] Jauin, P., Agogino, A.M.: Stochastic sensitive analysis using fuzzy inference diagrams. In: Schachter, M. (ed.) Uncertainty in Artificial Intelligence, pp. 79–92. North-Holland, Amsterdam (1990) [10] Nguyen, H.T., Prasad, N.R. (eds.): Fuzzy Modeling and Control: Selected Works of M. Sugeno. CRC Press, Boca Raton (2000) [11] Sheldon, R.: A First Course in Probability, 6th edn. Prentice Hall, Upper Saddle River (2002) [12] Sugeno, M.: Fuzzy measures and fuzzy integrals: A survey. In: Fuzzy Automata and Decision Process, pp. 89–102. North-Holland, Amsterdam (1977) [13] Vitanyi, P.M.B.: Andrei Nikolaevich Kolmogorov. CWI Quarterly 1, 3–18 (1988) [14] Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965) [15] Zadeh, L.A.: Probability measures of fuzzy events. Journal of Mathematical Analysis and Applications 23(2), 421–427 (1968) [16] Zadeh, L.A.: Outline of a new approach to the analysis of complex systems and decision processes. IEEE Transactions on Systems, Man and Cybernetics SMC-3, 28–44 (1973) [17] Zadeh, L.A.: Fuzzy probabilities. Information Processing and Management 20(3), 363– 372 (1984)

7 Fuzzy Deontics Kazem Sadegh-Zadeh

7.1

Introduction

As normative systems, morality and law consist of rules, or norms, of particular structure. Both moral and legal norms are reconstructible as deontic sentences and are thus amenable to deontic logic. The inquiry into the syntactic, semantic, logical, and philosophical problems of deontic norms and normative systems has come to be known as deontic logic and philosophy, or deontics for short. In the present paper, the ordinary deontics is extended to a gradualistic, fuzzy deontics that considerably increases the practical relevance of deontic analyses in normative systems and their applications. To this end, a concept of deontic rule is introduced in Section 7.2. Based on this concept, in Section 7.3 a novel, third type of sets, termed deontic sets, is added to ordinary and fuzzy sets. The fuzzification of these new sets in Section 7.4 straightforwardly yields fuzzy deontic sets which may be used as a point of departure for inquiries into fuzzy deontics. We shall use the standard connectives and quantifiers of classical predicate logic (¬, ∨, ∧, →, ↔, ∀, ∃). Lower case Greek letters α , β , γ , and δ will serve as statement variables. Sets will be represented by Roman capitals A, B, C, etc., and their members by Roman lower case letters a, b, ..., x, y, z. The complement of a set A is written Ac . Fuzzy set membership functions are symbolized by μ or ω .

7.2

Deontic Rules

In this section, the notion of a deontic rule is introduced that will help us construct deontic sets. These sets will be fuzzified to yield fuzzy deontic sets as a basis of fuzzy deontics. Descriptive ethics informs us about the moral IS, i.e., about the actual morality in a particular community in the present or past. Its results are reports, and for that matter, not morally binding. To know that some people believe that something is morally good or bad, may be interesting. But you need not share their beliefs and need not behave as they do. It is only the moral rules promulgated by the normative ethics of a given community that are morally binding on members of that community. A normative-ethical system, briefly referred to as a normative system, comprising moral rules is concerned with the moral OUGHT, which tells us how to live by suggesting a more or less attractive conception of a good moral agent. R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 141–156. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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However, in this case it does not suffice to assert “what is morally good”, e.g., honesty is good. Normative-ethical items are usually put forward in terms of such value assertions. It is overlooked that a value assertion of this type has the syntax of a constative like “the Eiffel Tower is high”. Therefore, it sounds rather like a descriptive report, or like empirical knowledge, whose subject has a particular property, for example, goodness, which is identified as a ’fact’. This pseudo-constative character of normative-ethical sentences is the root of the philosophical-metaethical debate about moral realism and anti-realism ([2], [5], [10]). In order for a normative-ethical item to be discernible as something mandatory, it must place constraints on the pursuit of our own interests by prescribing what we ought to do, what we must not do, and what we may do. That is, it must suggest a rule of conduct that regulates human behavior. Examples are the following explicit rules or norms: 1. Everybody ought to tell the truth; 2. Everybody is forbidden to commit murder; 3. Everybody is permitted to drink water. These rules contain the three deontic operators ought to, forbidden, and permitted, which we shall represent by the following standard, deontic-logical operators: It is obligatory that . . . symbolized by: OB it is forbidden that . . . FO it is permitted that . . . PE respectively. If α is any sentence in a first-order language, e.g., “Mrs. Dorothy McNeil tells the truth”, then each of the strings OBα , FOα , and PE α is referred to as a deontic sentence because it contains a deontic-logical operator. Thus, the three sentences above say: 1a. For everybody x, it is obligatory that x tells the truth; 2a. For everybody x, it is forbidden that x commits murder; 3a. For everybody x, it is permitted that x drinks water. Rewritten in a formalized fashion, that means: 1b. ∀xOB(T x) 2b. ∀xFO(Mx) 3b. ∀xPE(W x) where OB, FO, and PE are the above-mentioned three deontic operators; and T x, Mx, and W x represent the action sentences “x tells the truth”, “x commits murder”, and “x drinks water”, respectively. Moral rules are obviously representable by deontic sentences. This is why we call them deontic rules. And since they constrain human behavior, they are also referred to as deontic norms, or norms for short. An ethic as a system of morality in fact consists of such deontic norms, and has therefore been called a normative system above. Examples are the Christian ethic and the moral codes of health care systems in Western countries.

7.2 Deontic Rules

143

Deontic operators also enable us to represent juridical laws, i.e., legal norms such as “theft is forbidden”, “practicing physicians must report new cases of tuberculosis to public health authorities”, or “as of age 6, children have to be enrolled in school”, as deontic rules. That is: a. For everybody x, it is forbidden that x steals; b. For everybody x, if x is a practicing physician, then it is obligatory that x reports new cases of tuberculosis to public health authorities; c. For everybody x, if x is a child of age 6, then it is obligatory that x is enrolled in school. All three examples are actually German laws. Obviously, legal norms too are deontic sentences and are thus amenable to deontic logic. We may therefore treat moral as well as legal rules as deontic norms to render morality as well as law subjects of deontics. It is worth noting that the three deontic operators above are interdefinable. Specifically, PE is definable by FO, and FO is definable by OB thus: Definition 7.2.1. 1. FOα ↔ OB¬α . 2. PE α ↔ ¬FOα . This definition implies that PE α ↔ ¬OB¬α . On this account, the obligation operator OB may serve as the basic and only deontic operator to formulate both types of deontic rules, moral and legal ones. As was pointed out above, a deontic norm is not a constative sentence like “Mrs. Dorothy McNeil has gallstone colic”. Thus, it is not a statement and does not assert something true or false. It is a prescription, a command, and as such, without empirical content. The above examples demonstrate that no syntactic, linguistic, or logical dividing line can be drawn between the moral and the legal because both types of rules are of deontic character. Moreover, legal-normative and moral-normative systems are not disjoint. They have non-empty intersections, e.g., rules of conduct such as “murder is forbidden”. Since the realization of any such rule usually depends on some factual circumstances, it is both philosophically and logically important to note that many, perhaps all, deontic rules are conditional norms because they are conditionals. Let α , β , γ , and δ be any sentences in a first-order language, a conditional norm is a sentence of the form: • α → OBβ called a: conditional obligation, conditional permission, • α → PE γ • α → FOδ conditional prohibition,

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which we call deontic conditionals. A commitment is a conditional obligation, i.e., an obligation such as OBβ , on the condition that something specified, α , is the case: α → OBβ . Examples are the rules (b-c) above. Rule (b) is a deontic conditional with the following syntax: ∀x(Px → OB(Qx))

(7.1)

and says: For all x, if x is P, then it is obligatory that x is Q. This is a simple conditional obligation. Written in a generalized form, conditional obligations, prohibitions, and permissions are of the following structure: • Q (α → OBβ ) • Q (α → FOβ ) • Q (α → PE β ) such that Q is the prefixed quantifier complex of the sentence, e.g., ∀x in (7.1) above, or ∀x∃y∀z, or something else; and α as well as β are sentences of arbitrary complexity. An example is the following conditional obligation: If a terminally ill patient has an incurable disease, is comatose, is dying, and has a living will that says she rejects life-sustaining treatment, then physicians and other caregivers ought not to sustain her life by medical treatment. That is: Q (α → OB (¬β1 ∧ ¬β2 )) where:

(7.2)

α ≡ α1 ∧ α2 ∧ α3 ∧ α4 ∧ α5 ∧ α6 ∧ α7

and:

α1 ≡ x is a terminally ill patient, α2 ≡ x has an incurable disease, α3 ≡ x is comatose, α4 ≡ x is dying, α5 ≡ there exists a living will of x which says that x rejects lifesustaining treatment, α6 ≡ y is a physician, α7 ≡ z is a caregiver other than y, β1 ≡ y sustains x’s life by medical treatment, β2 ≡ z sustain x’s life by medical treatment, Q ≡ ∀x∀y∀z. Using these notations, we may take a look at the micro-structure of sentence (7.2) to understand why we prefer abbreviations of the form (7.1):

7.2 Deontic Rules

∀x∀y∀z

145

(x is a terminally ill patient ∧ x has an incurable disease ∧ x is comatose ∧ x is dying ∧ there exists a living will of x which says that x rejects life-sustaining treatment ∧ y is a physician ∧ z is a caregiver other than y → OB(¬y sustains x’s life by medical treatment ∧ ¬z sustain x’s life by medical treatment)).

That is: ∀x∀y∀z (α1 ∧ α2 ∧ α3 ∧ α4 ∧ α5 ∧ α6 ∧ α7 → OB (¬β1 ∧ ¬β2 )) Note, first, that the ought-not component in the consequent of (7.2) is formalized as OB(¬β1 ∧ ¬β2 ) and not as ¬OB(β1 ∧ β2 ). The latter formulation would mean that it is not obligatory that β1 ∧ β2 , whereas the rule says “it is obligatory that ¬β1 ∧ ¬β2 ”. Note, second, that according to deontic-logical theorems, the sentence OB(¬β1 ∧ ¬β2 ) is equivalent to OB¬β1 ∧ OB¬β2 such that rule (7.2) above and the following rule (7.3) are equivalent: Q (α → OB¬β1 ∧ OB¬β2 )

(7.3)

Further, “ought not to γ ” is the same as “it is obligatory that not γ ”, i.e., OB¬γ . This is, by Definition 7.2.1 above, equivalent to “it is forbidden that γ ”, written FOγ . Thus, the consequent of our last formulation (7.3) says that FOβ1 ∧ FOβ2 , i.e., “it is forbidden that a physician sustains the patient’s life by medical treatment and it is forbidden that other caregivers sustain her life by medical treatment”. We eventually obtain the sentence: Q (α → FOβ1 ∧ FOβ2 )

(7.4)

as an equivalent of (7.2). We see that because of equivalence between OB¬α and FOα , it makes no difference whether a norm is formulated negatively as a negative obligation such as (7.3), e.g., “one ought to do no harm to sentient creatures”; or positively as a prohibition such as (7.4), e.g., “it is forbidden to do harm to sentient creatures”. The discussion in the literature on the distinction between moral rules, moral norms, moral principles, moral ideals, moral commitments, virtues, and values is often misleading because all of these entities are in fact one and the same thing, i.e., deontic rules as explicated above. For instance, the well-known four medical-ethical principles of respect for autonomy, non-maleficence, beneficence, and justice are deontic rules. To exemplify, the principle of autonomy says, in effect, “one ought to respect the autonomy of the patient”. Likewise, the principle of non-maleficence says “one ought not to inflict evil or harm”; and so on. Thus, Beauchamp and Childress’s principles of biomedical ethics are deontic rules of obligation (see also [1], p. 114 ff). We must be aware, however, that not every deontic sentence is a deontic rule. An example is the deontic sentence “it is obligatory that the sky is blue”. Although it is a syntactically correct deontic sentence, semantically it is not meaningful. Only

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human actions can be obligatory, forbidden, or permitted. Whatever is outside the sphere of human action, cannot be the subject of deontics. With the above considerations in mind, we will now introduce the concept of an ought-to-do action rule, which we shall use below. To this end, we will first recursively define what we understand by the term “action sentence”. A sentence of the form P(x1 , . . . , xn ) with an n-place predicate P will be called an action sentence if P denotes a human action such as “tells the truth” or “interviews”. Thus, sentences such as “Dr. Smith interviews Dorothy McNeil” and “Dorothy McNeil tells the truth” are action sentences, whereas “the sky is blue” is none. Our aim is to base the notion of a deontic rule on action sentences so as to prevent vacuous obligations such as “it is obligatory that the sky is blue”. Definition 7.2.2 (Action sentence) 1. If P is an n-ary action predicate and x1 ,. . . , xn are individual variables, then P(x1 , . . . , xn ) is an action sentence; 2. If α is an action sentence, then ¬α is an action sentence referred to as the omission of the action; 3. If α and β are action sentences, then α ∧ β and α ∨ β are action sentences; 4. If α is any sentence and β is an action sentence, then α → β is an action sentence, referred to as a conditional action sentence. For example, “Dorothy McNeil tells the truth” is an action sentence. “Dorothy McNeil is ill” is none. And “if Dr. Smith interviews Dorothy McNeil, then she tells the truth” is a conditional action sentence. We will now introduce the notion of a deontic rule in two steps. Definition 7.2.3 (Deontic action sentence) 1. If α is an action sentence and ∇ is a deontic operator, then ∇α is a deontic action sentence; 2. If α and β are deontic action sentences, then ¬α , α ∧ β , and α ∨ β are deontic action sentences; 3. If α → β is an action sentence, then α → ∇β is a deontic action sentence referred to as a deontic conditional; Definition 7.2.4 (Deontic rule) 1. If α is a deontic action sentence with n ≥ 1 free individual variables x1 , . . . , xn , then its universal closure ∀x1 . . . xn α is a deontic rule. 2. A deontic rule of the form ∀x1 . . . xn α → ∇β is called a conditional obligation if ∇ ≡ OB; a conditional prohibition if ∇ ≡ FO; and a conditional permission if ∇ ≡ PE. A deontic rule is also called a deontic norm. For example, according to Definition 7.2.3.1, the sentence “it is obligatory that x tells the truth” is a deontic action sentence because its core, “x tells the truth”, is an action sentence. And according to Definition 7.2.4.1, its universal closure is a deontic rule, i.e.: 1. For everybody x, it is obligatory that x tells the truth.

7.3 Deontic Sets

147

Additional examples are: 2. For everybody x, it is forbidden that x steals; 3. For everybody x, if x is a child of age 6, then it is obligatory that x is enrolled in school. These three rules may be formalized as follows: • ∀xOB(T x) • ∀xFO(Sx) • ∀x(Cx → OB(ESx)). The first two examples are unconditional deontic rules. The third example represents a conditional deontic rule, specifically a conditional obligation. Thus, an unconditional deontic rule is a sentence of the form: Q∇β and a conditional deontic rule is a sentence of the form: Q (α → ∇β ) , where Q is a universal quantifier prefix ∀x1 . . . xn ; ∇ is one of the three deontic operators obligatory, forbidden, or permitted; and β is an action sentence. We know from Definition 7.2.1 that the three deontic operators are interdefinable. Specifically, PE is definable by FO, and FO is definable by OB: FOα ≡ OB¬α PE α ≡ ¬FOα

and thus PE α ≡ ¬OB¬α

The obligation operator OB will therefore serve as our basic and only deontic operator. That means that all deontic rules prescribe ought-to-do actions. They may be viewed as ought-to-do action rules, ought-to-do rules, or action rules for short. These terms will be used interchangeably. According to the terminology above, an ought-to-do rule may be unconditional or conditional. An unconditional ought-to-do rule is an unconditional obligation such as “everybody ought to tell the truth”. It does not require any preconditions. A conditional ought-to-do rule, however, is a conditional obligation and has a precondition. An example is the conditional obligation “a child of age 6 ought to be enrolled in school”, that is, “for everybody x, if x is a child of age 6, then it is obligatory that x is enrolled in school”.

7.3

Deontic Sets

By obeying or violating deontic rules, human beings structure their society. The structuring consists in the partitioning of the society into different categories of individuals according to whether they obey or violate a particular rule, e.g., ‘the set

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of criminals’ and ‘the set of non-criminals’; ‘the set of honest people’ and ‘the set of dishonest people’; and others, which we call deontic sets because they are induced by deontic rules of the society. Deontic sets are not natural kinds, but social constructs, for there are no natural deontic rules that could be obeyed or violated. For example, ‘the set of criminals’ is not a natural category like apple trees or the set of those with blue eyes. It comes into existence by classifying the social conduct of people in relation to deontic rules which forbid some criminal conduct, and because deontic rules are society-made. It may therefore be viewed as a socially constructed class, set, or category. To elucidate, we will reformulate deontic rules extensionally. To this end, recall that deontic rules say what one ought to do, is forbidden to do, and so on. As a simple example, consider the following sentence: For everybody x, it is obligatory that x tells the truth.

(7.5)

This means, extensionally, “For everybody x, it is obligatory that x belongs to the set of those people who tell the truth”. If the set of those people who tell the truth is denoted by “T”: T = {y | y tells the truth}, then the deontic sentence (7.5) may also be represented in the language of set theory in the following way: For everybody x, it is obligatory that x belongs to set T . That is, or more formally:

∀x (it is obligatory that x ∈ T ) ∀x (OB(x ∈ T ))

where the extensionally written sentence “x ∈ T ” translates the intensionally expressed action sentence “x tells the truth”. Based upon the considerations above, the notion of a deontic set may be introduced as follows [7].1 Definition 7.3.1 (Deontic subset). Let Ω be a universe of discourse, e.g., the set of human beings or a particular community. And let A be a subset of Ω . This set A is said to be a deontic subset of Ω iff one of the following deontic norms exists in Ω : • ∀x ∈ Ω (OB(x ∈ A)) • ∀x ∈ Ω (OB(x ∈ Ac )) where Ac is the complement of set A. 1

We could take all three standard deontic modalities (obligation, prohibition, and permission) into account to introduce three corresponding notions of a deontic set, i.e., the set of those who do obligatory actions, the set of those who do forbidden actions, and the set of those who do permitted actions. However, for the sake of convenience we confine ourselves to the modality of obligation only. The reason is that the three modalities are interdefinable and we chose OB as our basic operator.

7.3 Deontic Sets

149

For instance, let Ω be the set of human beings and consider the following subsets of Ω : P L H M G

the set of those people who keep their promises the set of liars, the set of helpful people, the set of murderers, the set of gardeners.

According to Definition 7.3.1 the fifth set, G, is not a deontic subset of Ω because there is no deontic rule in the human community requiring that one ought to become a gardener or that forbids one from becoming a gardener. However, the first four sets are deontic subsets of Ω because the following four deontic norms are elements of common morality and thus exist in the human community: You ought to keep your promises, you ought not to lie, you ought to be helpful, you ought not to commit murder, that is: ∀x ∈ Ω ∀x ∈ Ω ∀x ∈ Ω ∀x ∈ Ω

(x ought to belong to P), (x ought to belong to Lc ), (x ought to belong to H), (x ought to belong to M c ),

or more formally: ∀x ∈ Ω ∀x ∈ Ω ∀x ∈ Ω ∀x ∈ Ω

(OB(x ∈ P)) (OB(x ∈ Lc )) (OB(x ∈ H)) (OB(x ∈ M c )).

Definition 7.3.2 (Deontic set). A set A is a deontic set iff there is a base set Ω of which A is a deontic subset. For instance, the sets T , P, L, H, and M mentioned above are deontic sets, whereas set G is none. From these considerations we can conclude that a deontic norm in a society defines, creates, or induces a deontic set (class, category). Those people who satisfy the norm are members of that deontic set, whereas those people who violate the norm stand outside of the set, i.e., are members of its complement. For instance, the deontic set of helpful people includes all those who satisfy the norm you ought to be helpful, whereas the deontic set of murderers includes all those who violate the norm you ought not to commit murder. Although deontic sets really exist in the world out there (see, e.g., the inmates of prisons), their existence is not independent of the norms of the social institution

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creating them. For this reason, we consider them as deontic-social constructs, or deontic constructs for short. This idea was used by the author to demonstrate that what is called ‘disease’ in medicine is a deontic construct and not a natural phenomenon ([8], [9]).

7.4

Fuzzy Deontics

In Section 7.2, we concerned ourselves with ordinary, two-valued, crisp deontics only and considered a deontic rule an all-or-nothing norm. So conceived, a deontic rule categorically declares an action either as obligatory or not obligatory, forbidden or not forbidden, permitted or not permitted. Taking into account the imperfection of classical logic [9], we have good reason to question the adequacy of a deontic logic built upon this traditional dichotomy and bivalence. We shall address that question in the following two sections. There are also other non-classical approaches to deontic logic, for example ([3], [4]). 7.4.1

Quantitative and Comparative Deonticity

To motivate our task, consider the following case report. The question was posed in a recent medical-ethical publication whether truth is a supreme value ([6], p. 325). It had been prompted by the conduct of a doctor, who at the request of a Muslim patient, had attested that she was not pregnant, even though she was. The physician’s aim at hiding and reversing the truth had been to prevent the divorced, pregnant woman from being killed by her relatives to maintain “the honor of the family”. The doctor’s preference for saving the patient’s life over telling the truth had later been supported by a medical ethicist who had confirmed that “truth is not the supreme value. [. . . ] the potential saving of life is more important and takes precedence to the truth” ([6], p. 325). The author of the article, however, had moral problems with this assessment. According to our concept of a deontic norm introduced in Definition 7.2.4, both telling the truth and saving life are required by the deontic norms of common morality and medical ethics: 1. For everybody x, it is obligatory that x tells the truth,

(7.6)

2. For everybody x, if x is a doctor, it is obligatory that x saves the life of her patients. The quotation above demonstrates that there are situations in which a deontic norm, such as the second one in (7.6), is given precedence over another deontic norm like the first one. Analogous examples are clinical settings where a diagnostic or therapeutic action A is to be preferred to another one, B, although both actions are declared as obligatory: “You ought to do A and you ought to do B”. Observations of this type give rise to the question of how the legitimacy of norm precedence and preference may be conceptualized. We shall suggest a comparative notion of

7.4 Fuzzy Deontics

151

obligation, “it is more obligatory to do A than to do B”, which we shall base upon a fuzzy concept of obligation. By fuzzifying the concept of obligation, we shall pave the way for fuzzy deontics, in which norms may be ranked according to the degree of obligatoriness of what they prescribe. The approach may be instrumental in human decision-making in areas such as medicine and medical ethics, bioethics and ethics in general, law, politics, and others. Our first step is to introduce the notion of a fuzzy deontic set by generalizing the notion of a deontic set, introduced in the previous section. To this end, recall that a deontic sentence of the form (7.6) above, e.g., “one ought to tell the truth”, is representable in the following way: ∀x(OB(x ∈ A))

(7.7)

where A = {y | y tells the truth} is the set of those people who tell the truth. We will conceive the phrase “ought to belong to”, i.e., the predicate “OB ∈” in (7.7), as a compound deontic predicate and will introduce below its characteristic function that will be written ω (x, A) to read “the extent to which x ought to belong to set A”. Let us abbreviate the binary function symbol ω to the pseudo-unary function symbol ωA with the following syntax:

ωA (x) = r

i.e., “the extent to which x ought to belong to set A is r”.

In other words: “The degree of deontic membership of x in set A is r”; or “the obligatory A-membership degree of x is r”; or “the degree of obligatoriness of Amembership of x is r”. On the basis of our terminology, we can define this crisp deontic membership function ωA as follows:

Definition 7.4.1

ωA (x) =

1, 0,

iff OB(x ∈ A) iff OB(x ∈ Ac )

Consider, for example, the following deontic norms from Section 7.3 above: • • • •

One ought to keep one’s promises, one ought not to lie, one ought to be helpful, one ought not to commit murder,

and these sets, which were used in the same context: P L H M

≡ the set of those people who keep their promises, ≡ the set of liars, ≡ the set of helpful people, ≡ the set of murderers.

(7.8)

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Then the sentences (7.8) may be rewritten as follows:

ωP (x) = 1 i.e., to the extent 1 x ought to keep her promises; ωL (x) = 0 to the extent 0 x ought to lie; ωH (x) = 1 to the extent 1 x ought to be helpful; ωM (x) = 0 to the extent 0 x ought to commit murder. For instance, like everybody else Mrs. Dorothy McNeil ought to keep her promises, i.e., ωP (Dorothy McNeil) = 1. However, she ought not to lie, and thus ωL (Dorothy McNeil) = 0. In all of our examples so far, the deontic membership function ω has taken values in the bivalent set {0, 1}. It is a bivalent function and partitions by the following mapping:

ω : Ω → {0, 1} the base set Ω of human beings into two crisp subsets, e.g., L and Lc , liars and non-liars. A crisp deontic set of this type with its all-or-nothing characteristic either includes an individual totally or excludes her totally. Someone is either a liar or a non-liar; a Samaritan or none; a murderer or none; and so on. There is no gradualness in deontic behavior, i.e., no degrees of deonticity. In the real world, however, we often have difficulties in determining whether someone does or does not definitely belong to a particular deontic class such as, for example, the class of those who tell the truth, keep their promises, are helpful, are guilty of murder, etc. Like a vague, non-deontic class such as that of diabetics or schizophrenics, the class of honest people, the class of murderers, and other deontic classes are also vague and lack sharp boundaries. It is to our advantage, then, to fuzzify the notion of a deontic set, which we may do by fuzzifying the deontic membership function above. We thereby obtain the notion of a fuzzy deontic set in two steps as follows: Definition 7.4.2 (Fuzzy deontic subset). Let Ω be a universe of discourse, e.g., the set of human beings or a particular community. A is a fuzzy deontic subset of Ω iff there is a deontic membership function ωA such that: 1. ωA : Ω → [0, 1], 2. A = {(x, ωA (x)) |x ∈ Ω }, 3. ωA (x) = ωA (y) for all x, y ∈ Ω . For instance, if Ω is a family consisting of the members {Amy, Beth, Carla, Dorothy}, then the following set is a fuzzy deontic subset of this family: HONEST = {(Amy, 0.8), (Beth, 0.8), (Carla, 0.8), (Dorothy, 0.8)}. We have, for example, ωHONEST (Beth) = 0.8. To the extent 0.8 Beth ought to tell the truth. Note that all members have the same degree of membership in the set (Clause 3). Definition 7.4.3 (Fuzzy deontic set). A is a fuzzy deontic set iff there is a base set Ω of which A is a fuzzy deontic subset.

7.4 Fuzzy Deontics

153

Using fuzzy deontic sentences on the basis of fuzzy deontic sets, it becomes possible to compare the deontic strength of different norms, for example, of telling the truth and saving the life of a patient. Let T be the set of those people who tell the truth, and let S be the set of those who save other people’s lives. Then the following sentence says that for Dr. You it is more obligatory to save other people’s lives than to tell the truth:

ωS (Dr. You) > ωT (Dr. You).

(7.9)

No doubt, this lowbrow example is not acceptable at first glance and per se. It only serves to show the way comparative deontic norms may be formulated. Surprisingly, however, the unconditional comparative norm (7.9) appears to be a meaningful constituent part of a comparative conditional norm. Consider, for instance, this comparative conditional norm: If the life of a patient is endangered, as was the case with the Muslim woman at the beginning of this section, then it is more obligatory that her doctor saves her life than tells the truth. That is: The life of the patient is endangered → ωS (Dr. You) > ωT (Dr. You). To keep it readable, we avoided completely formalizing this example. Nevertheless, it demonstrates how the moral dilemma that was quoted at the beginning of the present section, may be resolved by fuzzy deontics. Our considerations provide the nucleus of a conceptual framework that could contribute to a fuzzy deontics in ethics, to a fuzzy ethics so to speak. It also enables a method by which (i) to interpret and reconstruct situations concerning the superiority of one legally protected interest over another by introducing a rank order of norms that are relevant in a given circumstance, and (ii) to introduce a comparative relation of performance order for diagnostic-therapeutic actions in clinical medicine [7], [9]. 7.4.2

Qualitative Deonticity

As was emphasized above, the notion of condition, situation, or circumstance plays a central role in human decision-making in that the making of a particular decision depends on a given circumstance, for example, a patient’s specific health state. Accordingly, we reconstructed deontic rules as deontic conditionals, for instance, (i) if the life of the patient is endangered, then saving her life is more obligatory than telling the truth; (ii) if a patient has disease X, then you ought to do Y ; or (iii) if x is a child of age 6, then it is obligatory that x be enrolled in school. The if -component of such a rule is the circumstance under which its consequent is obligatory, forbidden, or permitted. We know, however, that most circumstances are vague states of affairs and admit of degrees to the effect that there is a gradualness between their presence and absence. For instance, a patient’s life may be endangered slightly, moderately, or severely. Similarly, the pneumonia of a patient may be slight, moderate, or severe. The lower the degree of existence of such a state of affairs X, the higher that of its complement not-X, and vice versa. X and not-X co-exist to particular extents. This brings with it that if under the circumstance X an action Y is obligatory (forbidden,

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or permitted) to a particular extent r, then it is not obligatory (not forbidden, not permitted) to the extent 1 − r, respectively. Thus, an action may be obligatory (forbidden, or permitted) and not obligatory (not forbidden, not permitted) at the same time, respectively. We shall capture this deontic pecularity below. To this end, we will introduce the terms “fuzzy conditional” and “fuzzy rule”. A fuzzy conditional is simply a conditional α → β that contains fuzzy terms in its atecedent α , or consequent β , or both. An example is the fuzzy conditional “if the patient has bronchitis, then she has a fever and cough”. It contains the vague terms “bronchitis”, “fever”, and “cough”. Definition 7.4.4. A deontic rule, as defined in Definition 7.2.4, is said to be a fuzzy deontic rule iff it is a fuzzy conditional. We will sketch a concept of qualitative deonticity with hopes of stimulating further discussion and research on this subject. Our sketch will be limited to a qualitative concept of obligatoriness only. The other two deontic modalities may be treated analogously. For our purposes, let OBL be a linguistic variable that ranges over actions, and let Y be an action (for the theory of linguistic variables, see [11], [12], [13]). The expression: OBL(Y ) = B reads “the obligatoriness of Y is B”, or “Y is obligatory to the extent B”. For example, “the obligatoriness of antibiotic treatment is weak”, or “antibiotic treatment is weakly obligatory”. The term set of the linguistic variable OBL may be conceived as something like: T (OBL) = {very weak, weak, moderate, strong, very strong, extremely strong}.

Let X be a linguistic variable that ranges over circumstances and takes values auch as A, A1 , A2 , etc. For example, X may be a patient’s disease state bacterial pneumonia that takes values such as slight, moderate, and severe: T (bacterial pneumonia) = {slight, moderate, severe}, e.g., Mrs. Dorothy McNeil’s bacterial pneumonia is slight. That is, Mrs. Dorothy McNeil has slight bacterial pneumonia. And let Y be an action as above. A fuzzy conditional of the following form is obviously a fuzzy deontic rule: If X is A, then OBL(Y ) = B where A ∈ T (X) and B ∈ T (OBL). Some examples are: • If the patient has slight bacterial pneumonia, then antibiotic therapy is moderately obligatory, • If the patient has moderate bacterial pneumonia, then antibiotic therapy is strongly obligatory,

7.5 Conclusion

155

• If the patient has severe bacterial pneumonia, then antibiotic therapy is very strongly obligatory. We have thus three related fuzzy deontic rules that regulate the treatment of bacterial pneumonia. A closer look reveals that they constitute a small algorithm that in terms of the theory of fuzzy control enable the deontic control of the variable bacterial pneumonia. The general structure of such fuzzy deontic algorithms may be represented as follows: X11 is A11 , . . . and X1k is A1k → OBL(Y11 ) is B11 , . . . and OBL(Y1p ) is B1p . . . Xm1 is Am1 , . . . and Xmn is Amn → OBL(Ym1 ) is Bm1 , . . . and OBL(Ymq ) is Bmq

with k, m, n, p, q ≥ 1. On the basis of the considerations above, ethical and legal issues become amenable to fuzzy logic. We have delieberately used medical examples above to demonstrate that even clinical decision-making may be treated as a subject of fuzzy deontics. To this end we have shown elsewhere that indication and contra-indication, as central concepts of clinical decision-making, are fuzzy deontic concepts [9].

7.5

Conclusion

A concept of fuzzy deontics was proposed that makes moral and legal normative systems amenable to fuzzy logic. To this end, fuzzy deontic sets are introduced that enable us to fuzzify deontic norms and to conduct numerical and comparative deontic analyses and judgments. Using linguistic variables, a sketch was given of fuzzy-qualitative deontic rules that may aid fuzzy deontic decision-making in ethics, law, and medicine.

References [1] Beauchamp, T.L., Childress, J.F.: Principles of Biomedical Ethics, 5th edn. Oxford University Press, Oxford (2001) [2] Cuneo, T.: The Normative Web: An Argument for Moral Realism. Oxford University Press, Oxford (2010) [3] da Costa, N.C.A.: New systems of predicate deontic logic. The Journal of Non-Classical Logic 5(2), 75–80 (1988) [4] Grana, N.: Logica deontica paraconsistente. Liguori Editore, Naples (1990) [5] Kramer Matthew, H.: Moral Realism as a Moral Doctrine. John Wiley & Sons Ltd., Chichester (2009) [6] Peleg, R.: Is truth a supreme value? Journal of Medical Ethics 34, 325–326 (2008) [7] Kazem, S.-Z.: Fuzzy Deontik: 1. Der Grundgedanke. Bioethica 1, 4–22 (2002)

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[8] Sadegh-Zadeh, K.: The prototype resemblance theory of disease. The Journal of Medicine and Philosophy 33, 106–139 (2008) [9] Sadegh-Zadeh, K.: Handbook of Analytic Philosophy of Medicine. Springer, Dordrecht (2011) [10] Shafer-Landau, R.: Moral Realism: A Defence. Oxford University Press, Oxford (2005) [11] Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning, I. Information Sciences 8, 199–251 (1975) [12] Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning, II. Information Sciences 8, 301–357 (1975) [13] Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning, III. Information Sciences 9, 43–80 (1976)

8 Soft Deontic Logic Txetxu Ausín and Lorenzo Peña

8.1

Deontic Logic

Deontic logic or logic of norms is a kind of special logics, so-called ’extended logics’.1 This branch of logics tries to analyze the formal relations established among obligations, permissions, and prohibitions. To this end, the language of logic is supplemented with a new vocabulary that consists of three operators which refer to the above deontic qualifications: ’o’ (obligation, duty), ’a’ (allowed, permission) and ’v’ (interdiction, forbidden). As Sánchez-Mazas says ([21], p. 25), the term ’Deontic Logic’ is intended to cover today, in a general way, all studies on the peculiar logical structure of systems of norms of any kind or, if you will, on the sets of values, laws, and deduction rules that govern those systems. For example, if it is an obligation to vote for any political party, then is it allowed? If I have the right to move across the European Union, are prohibited to impede me? Is there a duty to provide the means to such places? If I am required to hold choice to do this or that, how will fulfil that duty, joining the first, or the second, or both requirements? If I have the obligation to attend the institutional acts of my university and I also have the duty not to lose any of my graduate lessons, am I subject to the joint obligation of both actions? Are the causal consequences of an allowed action lawful too? Etc. Many issues of this kind arise when we think about the framework of deontic qualifications or regulations that appear in all human relationships. In sum, the terms that contain such qualifications are norms (moral, legal) whose structure and inferential relations analyzes deontic logic. Thus, we can say that deontic logic is the theory of valid inference rules, that is, the analysis of the conditions and rules in which reasoning including qualifications of prohibition, duty, or permission, is correct. We assume, therefore, that there are structural relations among expressions that include qualifications as required, prohibited, permissible, right, duty, etc. That is, there is a principle of inference between the norms so that, from a structured set of norms, it is possible to establish deductive inferences (logical consequences). In this way, we treat norms as entities like propositions, which can be negated and 1

The term ’deontic logic’ is generalized from von Wright’s work (1951)[25] in order to refer to the study of inferential relations among norms. The word ’deontik’ has been previously used by Mally (1926)[16] related to his ’logic of will’ and by Broad (1950)[6], speaking about ’deontic propositions’. Bentham also used the term ’deontology’ referring to ethics. The source of that expression is greek word ’tò deón’, translated as ’duty’ or ’obligation’.

R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 157–172. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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combined using logical connectives and quantifiers, clearly taking the view of the true possibility of the logic of norms.

8.2

Theoretical Assumptions of Standard Deontic Logic

There are three main assumptions in standard deontic logic that determine the usual logical analysis of norms: the modal conception of deontic logic (ideality), the gap between facts and norms (non-cognitivism/separatism), and the bivalence. We are going to analyze each of them. 8.2.1

The Modal Paradigm in Deontic Logic

All standard approaches consider deontic logic as a transcription of alethic modal logic, that is, the logic of necessity and possibility. Thus, deontic qualifications of obligation, prohibition and permission would be but the equivalent of alethic necessity, impossibility and possibility. This parallelism has come since the first studies about the logical inference of norms in the late scholastics and clearly in Leibniz: All the complications, transpositions and oppositions of the [Alethic] ’Modes’ shown by Aristotle and others in their treaties about logics, can be transcribed to our ’Modes of Law’. ([15], p. 84). Leibniz defined the permission (lawful) as what a good man is possible to do; and the obligation (duty) as what a good man is necessary to do ([15], p. 83). Such identification is inherited uncritically by modern deontic logic systems that arise simultaneously in the fifties of XX century ([14], [25], [4], [8] and [7]). This isomorphism is not an arbitrary or trivial question: It seems understood that when we speak of obligations, duties, we are referring to what is “necessary”, morally or legally speaking. Therefore, obligatoriness is interpreted as a kind of necessity, specifically, a normative necessity or requirement. Expressed in terms of Kripkean semantics for modal logics, an action or behaviour is mandatory if it occurs in all alternative worlds relative to ours, i.e., what is compulsory is true in every possible normative world. Therefore, the duty is defined as the normative required, as a kind of necessity: For ’op’ to be true in a world ’m’, we should check if ’p’ occurs in all the alternatives worlds of ’m’; that is, something is obligatory if it occurs in all the normative worlds relating to that used as reference -usually, the actual. This means that a duty has to be fulfilled in any normative world, context or situation -which is an idealized and maximalist view of the normative realm. Furthermore, it appears very odd to try to say that not all obligations (norms) obtaining in the actual world obtain in its deontic alternatives. ([11], p. 71). In this context, it is unquestionably the so-called ’Rule of Closure’, whereby the logical consequences of duties are also mandatory. It means that obligatoriness is closed under the logical conditional: (RC)p ⊃ q/op ⊃ oq.

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Its importance lies in expressing an important normative principle, namely, that one is committed to the logical consequences of her obligations. This [RC] seems right an indeed useful in moral philosophy, for by means of this axiom we may persuade moral agents that they are committed to the logical consequences of their moral principles. ([22], p. 151). 8.2.2

The Gap between Facts and Norms

One of the most controversial issues about arguments involving deontic matters (ethical, legal) is whether the statements of duty or right can be inferred from statements of fact, and conversely ([3]). Most philosophers, at least since Hume, have opted to give a negative answer to this question, arguing that obligations or permissions are not concerned with facts and that no combination of facts can imply a duty or a right, and no combination of duties or rights implies a fact. It is called the radical separation thesis between facts and norms (separatism), such that deontic expressions are a special kind of assertion whose content has nothing to do with the factual assertions. According to this thesis, factual assertions refer to states of affairs that exist or not in this world while deontic assertions refer to a particular type of entity -a right, an obligation-, whose existence or obtaining would be independent of the existence or obtaining of facts or states of affairs. This position is linked to so-called non-cognitive ethics (emotivism, precriptivism): ’good’ is not a property of X, ’X is good’ is not a descriptive term; the deontic qualifications do not bring new “knowledge” about X. This approach to deontic logic is a consequence of the above mentioned isomorphism that exists between alethic modal logic and standard deontic logic in terms of possible-world semantics. We have said that, from this perspective, A is obligatory if and only if every normative possible world contains A (correspondingly, B is allowed if and only if at least one normative possible world contains B). But there would be no inferential link between the content of those normative possible worlds (ideal) and the content of the real or actual world or of any bunch of designated worlds. Consequently, a duty exists whether or not the facts and conditions are so and so. What is obligatory and what is licit would not change with any change in the facts of the world ([3], p. 45). 8.2.3

Bivalence

The standard deontic analysis is, as we said, a sort of supplementary logic, extended to the realm of norms, by introducing deontic operators which relate to the qualifications of permission, obligation and prohibition ([9]). However, it is not an alternative or divergent system to classical logic, to the extent that it does not question any of axioms or theorems of classical logic. Specifically, as we are concerned, standard deontic logic assumes bivalent conception of classical logic inherited from Parmenides and Aristotle, according to that may be envisaged only two truth values: (entirely) falsehood and (entirely) truth. So, it is impossible that something be and

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not be at the same time; that is, stated as the basic principle the so-called ’principle of non-contradiction’: it is not true that A and notA : ∼ (A & ∼ A). The principle of non-contradiction has been, from Greek antiquity to the present, the most obvious of all the principles of logic. Medieval logicians bestowed supreme importance because they were convinced that reason could not accept contradictions; hence, a contradictory theory was invalid ipso facto. The contradiction would have such disastrous effects, when given, reason loses control over herself and, in particular on deductive processes, so anything could happen. This possibility was strongly expressed by the maxim Ex Contradictoriis Quodlibet ([20], p. 649, [5], p. 319 ff.). Thus, a system or theory that accepts as valid conflicting claims would not in any way differentiate between true and false. Therefore, such a system would be trivial as long as from a contradiction we can deduce anything. This is called the principle of Pseudo-Scotus (Cornubia) (ex contradictione sequitur quodlibet), which may be formulated in different ways: A & ∼ A ⊃ B; A ⊃ (∼ A ⊃ B); ∼ A ⊃ (A ⊃ B). Beyond logical, ontological and epistemological considerations, this principle of bivalence has favoured a dichotomous view of reality and a similar tendency in any discussion, debate or argumentation. That is, it has involved an approach to normative reasoning in terms of “all-or-nothing”, so that the normative definitions and qualifications have been treated in sharp and abrupt terms. Remember that ’dichotomy’ means to divide into two, separately or exclusively, as wholly possessing or not a particular property.

8.3

Shortcomings of Standard Deontic Logic

There are also three main challenges for a deontic logic that aspires to cope with normative reasoning in a significant way: the case of conditional and ensuant obligations (paradox of lesser evil), the gradualism of reality (factual and normative), and the normative conflicts. 8.3.1

Conditional and Ensuant Obligations

Many factual situations give rise to prescriptions which would not arise at all unless those factual situations existed in first place. This is the case in conditional norms: allow, ban or compel to the extent that certain facts exist [13]. Thus, many duties and permissions are contingent on facts. This kind of conditional norms are really abundant in normative systems and demands a proper rule of deontic detachment: How can we take seriously a conditional obligation if it cannot, by way of detachment, lead to an unconditional one? ([23], p. 263). But, standard deontic logic does not give currency to deontic detachment when the condition or factual premise is fulfilled: It is not possible to infer  oq from  p and  o(p ⊃ q) . – Here we adopt a

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representation of conditional obligation as a whole norm, in toto ([26]), by means of a deontic operator with a wide scope: It is mandatory that, if p, then q.2 Moreover, relating to conditional obligations, we detect an important issue: the ensuant obligations or duties of lesser evil; namely, duties that arise as a result of antecedent factual situation, one wherein another duty has been breached. They are “reparational” or “compensatory” duties in the sense that they arise when one has failed in fact a prior obligation. Thus, for instance, resorting to war is forbidden by current international law but, in case such a prohibition is transgressed, new obligations arise as regards how to conduct the war, in accordance with international human conventions such as the Red Cross agreements. This has lead us to define the paradox of lesser evil as the pivot around which all deontic paradoxes hinge:3 A general principle of morals and law lays down that, if we act wrongly, at least, we have to act so as to implement the lesser evil. However, to the extent that the lesser evil is realized, evil is indeed done; but then, by means of the inference rule of logical closure RC, evil -without conditions- must be done. 8.3.2

Gradualism

Virtually any term, whether properties, definitions, qualifications, or states of affairs, involved in the normative reasoning -as in everyday life and in most sciences – is likely to have fuzzy edges and as regards borderline cases, without precise lines of demarcation. From this perspective, a particular action can have a more or less degree of licitness and, to the extent that it is not completely licit, it will have some degree of illicitness. A property or a set is fuzzy in the sense that there are degrees of possession thereof or of belonging to such a set – so that the relationship of a member belonging to a group varies in different degrees, in the same way as a property may possess it or lack it in various amounts. This gradualism of reality (normative and factual) appears itself in two types of language expressions: Adverbs of intensity or decay: pretty, little, a lot, entirely, somewhat,. . . ; and, comparative constructions: more or less, so and so, likeness, similar, close, etc. – for example, used in analogical reasoning, a crucial resource in legal realm. Thus, there is a profound disagreement between a continuous and gradual reality, riddled with nuances and transitions, a reality in gray, and a logic (an analysis and description of it) bivalent, between sheer truth and complete falsehood, between utter duty and pure ban, in “all-or-nothing” terms, black or white (principle of bivalence, above mentioned). Consequently, similar behaviours and situations cannot 2

3

Representing conditional obligation as a hybrid formulation (an implication with a factual antecedent and a deontic consequent) is unsatisfactory. Representing as a whole formulation  o(p ⊃ q) allows a way to fulfil the norm by completely refrain from  p (the condition) – which is not accounted for the narrow-scope rendering of conditional obligation. Chisholm’s paradox (contrary-to-duty imperatives paradox), Good Samaritan paradox, knower paradox, gentle murder paradox, bi-conditional paradox, praise paradox, second best plan paradox, penitent paradox, and Ross’ paradox ([1], p. 59 ff).

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receive a similar normative (ethical and legal) treatment, transgressing the elementary principles of fairness and proportionality. 8.3.3

Normative Conflicts

As a consequence of gradualism of reality (something or someone is so and so, to some extent, and, at the same time, is not) we find in normative reasoning true conflicts or collisions but the normative ordering is not a trivial system (it does not follow everything from a contradiction). Moreover, apart from the gradualism of many terms and properties (coercion, fraud, damage, equality, . . . ), legal normative conflicts are unavoidable since they arise from a variety of reasons: the multiplicity of sources of law; the own dynamics of legal systems (derogations, amendments); the indeterminacy – syntactic and semantic – of legal language; the legal protection of mutually conflict interests in social complex contexts (rules and exceptions); and the presence of gaps. There are three criteria used in legal doctrine to avoid antinomies, but, in fact, they are not sufficient to avoid all kind of legal normative conflicts.

8.4

Elements for a Soft Deontic Logic

In view of that challenges and shortages of standard deontic logic, we espouse soft deontic logic with the following five main characteristics: abandonment of modal paradigm, paraconsistency, fuzzy approach, non-relevantism, and quantification. 8.4.1

Abandonment of Modal Paradigm

We reject, as Hansson (1988)[10] and Weinberger (1991)[24], the idealized viewpoint of normative realm that presupposes alethic modal paradigm in deontic logic and, thus, the rule of closure (both in the core of deontic paradoxes). Obligatoriness has nothing to do with realization in ideal or optimal worlds. First, a state of affairs may be primitively obligatory while being quite undesirable. Second, many obligations exist only because the world is in fact thus or so. Many an obligation arises only when some factual circumstances are met. This is the main reason why all ideal-world approaches to deontic logic are doomed. Then, an alternative is necessary to cope with the commitments of our rights, duties and prohibitions. A right – a licit course of action – is such that its owner may not be compelled or constrained to give it up. Rights imply an obligation for everybody else to respect those rights, and so a duty not to disturb the right’s owners’ enjoyment thereof. So, what forcibly prevents the exercise of a right is forbidden (principle of ensuant obligation). Here the key issue is causal, and no merely inferential, consequences that prevent the fulfilment of a right. What makes a right an entitlement for somebody to do some action is nothing else but: you are not entitled to force anyone not to enjoy one of her rights.

8.4 Elements for a Soft Deontic Logic

8.4.2

163

Paraconsistency

Contradictions, normative collisions and conflicts, do not trivialize normative orderings. Thus, we refuse the principle of Pseudo-Scotus which, if admitted, will become Law and normative sets something absurd and useless since, from a conflict, would collapse all normative distinctions. 8.4.3

Fuzzy Approach

Deontic descriptions of licitness, prohibition and duty should be treated as gradual notions. From this perspective, a particular action can have a more or less degree of licitness and, to the extent that it is not completely licit, it will have some degree of illicitness. The core of this type of analysis is the principle of graduation, according to which, when two facts are similar, their deontic treatment must also be similar.4 As an alternative to the ’principle of bivalence’ that permeates the standard approach to deontic logic, we maintain the ’principle of graduation’, which says that facts and deontic qualifications are a matter of degree and therefore a fuzzy-logic approach is an appropriate theoretical method in deontic logic. Suggesting the principle of graduation in relation to norms leads to the rejection of the idea that matters involving rights, duties and prohibitions, i.e., moral or juridical matters, are “all-or-nothing” questions. Consequently, graduation leaves room for flexibility and adaptability when dealing with particular and contingent circumstances, and it gives an important role to jurisprudence -understood as reasonableness in the normative domain. From a juridical point of view, this gradualist approach brings in an ingredient of malleability and flexibility which rehabilitates, in some ways, a certain spirit closer to the Anglo-Saxon common law. In addition to that, this approach also opposes the eagerness for legislative fixation which characterizes a codifier strand, in the line of Justinian’s, with the perverse effects it carries. So, the fuzzy approach to deontic logic entitles us to soften the sharp dichotomies usually stated in normative reasoning. The main consequence of the fuzzy approach to deontics is that it allows us to cope with thousands of dilemmas and conflicts that arise in a way less wrenching, traumatic, and arbitrary than the “all-or-nothing” approach. 8.4.4

Non-relevantism

Our approach recognizes deontic postulates that follow from any set of normative rules, by means of the axiological basis of normative systems. That is, soft deontic logic is not purely formal but, as any kind of deontic logic, involves assumptions about the theory of duty, the normative cognitivism, etc. ([12], pp. 393-410). 4

This idea is clearly related to the Leibnizian principle of transition or continuity: lex iustitiae [2].

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Quantification

Most deontic systems developed have omitted the quantifiers in their formulations. In addition, the few times that have been introduced, it was in a mechanical way, tracing the relevant modal principles without elaborate on their normative significance or not. In our view, it is absolutely essential the use of quantifiers in deontic logic. The introduction into the language of deontic logic of expressions like ’all’, ’some’, ’none’, etc. will provide a inestimable clarification of the content and meaning of rights, duties and prohibitions, and the relationships and commitments among them, especially when involving the existential quantifier as in the case of positive rights [19].

8.5

The Underlying Quantificational Calculus

Our present treatment is based on transitive logic, a fuzzy-paraconsistent nonconservative extension of relevant logic E, which has been developed in the literature [18]. Transitive logic is a logic of truth-coming-in-degrees, and of truth being mixed up with falseness (the truer a proposition, the less false it is, and conversely). Thus it is mainly conceived of as a logic applicable to comparisons involving ’more’, ’less’ and ’as as’. The implicational functor, ’→’, is here construed as a functor of alethic comparison: p → q is read as “To the extent [at least] that p, q”. Since our main idea is that an action can be both (to some extent) licit and (up to a point) also illicit, and that one out of two actions, both licit (both illicit), can be more licit (less licit) than the other, our logic’s intended use is to implement valid inferences involving ’more’, ’less’ and ’as as’ in deontic and juridical contexts. Transitive logic (system P10) is built up by strenthening the logic of entailment E. (Our notational conventions are à la Church: no hierarchy among connectives; a dot stands for a left parenthesis with its mate as far to the right as possible; remaining ambiguities are dispelled by associating leftwards.) Transitive logic introduces a distinction between strong (¬) and simple (∼) negation. Simple contradictions are nothing to be afraid of, whereas contradictions involving strong negation are completely to be rejected. Alternatively, transitive logic has a primitive functor of strong assertion, ’H’, such that ¬p abbreviates H ∼ p (“Hp” can be read as “It is completely (entirely, wholly, totally) the case that p”). Within transitive logic disjunctive syllogism holds for strong negation; thus we define a conditional ’⊃’, such that p ⊃ q abbreviates ¬p ∨ q. Notice that conjunction & is such that p&q is as true as q, provided p is not utterly false; we read p&q as ’It being the case that p, q’. ’∧’ is simple conjunction, ’and’ (p&q is defined as ¬¬p ∧ q). We also define p \ q as p → q ∧ ¬(q → p) (“p \ q" means that it is less true that p than that q). Our reading of ’→’ is as follows: p → q is read: “To the extent [at least] that p, q”. We add a further definition: let p abbreviate ∼ (p →∼ p) – p meaning that p is true enough. (p could be read ’It is sufficiently (or amply or abundantly, or the like) true that p’, or something

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like that.) Last let ’α ’ be a primitive sentential constant meaning the conjunction of all truths. Here is an axiomatization of system P10. Primitives: ∧, → , ∼, H, α . Definitions of ’\ ’, ’¬’, ’’ are as above. p ∨ q abbreviates ∼ (∼ p∧ ∼ q. 8.5.1

Axioms

P10.01 α P10.03 α \ ∼ α P10.05 H p → q ∨ . ∼ p → r P10.07 p → q → .q → r → .p → r P10.09 p → q → . ∼ Hq → ¬H p P10.11 ∼ p → q → . ∼ q → p P10.13 p ∧ q → p

P10.02 (p → p) → .p → p P10.04 α → p ∨ .p → q P10.06 p → q → r ∧ (q → p → r) → r P10.08 p → q → .p → r → .p → .q ∧ r P10.10 p → (q∧ ∼ q) →∼ p P10.12 p →∼∼ p P10.14 p ∧ q → .q ∧ p

Sole primitive Rule of Inference: DMP (i.e. disjunctive modus ponens): for n ≥ 1: p1 → q ∨ (p2 → q) ∨ . . . ∨ pn → q, p1 , . . . , pn  q MP [Modus Ponens] for implication ’→’ is a particular case of the rule – the one wherein n = 1. Adjunction is a derived inference rule. This axiomatic basis seems to us reasonably clear, elegant and functional. It may however contain some redundance. Modus ponens for the mere conditional, ’⊃’, is also a derived inference rule: from p ⊃ q and p to infer q. The fragment of P10 containing only functors ∧, ∨, ⊃, ¬ is exactly classical logic (both as regards theorems and also as regards rules of inference). That’s why P10 is a conservative extension of classical logic. 8.5.2

Quantification

Our quantificational extension of system P10 is obtained by adding further axiomatic schemata plus three inference rules. We introduce universal quantifier as primitive and define ∃xp as ∼ ∀x ∼ p. By r [(x)] we mean a formula r with no free occurrence of variable ’x’. p [(x/z)] expresses a formula resulting from substituting “z” to “x”. Additional axiomatic schemata: ∃x(p ∧ ∀xq) ↔ ∀x(∃xp ∧ q) ∀x(p ∧ q) → .∀xp ∧ q ∀x(s \ r [(x)]) ⊃ ∃x(s \ r) ∀xp ∧ ∃xq → ∃x(p ∧ q) H∃xp → ∃xH p

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Quantificational Inference Rules

rinfq01: universal generalization rinfq02: Free variables change rule rinfq03: Alphabetic Variation

8.6

DV System of Soft Deontic Logic

We introduce as primitives these symbols: ap is read as ’It is allowed that p’ or ’It is licit that p’; op abbr. ∼ a ∼ p; ’o’ means obligation; vp abbr. ∼ ap; ’v’ means interdiction (unlawfulness); ’’ means a causal relation between facts, whereas ’’ means a relation of thwarting (obstructing, hindering) – not necessarily by brute force or violence, but in any case by material actions which make it practically impossible or very hard for the thus coerced person to perform the action she is prevented from doing. Hindering is of course a particular case of causation, but while an appeal can prompt someone to refrain from acting in a certain way, the link is not one of coercion. Preventing is causing an omission against the agent’s will. Thus, a state of affairs, that Peter’s house’s door is key-locked by John, blocks another state of affairs, namely for Peter to leave his house. We are aware that it would be nice to have axiomatic treatments of the two relations of causation and prevention, but no interesting set of such axioms has occurred to us. We also take for a common assumption, namely the equivalence between the obligatoriness of A and the illicitness of not-A. 8.6.1 [DI] [RI] [Sa] [RA] [RD] [DD] [DS] [RS] [DE] [RE] [LC] [NH]

8.6.2

Axioms a(p → q) → .op → q o(p → q) → .ap → q o(op → ap) ap ∧ aq → a(p ∧ q) a(p ∨ q) ∧ ¬p → aq o(p ∨ q) ∧ ¬p → oq o(p ∨ q) ∧ p → oq a(p ∧ q) ∧ p → aq oop → op oap → ap p  q&ap → aq p  q&ap → vq

(duty implication) (right implication) (deontic subalternation) (right aggregation) (right disjunction) (duty disjunction) (conditional duty simplification) (conditional right simplification) (duty enforcement) (right enforcement) (licit causation) (non-hindrance)

Inference Rules

[RP] If op is not a theorem, then  a ∼ p (Rule of Permission) [Eq]  p ↔ q ⇒ op ↔ oq (Rule of Equivalence)

8.6 DV System of Soft Deontic Logic

8.6.3

167

Comments on Deontic Postulates

Most of those axioms are to be understood with an implicit clause: provided p and q are separately and jointly contingent – that is to say, each expresses a contingent state of affairs and neither necessarily implies the other or its negation. That restriction applies to [DI], [RI], [RD] and [DD]. [RI] (or, equivalently, o(p → q) → .p → oq)) means that whenever you have a duty to do A-only-to-the-extent B is the case, then you only are entitled to A to the extent B happens. Suppose the consequent is false: you are entitled to A to a higher extent than B is the case. And suppose the antecedent is true: it is mandatory for you to realize A only to the extent B happens to be the case. Then you are put under two mutually incompatible deontic determinations: on the one hand you are allowed to do A to some high degree – higher than the one of fact B being realized; but on the other hand you are obliged to do A in at most as high a degree as that of B’s being realized. Thus supposing you choose to perform A to such a high degree, you are both enjoying your right and yet breaching your duty. In order to avoid such a deontic incongruity, [RI] ought to be accepted as a principle of deontic logic. The case fo [DI] (which is equivalent to a(p → q) → .p → aq) is quite similar – the reasoning being exactly parallel. A usual objection against [RI] and [DI] is that they allow us to draw factual inferences from deontic premises, which seems to be odd or inconvenient. Yet those inferes are legitimate. What duties and what rights exist depends not only on what enactments have been passed by the legislators but also on what factual situations exist. Under certain factual circumstances, A, once a right has been granted with an implicative content, a(A → B), B is automatically licit. If the legislator pronounces A → B licit and A obligatory while B is not realized at all, either of the two rules is null and void. Thus his enacting the obligation to A may be taken to imply repealing the right to A → B. The subalternation principle [Sa] is a deontic version of Bentham’s well-known thesis that whatever is obligatoty is also licit. Our particular version makes it into a deontic norm. It is an implicit rule of any deontic system that the system itself is bound to regard as rightful what it renders mandatory. In other words, any ruleer has to be aware that the system of norms as such contains a rule to the effect that, obligatorily, what is mandatory is also, to that extent, licit. Otherwise the ensuing set of rules would not count as a normative system at all, but would be an incongruous ensemble. [RA] is one of the most significant features of our approach. Standard systems of deontic logic tend to embrace duty-aggregation, which we reject, since for a system to make separately mandatory two courses of action – e.g. one by agent X and another by agent Y – does not necessarily entail that the system makes the (perhaps impossible) conjunction of those two courses of action mandatory. On the other hand, unless right-aggregation, [RA], is embraced no right is unconditional. For a right to A to be unconditional the agent has to be entitled to A whether of not s/he chooses to do B or to refrain from doing B. If the agent is both unconditionally entitled to A and unconditionally entitled to B, he is unconditionally entitled to do

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both, even if doing both is impossible. Being obliged to perform an impossible action is indeed quite irksome, but being entitled to do it is absolutely harmless. In fact once a course of action is commony held to be impossible, any prohibition to perform it tends to be automatically dropped or otherwise voided. Notice that [RA] escapes usual objections, such as that a man can be allowed to marry a woman while also being allowed to marry another woman without thereby being allowed to marry both. In fact his right to marry either is conditional – upon a number of assumptions, among others his refraining from marrying any other woman. This shows that [RA] applies to unconditional rights only and that many rights are implicitly or silently conditional. [DS] is merely a weakened version of a principle of standard deontic logic, namely that whenever a joint action, A-and-B, is obligatory, each conjunct, (A, B) is obligatory, too. As such the principle is wrong, since for you to have an unconditional duty to jointly perform those two actions, A and B, does not imply for you to be unconditional allowed – let alone obliged – to realize A in case you fail to perform B (for whatever reason, whether by your own will or because you are prevented), since perhaps doing B without A may be a worse course of action falling afoul of the normative system’s purposes. Yet our version avoids such a difficulty, since deontic simplification is made conditional on one of the two jointly-obligatory actions being realized. As for [RS] quite similar considerations apply, replacing “duty” by “right”. (In standard deontic logic [RS] is not needed as a primitive axiom, but our system needs both.) [RD], or right disjunction, is also a principle of deontic congruence, which means that, whenever you are entitled to realize either A or B, and in fact B is not the case at all (whether in virtue of your own choice or for whatever reason), the only way you can enjoy your right to A-or-B is by realizing A; hence, under such assumptions, you are unconditionally entitled to A. In deontic or normative contexts, being entitled to A-or-B is usually confined to free choices. Your right to A-or-B entails that you are free to choose either. Hence, if you refrain from A, you are entitled to B. Notice, though, that you may be allowed to A-or-B without being permitted to A-and-B. (For instance, you may either pay a higher tax or else renounce a pay increase, not both.) [DD] is exactly similar to [RD], only replace “right” with “duty”. For A-or-B being mandatory implies that, in case A completely fails to be realized – for whatever reason, whether by your own choice or not – B is mandatory, since, under such assumptions, doing B is the only way for you to abide by your duty to A-or-B. Notice, though, that neither [RD] nor [DD] apply when one of the disjuncts, A, is partly realized. Then even if A is not completely realized and thus it is partly true that ∼ A, the right or the duty to B cannot be derived. This is why we have chosen strong negation, ¬, for those two axioms. Thus our system of axioms fails to countenance your claiming a right to B, in case you are antitled to A-or-B, when you have already chosen B, even if only in a low degree. It is only by wholly refraining from the alternative course that your disjunctive right entitles you to embrace the remaining disjunct.

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[DE], oop → op (or, equivalently, ap → aap) is a principle of (dis)iteration. According to it, when a normative system makes an obligation mandatory, it implicitly makes the content of the obligation also mandatory; or, whenever the system makes an action rightful, it makes the right to perform that action also licit. This means that there are no unlawful rights. Notice, though, that for a legal situation, oA, to exist is not the same as for the law-giver to enact an act making A mandatory. The legislator’s enactment may be a sufficient cause for the obligation to arise (under certain conditions), but the two states of affairs are different. Thus our axiom does not mean that, whenever the legislator is bound to enact a law, the law is already implicitly enacted – which is of course false in a number odd cases. What alone is ruled out is that a situation should be made licit while the right to realize it is forbidden. To forbid the right implies to forbid its content. Likewise [RE], oap → ap – or, equivalently, op → aop – means that the normative system only contains lawful duties. To the extent the system, as such, makes an action mandatory, it makes the duty to perform that action licit. You cannot be illicitly obliged to do A. In other words, to the extent the obligation is unlawful, its content is not mandatory. Or: nothing is forbidden unless it is rightfully forbidden. Of course there can be sets of rules lacking such a principle, but they are not normative systems, but un-congruous amalgams, which cannot discipline society’s set of activities. [LC] p  q&ap → aq means that the causal consequences of licit actions are also licit. Again the legislator is not almighty. He can forbid a course of action, B. But if, at the same time, he allows A and, as a matter of fact, A causes B, that entails that either of his rulings is null – usually we must take the earlier one to be repealed by the later. [NH] p  q&ap → vq means that only to the extent an action is forbidden can be the case that a licit action hinders or prevents it. Suppose again that the legislator allows A while, as a matter of fact, A impedes B. Now suppose that the same legislator also allows B. Either of the two rulings is voided by the other (usually the later one will repeal the earlier). Notice that for an action to be unlawful is not the same as for it to fall afoul of, e.g., the criminal code or any other particular regulation. By proclaiming B legal, while A prevents B, the legislator is making A unlawful, even if A is not considered by the criminal code – a general rule for unlawful damage compensation will apply. Principle [NH] has to be restrictively interpreted. The adequate notion of hindrance is not a purely factual one, but a concept which is partly defined and construed by the normative system. It is not empty or idle, though. The principle does not mean that unlawful hindrances of licit actions are forbidden. What it means is that all hindrances of legal actions are forbidden, yet the apposite concept of hindrance is not wholly determined by nature but partly by law, which sorts out allowed and forbidden ways of interfering with other people’s behaviour. The Rule of Permission is a peculiar inference rule, which makes our system non-recursively axiomatizable, since it entitles us to infer the licitness of any course of action when the negation thereof cannot be proved to be forbidden. This rule is in effect a presumption rule which imposes the burden of proof upon such as

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claim that the considered course of action is forbidden. Unless and until it is proved to run against the law, it has to be assumed to be licit. This is one of the many differences between modal and deontic logics. Deontic logics have a practical job to perform. Practice cannot wait for logicians to carry their inferential work on and on. When legal operators are satisfied that the prohibition cannot be proved, they are automatically entitled to regard its content as in compliance with the law. Finally, the Rule of Equivalence is pretty obvious: any two logically (or, more generally, necessarily) equivalent situations are liable to share their deontic determinations or qualifications. Of course that does not apply to other sorts of loose equivalences, such as practical equivalence or the like; only to strict equivalences which exist in virtue of logic, or at least are metaphysically necessary. Thus A and ∼∼ A, A ∧ A, A ∨ A and so on, since they express equivalent states of affairs, are to be ascribed the same deontic attributes. 8.6.4

Quantificational Axioms

Since existential quantification is similar to infinite disjunction and universal quantification is like an infinite conjunction, axioms for quantifiers are easily guessed: [URA] [UDS] [URS] [UDD]

∀xap → a∀xp o∀xp ∧ ∀x(y = x ⊃ p [x/y]]) → op a∀xp ∧ ∀x(y = x ⊃ p [x/y]]) → ap o∃xp ∧ ∀x(y = x ⊃ p) → op [x/y]]

(universal right aggregation) (universal duty simplification) (universal right simplification) (universal duty disjunction)

Other universal generalizations of our axioms are left as an exercise for the reader. As particular examples of those axioms we shall mention the following ones. As for [URA]: to the extent everybody is entitled to do A it is also lawful for all to do A. (Otherwise, please notice, those separate courses of action are not unconditionally allowed, but only permitted to the extent other people refrain from following the same course of action. (Thus it is no the case that everybody is allowed to visit the museum on Monday.) As for [UDS], to the extent it is obligatory for all children of Rosa to help their mother, to that extent Rosa’s son, Jacob, is bound to help his mother if in fact all his brothers and sisters do help her. As for [UDD] take this example: to the extent it is mandatory for at least one of the co-debtors to pay the borrowed amount and John is one of them and in fact all other co-debts completely fail to pay, John is bound to pay. Of course the same restrictions apply to universal generalisations as do to its sentential-logic counterparts (the facts under consideration have to be contingent and logically independent of one another).

8.7

Conclusion

The DV system allows for degrees of licitness and obligatoriness, and also for degrees of compliance or realization. It countenances partial contradictions. But it puts limits on incongruity, beyond which the purported normative system does not

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count as one. One of the practical purposes this logic serves it to provide us with a criterion to know when a fresh regulation or statute repeals earlier regulations. We have already mentioned several cases. Our system is fact-sensitive. Norms and facts are partly interdependent ? despite Hume’s and Moore’s qualms or strictures. What norms are in operation is not an issue entirely independent of what facts happen in the world. And the other way round. There is a certain solidarity between facts and norms. Facts can abolish norms.

References [1] Ausín, T.: Entre la lógica y el derecho. Paradojas y conflictos normativos. Madrid/México, Plaza y Valdés (2005) [2] Ausín, T.: Weighing and Gradualism in Leibniz as Instruments for the Analysis of Normative Conflicts. Studia Leibnitiana XXXVII/1, 99–111 (2005) [3] Ausín, T., Pe´na, L.: Arguing from facts to duties (and conversely). In: van Eemeren, F.H., et al. (eds.) Proceedings of the Fifth Conference of the International Society for the Study of Argumentation, pp. 45–47. Sic Sat, Amstedam (2003) [4] Becker, O.: Untersuchungen über den Modalkalkül. Anton Hain, Meisenheim am Glan (1952) [5] Bobenrieth, A.: Inconsistencias,¿por qué no? Un estudio filosófico sobre la lógica paraconsistente. Tercer Mundo editores, Colombia (1996) [6] Broad, C.D.: Imperatives, categorical and hypothetical. The Philosopher 2, 62–75 (1950) [7] Castañeda, H.-N.: La lógica general de las normas y la ética, vol. 30, pp. 129–196. Universidad de San Carlos, Guatemala (1954) [8] García Máynez, E.: Los principios de la ontología formal del derecho y su expresión simbólica. México, Imprenta universitaria (1953) [9] Haack, S.: Filosofía de las lógicas. Madrid, Cátedra, (1978, 1982) [10] Hansson, S.O.: Deontic Logic without Misleading Alethic Analogies, Logique et Analyse, pp. 123–124, 337–370 (1988) [11] Hintikka, J.: Some Main Problems of Deontic Logic. In: Hilpinen, R. (ed.) Deontic Logic: Introductory and Systematic Readings, pp. 59–104. D. Reidel, Dordrecht (1971) [12] Innala, H.-P.: On the non-neutrality of deontic logic. Logique et Analyse, 171–172, 393–410 (2000) [13] Jackson, F., Robert Pargetter, R.: Oughts, Options, and Actualism. Philosophical Review 95, 233–255 (1986) [14] Kalinowski, G.: Théorie des propositions normatives. Studia Logica 1, 147–182 (1953) [15] Leibniz, G.W.: Elementa Juris Naturalis. Madrid, Tecnos (1671, 1991) [16] Maly, E.: Grundgesetze des Sollens. Elemente der Logik des Willens. Reidel, Dordrecht (1926, 1971) [17] Mazzarese, T.: Forme di razionalitá delle decisioni giudiziali. In: Torino, G.G, ed. (1996) [18] Peña, L.: A Chain of Fuzzy Strenghtenings of Entailment Logic. In: Barro, S., Sobrino, A. (eds.) III Congreso Español de Tecnologías y Lógica Fuzzy,Santiago de Compostela, Universidad de Santiago, pp. 115–122 (1993) [19] Peña, L., Txetxu (eds.): Los derechos positivos. Las demandas justas de acciones y prestaciones. Madrid/México, Plaza y Valdés/CSIC (2006)

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[20] Miró Quesada, F.: Paraconsistent Logic: Some Philosophical Issues. In: Priest, G., Routley, R., Norman, J. (eds.) Paraconsistent Logic. Essays on the Inconsistent, pp. 627–652. Philosophia Verlag, Munich (1989) [21] Sánchez-Mazas, M.: Cálculo de las normas, Barcelona, Ariel (1973) [22] Schotch, P.K., Jennings, R.E.: Non-kripkean Deontic Logic. In: Hilpinen, R. (ed.) New Studies in Deontic Logic, pp. 149–162. D. Reidel, Dordrecht (1981) [23] Van Eck, J.A.: A System of Temporally Relative Modal and Deontic Predicate Logic and Its Philosophical Applications 1. Logique et Analyse 99, 249–290 (1982) [24] Weinberger, O.: The Logic of Norms Founded on Descriptive Language. Ratio Juris 3, 284–307 (1991) [25] von Wright, G.H.: Deontic Logic. Mind 60, 1–15 (1951) [26] von Wright, G.H.: On conditional obligations. Särtryck ur Juridisk Tidskrift 1, 1–7 (1994, 1995)

Part IV

9 Retrieving Crisp and Imperfect Causal Sentences in Texts: From Single Causal Sentences to Mechanisms Cristina Puente, Alejandro Sobrino, and José Ángel Olivas

9.1

Introduction

Causality is a key notion in science. In empirical sciences, causality is a typical way to generate knowledge and to provide explanations. For example, when a quantum physicist calculates the probability of an atom absorbing a photon, he analyzes this event as the cause of the atom’s jump to an excited energy level; e.g., he tries to establish a cause-effect relationship [6]. Causation is a type of relationship between two entities: cause and effect. The cause provokes an effect, and the effect is followed from the cause. Causality can be viewed as a static process (A caused B) or as a dynamic one (A causes B, B causes C,. . . , E causes F). In this work, we will use causality in both senses, but mainly in the second one. Our approximation will adress a practise-oriented one: we are interested in causes as the means by which they operate; e.g., we will focus on the productive mechanism by means of which a cause generates an effect. So, we embrace the view that ’A causes B’ is the same as grasping the link through which the cause A operates the effect B [19]. Causality and conditionality are strongly related. As the effect is followed from the cause, causal statements are frequently verbalized as B is a consequence of A, B is deduced from A or if A then B. So, saying that Drinking two grams of cyanide caused someone’s death is similar to say that If somebody drinks two grams of cyanide, he will die. But neither every conditional statement is a causal statement nor is every causal statement a conditional one. Thus, a material conditional may not express any causal relation, like in If Marseille is in France, two is an even number. The phrase I stumbled and fell is causal but not, at least on the surface, conditional. Nevertheless, as the first example shows, there are many conditional sentences conveying causal meaning. Hence, in order to retrieve causal content in phrases, it is necessary to consider not only causal sentences, but conditional sentences too. For this purpose, one section of this work will be devoted to the extraction of conditional sentences in texts. A causal sentence usually fulfils at least three conditions [1]: 1. The cause must precede the effect. 2. Whenever the cause takes place, the effect must be produced. 3. Cause and effect must be closely related. R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 175–194. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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Not all the conditional sentences meet these three prerrequisites: in particular, as we previously said, in material conditionals antecedent and consequent may not be related. Besides, antecedent does not always precede the consequent, as in If 4 is an even number, then is divisible for 2 with remainder 0. More remarkably, not always the condition is satisfied, the consequent absolutely holds, as approximate quantification shows: Overpopulation almost always causes poverty. But, frequently, conditional and causal sentences are related as well as many causal sentences admit a conditional formulation (Friction causes heat / If two objects are rubbed against each other repeteadly, then they increase their temperature) and a lot of conditional sentences show causal content (If it is cloudy, visibility disminishes). The canonical form of causality is A caused B and the typical form of conditionality is If A then B. But causality and conditionality are not restricted to these forms. Thus, synonyms of ’cause’ or ’effect’ often announce causality: B is due to A, A produces B, etc. are some alternative ways to express a causal link. In the same vein, conditionality may be posed by other forms: B follows from A, A if only B and so on. Thus, in order to locate causality in its diversity, it is convinient to look for an extended causal lexicon. This work attemps to find automatically conditional and causal words in texts, as well as the links that causally connect them.

9.2

Causality and Information Retrieval Systems

One goal of the information extraction research is to develop systems able to identify passages or pieces of texts containing relevant information for a certain task by filling a structured template or a database record, as in Cardie [2], Cowie and Lehnert [3], and Gaizauskas and Wilks [4]. In these works they use templates indicating the slots to be filled and the type of information to be extracted in order to fill them. These set of slots define entities, aspects and roles relevant to a certain task or topic of interest and can be used for automatic summarization, information mining or knowledge adquisition [10]. In this paper, we will develop this point of view attending to causality. Information extraction systems have been developed for several tasks. However, few of them have focused on extracting cause-effect information from texts. Many of these studies used knowledge-based inferences to infer causal relations, as in Selfridge and Daniell [18] and Joskowsicz and Ksiezyk [8], which developed a computer prototype to extract causal knowledge from short messages entered in the knowledge base of an Expert System. Information Retrieval is a field that could highly benefit from these works related to causality. As Zadeh [20] points out, current Information Retrieval Systems have not deduction capabilities. This lack comes in part from the difficulties to find relationships among concepts. As we previously said, causality relates cause and effect and the information so linked shows frequently a relevant content. Most of the search engines on the Internet find answers using lexicographical resources. Improvements are provided expanding the lexical approach to a semantic or conceptual level, including fuzzy relations as synonymy, hyperonimy, etc., which allow

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for query expansion or multiplicity in answers. But progress towards developing more eficient search engines lead to implement inference skills. Causal links may be useful in this task. Thus, a challenge in modern IR is to develop a search engine able to obtain answers by reasoning, using the propositional part of the question as a fact and dispatching the conditional sentences stored in the database as rules. If the answer does not match any phrase or word stored into the database, an explanatory process may, nevertheless, help to find it. Causal explanation is perhaps the typical way to go ahead in this task. In this work we only notice this aspect. Figure 9.1 [5] shows the difference between answering a causal query whether using an inference module or not (cause-effect module manually performed). Without a causation module the system extracts paragraphs where the words involved in the question appear. With the causation module, relationships among concepts that do not necessary appear in the query, but have been obtained via an inference process, are stablished, displaying a more accurate answer to the requested query.

Fig. 9.1. Examples of queries in a QA system using a causation module [5].

If all the causes and effects included in a domain are known and all the causal links connecting them are identyfied, a causal theory can be provided. But, in fact, in many domains of knowledge it is not possible to fix a closed universe of causes and effects, neither to determine with precision the causal connections among them. In these cases, we should replace ’causal theories’ by ’causal mechanisms’. Relying on the proportion of the founded causes/effects links and on the strength of their conexions, we will have ’systems’ or ’structures’, weaker concepts than ’theories’. While a theory is a logical and deductive matter, the study of mechanisms associated to systems or structures is empirical and gradual: sometimes, frequently, almost always a cause determine in a degree an effect. Here plays a role the Fuzzy Logic, providing models to calculate the degree of influence of a cause in an effect. Fuzziness in causality may emerge from i) representing causes/effects with vague predicates, ii) tunning causal links by approximate quantifiers; iii) qualifying nodes with vague qualifiers (semantic hedges).

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Causal mechanisms offer not only a possible explanation for an indirect answer, but a contextualization in the system or structure that support it -something like to provide an ’intuitive theory’-. Mining a system or structure can be a good indicative of the relevant content of the text where the answer to the causal question is hosted. Retrieving causal mechanisms allows not only to supply for relevant information as an answer, but most important, to offer an explanation of it showing how it is connected with other data. Taking all these premises into account, this paper deals with the extraction of conditional and causal sentences in texts, as well as its contextualization in the net of concepts that configures the structure or system in which they are imbeded. In this work a preliminary study is presented on how to approach the detection of conditional and causal phrases in texts from their syntactic atoms and the structure in which they are embedded. The detection process, the classification phase and the fulfilled tests are described. Other purpose of this paper is to detect vague quantifiers and qualifiers granding the strenght of the cause, the strenght of the effect, or the strenght of the causal link. Finally, we will show a tool for automatically grouping causes, effect and links, obtaining a causal graph or mechanism. This achivement may be an intermediate step in the challenge to obtain answers in Information Retrieval Systems with inferential skills.

9.3

Retrieving Conditional Sentences in Texts

To detect conditional and causal statements in text documents, we have first to divide a phrase into its basic components (verbal tenses, adverbs, pronouns, nouns, etc.) in order to develop an algorithm able to detect and classify conditional and causal sentences based on structured patterns. According to the English1 grammar, there are two types of lexicon that characterize conditional sentences: (i) some selected verbs (ii) some discourse makers. Based on them, we present a set of labelled structures defining the types of conditional sentences, showing in each case the verbal tenses that must display.

Fig. 9.2. Example of conditional structure. 1

http://www.britishcouncil.org/ (last visit 14/10/2010)

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Since the number of analyzed structures was high, we implemented a representative subset as a first step to perform a prototype able to detect and classify them. Specifically, we analyze the structures from the English language belonging to the conditional in its classic form, if X then Y, corresponding to the first, second and third conditional. Other structures, that contain conditional lexicon related to causality, are also identified. 20 structures were selected, serving as input and basis for the detection and classification processes:

Fig. 9.3. Selection of conditional and causal sentences procedure.

Once these patterns have been selected, we designed an algorithm to extract and identify conditional and causal sentences in text documents. The algorithm goes through two phases: (i) detection and (ii) classification. As seen in figure 9.5, a base of knowledge containing all the retrieved conditional and causal sentences is obtained: The detection algorithm is in charge of filtering the conditional sentences whose fomat is within the 20 previously mentioned. To perform this process, the morphological analyzer Flex, created by the GNU project and freely distributed, was used. The key motive for the selection of this morphological analyzer is its compatibility with the C programming language. Flex allows for the definition of tokens and elements to be detected (in Lex language) and thus, facilitate their processing using the C language. In order to analyze the syntactic components of a sentence, a regular grammar is provided (figure 9.4) and a finite state automaton drawn from it. Flex allows for simple definition of each one of the states. In order to develop the detection algorithm, only three states were used: the initial (I) one and two more for controlling the conditional forms (Conditional and Conditional2). The procedure begins in the initial state, where syntactic units are processed and involve two steps: (a) If the processed token matches the conditional particle, the automaton will move on to the detect state, labelling the sentence as conditional; (b) if a token matches anything defined in the other structures as a conditional token, the automaton will move to state detect2 (figure 9.6) also labelling the sentence as conditional. If (a) or (b) are not satisfied, the process will continue analyzing syntactic units, storing them in a buffer, waiting for a conditional conjunction to come up. If the procedure receives a syntactic unit with a full stop, it will reject the sentence, emptying the buffer where it had been stored. Once the syntactic components of a

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Fig. 9.4. Detection process regular grammar.

Fig. 9.5. Set of conditional structures.

conditional sentence are found, the next step is to classify them. The generated algorithm recognizes the conditional type detected as a function of the verbal form and the syntactic structure, using as a reference the 20 causal templates defined before (figure 9.3).

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Fig. 9.6. Detection and classification automaton.

To perform the classification process, the automaton has been improved by the addition of two new states, cw and wd (figure 9.6), for checking the composed verbal forms. This module contains a vector with the most commonly used verbs in the English language, such as to have, to do, or to be, and their conjugated forms. So, if the verbal form to be processed dealt with would have, the automaton would change to the status wd on detecting would and remain there waiting for a verb that could form a composed tense, such as be or have. If the following analyzed token matchs with have, the process would activate the position associated to would have – instead of that of would – in the verb vector and would return to the previous status. Analysing the composition of the conditional patterns, it was noted that a verbal form almost always appeared -whether in the antecedent or in the consequent of the sentence- causing their membership to some structure. We also observed that the verbal form was generally composed of some type of the verbs; therefore, in order to classify a sentence into a conditional structure it would be enough to run the aforementioned vector analyzing the verbal forms in the sentence.

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For example, in the sentence, If the weather is fine, we’ll go to the beach, the verbal form vector would register is in position 2 and ‘ll in position 21. If we run the phrase through the vector, the algoritm detecting positions 2 and 21 as activated would determine that we are dealing with a type 2 structure, with the form if + present simple + future simple: To calculate the degree of success of the implementation within the structures classified, a Gold standard test has been performed with the file causal.txt (output file), of the Stephen Hawkins’s speech Quantum Cosmology. With these results, recall, precision and F-measure indicators have been calculated. Recall (R), is the number of correct causal sentences classified by the system divided by the number of causal sentences classified by manual analysis. Precision (P), is the number of correct causal sentences classified by the program, divided by the total amount of sentences retrieved. F-measure (F) is a combination of recall and precision, F = (2 ∗ P ∗ R)/(P + R), and can be interpreted as a weighted average of precision and recall.

Fig. 9.7. Golden test for the text Quantum Cosmology (S. Hawking).

So, according to these values, the algorithm was able to correctly classify 16 out of 18 phrases, i.e., showing a precision of 88,89%. The total number of retrieved sentences was 24, with a 66,66% recall factor of correctly classified structures from the total retrieved ones (16/24). F-measure value is 0,76. So, we can conclude that the classification process for this type of text was quite good. This same Golden test was done analyzing 50 pages of texts from different areas such as news or medicine.

Fig. 9.8. Golden test for different genres texts.

These results show better performance with medical texts -with a recall factor of 83% and the highest value for precision 91%- and scientific texts than in general purpose texts (novels), Gospel texts and the news, where language is not as direct and concise. Based on these data, later experiments were done using medical texts.

9.4 Retrieving Crisp and Imperfect Causal Sentences in Texts

9.4 9.4.1

183

Retrieving Crisp and Imperfect Causal Sentences in Texts Introducing Imperfect Causality

In the scientific area, causality is usually qualified as a precise relation: the same cause provokes always the same effect. But in daily life, the links between cause and effect are frequently imprecise or imperfect in nature. Fuzzy logic offers an adequate framework for dealing with imperfect causality. In this realm, some key questions are [13]: • To what degree does c cause e? • Does the relationship between cause and effect always hold the same strength in time and in every situation? • Does the causal strength represent the degree of influence that the cause has on the effect? These are several views about imperfect causality. All of them are largely inspired in the Kosko´s seminal work on fuzzy cognitive maps. Kosko introduces a definition of causality between concepts represented by fuzzy sets. A causes C can causally increase or causally decrease the effect E, being increased or decreased a degree concept. In short, C causally increases E if C ⊂ E and C  ⊂ E  and C causally decreases E iff C ⊂ E  and C ⊂ E, denoting ⊂ fuzzy subsethood and ’ complement. Later, Kosko generalized the notion of degree of subsethood in terms of inclusion in the power set. In his view, the association of set theoretical notions with geometrical properties of the hypercube provides causal models with both an intuitive presentation and a more general foundation. [11],[12] Based on Kosko’s models of causality, Ragin [16] argues that in social sciences conditions are best represented by qualitative properties formalized by fuzzy sets and that necessary and sufficient causes are suitably interpreted in terms of subsethood. For a combination of necessary causes, he proposes the t-norm min and for a combination of sufficient causes, he suggests the t-norm max. Finally, Hegalson & Job cit:Helgason2003 present a model of fuzzy causality as an alternative to population-statistics-based medicine. Their objective is to provide a perception-based representation of the psychological interaction of elements in an individual patient and to monitor their evolution over time starting from some initial conditions. They provide a numerical causal groundwork based on the following moral: the smaller the degree of causal relevance of A to B, the smaller the ’causal flow’ of A into B, and the larger the ’causal flow’ of extraneous influences on B. Many studies have looked at the analysis of causality in text documents. Some focused on the extraction of causal statements, like Girju’s [5], and some others linking causality and interrogative pronouns, like Black and Paice’s [14]. Using these studies as references, we suggested a new algorithm which is able to dispatch a causal process, using a target concept as input and the set of conditional and causal sentences retrieved as a base of knowledge. We also provide techniques to extract vague quatifiers and vague qualifiers, showing that imprecision is an usual property of causal relations.

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9 Retrieving Crisp and Imperfect Causal Sentences in Texts

Causal Sentence Analysis

The causal sentence analysis procedure presents a set of algorithms to filter those conditional sentences retrieved in the previous step related to a given concept. The mechanism is divided into three major parts, the input concept selection algorithm, in charge of the isolation of a concept involved in a query, the summary method, to select those causal sentences related to the input concept and extract the relevant information to be represented, and the graph summary algorithm, which displays the obtained information by means of a causal graph. A. Input concept selection algorithm A concept related to the scope of the sentences retrieved is selected. The objective is to establish a cause-effect relationship with some other concepts that may appear in the base of knowledge. In [15] we suggested a program using Flex, C programming language and the Stuggart tree tagger POST [17] to select a concept from an input query and to know whether the user is asking for causes or consequences.

Fig. 9.9. Causal sentence analysis and graph summary algorithms.

For example, if the user asks What provokes lung cancer?, the POST tagger would return what figure 9.10 shows. This POST output shows that the nominal clause, composed by two nouns (NN), is lung cancer. So, processing this clause with the morphological analyzer, the program, that detects the word provokes plus the interrogative pronoun what, would

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Fig. 9.10. POST tagger output.

assume that the user is asking for the cause of lung cancer. Once the nominal clause has been selected and isolated, another program has been developed to extract those sentences in which these concepts are contained. The searching set is the previous file created with the conditional and causal sentences. As a result, another file (smaller in size) will be produced with those causal and conditional sentences containing the sought concept. These retrieved set of sentences will serve as the input for the sentence summary process. B. Parsing the retrieved set To retrive conditional sentences related to a concept, we follow a similar procedure to that one employed to extract conditional and causal statements from a text, locating the sought concept into a sentence (in this case, instead of the causal clause) by means of a morphological analyzer, and selecting the matching sentences into a new file. Next we parse every sentence and the concepts contained in them. In [15], we started by analyzing the words and selecting the nominal clauses using a C program and the POST analyzer as input. Despite the fact that some sentences worked fine with this program, many others were rejected because the algorithm was unable to deal with complex sentences. We use a new version the Stanford Parser to overcome this gap [9]. Sentences poorly parsed by the POST processing (specially retrieving quantifiers and modifiers associated to the antecedent and consequent of the sentence) are analyzed with more detail. This parser allows for more complex analysis of a sentence and returns a set of tags that determine a grammatical relationship between two words involved in the same sentence. For example, in the phrase Lung cancer can cause severe complications such as shortness of breath, the parser returns the following relationships:

Fig. 9.11. Stanford Parser [9] output.

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Another C program is suggested for receiving as input the set of tags created by the parser and, hence, producing a summary of the useful information according to the following algorithm: • Load all the tags into a matrix to operate with them. • Locate the causal connector (if, due to, cause, etc.), and find the tag that points to the antecedent node. In the example it would be the tag nsubj (cause-4, cancer-2). • Jump backwards (the number of jumps can be entered as a parameter) and forwards to locate all the relevant tags (nouns, adjectives, in the example nn(cancer2, Lung-1)) that modify the word involved in the causal tag (in the example it would be cancer, with number indicator 2), and create the string of the antecedent node with the retrieved information. • Detect the prepositions and words that indicate locations by means of the tag prep_in, and the words near, close, around, next, and some others. • Find words, usually introduced by the tag such_as, that suggest specification or clarification of the node. • Retrieve the words related to those appearing in the node title that indicate intensity, frequency, amount, such as modifiers and quantifiers (more, less, high, low, much, many, etc.) denoted by the tags quantmod, amod, part-mod, advmod and advcl (in the example amod(complications-6, severe-5)). • Store all the information retrieved for the antecedent node in the database. • Store the type of connective that defines the causal relationship (in the example the word cause). • If they exist, retrieve the modifiers of the causal link (such as can or could), usually denoted by the tag aux. • Repeat all the processes dispatched to the antecedent node, but locating the tag that indicates the consequence. In the example it would be the tag dobj(cause-4, complications-5), dispatching the algorithm with the word complications. Following this algorithm, the relevant causal information of a sentence is extracted and summarized, being stored into a database and ready to be represented into a causal graph. 9.4.3

From Single Causal Sentences to Mechanisms

This section deals with a method for representing in a graph causal relationships extracted from the selected sentences; therefore, showing the causal mechanism that links the retrieved concepts. A. Storing Causal Relationship Summaries As previously shown, the causal relationship summary represents the output of the program’s analysis of a single sentence. So, it is necessary to save this output for employing it in further work. The summaries are stored into a relational database. That allows us to match data by using common characteristics found in the data set (for example, retrieving all the concepts causally related with a particular one). The database structure is as follows:

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• Relationship: this table stores the information of the relationship itself, namely the identifiers of both cause and effect nodes, the type of relationship, and the relationship modifier, if any. The cause and effect nodes can be a concept or an association. • Concept: each entry of this table is the representation of a concept involved on a causal sentence, be it the effect or the cause (or part of one of them). It stores the extracted concept itself along with three possible modifiers: location, specification and intensity. • Association: they act as intermediary nodes when there is a compound cause or effect on a relationship. Compound causes or effects are those where there are two or more concepts linked by an AND or an OR.

Fig. 9.12. Database representation of the causal relation summary in figure 9.11.

Compound-clause relationships are expanded into n relationships following these steps: • Each concept involved in the compound clause has a direct relationship with the corresponding association node. • The association node has a causal relationship (modifiers included) with the other clause of the causal sentence (which can be a single concept or another compound clause).

Fig. 9.13. Conceptual representation of a compound-clause causal relationship.

B. From Causal Relation Summaries to Graph Representation This process translates the causal relation summaries extracted from the processed sentences into a causal graph. For this purpose we have employed the Java programming language (object-oriented programming), along with the JGraph2 library, to 2

http://www.jgraph.com (14/10/2010)

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display the causal graph. In this process the concept is represented by an ellipse with a text string plus three possible modifiers: location, specification and intensity. The graphical representation of this structure and the way the concept nodes are going to be displayed on the graph is as follows:

Fig. 9.14. Graphical representation of a node with its modifiers.

To draw the graph, the stored summaries are retrieved from the database and loaded into memory as objects (relationships, concepts and associations). Then, the representation algorithm follows these steps: • Those concepts that do not take part in any relationship with the “effect” role (and therefore don’t have any other node above them) are selected to be placed on top of the graph (forming the first row of nodes). • If any node in this row takes part in an association and the other nodes taking part in it are also on the row, they are placed together. The graph node corresponding to that association is placed between the concepts row. • All these concept nodes as well as the association node if so (AND, OR nodes) are internally grouped and managed as a single node for the next step. • To place the remaining concepts in the row, it is needed to look at the relationships they take part on, and the effect nodes to place the nodes that are causally related to the same concept next to each other. • When the whole row of concept nodes is positioned, then the next row is formed with these effect nodes (concepts) involved in the relationships. • The algorithm goes back to the second point to place the new row, and is repeated until all the nodes (with their relationships) have been positioned. This representation algorithm is heavily focused on achieving a clear graph, avoiding causal links (graph edges) going from side to side of the representation hindering the perception of the graph. If the causal relationship summary set is very complex, the facilities provided by the JGraph library allows for a human supervisor to modify the final graph (reallocating nodes, moving edges, resizing nodes), sacrificing the degree of automation for a better representation quality (thus becoming a semi-automatic process). Figure 9.16 represents a schema of the overall process in a simplified way.

9.5 Generating an Example of Imperfect Causal Graph

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Fig. 9.15. Graph representation of the causal relation summary in figure 9.11.

Fig. 9.16. Description of the process.

9.5

Generating an Example of Imperfect Causal Graph

This section presents a complete example establishing the relationship between two concepts such as smoking and lung cancer. The set of documents selected for this experiment were taken from the Mayo clinic3 web site, the American society for clinical oncology4, the centers for disease control and prevention5, and the sites emedicine health6 and lung cancer online7. 3 4 5 6 7

http://www.mayoclinic.com (14/10/2010) http://www.cancer.net/patient/Cancer+Types/Lung+Cancer/ (14/10/2010) http://www.cdc.gov/cancer/lung/basic_info/ (14/10/2010) http://www.emedicinehealth.com/lung_cancer/article_em.htm (14/10/2010) http://www.lungcanceronline.org/info/index.html (14/10/2010)

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1. Detection and classification of conditional and causal sentences This process extracts and classifies all the sentences that match any of the 20 patterns considered as conditional or causal. The results of this part of the process were quite good in part motivated by the language used in medical texts, which is more clear and concise than in other areas. The results were the following:

Fig. 9.17. Report of sentences retrieved and rejected.

2. Summary process This part is in charge of filtering those sentences containing the words lung cancer or smoking. The system retrieved 82 sentences and we selected 15 of them to represent a middle size and, therefore, manageable graph. Some retrieved sentences are presented in table I.

Fig. 9.18. Table I: Causal sentences retrieved.

Once the phrases have been selected, the program parses each one of them with the API of the Stanford parser. For example, the first sentence would retrieve: In this case the summary process would locate the tag due_to as the causal indicator. This tag associates the words die and smoking as belonging to the nodes ’effect’ and ’cause’ respectively. To create the two nodes, the process searches through the labels using the number associated to each word. For instance, to create the ’cause’

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Fig. 9.19. Stanford parser output for the phrase Every year, about 3.000 nonsmokers die from lung cancer due to secondhand smoke (collapsed mode).

node, the algorithm would search through the tags the word smoke plus the number 14. If the tag is an adjective, adverb, or a modifier – in this case, amod(smoke-14, seconhand-13) –, the program would include this word in the node name. So, the algorithm retrieves a brief schema of the useful information to be represented into a causal graph:

Fig. 9.20. Summary of the phrase Every year, about 3.000 nonsmokers die from lung cancer due to secondhand smoke.

3. Generating the graph This schema is stored in a database which is accessed by the graphing process. Once all the phrases have been processed and schematized, the graph algorithm creates the causal graph. Some of these sentences share the same antecedent or consequence node, as happen with risk lung cancer, which appears several times. This node inherits all the modifiers that could appear within these sentences. A relationship is specified by a modifier if the arrow is inside of the modifier cell, otherwise the arrow will be pointing at the border of the node. Causes or effects may be described generically (pulmonary hypoplasia) or more specifically: doctors often speak about pulmonary hypoplasia in ’right lung’ or ’right upper lobe’ pulmonary hypoplasia. Thus, in medical texts it is frequently to refer to locations (pulmonary) or specifications (upper, right) in nominal phrases. Locations or specifications are extracted from texts using parsers. If the relationship has more than one modifier, the arrow will point to an intermediate node, aiming the arrows to the corresponding modifiers. The intensity of the relationship’s arrow (quantifiers) if so, is marked in red, and the type of causal connective in black. As we can see in the graph, there are four nodes with

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the word smoking – or other related words –, and another four with the words lung cancer. The rest of the nodes have been retrieved in the process. So, using a graph like this, we could establish new relationships hidden at first sight – for instance, between smoking and fluid being accumulate around the lungs –, with a certain degree, taking into account the quantifiers and qualifiers involved in the causal paths.

Fig. 9.21. Automatically retrieved graph.

9.6

Conclusions and Future Works

The aim of this paper was twofold: (i) to show some programs capable of retrieving causal sentences or conditional causal sentences with causal content from texts and (ii) to automatically represent them on a causal graph. Retrieved sentences are used as a database to study imperfect causality. Sentences extracted from both scientific and humanistic texts show that causality is frequently imperfect or approximate in nature. This feature comes from the linguistic hedges or fuzzy quantifiers included in them. A Flex and C program analyses causal phrases denoted by words like ’cause’, ’effect’ or their synonyms and highlight vague words that qualify the causal nodes or the links between them. Frequently, vague words are fuzzy quantifiers or linguistic hedges. Another C program receives as input a set of tags from the parser and generates a template with a start node (cause), a causal relation (denoted by lexical words), possibly qualified by fuzzy quantification, and a final node (effect), possibly modified by a linguistic hedge showing its intensity. Finally, a Java program automates this task. Future work will include the determination of how to perform the fusion of causal sub-graphs with a similar meaning. In many cases, concepts are repeated many times and the resulting graph is enormous. A challenge is to summarize some parts of it without losing relevant content. Another task is related with the aggregation or propagation of fuzzy quantifiers in the causal graphs. As figure 9.21 shows, some effect nodes are causally linked by more than one cause node, each of them having a different degree of influence. The evaluation of to what extent the effect is influenced by various causes requires investigation.

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Causal graphs derived from stories often reflect indirect causality: A node is connected to another through intermediary nodes. Another challenge is to evaluate how fuzzy quantifiers are spread in causal chaining that includes several links. These challenges make the study of fuzzy causality a fascinating subject. Making progress on it should enable us to advance in the natural or scientific language processing, contributing to automatically recover hidden causal mechanisms included in texts.

References [1] Bunge, M.: Causality and modern science. Dover (1979) [2] Cardie, C.: Empirical methods in information extraction. AI Magazine 18(4), 65–79 (1997) [3] Cowie, J., Lehnert, W.: Information extraction. Communications of the ACM 39(1), 80–91 (1996) [4] Gaizauskas, R., Wilks, Y.: Information extraction beyond document retrieval. Journal of Documentation 54(1), 70–105 (1998) [5] Girju, R.: Automatic detection of causal relations for question answering. In: Proc. Of the 41st ACL Workshop on Multilingual Summarization and Question Answering (2003) [6] Hausman, D.M.: Causal Asymmetries. Cambridge University Press (1998) [7] Helgason, C.M., Jobe, T.H.: Perception-Based Reasoning and Fuzzy Cardinality Provide Direct Measures of Causality Sensitive to Initial Conditions in Individual Patient. International Journal of Computational Cognition 1, 79–104 (2003) [8] Joskowicz, L., Ksiezyk, T., Grishman, R.: Deep domain models for discourse anaysis. In: The Annual AI Systems in Government Conference (1989) [9] Klein, D., Manning, C.D.: Fast Exact Inference with a Factored Model for Natural Language Parsing. In: Advances in Neural Information Processing Systems 15 (NIPS 2003), pp. 3–10. MIT Press, Cambridge (2003) [10] Khoo, C.S.-G., Kornfilt, J., Oddy, R.N., Myaeng, S.H.: Automatic extraction of causeeffect information from newspaper text without knowledge-based inferencing. Literary and Linguistic Computing 13(4), 177–186 (1998) [11] Kosko, B.: Fuzzy Cognitive Maps. International Journal of Man-Machine Studies 24, 65–75 (1986) [12] Kosko, B.: Fuzziness vs. Probability. International Journal of General Systems 17, 211– 240 (1990) [13] Mazlack, L.J.: Imperfect causality. Fundamenta Informaticae 59, 191–201 (2004) [14] Paice, C., Black, W.: The use of causal expressions for abstracting and questionanswering. In: Proc. of the 5th Int. Conf, on Recent Advances in Natural Language Processing, RANLP (2005) [15] Puente, C., Sobrino, A., Olivas, J.Á.: Extraction of conditional and causal sentences from queries to provide a flexible answer. In: Andreasen, T., Yager, R.R., Bulskov, H., Christiansen, H., Larsen, H.L. (eds.) FQAS 2009. LNCS, vol. 5822, pp. 477–487. Springer, Heidelberg (2009) [16] Ragin, C.: Fuzzy Set Social Science. Chicago University Press, Chicago (2000) [17] Schmid, H.: Probabilistic Part-of-Speech Tagging Using Decision Trees. In: Proceedings of International Conference on New Methods in Language Processing (1994)

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[18] Selfridge, M., Daniell, J., Simmons, D.: Learning causal models by understanding real-world natural language explanations. In: Second Conference on Artificial Intelligence Applications: The Engineering of Knowledge-Based Systems, pp. 378–383. IEEE Computer Society, Silver Spring (1985) [19] Vaerenbergh, A.: A.M. Explanatory models in suicide research: explaining relationship. In: Frank, R. (ed.) The explanatory power of models, pp. 51–66. Kluwer, Dordrecht (2002) [20] Zadeh, L.A.: From Search Engines to Question-Answering Systems - The Problems of World Knowledge, Relevance, Deduction and Precisiation. In: Fuzzy Logic and the Semantic Web, pp. 163–210. Elsevier (2006)

10 Facing Uncertainty in Digitisation Lukas Gander, Ulrich Reffle, Christoph Ringlstetter, Sven Schlarb, Klaus Schulz, and Raphael Unterweger

10.1

Introduction

Many library collections contain assets, like newspapers, books, or manuscripts, for which no adequate digital version has been created so far. This is especially the case for historical materials from the 16th to 19th centuries where the quality of text recognition results is still far from meeting the expectations of European libraries evaluating the feasibility of large scale digitisation of their historical materials. The reason for this is twofold: State of the art text recognition software is mainly optimised for printed versions of modern digital-born text documents, and, digitising historical material has its own, very specific challenges, such as historical fonts and language, different fonts and languages on a single book or newspaper page, bad paper quality or deterioration of the material due to unfavorable storage conditions, page distortion and skew because of the scanning process, variable inking, etc. Modern software is addressing all these issues to some extent, but the results are not yet altogether satisfactory. IMPACT (IMProving ACcess to Text) is a European project, partially supported by European Community, which brings together national and regional libraries, research institutions and commercial partners to work on software for digitisation with the special focus on enabling cultural heritage institutions to conduct mass digitisation projects in a standardised, efficient and reliable way. Digitisation, in this context, refers to a processing chain, starting with the scanning the original asset (newspaper, book, manuscript, etc.) and the creation of digital images of the asset’s pages, from which digital text is produced. The core process is Optical Character Recognition (OCR) which is preceded by image enhancement steps, like deskewing, denoising, etc., and is followed by post-processing steps, such as linguistic correction of OCR errors or enrichment of the OCR results, like adding layout information and identifying semantic units of a page (e.g. page number). In this paper, we will focus on the post-processing steps and present two selected research areas of the IMPACT project. First, we will detail a technology for OCR and information retrieval on historical document collections, developed at the Center of Information and Language Processing of the University of Munich (CIS). Second, we will discuss the so called Functional Extension Parser (FEP), a layout analysis and OCR enrichment software developed at the Innsbruck University. We will then R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 195–207. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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briefly outline the main features of a fuzzy rule-based system for each of these research areas. It should be noted that IMPACT only plans to apply the rule-based approach with regard to the Functional Extension Parser. With regards to the first application scenario, the intent is to make a general comment on the concept and potential uses of a fuzzy rule-based approach. “Rule-based”, in this context, means that rules are used for representing conditional relations expressing dependencies of facts typically present in the area of digitisation. The attribute “fuzzy” refers to a typical fuzzy rule-based approach with the following main characteristics: The constituent variables of the rule’s antecedent are linguistic variables and the values of the variables are fuzzy terms which are mathematically represented by membership functions indicating the continuous-valued degree of membership for all values of the linguistic variable’s domain of reference (universe of discourse), in the following called “domain”. A basic algebra defines negation, conjunction, and disjunction which are the operations for evaluating the antecedent of a rule. The evaluation of the antecedent gives the degree of fulfilment of the rule’s condition which is the input for the inference step indicating the degree to which the conclusion applies. (See [6])

10.2

Language Technology for OCR and Information Retrieval on Historical Document Collections

Optical Character Recognition (OCR) and Information Retrieval (IR) are two disciplines that need some basic form of language technology to reach their goals. In both disciplines, the main focus has so far been on modern document collections. After several decades of interdisciplinary work there is now a good understanding of the uses of language technology for OCR and IR on modern texts. When facing historical texts, the problem becomes much harder and many new questions are raised. When it comes to the OCR of historical texts, questions such as “what is a correct word?”, or “how can we detect and correct misrecognised words?” become much more difficult to answer. Similar challenges arise in IR of historical documents. For example, notions of synonymy have to be expanded, taking language change into account. In this section we review useful language technology for IR and OCR and we describe some of the central challenges and possible solutions in the context of historical document collections. 10.2.1

Building and Applying Lexica

Standard OCR uses lexica in a simple way. For a given token (sequence of characters representing a possible word) w, recognised by the OCR kernel engine, the engine checks if w occurs in the dictionary D. In the positive case, w is usually considered to be correctly recognised. In the negative case, words in the dictionary that are sufficiently similar to w are treated as correction suggestions. The notion of

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word similarity has to be made precise, using, e.g., the Levenshtein metrics. [5] If there are reasons to assume that the character segmentation suggested by the OCR engine is correct, then we may use a restriction of the Levenshtein metrics where only substitutions are possible edit operations. Various refinements and extensions of the Levenshtein metrics can be useful for judging word similarity in distinct application scenarios. For example, probabilistic approaches assign non-uniform costs in the interval [0, 1] to edit operations, depending on the symbols involved. “Cheap” substitutions, such as “i → l”, might come with costs of 0.1; “expensive” substitutions (e.g. “i → m”) usually have cost 1. Besides the standard edit operations (i.e., deletion, insertion of a single symbol, substitution of symbols), other operations such as transposition of symbols, merges and splits of symbols are also considered. Even for modern language, the selection of an appropriate lexicon is a delicate matter which heavily influences the OCR accuracy that can be attained (would you use “ceul” or “groudy” in your English dictionary?). When processing historical documents, the relative absence of normalised orthography and the change of language over historical periods force us to ask what a lexicon for such documents should consist of. There are at least two options. First, we can take a corpus based perspective, collect as many proofread historical documents as possible and use (perhaps after some cleaning steps) the tokens found in the corpus as our dictionary. We can even go further in this direction and add morphological variants of the words. To this end, appropriate models for modern and historical morphology need to be available. The main problem with the corpus-based approach is caused by the large number of spelling variants that are found in historical documents. In practice it is impossible to collect a corpus of proofread historical documents that captures all (or most of) the orthographic variants to be found in historical documents. A possible alternative is a matching-based approach. In most cases, historical variants of modern words are phonetically similar to the modern word. We can try to find a rule set of special rewrite operations or patterns that covers this notion of similarity. Instead of a lexicon of “proofread” words we may then use a modern lexicon and a matching procedure which “explains” the historical spelling, offering a “trace” as to how the historical word and the modern word are related using rules. As an example, the trace “t → th” shows how the historical German word “thun” (Engl.: to do) is derived from its modern form “tun”. Empirical evidence is needed to judge the usefulness of each type of lexicon. Several experiments conducted at the Center of Information and Language Processing of the University of Munich (CIS) indicate that texts from distinct centuries need distinct lexical techniques.[2] Puristic matching based approaches give good (acceptable) results for German texts from the 19th (18th) century. For older periods, where historical language change is more of an issue, conventional lexica with words from proofread texts represent an important corrective. The experiments also showed that a significant gain in OCR accuracy is attained when using special language resources for historical language as opposed to a conventional lexicon of modern words. Similar observations can be made for IR on historical texts. Here we have the need to connect the modern words used in queries to synonymous historical variants

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found in the texts. Note that the matching approach automatically established such a connection between old and modern words. However, these matching suggestions can turn out to be misleading. The construction of a lexicon where tokens from proofread historical texts are manually associated with synonymous modern lemmas is a time-consuming task. Yet, experiments at CIS show that the matching approach alone does not lead to satisfactory results in Information Retrieval for document collections from before the 19th century. 10.2.2

Use of Language Models

Language models of the form used in OCR engines list possible sequences of n consecutive words (n-grams) and their probabilities. An n-gram statistic helps to detect “false friends”, e.g., misspelled words which accidently have the form of a correct word. Using the statistics, we can check if a word triple uyv is more likely to be correct than a triple uxv found in the OCR output. For similar reasons, n-gram statistics are very useful to find an optimal correction suggestion for a misspelled word x, assuming that the context words of u and v are correctly recognised. In text correction tasks the use of word trigrams is now standard. When moving to historical document collections, due to the relative paucity of orthographic standardisation, we found ourselves in a situation with two possible means of progress. A first option is to calculate n-gram probabilities for all triples of consecutive words found in a background corpus, leaving the words found in the texts in the original spelling. Alternatively we might try to normalise each word of a historical corpus, using a canonical spelling, and to calculate probabilities for normalised word ngrams. Regardless of which is chosen, due to the small size of available proofread historical corpus resources, smoothing techniques that assign a positive probability to unknown n-grams are important. It is fair to say that currently almost no practical experience exists as to the use of language models for historical texts. 10.2.3

Adaptivity, Postcorrection and the Idea of Profiles

One interesting line of research in the IMPACT project is centered around the idea of “profiling” OCRed input texts. Ideally, the profile of a recognised text should precisely characterise the type of historical language of the document in terms of the rewrite patterns used and it should list the most frequent recognition errors of the OCR engine in the text. Clearly, in the absence of complete ground truth data, the best we can hope for is a “reasonable guess” of patterns and recognition errors based on some form of (rather restricted) statistical evidence. Nevertheless, even imperfect profiles of this form have many interesting applications. They may be used in adaptive OCR, providing feedback to the OCR engine on how to improve recognition in the given text, possibly revisiting specific characters and words. Furthermore, profiles are useful for interactive postcorrection, pointing to tokens of the text that are likely to represent recognition errors. Finding intelligent ways of how to compute profiles such as these is currently one of the main research activities at

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the CIS. In this work, profiles are basically obtained by accumulating local “interpretations” of tokens (word candidates) and merging the local pictures to a global view of the text. An interpretation lists historical patterns and OCR errors occurring in the tokens. Difficulties in computing interpretations and profiles arise from imperfect lexical ressources and the large number of possible interpretations of garbled tokens. A special source of problems comes from imperfect word segmentation computed by the OCR engine. Perhaps surprisingly, parts of words (even accidentally merged words) often have plausible – but wrong – interpretations of identified correct words, historical patterns and OCR errors. 10.2.4

The Fuzzy Rule-Based Approach

Regarding the questions “how can we detect misrecognised words?” and “what is a correct word?”, there is underlying uncertainty because there is lack of information. In order to decide if a token should be replaced or not, additional information is required, such as the fact that a specific token can be found in a dictionary, for example. This additional information decreases the degree of uncertainty (or increases the degree of confidence) related to the corresponding fact. Following the “fuzzy rule-based” approach, briefly outlined in the introduction, let us assume the following examples of linguistic variables of this application area as relevant factors for deriving the degree of confidence for the substitution of a token: • Variable: EC := OCR Engine Confidence – Terms: low, quite low, medium, quite high, high – Domain: Confidence value in % [0,100%] – Note: The ABBYY OCR engine used in IMPACT provides recognition confidence at word and character level. Recognition confidence of a character or word image is an estimate of the similarity of this image compared to the “ideal” representation of the item that it is supposed to be. • Variable: T S := Token similarity – Terms: low, medium, high – Domain: Token similarity – Note: Token similarity is based on a metric indicating the distance of two tokens. • Variable: HC := Historical Context – Terms: 16th, 17th, 18th, 19th century – Domain: Integer based year interval [1500,2100] – Note: • Variable: SC := Substitution Confidence – Terms: low, medium, high – Domain: Continuous-valued interval [0,1] – Note: These variables are then used to create a set of rules while each literal of the rules’ antecedent is a variable with a fuzzy term assigned, as in the following rules:

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if EC is high and T S is high then SC his very high

(10.1)

if HC is 17th and T S is low then SC is low

(10.2)

The question of whether substitution should be applied is a boolean variable. The actual replacement, however, can still be controlled by a treshold parameter, the fuzzy terms of the conclusion variable SC representing the different threshold options. In this example, “fuzzy reasoning” takes only place on the meta level where conditional interdependencies of fine-grained numerical variables are expressed by means of fuzzy variables and rules. It must be noted that only the variable HC has a genuine numerical domain where the fuzzy membership functions can be directly based upon. All the other domains are either inherently statistical in the sense that the values are based on frequency information, such as token occurrence, for example, or represent a string similarity measure, such as the Levenshtein edit distance, for example.1 A special case is the EC variable which is computed by an undisclosed algorithm of the ABBYY OCR engine. As a consequence, the confidence value is a fuzzy measure for which we only know that it is somehow based on statistical information available in the dictionaries used by the OCR engine. The advantage of this approach is that the configuration of interdependencies between units of knowledge is separated from the numerical facts hidden in the membership functions. As a consequence, the rule-base is reader friendly in the sense that the interpreation of the domain by means of fuzzy membership functions is valid for all rules contained in the rule-base and can easily be adapted by an expert of the knowledge domain while the application of rules does not require to know the precise values that separate the terms of the linguistic variable.

10.3

The Functional Extension Parser (FEP)

OCR technology plays a significant role in the context of digital library applications. The following are some application scenarios where further processing of OCR results might be of use: • eBooks: Mobile devices are more and more being used for displaying books and articles, and it is highly probable that they will play an important role in the “digital reading market” within in the next decade. eBook formats, such as Mobipocket or EPUB have special requirements which need to be addressed in an automated way. • Print on Demand: Companies such as Amazon and libraries such as the Cornell University Library or the University Innsbruck Library are offering services where historical books are reprinted on demand. • Digital library applications: Correct page numbering and consistent creation of content tables, as well as providing precise full-text searchability based on the 1

A method of using fuzzy logic for string matching taking the Levenshtein edit distance and other string similarity measures into account is presented in [1].

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logical structure of the books, is very important for digital libraries. As a consequence, characteristics of books, like page header, footnotes, and signature marks, have to be taken into account already during the indexing process. The Functional Extension Parser (FEP) is a prototype developed within the IMPACT Project with the goal of detecting and reconstructing some of the main features of a digitised book based on the OCR results of the digitised images. As a result the FEP provides the following features to support libraries in the application areas stated above. Page numbers: The reconstruction of the original pagination has two main benefits. First, it guarantees that the mapping from images to the corresponding pages is correct according to the page numbering. Second, the reconstruction of the original pagination provides the means for quality assurance, because errors, such as missing or duplicated pages, can be detected automatically. Print space: The print space determines the content area on the paper of a book, journal or other printed pages. The print space is limited by the surrounding borders, or in other words the gutters outside the printed area. Usually, a page is scanned and then cropped during digitisation. This improves the look and feel of an image and makes it more appropriate for further processing. The disadvantage of the cropping is that there might be a loss of relevant information regarding the size of the borders. The borders can be reconstructed automatically by detecting the content within the print space of the scanned image and applying a common schema for the margins (e.g. Golden section). This step is especially important in the context of Print on Demand applications. Logical Structure: The FEP is able to detect the most important features of the logical structure of a book, such as page-headers, headlines, footnotes, signature-marks, catchwords and marginalia. This additional information can be used by digital library applications to improve full-text search. In a further step it is possible to reconstruct the table of contents by combining the detected headlines with the reconstructed page numbers. The reconstructed table of contents offers the possibility of linking items from the table to the relevant content sections of the book. This feature is especially interesting for books which originally didn’t provide a table of contents. 10.3.1

FEP Architecture

The FEP Architecture comprises six interacting modules which are represented in figure 10.1. The Ingest Module, implemented as a Java-based web service, takes images and OCR XML result files as input. Different OCR XML formats (e.g Abby, Omnipage) are normalised to the so called FEP-XML by applying an XSL transformation. Consequently, original OCR XML result file and the images are stored on the server’s file system, and FEP-XML files are stored in the XML Database (XML DB).

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The XML DB is the “knowledge base” of the FEP. It contains FEP XML instances and receives new facts from the rule engine. Moreover, it provides basic data to the Visualization and Correction Module and to the Export Module. The Visualisation and Correction Module is a WEB-Application created by means of the Google Web Development Toolkit. This module allows users to run an FEP analysis process and to visualise analysis results. It provides the possibility of inspecting and correcting facts gained by the FEP analysis before adding them to the FEP knowledge base. At the end of a working session, the enhanced data can be exported in several ways: results can either be saved in an updated FEP XML instance, or they will be transformed into instances of other XML-Schemas, such as METS/ALTO for digital library applications.

Fig. 10.1. Component diagram (FEP)

The FEP uses a rule-based approach for representing interdependencies of knowledge units related to logical structure items. The rule-based system is realised as part of the FEPCore Module which will be described in more detail in the next section. 10.3.2

FEPCore

The FEPCore Module is the heart of the FEP and it implements a rule-based system to derive the asserted facts found in a book based on the OCR output (OCR XML

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result). The FEPCore module consists of several components outlined in the next sections. Rule Base The rule base is the knowledge representation layer and, as the basis for the rule engine or semantic reasoner, it is a temporary working memory containing the fact base and the conditional relations between facts. Facts State-of-the-art OCR engines, such as the ABBYY FineReader “know” much more about the image and the text than one would suppose by only considering the simple text output. As part of the recognition process an advanced layout and structure analysis process is carried out on the macro as well as the micro level of images and pages. As a result, most of the OCR engines available today usually provide information on the following document and text features: • Coordinates of regions and blocks. • Types of blocks, e.g. text, picture and table. • Coordinates of lines, their left and right indent, their baseline, their font size, etc. • Languages used in the document, and if words have been found in the internal dictionary of an OCR engine. • Coordinates and confidence for each single character. • Formatting information, like bold, italic or superscript, etc. All of these features are stored in the database and can be transformed on demand into Facts and handed over to the inference engine. Rules As a preliminary, one should keep in mind that from around 1500 to 2000, better printing technology produced no significant change in the structure of printed items. As a consequence, as far as the logical structure is concerned, a book from 1650 looks pretty similar to a modern book from 1980. Furthermore, book printing always was a normalised craft, so that the standards of how a book should look were quite determined. Several strict rules were set up in order to define the main features of a page and a printed text. This domain knowledge is encoded into declaratives rules with respect to the available facts from the OCR output. Rule Engine A rule engine is software which is able to derive new facts from given facts and ˙ FEP uses Jess as the rule engine, which rules, by a process called “reasoning”The

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has a Java’s API and can therefore be easily embedded into the Java-based FEP. In order to explain the core functionality of the rule engine, we provide the following example: • Boolean variable NP := There is a numeral in the first line of the page. • Boolean variable NC := The numeral in the first line of the page is centred. • Boolean variable PN := The numeral is a page number. if NP and NC then PN

(10.3)

Now let us assume that we add the following facts gained from the OCR result into the working memory of the rule engine: • The width of the page is 2,500 pixel. • There is a String within the first line with the value “17” which starts at horizontal position 1230 and ends at horizontal position 1280. The rule engine checks if some of the facts loaded into the working memory match the premises of the rule formulated above. The rule engine recognises that there is a numeral (17) within the first line of the page. Since the width of the page is 2,500 pixels even the second condition is fulfilled (17 is centred within the page). All the premises of our rule are fulfilled and the rule engine begins to execute the action defined in the THEN-part of the rule and asserts a new fact which states that 17 is a possible page number. 10.3.3

FEPController

The FEP Controller serves as the interface which communicates with the outside. It monitors the whole analysis process and allows users to employ or alter rules according to visual inspection. This means that the FEPController loads the OCR output on demand from the database and to convert this output into Facts which can be used by the rule engine. In a next step the controller has to load the correct rule set needed for the specific analysis step. As soon as the rule engine has finished the analysis the controller has to fetch the derived facts from the rule engine and to store them in the database. These facts may be a final output fact (e.g page number or print space) or an intermediate result fact which can be used by further computations. 10.3.4

The Fuzzy Rule-Based Approach

The use of a fuzzy logic in layout analysis is not new and has been studied by various authors in the last decade. (See [3], [4], [7]) In this section we will briefly outline how we would make use of a fuzzy rule- based approach in the FEP. As we have seen so far, in the area of layout recognition, we are dealing with different types of numerical properties, and rules are used to express logical interdependencies between assertions, as between the assertions “the text line is a headline” and “the distance to the next line is 25 pixels”. Generally, we assume that these kinds of interdependency are not well described by crisp sets with sharp

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transitions between different semantic properties that have a fine-grained numerical value domain (i.e. either continous valued or integer based with a reasonable set of values). Font size, for example, is a character property with the number of pixels as value domain. Let us assume that we distinguish between “small”, “normal” and “big” font size, and that a “big” font size means that the textline is the headline of the chapter. This is to say that the confidence in the conclusion that a line is a chapter headline depends on the confidence we have in assigning the label “big” to the mean font size of the corresponding text line. Another example is the property “line spacing” where the distance to the subsequent line together with the position of the line on the page is the basis for drawing conclusions regarding the label of a text line. From the OCR results of a sample data set containing the information about line positions and distances, we are able to find the distribution of values in pixels. Figure 10.2 displays the number of instances present in the sample data set for each single line distance value in pixels. The value distribution has multiple peaks which correspond to distinguishable layout items, like a headline and a paragraph line, for example.

Fig. 10.2. Line distance value distribution

On the basis of this distribution, our suggestion is to take the number of items for each pixel value as domain of the fuzzy membership function and take the local maxima as the basis for defining the core area of a membership function. Let us assume that we consider the division into “small”, “normal” and “big” as appropriate, we can use the statistical information to define the membership functions of these fuzzy terms like shown in figure 10.3.

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Fig. 10.3. Membership functions. Note that beginning and end of the core areas corresponds to the peaks shown in figure 10.2

An obvious question is: if we have statistics about the value distribution available, why do we not simply derive probabilities for each pixel value based on the value frequency? The answer is that we use the statistical information simply as the grounds for defining the linguistic terms which can also be modified. An expert in digitisation reviews the fuzzy terms created on basis of statistical information gained from the sample data sets for the various linguistic variables and adapts the membership functions when appropriate. Furthermore, the expert reviews rules, adds weighings, and generally guarantees that the rules are in compliance with his experience.

10.4

Concluding Remarks

In this paper we have provided an insight into two selected areas of the IMPACT project where uncertainty plays a major role. In a broad outline, we described some basic ideas on how a fuzzy rule-based approach could be employed for modelling uncertainty in each of these areas. As already stated, there is an implementation plan for the second application scenario, where we can expect concrete results regarding the appropriateness of the approach.

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Acknowledgment The IMPACT Research and Development work presented here is partially supported by European Community under the Information Society Technologies Programme (IST-1-4.1 Digital libraries and technology-enhanced learning) of the 7th framework programme - Project FP 7-ICT-2007-1.

References [1] Astrain, J.J., Villadangos, J.E., de González Mendívil, J.R., Garitagoitia, J.R., Fariña, F.: An Imperfect String Matching Experience Using Deformed Fuzzy Automata. In: Abraham, A., Ruiz-del-Solar, J., Köppen, M. (eds.) Soft Computing Systems - Design, Management and Applications, Hybrid Intelligent Systems: HIS 2002, Santiago, Chile, December 1-4, vol. 87, pp. 115–123. IOS Press (2002) [2] Gotscharek, A., Neumann, A., Reffle, U., Ringlstetter, C., Schulz, K.U.: Enabling information retrieval on historical document collections: the role of matching procedures and special lexica. In: AND 2009: Proceedings of The Third Workshop on Analytics for Noisy Unstructured Text Data, pp. 69–76. ACM, Barcelona (2009) [3] Klink, S., Kieninger, T.: Rule-based Document Structure Understanding with a Fuzzy Combination of Layout and Textual Features. IJDAR - International Journal on Document Analysis and Recognition 4(1), 18–26 (2001) [4] Kuiper, R., Wieringa, R.: Fuzzy Spatial Relations for Document Layout Analysis, Groningen (1999) [5] Levenshtein, V.I.: Binary Codes Capable of Correcting Deletions, Insertions, and Reversals. Soviet Physics - Doklady 10, 707–710 (1966) [6] Mendel, J.M.: Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions. Prentice-Hall, Upper-Saddle River (2001) [7] Palmero, S., Ismael, G., Dimitriadis, Y.A.: Structured Document Labeling and Rule Extraction Using a New Recurrent Fuzzy-Neural System. In: Fifth International Conference on Document Analysis and Recognition (ICDAR 1999), icdar, p. 181 (1999)

11 The Role of Synonymy and Antonymy in ’Natural’ Fuzzy Prolog Alejandro Sobrino

11.1

Introduction

Up till now, the most usual way to develop Fuzzy Prolog was adding degrees to facts or rules of classical Prolog. The main features of classical Prolog are unification and resolution. Resolution is the only rule of the resolution calculus, a version of the first-order predicate logic developed to automate reasoning. Unification is the procedure to match goals or sub-goals in order to progress in the resolution process. Resolution and unification are crisp in nature and based only on syntactic features: A Prolog interpreter resolve with full contradictions and match exclusively clauses that have the same predicates and arguments (at once or under appropriate substitutions). Traditional Fuzzy Prolog are mainly concerned with the flexibility of the resolution rule or the relaxation of the matching process in unification. Resolution rule triggers the search space for solutions isolating the propositional part of the question as a negated fact. In currently fuzzy Prologs, relaxation in resolution comes from the weakening of the hard negation (not_p = 1 − p). The use of negation functions in the frame of fuzzy set theory provides other alternative values to the only one provided by the hard negation. The flexibility in the matching process is reached through similarity functions, determining to what extent a predicate is interchangeable by other without a remarkable loss of meaning. Both interventions leads to manage some kind of fuzzy arithmetic, that deals with simple or complex degrees of truth, in order to obtain fuzzy qualified solutions to the posed goal. This view has deep theoretical bases in fuzzy logic. Pioneers have provided us with works about connectives and similarity functions. It deserves special mention the papers by E. Trillas about negation and indistinguishability functions [25]. Similarity functions received also contributions from other fuzzy researchers, as Dubois & Prade [6], Bouchon-Meunier et alii [2] or Cross & Sudkamp [3]. Traditional fuzzy Prolog nourishes up till now from these sources, providing it with theoretical support to explain and control the diversity of the fuzzy phenomena. But even if fuzzy Prolog is theoretically well founded, required adaptation to the peculiarities of computers in the problem solving task and implementing a fuzzy arithmetic module characteristic of the mechanical vague information processing are needed. Degrees of truth are a common way to represent the imprecision that vague statements show. It is a legacy of multiple-valued logic, which extends the semantic of R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 209–236. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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classical logic, focused on two values {1, 0}, to finite or infinite values represented in the closed interval [0, 1]. Thus, in traditional fuzzy Prolog we can say that p and p∗ match in degree x, being x the similarity between p and p∗ , or that ? − p fires a resolution process by virtue of his similarity with p∗ , being p∗ a fact or the head of a rule. But human beings act differently. It seems that people usually do not reason in a similar way as it does a Prolog machine. Questions tags seem to trigger an indirect proof, but using them we only want to ask a question to which we already know the answer – or we think we know the answer –, e.g. We need money to live, don´t we? But to progress in our discourse, let suppose that people and Prolog machine have a similar behavior. If so, machines should look for answers denying the goals in a human way. Negation in natural language is not always denoted by the prefix ’not’ – i. e., the negation of P is not always not_P –, but by a linguistic negation (the antonymy of P). Similarly, ordinary people match predicates with similar meaning without specifying the extent – the numerical degree – in what a predicate looks like another, but rather using their linguistic competence to check synonymy. Of course, in a lot of cases covering imprecision means to specify degrees and operate with them. But this procedure is not innocuous in a kind of fuzzy natural logic; it carries a debt. The cost is what is lost in the process to convert language to numbers (degrees) and, once operated, to reconvert numbers in words understandable by human beings. The efficient management of numbers contributes to a loss of meaning. Words and numbers are two poles of tension for fuzzy logic, a logic that deals with the way ordinary people argue. In this paper we will try to make a simple test of a natural language fuzzy Prolog; i. e., a Prolog that not only use a matching process based on a similarity function, but also on linguistic synonymy; that not only trigger resolution based on fuzzy negation functions, but rather employing linguistic antonymy. Sobrino & Fernández & Graña [23] have developed an electronic dictionary of synonyms and antonyms of Spanish language (DESALE), a software able to retrieve synonyms or antonyms of a word in its context (i. e., sensitive to senses). Sensitivity to senses is necessary to maintain consistency. Thus, if we ask for synonyms of wood as a piece of a tree (sense_1) and the system answers brassie (a golfing tool) (sense_2), the match would not be right as they do not share the same context. Thesaurus and other electronic linguistic tools, as Word-Net [17] or Concept-Net [15], can be useful resources for performing a more natural fuzzy Prolog. Nevertheless, our aim is not to substitute or revoke the standard fuzzy Prolog initiatives. We value the importance of fuzzy calculus as the typical way for performing imprecise reasoning using arithmetical facilities, but in order to bring fuzzy Prolog near to the way usually human beings argue, it seems appropriate to use linguistic relations instead of numbers. We are not ignorant that this naive approach will need further developments to provide large and consistent solutions. In this paper we just sketch a new way of approaching fuzzy Prolog resolution using linguistic tools as synonymy and antonymy. Thus, we organized this contribution as follows: The next section will be devoted to study of the fuzzy Prolog development, from the management of degrees to the specification of linguistics values. In section 11.3 we analyze some characteristics

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and properties of synonymy and antonymy, focusing our attention on antonymy as a way of negation that permits to obtain contradictions, and distinguishing between hard contradictions -by antonymy-, and soft-contradictions, due to negation and contrastive relations. In section 11.4 we sketch, using examples, a possible way to conduct a linguistic and approximate resolution based on synonymy and antonymy. Finally, a summary of the main conclusions and an outline of the future work will close this work.

11.2

Fuzzy Resolution

As Shönning points out in [27], the use of the resolution rule in classical Prolog is based on a complete functionally set of connectives; in particular, on the unambiguous definition of → in terms of (¬ ,∨), ¬p = 1 − p. Resolution rule says that if we have two clauses C1 = p → q (¬p ∨ q is its conjunctive normal form) and C2 = p, it is authorized to delete ¬p from C1 and p from C2 and resolve q. In scheme:

In [21], Rescher shows that there are not a single definition of implication in many-valued logics: thus, in the 3-valued logic of Kleene (K3 ) and Lukasiewicz (L3 ), implication have different formulas: (p →K q) = max(1 − p, q), (p →L q) = min(1, 1 − p + q). Note that while in K3 (p → p) is equivalent to (¬p ∨ p) when v(p) = 1/2, this equivalence does not holds in L3 . Therefore, in L3 there is not a generalization of the resolution principle, as it is not possible to turn implication formulas into a prenex form. But while L3 preserves classical laws (e. g., the identity principle) for intermediate truth-values, K3 does not. So, it would be interesting to have a version of the resolution principle for L3 . In order to overcome this problem, in [24] Tamburrini and Termini suggested an adaptation of the L3 definition of implication that contributes to solve this puzzle. Fuzzy logic, as multiple-valued logic, assume that there are different varieties of the resolution rule, but goes one step further in complexity by adding ways to value others than degrees. In the next subsections, we will show some modalities of fuzzy resolution that uses fuzzy values as: • single degree values, • intervals of degree truth-values, with or without threshold, • linguistic values. Next, we will describe – through examples – some of the more relevant fuzzy Prologs that use values in any of these modalities.

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11 The Role of Synonymy and Antonymy in ’Natural’ Fuzzy Prolog

Lee’s Resolution with Degrees of Truth

In [12] Lee proposed a resolution model for propositional formulas in prenex form, with variables qualified by degrees and the max, min and 1 − p logic for calculating complex truth-values using connectives. Example: Let C1 = ¬p ∨ q be and C2 = p. Let v(p) = 0.3 be and v(q) = 0.2. q is the resolvent of C1 and C2 . Then, v(C1 ∨ C2 ) = v[(¬p ∨ q) ∧ p] = min[max[1 − v(p), v(q)], v(p)] = 0.3 This example shows a remarkable difference between classical and Lee resolution: in classical resolution, if G (goal) is a logical consequence of Ci (clauses), v(G) ≥ v(Ci ), v(C1 ∧ C2 ) = min(v(C1 ), v(C2 )); instead, in Lee resolution it follows that v(G) = v(R(C1 ,C2 )) ≤ v(C1 ∧ C2 ), i.e., the value of C1 ∧ C2 (clauses) is not necessarily smaller than or equal to one of its resolvents. There is, nevertheless, an exception: if v(C1 ∧C2 ) ≥ 0.5, then max[v(C1 ), v(C2 )] ≤ v(R(C1 ,C2 )) ≤ min[v(C1 ), v(C2 )]. An example illustrates this: Example: Let be the clauses C1 = p ∨ q and C2 = ¬p. Let be v(p) = 0.2 and v(q) = 0.7. q is the resolvent. The joint truth-valued of the clauses is, v(C1 ∧C2 ) = min[max[v(p), v(q)], v(¬p)] = = min[max[0.2, 0.7], 1 − 0.2] = 0.7 v(C1 ∧C2 ) = 0.7 > 0.5. Note that, v(q) = v(R(C1 ,C2 )) ≤ v(C1 ∧C2 ); in effect, v(q) = 0.7 = v(C1 ∧C2 ) Thus, min[v(C1 ), v(C2 )] ≤ v(R(C1 ,C2 )) ≤ max[v(C1 ), v(C2 )] min(0.7, 0.8) ≤ 0.7 ≤ max(0.8, 0, 7); i.e., 0.7 ≤ 0.7 ≤ 0.8. This example verifies the following general principle: Let S be a set of clauses. If each clause in S has a truth-value greater than 0.5 and the clause with the greatest truth value has value a, logical consequences obtained applying repeatedly the resolution rule will reach a value at most equal to a, but never exceeding the value of the clause more feasible. 11.2.2

Atanasov & Georgiev’s Resolution with Intervals of Degrees of Truth

In [1], Atanasov & Georgiev suggested a type of resolution based on a type of logic with intervals of degrees of truth in which the negation of p as 1 − p not necessarily holds. Fuzzy propositions are represented by Inf-clauses (intuitionistic fuzzy   An inf-clause with  an inf-interval    matches the following pattern:  h clauses). μi , μsh , γih , γsh H : −B1 , ..., Bn μib , μsb , γib , γsb , H the head and B the body of

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the clause. Atanasov & Georgiev introduced a modality of resolution rule for interval Inf-clauses that carry two values, one of them representing the degree of truth and the other the degree of falsity. Example: Let P be the following program: Clauses: (1) (2) (3)

[0.6, 0.8][0.1, 0.2]d(X) : −p(X), l(X)[0.4, 0.7][0, 0.2]. [0.4, 0.7][0.1, 0.2]d(X) : −c(X)[0.5, 0.8], [0, 0.1]. [0.5, 0.8][0.15, 0.2]p(X) : −e(X), r(X)[0.3, 0.75][0, 0.2].

(4) (5) (6) (7) (8) (9) (10) (11)

[0.7][0.2]r(a). [0.8][0.2]l(a). [0.6][0.1]c(a). [0.6][0.2]e(a). [0.8][0.1]r(b). [0.9][0.0]l(b). [0.9][0.1]e(b). [0.3][0.6]c(b).

Facts:

and the question: ? − d(X) Following a resolution procedure based on this formulas: (1) μH = μih + αμ (μsh μih ) (2) γH = γih + αγ μ (γsh γih ) where αμ = μB − μib ÷ μsb − μib ifμsb > μib ; 1/2 otherwise

αγ = γB − γib ÷ γ μsb − γib ifγsb > γib ; 1/2 otherwise (÷ denoted division) the interpreter will answer: YES 0.5 : 0.15 X = a; no. To achieve this conclusion, Inf-Prolog acts so: The first inspected clause is 1, but the interpreter don’t fire the resolution process because the sub-goal l(X) can not be matched as μa = 0.8 and μb = 0.9 are out of the range of admissible truths ([0.4, 0.7]). The other alternative is to match the clause 2. Sub-goal c(a) has degrees of truth [0.6, 0.1], that holds in the intervals [0.5, 0.8][0, 0.1]. Thus, the interpreter calculates the value of d(X) accordingly to the rule. This is the calculus for the sub-goal c(a):

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μH = 0.4 + [0.6 − 0.5 ÷ 0.8 − 0.5] × (0.7 − 0.3) = 0.4 + 0.33 × 0.3 = 0.5   (Note that μsb < μib ).

γH = 0.1 + 0.5(0.2 − 0.1) = 0.15.   (Note that γsb < γib ). The interpreter offers d(A) as a result, specifying that the conclusion has a confidence range of ([0.50, 0.15]). 11.2.3

Weitgert et al.’s Resolution with a Threshold of Truth

In [26], Weigert et al. proposed a modification of the resolution principle based on the use of a threshold (ˆ). This feature introduces some particularities in the definition of negation and, consequently, in the deleting process of literals in resolution. Let C be a clause such that C is the disjunction of literals l1 A1 ∨ l2 A2 ∨ ... ∨ ln An , n ≥ 0. If for every li , Ai ∈ C, li ≥ 1 −ˆor for every li , Ai ∈ C, li ≥ˆthere is a substitution s such that A1 s = A2 s, ...., = An s, then Cs is a ˆ-clause. Let l1 A1 and l2 A2 be two literals. If there exist a substitution s such that (l1 A1 ∨ (1 − l2 )A2 ) is aˆ-clause unit, then we can say that they areˆ-complementary under s substitution. Weigert et al. suggested the following combination function of confidences λ . Let be λ1 , λ2 ∈ [0, 1]. λ1 ⊗ λ2 = (2λ1 − 1) × λ2 − λ1 + 1. Therefore, it is possible to define a resolution adapted to ˆ-operator based on this relative or threshold supported notion of complementary. Example: Let be the following clauses: C1 = 0.2A ∨ 0.9B and C2 = 0.98A ∨ 0.78A, and the threshold ˆ= 0.75. Thus, we can take as complementary all couple of literals with a value greater than 0.75 and smaller than 0.25. In this example, we can delete 0.2A and 0.78A, so that Rˆ(C1,C2) = 0.9B. Negation relative to a threshold is relevant to define refutation, because entail a new notion of inconsistency: A formula F isˆ-valid? iff 0F is (1 −ˆ) inconsistent. Example: Let C be = {1A, 1(0A → B), 0.6(0.1C → 1D)} and G = 1B ∧ 1D ∧ 0.43C. Does G follow from C? Two steps: 1. To obtain the prenex form C ∪ 0G, 1. 1A 2. 0A ∨ 1B 3. 0.42C ∨ 0.6D 4. 0B ∨ 0D ∨ 0.57C

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2. Usingˆ-resolution (ˆ= 0.6), we can infer, 5. 1B (1, 2) 6. 0D ∨ 0.57C (4, 5) 7. 0.42C ∨ 0.57C (3, 6) Therefore, C ∪ 0G is ˆ-inconsistent and C is (1 −ˆ)-valid. Dropping the threshold to 0.57, the step 7 resolves the empty clause. In this case, we can conclude that C is 0.57-valid. 11.2.4

Li & Liu Resolution with Linguistic Labels

Based on the normal definitions of linguistic hedges and the max, min, 1 − p fuzzy logic, Li & Liu proposed a resolution with linguistic labels in [13]. They distinguished three types of fuzzy clauses: • f-fact: H(xi ) : −[ f ] − ., i = 1, 2, ..., n, denoting that H(xi ) is true with truth value f or, alternatively, the confidence of atributing H to xi deserves a degree of belief f . • f-rule: H(xi ) : −[ f ] − Ti ., i = 1, 2, ..., n, the truth-value of H(xi ) is the min of Ti , i = 1, 2, ..., n. When i = 0, f = 1. • f-question: It has two modalities: o o

? − [ f ] − Ti., i = 1, 2, ..., n, ( f constant) What is the satisfaction degree of Ti , i = 1, 2, ..., n? ? − [F] − Ti ., i = 1, 2, ..., n, (F is a variable.) What is the linguistic qualification of the satisfaction of Ti , i = 1, 2, ..., n?

Example: Let the following program be: Clauses: has big shoes(X):−[0.9] tall(X). tall(mark):−[1]. tall(john):−[0.8]. tall(tom):−[0.9]. tall(bob):−[0.6]. Question: ? − [0.81]- has_big_shoes(tom). Yes Question: ? − [0.81]- has_big_shoes(bob). no Note that, for f-questions, the system provides boolean answers and that, for Fquestions, the interpreter will provide substitutions for variables (individuals, degrees of truth or both of them):

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? − [0.85]-has_big_shoes(X). X = mark; X = tom; no ? − [F]- has_big_shoes(tom). F = 0.90; no ? − [F]- has_very_big_shoes(X). X = mark F = 1; X = john F = 0.64; X = tom F = 0.81; X = bob F = 0.36; no As we can see, answers includes boolean and not boolean values. But this is not a distinctive with respect to other fuzzy Prologs previously described. In order to furnish an advanced fuzzy Prolog, Li & Liu extended the previous formalism to f l rules – fuzzy linguistic rules – : H : −[ fl ]Ti ., i = 1, 2, ..., n, where f l is instead of a linguistic truth value. Linguistic values are taken from the specification of the linguistic variable Possibility, showing the confidence in facts or rules: T(Possibility) = {definite, very possible, possible, reasonably possible,...}; shortly, {d, VERY_p, p, REASONABLY_p, ...}. For performing the calculus, numerical correlations are provided: d = 1, V ERY _p = 0.8, p = 0.7, etc. Authors assumed the usual definitions of linguistic hedges: μvery(x) = μ 2 (x), μextraordinarily (x) = μ 3 , etc. Next, we illustrate this idea with an example: Example: [_]-drive(X, big_car):−[0.9] -american(X). american(robert) : −[0.9]. ? − [μ ]-drive(X, big_car). X=robert; μ = 0.81. ? − [F]-drive (robert, big_car). YES; F = VERY possible Thus, depending on the question, the Lee et al. interpreter will offer an answer that is a boolean value, a degree of truth or a linguistic truth-value, although this later seems to be an artificial solution because it is a predefined translation of a numerical degree.

11.2 Fuzzy Resolution

11.2.5

217

Further Advances

Distinguishing ’No’ from ’Unknown’ Answers. Guadarrama’s et al. Approach Guadarrama et al. proposed in [7] a fuzzy Prolog with the aim to subsume in their proposal previous contributions. In my view, the main contribution of their approach is to combine crisp and fuzzy logic in order to highlight the difference between a negative answer (answer = 0) and an unknown answer (answer =?), distinction that collapses in classical Prolog. Example: Let be the following fuzzy rule (denoted by: ∼) teenage_student(X,V) :∼ student(X), age_about_15(X,V ). The question ? − teenage_student(X,V) should have the following answers: V =0 V =v

% if X do not match with ’student’. % if X match with ’student’ and we can calculate the value for age_about_15(X). Unknown, % if X match with ’student’, but we do not know anything about her or his age. As it is well known in Prolog, everything that is is not explicitly true, is false (by the Close Word Hypothesis, CWH). But between true and false hold other values. Ordinary knowledge and modern physics show facts that are not definitively true or definitively false. The approach by Guadarrama et al. contributes to overcoming this overlap, as the following example, with crisp and fuzzy facts added to the rule posed above, shows: student(john). student(peter). student(nick). student(susan). age_about_15(john, 1):∼. age_about_15(susan, 0.7):∼. age_about_15(nick, 0):∼. The goal ?-teenage_student(X, V)

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will convey the following answers: ?-teenage_student(john, V). V. = .1 ?-teenage_student(susan, V). V. = .0.7 ?-teenage_student(nick, V). V. = .0 ?-teenage_student(peter, V). no. This answers distinguish people we know their age – compatible or incompatible with age_about_15 – from those about which we do not know anything (Peter). 11.2.6

Distinguishing Symmetry and Asymmetry in Fuzzy Relations: Julián et al.’s Bousi-Prolog

Bousi-Prolog system, from P. Julián, C. Rubio and J. Gallardo [10], focus on the concepts of linguistic variable and term-set, highlighting that the fuzzy inclusion relation between linguistic truth-values are reflexive but not necessarily symmetric or transitive. This is based on the following intuition: let Age be a linguistic variable and young (t1 ) and between_19_22 (t2 ) two terms in T (Age). It is clear that the meaning of t2 is included in the meaning of t1 , but the converse does not hold. Linguistic variables as Age can be represented in Bousi-Prolog by a directive, defining a list of fuzzy subsets whose universe of discourse is the domain age. : − f uzzy_set(age, [young(0, 0, 30, 50), middle(20, 40, 60, 80), old(50, 80, 100, 100)]). For generating fuzzy relations between linguistic terms, we have to take into account the distinction between general and specific knowledge. Example: Let be a simple program containing only one fact: person(mary, age#28). If we launch the goal: ? - person(mary, young) an admissible answer should be Yes with degree α , where α = μyoung (28) = 1.0 Resolution principle is able to provide this answer if the original program is completed with the entry R(young, age#28) = μ _young(28). Now assume another program with the only fact person(mary, young). If we launch the goal ? - person(mary, age#28)

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it should expect a positive answer with a degree β lower than the obtained in the previous example, because although a 28 years old person must be considered as young, the specific knowledge of ’young’ involves other age values besides 28 years old. For preserving the rationale that the degree β is lower than μyoung (28), Bousi Prolog system fuzzify the domain point age#28(x) = 1 into a singleton fuzzy set characterized by a trapezoidal function f (x, 28, 28, 28, 28) = μage#28 (x) = 1 if x = 28 and μage#28 (x) = 0 if x = 28. An adequate matching function R(age#28, young) = matching(μyoung, μage#28 ) provides that value. Those are some solutions supported by several fuzzy Prologs, progressing from early approaches, based on the management of single truth degree values, to more advanced proposals, that attempt to deals with linguistic truth values. Traditional fuzzy Prolog made progress moving from degrees to linguistics values. But in both cases the underlying mode of resolution are numbers, degrees or fuzzy numbers. In the surface, answers qualified with linguistic values are friendly than answers qualified by a degree. But at the far end, all are numbers. In order to overcome this drawback, next we propose some ideas concerning a genuine linguistic f -prolog.

11.3

Towards a Linguistic Fuzzy Prolog: Linguistic Relations of Synonymy and Antonymy

The purpose of this section is to approach a fuzzy Prolog that solves conclusions from facts and rules using linguistic sources. If facts or rules include vague predicates, resolutions will be imprecise or approximate in nature. In daily life, human beings frequently argue in inexact environments. People reason employing logical rules but also using natural language tools, as linguistic quantifiers or predicates. Aristotelian syllogistic is an example of this. Resolution calculus is a kind of reasoning adapted to computers, but it is possible to adjust linguistic sources employed in a natural style reasoning to a more formal frame, as resolution presupposes. Some well-known linguistic tools linked with the presence of similarity in natural language are synonymy, antonymy, hyperonymy or hyponymy, the last two related with borderline cases of vagueness, as generality or specificity. Our proposal will attempt to introduce linguistic relations in a fuzzy resolution frame, in order to advance towards a kind of ’natural’ fuzzy Prolog. The following section is devoted to analyze some aspects related with vague or imprecise relations between words, as those previously quoted. We will mainly focus on synonymy and antonymy, the last susceptible to be used as a resolution mechanism. 11.3.1

Synonymy

From linguistics, a couple of words are synonymous if they are identical or very similar meaning. However, as Thesauri show, the vast majority of synonyms words denote approximate instead of sameness in meaning.

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In [16] Lyons distinguishes absolute synonymy, complete synonymy, descriptive synonymy and near-synonymy. In his view, two words are considered complete synonymous if they have identical descriptive, expressive and social meaning in the range of the given contexts. He admits that complete synonymy is rare and that absolute synonymy hardly exists. Taken as a crisp relation, the only true synonyms would be terms having precisely the same denotation, connotation and range of applicability. Perhaps only in technical contexts, as in botany, we can found pairs of words – as e.g., oak tree and quercus – approaching a complete synonymy. It is a matter of fact that, in the vast majority of cases, there are not two words identical in every aspect of meaning so that they can be applied interchangeably. English is a language rich in synonymous: car-automobile, smart-intelligent, babyinfant, student-pupil, and so on. Traditionally, synonymy can only hold between words and, more precisely, between words belonging to the same part of speech: enormous-huge; gaze-stare. Words susceptible to be synonymous are nouns, adverbs or adjectives. So, we can consider that synonymous terms are those having not the same, but nearly the same denotation. It is usual that speakers can choice among a set of words different in form but similar in denotation. So, for denoting a certain landform we can use shore (from Old English), coast (from Latin), or littoral (from Latin). Choose one word or another rest frequently in a stylistic option: to use simpler vs. complex; formal vs. informal terms. The fact that these terms share the same denotation makes them available as substitutes, allowing to choose the more precise, the more colorfully or, simply, any of them to avoid repetition. In [4] Cruse suggests the ’normality test’ as a way of determining the absence of absolute synonymy. This test is based on the observation that one of the two terms is normal in a given context and the other less normal, as (1) and (2) show: (1) He told me the concert starts at 20.00 (+ normal) (2) He told me the concert commences at 20.00 (- normal) But if we add: (3) He told me the concert begins at 20.00 and compare it with (1), it is difficult to attribute greater normality to one or the other. In this case, synonymy is affected by similarity or indistinguishability. Palmer in [19] suggests other trial for synonymy. He proposes to substitute one word for another and checking (i) mutually interchangeability in all contexts -for absolute synonymy – (ii) the overlaped set of antonyms -for relative synonymy –. For instance, superficial is the opposite of deep and profound, but only shallow is the opposite of deep. Thus, superficial and shallow are synonyms in a degree.

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It is possible to find synonyms automatically using the distributional hypothesis from Harris [9]. This procedure suggested that words with similar meaning tend to appear in similar contexts. The following list shows the top-20 distributionaly similar words of adversary, using the Lin et alii’s method [6][14] on a 3GB newspapers corpus. Adversary: enemy, foe, ally, antagonist, opponent, rival, detractor, neighbor, supporter, competitor, partner, trading partner, accuser, terrorist, critic, Republican, advocate, sleptic, challenger. Compared with manually compiled thesauri, distributionally similar words offer better coverage. Thus, in Webster Collegiate Thesarus, adversary has only the following related words: Adversary: Synonyms: antagonist, anti, con, math opposer, oppugnant; Related words: assaulter, attacker; Contrasted words: backer, supporter, upholder; Antonyms: ally. But distributionally procedure has a deficiency: some of the collected words are antonyms; e. g., ally and supporter in the above example. To distinguish synonymy from antonymy, Lin et al. proposed to check the following phrasal patterns as indicative of antonymy: • from X to Y • either X or Y Thus, if two words X and Y appear in once of these patterns, they are very likely to be semantically incompatible. Lin et al. confirmed this hypothesis putting some queries in a web searcher. The right column shows the number of pages retrieved for each query: Table 11.1. Queries in a web searcher and number of pages retrieved for each query. Query adversary NEAR ally “from adversary to ally” “from ally to adversary” “either adversary or ally” “either ally or adversary” adversary NEAR opponent “from adversary to opponent” “from opponent to adversary” “either adversary or opponent” “either opponent or adversary”

Hits 2469 8 19 1 2 2797 0 0 0 0

Note that with the NEAR proximity operator, the target words appear together in a lot of pages, but the number of records tends to be small if words are embeded in the squemes pointed above. 11.3.2

Antonymy

Two words are antonymous (from the Greek anti (“opposite”) and onoma (“name”)) if they are opposite in meaning. Opposite and similar meaning are related: Although

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terms that are semantically similar (plane-glider, doctor-surgeon) are also semanticaly related, terms that are semantically related may not always be semantically similar (plane-sky, surgeon-scalpel). Thus, surgeon is a type of doctor, but sky and plane, as isolated words, do not share any meaning. Antonymous concepts are semantically related but not semantically similar (black-white); thus, following Cruse we can distinguish antonymy from other semantic relations (synonymy, hyponymy,. . . ) because it shares both a sense of closeness and of distance. Antonyms are opposites and lie at the ends of a continuum: boiling-freezing / boil-hot-warm-temperate-mild-cool-freeze. boil-freeze are perhaps contradictories, but also, although in a lesser extent, boiling-cooling. So, antonymy is a matter of degree. Antonymy denotes contradiction. In order to detect contradictions is useful to ask if affirmation and negation are opposites in meaning. In the gradation of negation we distinguish antonymy, negated and contrastive relations. Antonymy pointed at contradiction; negated and contrastive relations, in turn, determine contraries. Let’s go back to something we have said in the previous section: the negation of the goal is what fires resolution. Negation differs in meaning with affirmation, but there are more varieties of negation that those marked by interrogative particles as not or nor. Not is only a way for denying, but some predicates are denied using antonyms or contrastives. So, we can distinguish three types of negation: • Antonymy, characterized by the opposite poles of a continuum. • Negated, marked by the use of prefixes or negative particles. • Contrastives, characterized by words more dissimilar that similar. Antonymous without lexical relations between them occur frequently in text (hotcold, dark-light). On the contrary, negatives are located checking prefixes such un(clear, un-clear) and -dis (honest, dis-honest). The following table lists sixteen morphological rules for obtaining the negated term. Table 11.2. 16 morphological rules for obtaining the negated term. w1 X X X X X X X X

w2 abX antiX disX imX inX malX misX nonX

example pair normal-abnormal clockwise-anticlockwise interest-disinterest possible-impossible consistent-inconsistent adroit-maladroit fortune-misfortune aligned-nonaligned

w1 w2 example pair X unX biased-unbiased lX illX legal-illegal rX irX regular-irregular imX exX implicit-explicit inX exX introvert-extrovert upX downX uphill-downhill overX underX overdone-underdone Xless Xful harmless-harmful

In [18], Mohammad et al. apply these rules to the words of the Macquarie Thesaurus, generating 2.734 word pairs. He founded that not all of the achived pairs are antonymous (sect-insect, coy-decoy), although exceptions are few in number.

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People consider certain contrasting pairs of words to be more antonymous (largesmall) – contradictories – than others (large-little) – opposites –. Thus, for speakers antonymy is not an all/none phenomenon, but a matter of degree. We can say that there are different degrees of antonym, as shows this categorization: • pair of words antonymous (hot-cold, white-black, inferior-superior) • pair of words one is the negated of the other (legal-illegal, interest-disinterest) • pair of words semantically contrasting (inferior-first_class, enemy-fan, idealwrong) • pair of words not antonymous at all (cold-chilly, goose-clown, ideal-rational) This gradation is theoretically founded. In [4] Cruse points out that if the meaning of the target words is completely defined by one semantic dimension and the words represent the two poles of it, then they tend to be considered antonyms. Other favorable note is the co-ocurrence of antonyms in a sentence more often than chance. More semantic dimensions or added connotations are unfavorable to diagnose antonymy. Let remember us that negations, in any of its forms, denotes words that are semantically contrasting and, therefore, contradictories in meaning – Santiago de Compostela has a predominantly rain weather/It is mostly dry in Santiago de Compostela, I always liked the paella/I never liked the paella. But contradictions are not only related with lexicon (rain/dry, always/never), but with inference. The use of textual inference to discover contradictions leads also to identifiy and remove negations of propositions and testing for textual entailment. Harabagiu et al. in [8] suggested that we can detect contradictions in an argument ckecking if the conclusion follows or not from premises. Thus, the conclusion John was not lucky the first two days of this week are contradictory with the premises Monday, John bet and won 5000 dollars and Tuesday, John bet again and won 5000 dollars. but or although are also words used to diagnose contradiction in text (But/Although John was not lucky the first two days of this week) Contradiction is used for performing resolution in Prolog. The modality of reasoning and proof known as Reductio ad absurdum exemplifies the use of contradictions as a tool for proving conclusions. According to Rescher in [22], Reductio ad absurdum leads to three progressively weaker modalities of contradiction: • Self-contradiction (ad absurdum) • Falsehood (ad falsum) • Implausibility (ad ridiculum) Self-contradiction is syntactic in nature and matches with the logic falsum (⊥): A contradiction states that p and ¬p holds at the same time. Falsehood involves a semantic definition of negation (¬p); in classical logic ¬p = 1 − p. In this sense, a contradiction means that p is true and false at the same time. Implausibility refers to an apparent, but not definitive incompatibility of p and ¬p, as ¬p may involve the opposite, not the contradictory, of p. In this last sense, a soft contradiction means that p and not-p are contrastive. In a linguistic context, contradictions happen when information included in two different texts are incompatible, i.e., when a sentence

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contains a predicate and its antonym, a predicate and its negation or a predicate and its contrastives. Next, we will focus on synonymy and antonyms as a way to perform a kind of more natural resolution procedure in order to look for answers to Prolog questions.

11.4

Resolution with Synonymy and Antonymy

If we wish a fuzzy Prolog based on the synonymy and antonym, we should provide a resolution principle that, instead of resolving with numerical negations functions, use linguistic resources as antonymy, and that instead of requiring a degree of matching, allow the semantic overlapping provided by synonymy. Our proposal has resemblance with the intuition under the Aristotelian syllogistic reasoning. Aristotelian figures – as e.g., Barocco: If every G is M and a P is M, then P is a G –, outline a sound reasoning using only pieces of language, not numerical substitutes. To avoid dealing with alternatives of language, be formulas or numbers, may seem strange. Although numbers are more precise than language, they are also less natural. And for certain purposes we may not require more precision than that the common language terms convey. As Popper said in [20], to wish more precision that the situation required, even in a scientific scenario, is a mistake. Computing with words leads to employ available linguistic resources. Classical Prolog uses only the spellings of the words in the matching process, not their meaning; thus, the use of language in resolution is purely lexical and linguistically poor. Sources containing linguistic knowledge, such as dictionaries, may introduce semantic in the resolution process and a fuzzy Prolog should benefit from it. Next, a genuinely linguistic fuzzy Prolog is suggested based on the concepts of synonymy and antonymy. Our approach is naïve and guided by examples; thus, we are aware that a lot of work remains still to do. Synonymy allows flexible matching. The use of antonymy in the resolution process is more complex. We will refer to it mainly in the area of resolution with linguistic negation, but also introducing other two very small applications: (i) to the detection of inconsistencies in databases; (ii) managing linguistic negations in the heads of the clauses, helping to perform a defective Prolog. A linguistic fuzzy Prolog should work with an associated eThesaurus, which passes on to the interpreter when two terms are synonyms or antonyms. e-Thesaurus should be sensitive to senses. 11.4.1

The Role of Synonymy

We start showing classical facts and rules that will be modified later using synonyms and antonyms.

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Let be the following program: like(mary, rock). like(mary, pop). hire(paul, X):like(X, pop). The program says that ’Mary likes rock’, ’Mary likes pop’ and that ’Paul hire all people who like pop’. Let now the question be: ?-hire(paul, Y). The interpreter translates this query to the formula: ∃y¬hire(paul,Y ). By Modus Tollens, the goal ¬hire(paul,Y ) is reduced to the sub-goal ¬like(Y, pop) − −X matches Y , as they are two variables –. Checking the database, we find that like(mary, rock); so, the interpreter find a contradiction under the substitution [Y/mary] and resolves the empty clause suggesting the answer: Y=mary; No Schematically, this is the Prolog reasoning process: Lets F be the set of clauses: F = {{like(mary, rock)} , {like(mary, pop)} , {hire(paul, x), ¬like(x, pop)}}, and the query Q : {¬hire(paul, y)}. The following diagram shows the process for obtaining the derivation of the empty clause:

Fig. 11.1. Process for obtaining the derivation of the empty clause.

Lets look at some aspects of a resolution based on synonymy. Synonymy affects specially to the matching process between predicates that are similar in meaning. In the searching process, Prolog denies the main objective and look for a predicate in the database that exactly matches with it. But in an approximate scenario, the

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negated predicate can match precisely with the same predicate or with some other predicate similar in meaning. This is the place where synonymy comes in. Next, we will suggest a toy-program to exemplify the relation between synonymy and resolution. We start extending the previous program using synonyms of the predicate ’like’, as ’enjoy’, ’love’ or ’feel’. New clauses are: love(anna, pop). enjoy(carol, pop). Let suppose that an e-thesaurus provides sense-sensitive synonyms of ’hire’ and ’like’ and, optionally, calculates the degree of synonym between them. sin(hire, Sense_A, contract, Sense_A, 1). sin(like, Sense_A, enjoy, Sense_A, α ). sin(like, Sense_A, love, Sense_A, β ). where Sense_A denotes sense (context) and 1, α , β , γ , ... denotes degrees of synonymy. Let see the role of synonym in several phases of resolution: • In questions that doesn’t match with the head of the rule. Let be the following question: ?- contract(paul, Y). The interpreter generates the formula: ∃y¬contract(paul,Y). which does not match with the head of the rule. Thus, the resolution does not fire. But if the interpreter access to an e-thesaurus, this should detect that: sin(hire, Sense_A, contract, Sense_A, α ). matching ?- contract(paul, Y) with hire(paul, Y) - the head of the rule – and thus executing resolution with a confidence degree relative to α ; i.e., the degree in which hire and contract are synonyms in the appropriate senses – contexts –. • In sub-goals able to match with the tail of the rule. Let the question be ?-like(paul, Y). It matches with the head of the rule, triggering the resolution and redirecting the satisfaction of the goal to the sub-goal like(Y, pop). Retract now the clause like(mary, pop) while remaining asserted love(anna, pop) and enjoy(carol, pop). The question ?-hire(paul, X). will be answered No by the interpreter; response which it is not in consonance with common sense. But using: sin(like, Sense_A, love, Sense_A, α ). sin(like, Sense_A, enjoy, Sense_A, β ). the sub-goal matches with the tail of the rule, obtaining Y = annaα ;Y = carolβ , α and β are the degree of similarity between ’like’ and respectively ’love’ and ’enjoy’ in the appropriate senses.

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• In sub-goals whose arguments do not match with the arguments of the question. Let the previous program be plus the following new rule: hire(paul, X):love(X, party). and the following clauses added: love(donna, cocktail). enjoy(emma, orgy). If the question is, ?-hire(paul, X). a classical Prolog interpreter reply No. But this answer, other time, is not in consonance with common sense. Note that although the predicate of the goal and the predicate of the sub-objective are the same (by simple matching or by synonym); the interpreter do not offer any answer because does not match the second argument of the tail. If the interpreter had the following added information: sin(party, Sense_A, cocktail, Sense_A, τ ). sin(party, Sense_A, orgy, Sense_A, υ ). then should answer: X=donnaτ ; X=emmaυ where τ indicates the degree of synonym between party and cocktail and υ degree of synonym between party and orgy. • In sub-goals where some of the argument is an exemplar of a class (hyperonymy) Let the following added rule be: has_affinity(paul, X):like(X,citrus). and the following facts: like(candy, grapefruit). like(bonnie, lemon). The question: ?-has_affinity(paul, X).

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is answered by the interpreter with No because the second argument of the subgoal like do not matches with any argument of any fact. But this conclusion is not reasonable. An ideological dictionary should detect that citrus is the name of a class of fruits including lemon or grapefruit. Let suppose that we have available this information as facts: is_a_type_of(lemon, citrus). is_a_type_of(grapefruit, citrus). is_a_type_of(orange, citrus). Reformulating the previous rule as: has_affinity(paul, X):like(X, Y), is_a_type_of(Y, citrus). the interpreter should answer: X=candy; X=bonnie; No answers now according to our intuitions. If some fruit is considered a prototype (orange, lemon) of a class (citrus) or an exemplar less representative of it (grapefruit), is_a_type_of (A, B) it should change its arity to is_a_type_of (A, B, α ), where α is the degree in which A represents B. If A is a prototype of B, α = 1. So, the interpreter should provide answers qualified by degrees. 11.4.2

Resolution Based on Antonym

Kim et al. noticed in [11] a fuzzy Prolog based on the linguistic notion of antonymy. Nevertheless, their contribution should not be strictly characterized as ’linguistic’ because antonymy was reduced to strong negation (i.e., v(¬p) = 1 − v(p)). Up till now, the diverse modalities of f-Prolog use single degrees or intervals degrees, with or without thresholds. Attending to this fact, it would be more appropriate to name it ’multiple-valued Prolog’ instead of ’fuzzy linguistic Prolog’. A resolution process aspiring to have a truly linguistic nature must deals with words, not numbers. This section is devoted to insinuate a possible way to perform this. The implementation of antonymy in resolution is complex. As we previously seen, antonymy is related, roughly speaking, with several modalities of negation. In classical logic, ¬p = 1 − p and, in fuzzy logic, negation is manifold and it depends on the negation function chosen. But in a linguistic frame, the negation of a

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predicate is denoted by a negative particle, as not or never, a contrastive predicate or an antonym. A word may have one antonym, none or more than one. Thus, Thesaurus.com returns downfall as the only antonym of prime (sense: best part of existence) and evil, immoral, noxious as antonyms of good. But there is not an antonym for green (sense: colour) and it is also debatable which is, if any, the antonym of middle age. In this section we deal with the use of negation that covers antonymy, negated and contrastive predicates. As previously, we show our ideas through examples. These examples use synonymy too, because although the antonymy is what fire resolutions, synonyms of antonyms may be decisive in determining the scope of the trigger through matching and unification. Resolution with Different Linguistic Negations Let be the program: love(mary, rock). like(mary, pop). love(anna, pop). enjoy(carol, pop). hire(paul, X):love(X, pop). and the question ?-hire(paul, X). A Fuzzy Prolog interpreter translate this question to not hire(paul, X) - in formula, ¬hire(paul,Y ) -. Let suppose that a Prolog interpreter argues as human beings. It should convert the predicate of the question in its negation. But, in natural language, the negation of a predicate is not always denoted by the particle not, as classical Prolog does. Let suppose that Reductio ad absurdum proof starts with a hard-contradiction as the assumption the goal does not hold. So, a human being would convert ¬hire in its antonym: lay_off (lay_off (paul, X )). Due to the database may include the predicate lay_off(_,_), the linguistic interpreter should inhibit all the possible answers possibly obtained by a direct matching with this predicate. Once done, the goal is derived to the subobjective neg(love(X, pop)), denoting neg antonymy, negative or a contrastive words. Let suppose that for love we have the following gradation, that moves from the two poles in which love and its antonym, hate, hold: Table 11.3. Gradation for the word love. love like enjoy forget neglect disregard dislike hate

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So, neg(love(X, pop)) may adopt some of this linguistics equivalents: hate(X, pop) antonym dislike(X, pop) negated disregard(X, pop) negated disrespect(X, pop) negated neglect(X, pop) contrastive forget(X, pop) contrastive Thus, it is possible to oppose love to hate (antonyms), love to dislike, disregard or disrespect (negated words) or, finally, love to neglect or forget (contrastive words), marking progressively weaker degrees of inconsistency. The steps in order to perform a linguistic resolution based on linguistic negation are: • Search in the text the goal predicate involved in the query. • Search in the text its antonym. • Inhibit (denoted by a dash line in the following figure) the possible answers obtained by direct matching with the antonym predicate. • Takes back the search to the sub-objectives. • If sub-objectives are facts, look for inconsistencies absolutes or relatives. • If sub-objectives are rules, negate the sub-goals predicates until the sub-goal that matches with facts are reached. • Look for inconsistencies absolutes (based on antonymy) and relatives (based on negated and contrastive predicates). • Resolve with three levels of confidence: high for resolutions under antonymy, medium for resolutions under negated predicates and low for resolutions under contrastive predicates. The following example illustrates these ideas. Let be the following program: regard(laura, pop). lay_off(paul, ringo). like(mary, pop). love(anna, pop). like(katy, pop). neglect(harry, pop). dislike(katy, pop). hate(ana, pop). enjoy(carol, pop). hire(paul, X):love(X, pop). and the question: ?-hire(paul, X).

11.4 Resolution with Synonymy and Antonymy

This is the search space of solutions:

Fig. 11.2. Search space of solutions.

231

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being α the level of contraposition between love and hate,being υ the level of contraposition between love and dislike and being ω the level of contraposition between love and neglect. Arrows marked with 1, 2, and 3 denote alternative ways for obtaining answers using the different modalities of ’neg’. Thus, for the question ?-hire(paul, X) the interpreter should be provide the following answers: X = annahigh ; X = katymedium ; X = harrylow ; No Antonymy for Diagnosis of Inconsistencies Resolution based on antonymy may be used for debugging inconsistencies in databases. Suppose we have the above program with the following clause added: dismiss(paul, mary). If we ask for: ?- dismiss(paul, X). The interpreter will answer: X=mary; No Next, if we ask: ?- hire(paul, X). The interpreter will answer again: X=mary; No That is, Paul hires and dismisses Mary and this is a kind of contradiction. A classic Prolog has no way to avoid it, because it do not know that dismisses is opposite in meaning to hire. It only knows that both are different words. An antonymy based Prolog can contribute to solve that paradox. Indeed, if we ask: ?-hire(paul, X).

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the interpreter has two possible pathways. First, find an answer to the question with the substitution X=mary. Second, detect that dismiss is an antonym of hire, inhibiting any substitution allowed by substitution. Thus, we can conclude with the following moral: if for predicates that are antonyms two routes of resolution provides the same answer, then the database is inconsistent. Antonymy Supporting Negations in the Heads of the Clauses Let the following program be: attend_mass(charles). attend_mass(camila). not_atheist(X):= attend_mass(X). where := denotes a defective rule; i. e., a rule that states that people usually go to church is not an atheist. To ask for an answer, the interpreter should admit negated queries, i.e. ?-not_atheist (X). % Who is not an atheist? reducing the search to the satisfaction of the subgoal attends_mass(X). But, (i) it is uncommon to ask using denied questions, (ii) there are formal settings, as the intuitionist, in which the principle of double negation does not hold – in this case, the resolution would be triggered by the formula ¬(¬ atheist(X))-. Turning the symbolic negation into its linguistic forms help to circumvent this problem. It is known that everyone who is not an atheist is a believer or an agnostic, although ’believer’ is more opposite in meaning than ’agnostic’ to ’atheist’. The knowledge that antonym(atheist, believer) and contrastive(ateist, agnostic) leads to formulate the following rule that captures this fact. not atheist(X):believer(X); agnostic(X). So, the defective rule may be represented by these two new rules: believer(X):= attend_mass(X). agnostic(X):= attend_mass(X).

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the first one offering a high level of confidence that the second one. Suppose now that the program includes a qualification of the facts as, e.g.: attend_massalways(charles). attend_masssometimes (camila). then, the first fact will contribute in a larger extent than the second to satisfy the head of the first rule and, similarly, the second fact will contribute more than the first one to satisfy the head of the second rule.

11.5

Conclusions

This paper deals with a new, although still rudimentary, approach to a fuzzy linguistic Prolog based on antonymy as a way to perform a fuzzy resolution and synonymy as a tool for imprecise matching. First, we presented the basics of Prolog programming language. Next, some of the most representative fuzzy Prologs are described, focusing on the nature of the truth values adopted in the resolution process. We showed several approaches that makes resolution with degrees of truth, intervals of degree of truth -with or without a threshold- or, finally, with linguistic truth-values. Some recent advances were also pointed out. Next, synonymy and antonymy were tackled and some of its main linguistic properties described. Finally, we proposed a naive approach of a kind of genuine linguistic Prolog based on synonymy and antonymy. Our exposition was organized around examples. A lot of work remains to be done in order to have a founded approach, but we think that this paper faced up to some initial concerns. The answers provided by antonymy in the resolution process can be obtained using classical resolution and synonymy. Thus, why to use antonymy? We think that a Prolog will be genuinely fuzzy when the resolution be fuzzy. And that it will be genuinely linguistically fuzzy when resolve using vague linguistics tools, as antonymy or other linguistic modalities of negation, as negated or contrastive predicates. Human beings not always, or even frequently, negate using not. So, it seems convenient to expand linguistic negation to alternative forms as previously seen. Our aim is to approach the possibility to automatically perform fuzzy reasoning in texts. A text may include phrases susceptible to be mechanically translated to clauses. If a program detect two sentences one of them having a predicate that is the negation (antonym, negated or contrastive) of the other, a fuzzy resolution process may be automatically fired and perhaps new knowledge inferred. This is one of the challenges pointed out by Zadeh in [28] to increase the deductive capabilities of the current searchers, so they could answer questions as ’What is the second longest river in Spain?’

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Acknowledgments This work has been partially supported by the Spanish Ministry of Science and Innovation (MICINN, TIN2008-06890-C02-01) and by the Spanish Ministry of Education (MEC, HUM2007-66607-C04-02) projects.

References [1] Atanassov, K., Georgiev, C.: Intuitionistic Fuzzy Prolog. Fuzzy Sets and Systems 53, 121–129 (1993) [2] Bouchon-Meunier, B., Rifqi, M., Bothorel, S.: General measures of comparison of objects. Fuzzy Sets and Systems 84(2), 143–153 (1996) [3] Cross, V., Sudkamp, T.A.: Similarity and Compatibility in Fuzzy Set Theory. PhysicaVerlag, Heidelberg (2002) [4] Cruse, D.A.: Lexical Semantics. CUP, Cambridge (1986) [5] De Soto, A.R., Trillas, E.: On Antonym and Negate if Fuzzy Logic. International Journal of Intelligent Systems 14, 295–303 (1999) [6] Dubois, D., Prade, H.: Similarity-based approximate reasoning. In: Zurada, J.M., Marks, R.J., Robinson, C.J. (eds.) Computational Intelligence Imitating Life, pp. 69–80. IEEE Press, New York (1994) [7] Guadarrama, S., Muñoz, S., Vaucheret, C.: Fuzzy Prolog: a new approach using soft constraints propagation. Fuzzy Sets and Systems 144, 127–150 (2004) [8] Harabagiu, S., Hickl, A., Lacatusu, F.: Negation, Contrast and Contradiction in Text Processing. In: Proceedings of then 23rd National Conference on Artificial Intelligence. AAAI, Boston, Mass (2006) [9] Harris, Z.S.: Distributional structure. Word 10(23), 146–162 (1954) [10] Julián-Iranzo, P., Rubio-Manzano, C., Gallardo-Casero, J.: Bousi Prolog: a Prolog Extension Language for Flexible Query Answering. Electronic Notes in Theoretical Computer Science 248, 131–147 (2009) [11] Kim, C.S., Kim, D.S., Park, J.S.: A new fuzzy resolution principle based on the antonym. Fuzzy Sets and Systems 113, 299–307 (2000) [12] Lee, R.C.T.: Fuzzy Logic and the Resolution Principle. Journal of the Association for Computing Machinery 19(1), 109–119 (1972) [13] Li, D., Liu, D.: A Fuzzy Prolog Database System. Research Studies Press and John Wiley and Sons (1990) [14] Lin, D., Zhao, S., Qin, L., Zhou, M.: Identifyng Synonyms among Distributionally Similar Words. In: Proceedings of then IJCAI, pp. 1492–1493 (2003) [15] Liu, H., Singh, P.: ConceptNet: A practical commonsense reasoning toolkit. BT Technology 22(4), 211–226 (2004) [16] Lyons, J.: Language and Linguistics: An Introduction. CUP, Cambridge (1981) [17] Miller, G.A., et al.: WordNet: An online lexical database. Int. J. of Lexicograph 3(4), 235–244 (1990) [18] Mohammad, S., Dorr, B., Hirst, G.: Computing Word-Pair Antonymy. In: Proceedings of the Conference on Empirical Methods in Natural Language Processing, Honolulu, Hawaii, pp. 982–991 (2008) [19] Palmer, F.R.: Semantics: A New Outline. CUP, Cambridge (1977) [20] Popper, K.R.: Conjectures and Refutations: The Growth of Scientific Knowledge. Routledge Classics (1963)

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[21] Rescher, N.: Many-valued Logics. McGraw-Hill (1969) [22] Rescher, N.: Reductio ad absurdum, The Internet Encyclopedia of Philosophy (2006), http://www.iep.utm.edu/reductio/ [23] Sobrino, A., Fernández-Lanza, S., Graña, J.: Access to a Large Dictionary of Spanish Synonyms: A tool for fuzzy information retrieval. In: Herrera, E., Pasi, G., Crestani, F. (eds.) Soft-Computing in Web Information Retrieval: Models and Applications, pp. 229–316. Springer, Berlin (2006) [24] Tamburrini, G., Termini, S.: Towards a resolution in a fuzzy logic with Lukasiewicz implication. In: Valverde, L., Bouchon-Meunier, B., Yager, R.R. (eds.) IPMU 1992. LNCS, vol. 682, pp. 271–277. Springer, Heidelberg (1993) [25] Trillas, E., Valverde, L.: An Inquiry into Indistinguishability Operators. In: Skala, H.J., Termini, S., Trillas, E. (eds.) Aspects of Vagueness, pp. 231–256. Reidel (1984) [26] Weigert, T.J., Tsai, J.-P., Liu, X.: Fuzzy Operator Logic and Fuzzy Resolution. Journal of Automated Reasoning 10, 59–78 (1993) [27] Shönning, U.: Logic for Computer Scientist. Birkhäuser, Basel (1989) [28] Zadeh, L.A.: From search engines to question-answering systems – The role of fuzzy logic. Progress in Informatics 1, 1–3 (2005)

12 On an Attempt to Formalize Guessing Itziar García-Honrado and Enric Trillas

Deduction is a necessary part of Induction William Whewell [21]

12.1

Introduction

In Science, the method of reasoning is the so called empirical method, based in experiments and different kind of proofs. It allows to build theories, or mathematical models, always subjected to test its provisory validity, or to refuse them. So, this method allows the sequential development of theoretical models in order to get the one that currently better explains the reality. The empirical method at least englobes the following tree general types of reasoning, going from a given body of knowledge to some conclusions, • Deduction, allowing to go from a general to a particular case, by applying known laws, models or theories. So, the conclusions can be called logical consequences, in the sense that they necessarily follow from the available information. Therefore, deduction does not allow to get “new" information, but to clearly deploy the known information. Deduction is typical of formal theories. • Abduction, allowing to find contingent explanations for the information. Is a kind of reasoning in which one chooses the hypothesis that could best explain the evidence. It is used to look for hypotheses of a given information that, then, can be deduced from them. For instance, in a medical diagnose’s problem in which the available information consists in the symptoms: ‘fever’ and ‘sore throat’, our hypothesis could be ‘anginas’, since some symptoms of having anginas are fever and sore throat. • Induction, which takes us beyond our current evidence, or knowledge, to contingent conclusions about the unknown. From particular observations, induction allows us to provisionally establish a ‘law’ that can explain these observations, and that is a (contingent) conjecture in the sense that it is not contradictory with the observations. This kind of reasoning is typical of Experimental Science.

R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 237–255. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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Once a single hypothesis is selected as the explanation of some evidence, following Popper (see [7], [6]), it is only a provisional explanation that should be submitted to the strongest than possible tests trying to refute it. Before, Popper it was C.S. Peirce who described (see, for instance, [5]) the processes of science as a combination of induction, abduction and deduction. Hence, the real process to built up science’s models consists on working with conjectures. That is, building up possible explanations (conjectures called hypotheses) from observations, that can change with new observations. Then, after deducing some necessary consequences of the hypothesis, they must be checked by repeated experiments to test its suitability. 12.1.1 Most of ordinary, everyday, or commonsense reasoning is nothing else than conjecturing or guessing. Often, human reasoning consists in either conjecturing or refuting hypotheses to explain something, or in conjecturing speculations towards some goal. Adding to guessing the reasoning done by similarity, or analogical reasoning, a very big part of ordinary reasoning is obtained. Only a little part of everyday reasoning could be typified as deductive reasoning, that is typical of formal sciences in the context of proof, like in the case of mathematical proof. Can deductive reasoning be seen as a particular type of conjecturing? Since human evolution is in debt with the people’s capacity for conjecturing and, even more, scientific and technological research is based on systematic processes of guessing and of doing analogies, it seems relevant to study what is a conjecture. How the concept of a conjecture can be described and where and how can it be formalized? Is deduction actually a pre-requisite for the formalization of conjectures? This paper deals with these questions, and to begin with let us pose a very simple but typical example of an everyday life decision taken on the base of conjecturing. Why each year many people decide to buy a ticket of the Christmas Lottery? Since this lottery counts with more than 90, 000 different numbers, the probability of winning the first award is smaller than 1/90, 000 = 111·10−7. Such actually small probability does not seem what conducts to the decision of buying a ticket. Instead, it comes from the fact that what is known on the lottery (the previous information) is not incompatible with the statement ‘I can win the first award’. Hence, this statement is the conjecture on which the decision of buying a ticket is primarily based and that, as the small probability of winning shows, has a big risk. Human ordinary reasoning and scientific reasoning can be considered as a sum of different kinds of reasoning, induction, deduction, abduction, reasoning by similarities, and also by some intrinsic characteristics of humans [21] such as imagination, inspiration,... In order to show how the model of formalizing guessing can work, it follows an example collected in [10]. Let L be an ortholattice with the elements, m for midday, e for eclipse, and s for sunny, and its corresponding negations, conjunctions and disjunctions. It is known that it is midday, midday and not sunny, and neither is an eclipse nor it is sunny.

12.2 Towards the Problem: Where Can Knowledge Be Represented?

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Representing and by product, ·, or by sum, +, and not by  , the set of premises is P = {m, m·s , (e·s) }. So, the résumé of this information can be identified with p∧ = m·m·s ·(e·s) = m·m·s ·(s + e ) = m·s . Among conjectures we can distinguish consequences, hypothesis and speculations. Then, once understood that a ≤ b means that b is a logical consequence of a (see [3] for the equivalence of this two notions), it follows, • It is not sunny, s , is a consequence of P, since p∧ = m·s ≤ s . • The statements “it is midday and not sunny and there is an eclipse", m·s ·e, and “there is an eclipse", e, are conjectures of P. Then, since they are not contradictory with m·s . – m·s ·e is a hypothesis of P, since m·s ·e < m·s . Therefore, if it is known m·s ·e, it can be deduced all the given information, since: m·s ·e ≤ m, m·s ·e ≤ m·s , and m·s ·e ≤ s ≤ s + e = (e·s) . – e is a speculation of P, since neither e follows from m·s (m·s  e), nor m·s follows from e (e  m·s ). However, asserting that there is an eclipse is non contradictory with the information, P. • The statements sunny and not midday are refutations of P, since m·s ≤ s = (s) , and m·s ≤ m = (m ) . The elements s and m , contradictory with the résumé p∧ , refute the information given by P. So, and of course in a very restricted and closed framework, this could be a formalization in an ortholattice of a human reasoning.

12.2

Towards the Problem: Where Can Knowledge Be Represented?

Without representation it cannot be done any formalization process. Although the precise classic and quantum reasoning can be formalized through representations in boolean algebras and orthomodular lattices, respectively, everyday reasoning is neither totally formalizable in these algebraic structures, nor in De Morgan algebras. A reason for this is that the big number of properties they enjoy give a too rigid framework for a type of reasoning in which context, purpose, time, imprecision, uncertainty, and analogy, often play jointly an important role. For instance, when interpreting the linguistic connective and by the operation meet of these lattices, it is needed a big amount of (not always available) information on the two components of the conjunctive statement to be sure that and is its infimum. In addition, the meet is commutative but the Natural Language and is not always so, since, when ‘time’ intervenes this property is not always preserved. Hence, for representing everyday reasoning, usually expressed in terms of natural language, more flexible algebraic structures are needed. Standard algebras of fuzzy sets (see [8], [15]) are a good instance of such flexible structures, of which the following abstract definition of a Basic Flexible Algebra seems to be a good enough algebraic structure. Definition 12.2.1. A Basic Flexible Algebra (BFA) is a seven-tuple L = (L, ≤ , 0, 1; ·, +, ), where L is a non-empty set, and

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1. (L, ≤) is a poset with minimum 0, and maximum 1. 2. · and + are mappings (binary operations) L × L → L, such that: a) a·1 = 1·a = a, a·0 = 0·a = 0, for all a ∈ L b) a + 1 = 1 + a = 1, a + 0 = 0 + a = a, for all a ∈ L c) If a ≤ b, then a·c ≤ b·c, c·a ≤ c·b, for all a, b, c ∈ L d) If a ≤ b, then a + c ≤ b + c, c + a ≤ c + b, for all a, b, c ∈ L 3.  : L → L verifies a) 0 = 1, 1 = 0 b) If a ≤ b, then b ≤ a 4. It exists L0 , {0, 1} ⊂ L0  L, such that with the restriction of the order and the three operations ·, +, and  of L , L0 = (L0 , ≤, 0, 1; ·, +, ) is a boolean algebra It is immediate to prove that in any BFA it holds: a·b ≤ a ≤ a + b, and a·b ≤ b ≤ a + b, for all a, b ∈ L. Hence, provided the poset (L, ≤) were itself a lattice with operations min and max, it follows a·b ≤ min(a, b) ≤ max(a, b) ≤ a + b. Lattices with negation and, in particular, ortholattices and De Morgan algebras are instances of BFAs. Also the standard algebras of fuzzy sets ([0, 1]X , T, S, N) are particular BFAs if taking, for μ , σ in [0, 1]X , μ ·σ = T ◦(μ × σ ), μ + σ = S ◦(μ × σ ), μ  = N ◦ μ , with 0 = μ0 , 1 = μ1 (the functions constantly zero and one, respectively), the partial pointwise order, μ ≤ σ ⇔ μ (x) ≤ σ (x) for all x ∈ X, T a continuous tnorm, S a continuous t-conorm, and N a strong negation (see [8], [15], [1]). Notice that although neither idempotency, nor commutativity, nor associativity, nor distributivity, nor duality, nor double-negation, etc., are supposed, ortholattices ([15]) (and in particular orthomodular lattices and boolean algebras), De Morgan algebras, and algebras of fuzzy sets (and in particular the standard ones), are particular cases of BFA. Nevertheless, it should be newly recalled that, for what concerns the representation of Natural Language and Commonsense Reasoning, their too big number of properties imply a too rigid representation’s framework. Notwithstanding, this paper will only work in the case the BFA is an ortholattice (see Appendix).

12.3

Towards the Concept of a Conjecture

The skeleton of the examples in 12.1.1 helps to pose the following definition and questions, relatively to a given problem on which some information constituted by a non-empty set, P = {p1 , p2 , ..., pn } of n premises pi is known. • Definition: q is a conjecture from P, provided q is not incompatible with the information on the given problem once it is conveyed throughout all pi in P. • Questions a) Where do the objects (‘represented’ statements) pi and q belong to? That is, which is L such that P ⊂ L and q ∈ L? b) With which algebraic structure is endowed L?

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241

c) How can the information on the current problem that is conveyed by P be translated into L? How to state that P is consistent? d) How to translate that q is not incompatible with such information? On the possible answers to these four questions depend the ‘formalization’ of the concept of a conjecture. Of course, the answer to question (a) is in strict dependence of the context and characteristics of the current problem, for instance would this problem deserve a ‘body of information’ given by imprecise statements, the set L could be a subset of fuzzy sets in the corresponding universe of discourse X, that is, L ⊂ [0, 1]X . Consequently, and once L is choosen, it will be endowed with an algebraic structure that could respond to the current problem’s context, purpose and characteristics, for instance, would the problem concern a probabilistic reasoning, the structure of L will be either a boolean algebra, or an orthomodular lattice, provided the problem is a classical or a quantum one, respectively. Although questions (c) and (d) deserve some discussion, let us previously consider the information’s set P and its consequences. Let us point out that with P = 0, / the authors adhere to the statement ‘Nothing comes from nothing’, attributed to Parmenides. 12.3.1

Bodies of Information

We will always deal with reasonings made from some previous information given by a finite set of statements, and once they are represented in a BFA L (suitable for the corresponding problem), by elements p1 , ..., pn in L. Each pi is a premise for the reasoning, and P = {p1 , ..., pn } ⊂ L is the set of premises. In what follows, it will be supposed that P is free of incompatible elements, that is, for instance, there are not elements pi , p j in P such that pi ≤ pj , or pi ·p j = 0. Provided there were pi ≤ pj , it would be p1 ·...·pn = 0, and to avoid that possibility we will suppose that the résumé r(P) of the information contained in P is different from zero: r(P) = 0. Analogously, provided this information is given by what follows deductively from P, and C is an operator of consequences, we will suppose that C(P) = L. A set P of premises that is free from incompatibility is a body of information, and it will be taken in a concrete family F of subsets in L like, for example (see [17]), F1 = P(L) F2 = {P ∈ P(L); for no p ∈ P : p ≤ p } F3 = {P ∈ P(L); for no pi , p j ∈ P : pi ≤ pj } F4 = {P ∈ P(L); for no finite subsets {p1 , ..., pr }, {p∗1 , ..., p∗m } ⊆ P : p∗1 ·...·p∗m ≤ (p1 ·...·pr ) }. 5. F5 = {P ∈ P(L); p1 ·...·pn  (p1 ·...·pn ) } 6. F6 = P0 (L) = {P ∈ P(L); p1 ·...·pn = 0}

1. 2. 3. 4.

Obviously, F4 ⊂ F3 ⊂ F2 ⊂ F1 , F5 ⊂ F6 ⊂ F1 , and if L is finite F4 ⊂ F6 . If elements in L verify the non contradiction law, it is F6 ⊂ F4 . If L is a boolean algebra, it is F3 = F4 = F5 = F6 ⊂ F1

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Once the family F is selected in agreement with the kind of incompatibility that is the one suitable for the current problem, a consequence’s operator in the sense of Tarski (see [17]) is a mapping C : F → F, such that, • P ⊂ C(P), C is extensive • If P ⊂ Q, then C(P) ⊂ C(Q), C is monotonic • C(C(P)) = C(P), or C2 = C, C is a closure, for all P, Q in F. In addition, only consistent operators of consequences will be considered, that is, those verifying / C(P). • If q ∈ C(P), then q ∈ Operators of consequences are abstractions of ‘deductive’ processes.

12.4

The Discussion

The discussion will be done under the supposition that L is endowed with an ortholattice structure L = (L, ·, +, ; 0, 1). 12.4.1 The information conveyed by the body of information P can be described, at least, by: 1. The logical consequences that follow from P, each time a consequence operator C is fixed. By the set C(P), deploying what is in P. 2. By a suitable résumé of P in some set. Let us call r(P) such a résumé. What is in (2) is not clear enough without knowing what is to be understood by r(P) or, at least, which properties is r(P) submitted to verify, as well as to which set r(P) does belong to. Three instances for r(P) are: • r(P) = p∧ = p1 ·...·pn ∈ L • r(P) = p∨ = p1 + ... + pn ∈ L • r(P) = [p∧ , p∨ ] = {x ∈ L; p∧ ≤ p ≤ p∨ }, with r(P) ∈ P(L) Anyway, and to state the consistency of P, in case (2) it is reasonable to take r(P) not self-contradictory, for instance r(P)  r(P) , (r(P)  r(P) ), for what it should be r(P) = 0 (r(P) = 0), / since r(P) = 0 ≤ 1 = 0 = r(P) . In case (1) the consistency of P can be stated by supposing C(P) = L. 12.4.2 In the case (1), the non incompatibility between the information conveyed by P and / C(P). In the case (2), and provided it is r(P) ∈ L, a ‘conjecture’ q is given by q ∈ there are three different forms of expressing such non incompatibility: r(P)·q = 0, r(P)·q  (r(P)·q) , and r(P)  q (see [12]). All that conducts to the following four possible definitions of the set of conjectures from P:

12.4 The Discussion

• • • •

243

Con jC (P) = {q ∈ L; q ∈ / C(P)}, provided C(P) = L. Con j1 (P) = {q ∈ L; r(P)·q = 0} Con j2 (P) = {q ∈ L; r(P)·q  (r(P)·q) } Con j3 (P) = {q ∈ L; r(P)  q }

With the last three definitions a problem arises: Are they coming from some operator of consequences in the form of the definition of Con jC ? For instance, to have Con j3 (P) = {q ∈ L; q ∈ / Cr (P)}, it is necessary that Cr (P) = {q ∈ L; r(P) ≤ q}, and provided r(P) verifies r(P) ≤ p∧ , P ⊂ Q implies r(Q) ≤ r(P), and r(Cr (P)) = r(P), Cr is an operator of consequences, since: • r(P) ≤ p∧ ≤ pi (1 ≤ i ≤ n), means P ⊂ Cr (P). • If P ⊂ Q, if q ∈ Cr (P), from r(P) ≤ q and r(Q) ≤ r(P), follows q ∈ Cr (Q). Hence, Cr (P) ⊂ Cr (Q) • Obviously, Cr (P) ⊂ Cr (Cr (P)). If q ∈ Cr (Cr (P)), from r(Cr (P)) ≤ q and r(Cr (P)) = r(P) follows r(P) ≤ q, or q ∈ C(P). Hence, Cr (Cr (P)) = Cr (P). In addition, Cr is consistent since if q ∈ Cr (P) and q ∈ C(P), from r(P) ≤ q and r(P) ≤ q , follows r(P) ≤ q·q = 0, or r(P) = 0, that is absurd. Hence, q ∈ Cr (P) ⇒ q ∈ / Cr (P). In particular, if r(P) = p∧ , Con j3 comes from the consistent operator of consequences C∧ (P) = {q ∈ L; p∧ ≤ q}, that is the greatest one if L is a boolean algebra and F = P0 (L) (see [19]). Remark 1. To have C(P) ⊂ Con jC (P), it is sufficient that C is a consistent operator of consequences, since then q ∈ C(P) implies q ∈ / C(P), and q ∈ Con jC (P). This condition is also necessary since, if C(P) ⊂ Con jC (P), q ∈ C(P) implies q ∈ Con jC (P), or q ∈ / C(P). Hence, the consistency of C is what characterizes the inclusion of C(P) in Con jC (P), that consequences are a particular type of conjectures. For instance, it is C∧ (P) ⊂ Con j3 (P), and Con j3 (P) = {q ∈ L; q ∈ / C∧ (P)}. 12.4.3 / C1 (P)} Concerning Con j1 (P) = {q ∈ L; r(P)·q = 0}, it is Con j1 (P) = {q ∈ L; q ∈ provided C1 (P) = {q ∈ L; r(P)·q = 0}. Let us only consider the case in which r(P) = p∧ = 0. It is P ⊂ C1 (P), since p∧ ·pi = 0 (1 ≤ i ≤ n). If P ⊂ Q, q ∈ C1 (P), or p∧ ·q = 0, with q∧ ≤ p∧ implies q∧ ·q = 0, and q ∈ C1 (Q), thus, C1 (P) ⊂ C1 (Q). Nevertheless, C1 can not be always applicable to C1 (P) since it easily can be r(C1 (P)) = 0, due to the fail of the consistency of C1 . For instance, if L is the ortholattice in figure 12.1, with P = { f , e} for which p∧ = b, it is C1 (P) = {1, a, b, c, d, e, f , g, a , c }, and r(C1 (P)) = 0. Hence, Con j1 is not coming from an operator of consequences, but only from the extensive and monotonic one C1 , for which the closure property C1 (C1 (P)) = C1 (P) has no sense, since C1 (P) can not be taken as a body of information.

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12 On an Attempt to Formalize Guessing

Fig. 12.1. Ortholattice

Notice that if L is a boolean algebra, and q ∈ C1 (P), from p∧ ·q = 0 follows p∧ = p∧ ·q + p∧·q = p∧ ·q = 0, that means q ∈ Con j1 (P): C1 (P) ⊂ Con j1 (P). In addition, p∧ = p∧ ·q is equivalent to p∧ ≤ q, that is, q ∈ C∧ (P) : C1 (P) ⊂ C∧ (P). Even more, in this case, if q ∈ C∧ (P), or p∧ ≤ q, it follows p∧ ·q ≤ q·q = 0, and q ∈ C1 (P). Thus, if L is a boolean algebra, C∧ = C1 , and Con j1 = Con jC∧ . 12.4.4 Concerning Con j2 (P) = {q ∈ L; p∧ ·q  (p∧ ·q) }, to have Con j2 (P) = {q ∈ L; q ∈ / C2 (P)}, it should be C2 (P) = {q ∈ L; p∧ ·q ≤ (p∧ ·q ) }. Of course, if L is a boolean algebra, it is C2 (P) = {q ∈ L; p∧ ·q = 0} = {q ∈ L; p∧ ≤ q} = C∧ (P), Con j2 (P) = Con jC∧ (P), and Con j2 comes from the operator of consequences C∧ . In the general case in which L is an ortholattice, it is P ⊂ C2 (P) since p∧ ·pi = 0 ≤ 0 = 1. If P ⊂ Q, and q ∈ C2 (P), or p∧ ·q ≤ (p∧ ·q ) , with q∧ ≤ p∧ implies q∧ ·q ≤ p∧ q and (p∧ ·q ) ≤ (q∧ ·q ) , that is, q∧ ·q ≤ q∧ ·q ≤ (p∧ ·q ) ≤ (q∧ ·q ) , or q∧ ·q ≤ (q∧ ·q ) , and q ∈ C2 (Q). Hence, C2 (P) ⊂ C2 (Q), and C2 is expansive and monotonic. Notwithstanding, in the ortholattice in figure 12.1, with P = {g} (p∧ = g) is C2 (P) = {q ∈ L; g·q  (g·q) } = {a, b, c, d, e, f , g, b , a , c , d  , e , f  , g , 1} with r(C2 (P)) = 0. Hence, C2 is not applicable to C2 (P), since C2 (P) is not a body of information, and the closure property has no sense (notice that C2 is not consistent). Thus, unless L is a boolean algebra, Con j2 is not a conjectures operator coming from a consequences one.

12.4 The Discussion

245

12.4.5 Concerning Con j3 (P) = {q ∈ L; p∧  q }, to have Con j3 (P) = {q ∈ L; q ∈ / C3 (P)}, it should be C3 (P) = {q ∈ L; p∧ ≤ q} = C∧ (P), as it is said in Remark 1. 12.4.6 Let us consider again the operators • C4 (P) = {q ∈ L; q ≤ p∨ } • C5 (P) = {q ∈ L; p∧ ≤ q ≤ p∨ } As it is easy to check, only the second is an operator of consequences that is consistent unless p∧ = 0 and p∨ = 1. With it, it is Con j5 (P) = {q ∈ L; q ∈ / C5 (P)} = {q ∈ L; p∧  q or q  p∨ }. 12.4.7 When it is Con j(P) = 0? / It is clear that / C(P)} = 0, / if and only if C(P) = L • Con jC (P) = {q ∈ L; q ∈ • Con ji (P) = 0/ (1 ≤ i ≤ 3), if r(P) = 0, in which case C∧ (P) = L, and also C1 (P) = C2 (P) = L. Notice that C(P) = L implies that also C is not consistent. These cases are limiting ones, and facilitate a reason for supposing that C is consistent and r(P) = 0. Concerning the operator Con j5 , it is empty provided p∧ = 0 and p∨ = 1, that is when C5 (P) is not consistent. 12.4.8 It is easy to check that, if r(P) = 0, it is Con j1 (P) ⊂ Con j2 (P) ⊂ Con j3 (P), and that if L is a boolean algebra the three operators do coincide. 12.4.9 What happens if the information conveyed by P can be given by two different résumés r1 (P) and r2 (P)? (i) Let us denote by Con j j (1 ≤ i ≤ 2, 1 ≤ j ≤ 3) the corresponding operators of conjectures. If r1 (P) ≤ r2 (P), it is easy to check that (1)

(2)

Con j j ⊂ Con j j , for 1 ≤ j ≤ 3. Analogously, if C1 ⊂ C2 (that is, C1 (P) ⊂ C2 (P) for all P ∈ F), then Con jC2 (P) ⊂ Con jC1 (P).

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Remark 2. In the case in which the information conveyed by P is what can deductively follow from P, there can be more than one single consistent operator of consequences to reflect such deductive processes. If C is the set of such operators, it can be defined Con jC (P) =



Con jC (P),

C∈C

but a possible problem with this operator is that it can easily be a too small set. 12.4.10

The Goldbach’s Conjecture

Let N be the set of positive integers as characterized by the five Peano’s axioms, namely: p1 . p2 . p3 . p4 . p5 .

1 is in N. If n is in N, also its successor, s(n), is in N. It is not any n ∈ N such that s(n) = 1. If s(n) = s(m), then n = m. If a binary property concerning positive integers holds for 1, and provided it holds for n it is proven it also holds for n + 1, then such property holds for all numbers in N.

The proof of a single ‘not pi ’ (1 ≤ i ≤ 5) will mean a refutation of the Peano’s characterization of N. The majority of mathematicians believe (supported by the 1936 Gentzen’s proof on the consistency of P, based on transfinite induction up to some ordinal number), that the set P = {p1 , p2 , p3 , p4 , p5 } is consistent. The elementary theory of numbers consist in all that is deductively derivable in finitistic form from P. Let us represent by C (operator of consequences) such a form of deduction. Given the statement q: All even number larger than 2 is the sum of two prime numbers, it can be supposed q ∈ Con jC (P), since no a single instance of an even number that is not the sum of two primes (not q) has been found after an extensive search for it. This statement q is the The Goldbach’s conjecture, that will be solved once proven q ∈ C(P).

12.5

The Properties of the Operators of Conjectures

12.5.1 In all the cases in which there is a expansive operator C such that Con j(P) = {q ∈ / C(P)} and C(P) ⊂ Con j(P), since P ⊂ C(P), it is P ⊂ Con j(P), and also L; q ∈ Con j is an expansive operator. This is what happens always when Con j is given by one of such operators, as are the cases Con jC , Con j3 , and also that of Con j4 . In the case of Con j2 , it is p∧ ·pi = p∧  p∧ , and also P ⊂ Con j2 (P). In the case of Con j1 , it is p∧ ·pi = p∧ = 0, and also P ⊂ Con j1 (P). With Con j5 , provided C5 is consistent, it is also P ⊂ Con j5 (P).

12.5 The Properties of the Operators of Conjectures

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12.5.2 If P ⊂ Q, since C(P) ⊂ C(Q), provided q ∈ Con jC (Q), or q ∈ / C(Q), it is q ∈ / C(P), and q ∈ Con jC (P). Then Con jC (Q) ⊂ Con jC (P), and the operators Con jC are antimonotonic. Hence, Con j3 is also anti-monotonic. With Con j1 , if P ⊂ Q and q ∈ Con j1 (Q), or q∧ ·q  (q∧ ·q) , from q∧ ≤ p∧ follows q∧ ·q ≤ p∧ ·q and 0 < p∧ ·q, or q ∈ Con j1 (P). Thus, Con j1 (Q) ⊂ Con j1 (P), and the operator Con j1 is anti-monotonic. With Con j2 , if P ⊂ Q and q ∈ Con j2 (Q), or q∧ ·q  (q∧ ·q) , from q∧ ≤ p∧ follow q∧ ·q ≤ p∧ ·q and (p∧ ·q) ≤ (q∧ ·q) . Hence, provided p∧ ·q ≤ (p∧ ·q) (q ∈ / Con j2 (P)), will follow q∧ ·q ≤ p∧ ·q ≤ (p∧ ·q) ≤ (q∧ ·q) , that is absurd. Thus, q ∈ Con j2 (P), or Con j2 (Q) ⊂ Con j2 (P), and the operator Con j2 is anti-monotonic. 12.5.3 For what concerns r(P), it should be noticed that the idea behind it is to reach a ‘compactification’ of the information conveyed by the pi in P. Of course, how to express and represent r(P) depends on the current problem that, in some cases, offers no doubts on how to represent r(P). For instance, if the problem consists in doing a backwards reasoning with scheme If p, then q, and not q: not p, r(P) does represent the statement (p → q)·q , with which it must follow (p → q)·q ≤ p , to be sure that p follows deductively from P = {p → q, q }, under C∧ and provided r(P) = 0. That is, to have p ∈ C∧ ({p → q, q }) = {x ∈ L; (p → q)·q ≤ x} = [(p → q)·q , 1] = [r(P), 1]. Provided it were r(P) = 0, it will follow the non-informative conclusion p ∈ {x ∈ L; 0 ≤ x} = L. 12.5.4 After what has been said, it seems that any operator of conjectures does verify some of the following five properties, 1. 2. 3. 4. 5.

Con j(P) = 0/ 0∈ / Con j(P) / C(P)} It exists an operator C such that Con j(P) = {q ∈ L; q ∈ Con j is expansive: P ⊂ Con j(P) Con j is anti-monotonic: If P ⊂ Q, then Con j(Q) ⊂ Con j(P)

Let us reflect on properties 3, 4, and 5, in the hypothesis that C is consistent, that is, it verifies ‘q ∈ C(P) ⇒ q ∈ / C(P)’. Obviously, C(P) ⊂ Con j(P).

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• If Con j is anti-monotonic, C is monotonic. Proof. Provided P ⊂ Q, if q ∈ C(P) follows q ∈ / Con j(P) and, since Con j(Q) ⊂ Con j(P) it is q ∈ / Con j(Q), or q ∈ C(Q). Hence, C(P) ⊂ C(Q)  • If C is extensive and monotonic, Con j is extensive and anti-monotonic. Proof. It is obvious that P ⊂ Con j(P), since from C(P) ⊂ Con j(P) follows P ⊂ C(P) ⊂ Con j(P). Provided P ⊂ Q, follows that if q ∈ Con j(Q), or q ∈ / C(Q), it is also q ∈ / C(P), or q ∈ Con j(P).  Thus, provided C is consistent, a sufficient condition to have Con jC expansive and anti-monotonic is that C is expansive and monotonic. In addition Con jC is anti-monotonic if and only if C is monotonic. What, if C is also a closure? C2 (P) = C(P) implies Con jC (P) = Con j(C(P)). / C(P) ⇔ q ∈ / C(C(P)) ⇔ q ∈ Con jC (C(P)) : Con jC (P) = Thus q ∈ Con jC (P) ⇔ q ∈ Con jC (C(P)). Hence, if C is a consistent consequences operator the associated operator Con jC is extensive, anti-monotonic, and verifies Con jC ◦ C = Con jC .

12.6

Kinds of Conjectures

12.6.1 It is clear that with Con j3 (P) it is C3 (P) = C∧ (P) ⊂ Con j3 (P), as it was said before, but what with Con j1 (P) and Con j2 (P) that are not coming from an operator of consequences? Is there any subset of Con j1 (P) and Con j2 (P) that consists of logical consequences of P? Are always logical consequences a particular case of conjectures? Namely, given Con j2 (P) = {q ∈ L; p∧ ·q  (p∧ ·q) }, exists C(P) ⊂ Con j2 (P) such that C is a Tarski’s operator? Of course, such C is not C2 (P) = {q ∈ L; p∧ ·q ≤ (p∧ ·q ) }, but is there any subset of Con j2 (P) that consists of logical consequence of P? The answer is affirmative, since q ∈ C∧ (P), or p∧ ≤ q, is equivalent to p∧ ·q = p∧ , and as it cannot be p∧ ≤ p∧ , it is p∧ ·q ≤ (p∧ ·q) , or q ∈ Con j2 (P). Then, C∧ (P) ⊂ Con j2 (P). Thus, what can be said on the difference Con j2 (P) − C∧ (P)? Concerning Con j1 (P) = {q ∈ L; p∧ ·q = 0}, it is known that C(P) is not C1 (P) = {q ∈ L; p∧ ·q = 0}, but it is also C∧ (P) ⊂ C1 (P), that newly allows to ask on the difference Con j1 (P) − C∧ (P)? The idea behind the two former questions is to classify the conjectures in Con ji (P) − C∧ (P), i = 1, 2, that is, those conjectures that are not ‘safe’ or ‘necessary’ ones, but contingent in the sense that it could be simultaneously q ∈ Con ji (P) − C∧ (P) and q ∈ Con ji (P) − C∧ (P) . What is clear is that Con ji (P) − C∧ (P) = {q ∈ Con ji (P); q < p∧ } ∪ {q ∈ Con ji (P); q NC p∧ }, with the sign NC instead of non ‘order comparable’. Let’s call as follows these two subsets, • Hypi (P) = {q ∈ Con ji (P); q < p∧ }, and its elements ‘hypotheses for P’. • Spi (P) = {q ∈ Con ji (P); q NC p∧ }, and its elements ‘speculations from P’.

12.6 Kinds of Conjectures

249

Notice that since 0 ∈ / Con ji (P) (i = 1, 2), it is actually Hypi (P) = {q ∈ Con ji (P); 0 < q < p∧ }. Obviously, the decomposition Con ji (P) = C∧ (P) ∪ Hypi (P) ∪ Spi (P) is a partition of Con ji (P), and defining Re fi (P) = L − Con ji(P), as the set of refutations of P, the following partition of L is obtained, L = Re fi (P) ∪Con ji (P) = Re fi (P) ∪ Hypi (P) ∪ Spi (P) ∪C∧ (P). 12.6.2 If P ⊂ Q, from q∧ ≤ p∧ , if 0 < q < q∧ , follows 0 < q < p∧ , that is Hypi (Q) ⊂ Hypi (P). That is, the operators Hypi are, like Con ji , anti-monotonic. Concerning Spi , some examples (see [14]) show it is neither monotonic, nor antimonotonic, that is, if P ⊂ Q there is no any fixed law concerning how can Spi (P) and Spi (Q) be compared: they cannot be comparable by set inclusion. In fact, coming back to figure 12.1, and taking P1 = { f } ⊂ P2 = {e, f }, it is Sp3 (P1 ) = {a, e, a , b c , d  , e } and Sp3 (P2 ) = {a, c, a , b , c }, which are non-comparable. We can say that Spi are purely non-monotonic operators. Notice that it is Sp1 (P) = {q ∈ L; p∧ ·q = 0 & p∧ NCq}, and Sp2 (P) = {q ∈ L; p∧ ·q  (p∧ ·q) & p∧ NCq}, that, if L is a boolean algebra, are coincidental, since in such case ‘p∧ ·q  (p∧ ·q) ⇔ p∧ ·q = 0. Remark 3. Provided C is a consistent operator of consequences, and in a similar vein to Gödel’s First Incompleteness Theorem, let us call C-decidable those elements in C(P), and consider the set UC (P) = {q ∈ L; q ∈ / C(P)&q ∈ / C(P)}. UC (P) consists in the C-undecidable elements in L given P, those that neither follow deductively from P (under C), nor their negation follows deductively from P (under C). It is, UC (P) = C(P)c ∩Con jC (P) = [Sp(P) ∪ Hyp(P) ∪ Re f (P)] ∩Con jC(P) = Sp(P) ∪ Hyp(P) Thus, given a consistent set of premises (C(P) = L, or p∧  0), reasonably the C-undecidable elements in L are either the speculations or the hypotheses: C−undecidability coincides with contingency.

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If C∗ is a consistent and more powerful operator of consequences than C (C(P) ⊂ it is obvious that it holds UC∗ (P) ⊂ UC (P), but not that UC (P) ⊂ UC∗ (P): What is C-undecidable is not necessarily C∗ -undecidable. It happens analogously if C and C∗ are not comparable, but C(P) ∩C∗ (P) = 0. / The undecidability under C does not imply the undecidability under a C ∗ that is not less powerful than C.

C∗ (P)),

12.6.3

Explaining the Experiment of Throwing a Dice

Which are the significative results that can be obtained when throwing a dice? The reasonable questions that can be posed relatively to the results of the experiment are, for instance, • • • • •

Will an even number be obtained?, with answer representable by {2, 4, 6} Will an odd number be obtained?, with answer representable by {1, 3, 5} Will a six be obtained?, with answer representable by {6} Will fail the throw?, with answer representable by 0/ Will any number be obtained?, with answer representable by {1, 2, 3, 4, 5, 6}, etc.

Thus, the questions can be answered by the subsets of the ‘universe of discourse’ X = {1, 2, ..., 6}, with which the boolean algebra of events is P(X), and the body of information for the experiment of throwing a dice is P = {X}, with p∧ = X = 0. / Since L = P(X) is a boolean algebra, it can be taken the consistent operator of consequences C∧ (P) = {Q ∈ P(X); X ⊂ Q} = {X}. Hence, / and Con jC∧ (P) = {Q ∈ P(X); X  Qc } = {Q ∈ P(X); Q = 0}, Re fC∧ (P) = {0}, / that is, the conjectures on the experiments are all the non-empty subsets of X. In addition, Hyp(P) = {Q ∈ P(X); 0/ ⊂ Q ⊂ X}, and Sp(P) = {Q ∈ P(X); Q = 0/ & Q NC X} = 0, / show that Con jC∧ = C∧ ∪ Hyp(P) = {X} ∪ {Q ∈ P(X); 0/ = Q = X}. That is, the significative results of the experiment are those Q ⊂ X that are neither empty, nor coincidental with the ‘sure event’ X: those that are contingent. In fact, in the case of betting on the result of throwing a dice, nobody will bet on ‘failing’, and nobody will be allowed to bet on ‘any number’. Hence, the presented theory of conjectures explains well the experiment, and the risk of betting for an event can be controlled by means of a probability p : P(X) → P(X), defined by p({i}) = pi (1 ≤ i ≤ 6) such that 0 ≤ pi ≤ 1, and ∑ pi = 1, i∈{1,...,6}

with the values pi depending on the physical characteristics of the dice. For instance, the probability of ‘obtaining even’, is p({2, 4, 6}) = p2 + p4 + p6.

12.7 On Refutation and Falsification

12.7

251

On Refutation and Falsification

Let it be P = {p1 , ..., pn }, with p∧ = 0, and a conjecture’s operator Con jC (P) = {q ∈ L; q ∈ / C(P)} with C an, at least, expansive and monotonic operator. The corresponding operator of refutations is Re f (P) = L − Con j(P) = {r ∈ L; r ∈ C(P)}. When it can be specifically said that r ∈ L: 1) refutes P?, and 2) refutes q ∈ Con j(P)? The answer to these two questions depends on the chosen operator C, provided r ∈ Re f (P), r ∈ C(P), and that r is incompatible with the totality of the given information. In particular, and supposed r ∈ Re f (P): • For Con j3 (P), C = C∧ 1. r refutes P, if r is contradictory will all pi ∈ P, that is, p1 ≤ r ,..., pn ≤ r . Notice that this chain of inequalities implies p∧ ≤ r , or simply r ∈ C∧ (P). 2. r refutes q, if r ∈ C∧ ({q}) and q ≤ r , that is simply if r ∈ C∧ ({q}). • For Con j2 (P), C = C2 (C2 (P) = {q ∈ L; p∧ ·q ≤ (p∧ q ) }) 1. r refutes P, if all pi ·r are self-contradictory, that is, p1 ·r ≤ (p1 ·r) ,..., pn ·r ≤ (pn ·r) , implying p∧ ·r ≤ (pi ·r) , for 1 ≤ i ≤ n. From p∧ ≤ pi , follows (pi ·r) ≤ (p∧ ·r) , and p∧ ·r ≤ (p∧ ·r) , or simply r ∈ C2 (P). 2. r refutes q, if r ∈ C2 ({q}) and q·r ≤ (q·r) , that is simply if r ∈ C2 ({q}). • For Con j1 (P), C = C1 (C2 (P) = {q ∈ L; p∧ ·q = 0}) 1. r refutes P, if p1 ·r = 0,..., pn ·r = 0, implying p∧ ·r = 0, that implies p∧ ·r = 0, or simply r ∈ C1 (P). 2. r refutes q, if r ∈ C1 ({q}) and q·r = 0, that is simply if r ∈ C1 ({q}). Hence, for these three cases 1. r refutes P, if r ∈ C1 (P): r follows deductively from P 2. r refutes q, provided r ∈ Re f (P), and r ∈ C({q}): r is a refutation whose negation follows deductively from {q}. 12.7.1 In the particular case in which h ∈ Hyp(P) (0 < h < p∧ ), and in addition to the former answers in agreement with Popper’s ideas on the falsification of theories (C(P) = P) and hypotheses (see [7] [6]), it can be said what follows • If h ∈ HypC (P), then C(P) ⊂ C({h}) ⊂ Con jC (P), proven by the following sequences: 1) p∧ ≤ q & h ≤ p∧ ⇒ h < q. 2)h ≤ q & h ≤ q ⇒ h = 0. 3)h  q & p∧ ≤ q & h ≤ p∧ ⇒ h ≤ q which is absurd: p∧  q . Hence, in order to ascertain that some h ∈ L is not a hypothesis for P (falsification of h), it suffices to find q ∈ C(P) such that q ∈ / C({h}), or r ∈ C({h}) such that r∈ / Con jC (P). In these cases it is q ∈ Re f (P), and r ∈ Re f (P): both refute h. Thus:

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• Something that follows deductively from P, but not from {h}, makes h be false. • Something that follows deductively from {h}, but is not conjecturable from P, makes h be false. Remark 4. From p∧ ≤ pi and 0 < h < p∧ , it follows P ∪ {p∧ } ⊂ C({h}), hence P ∪ {p∧ } ⊂ C(P) ⊂ C({h}), that, although only in part, remembers the statement in [21], ‘Deduction justifies by calculation what Induction has happily guessed’.

12.8

The Relevance of Speculations

It is Sp3 (P) = {q ∈ Con j3 ; p∧ NCq} = {q ∈ L; p∧  q & p∧ NCq}, hence, if q ∈ Spi (P) (i = 1, 2, 3) it is not p∧ ·q = p∧ (equivalent to p∧ ≤ q, or q ∈ C∧ (P) ). Thus, if q ∈ Spi (P), it is 0 < p∧ ·q·q < p∧, that is p∧ ·q ∈ Hypi (P). This result shows a way of reaching hypotheses from speculations, and in the case the ortholattice L is an orthomodular one, for any h ∈ Hypi (P), it exists q ∈ Sp3(P) such that h = p∧ ·q (see [16]), there are no other hypotheses, and it is Hyp(P) = p∧ ·Sp3 (P). Of course, this result also holds if L is a boolean algebra. Analogously, since p∧ ≤ p∧ + q, it is p∧ + q ∈ C∧ (P), that shows a way of reaching logical consequences from speculations, and if L is an orthomodular lattice (and a fortiori if it is a boolean algebra), there are not other consequences (see[16]), that is, C∧ (P) = p∧ + Sp3 (P). Remarks 1 • p∧ ·Sp3 (P) ⊂ Hyp(P), could remember a way in which humans search for how to explain something. Once P and p∧ are known, a q ∈ L such that p∧  q and p∧ NCq, that is, neither incompatible, nor comparable with p∧ , gives the explanation or hypotheses p∧ ·q for P, provided p∧ ·q = 0, and p∧·q = p∧ . Of course, an interesting question is how to find such q ∈ Sp3 (P). In some cases, may be q is found by similarity with a former case in which a more or less similar problem was solved, and plays the role of a metaphor for the current one. • Out of orthomodular lattices, there are hypotheses and consequences that are not reducible, that is, belonging to Hyp(P) − p∧·Sp3(P), or to C∧ (P) − (p∧ + Sp3 (P)) (see [16]).

12.9

Conclusion

This paper represents a conceptual upgrading of a series of papers on the subject of conjectures, a subject christianized in [9] as ‘CHC Models’.

12.9 Conclusion

253

12.9.1 In the course of millennia the brain’s capability of conjecturing resulted extremely important for the evolution of the species Homo. Such capability helped members in Homo to escape from predators, to reach adequate food, to protect themselves from some natural events, or even catastrophes, as well as to produce fire, to make artifacts, and to travel through high mountains, deserts, forests, rivers and seas. Without articulate language and partially articulate guessing, possibly Homo would have neither prevailed over the rest of animals, nor constituted the social, religious and economic organizations typical of humankind. And one of the most distinguishing features of Homo Sapiens is the act, and especially the art, of reasoning, or goaloriented managing conjectures. Even more, scientific and technological research is a human activity that manages guessing in a highly articulated way. Actually, reasoning and conjecturing are joint brain activities very difficult to separate one from the other. Although consequences and hypotheses, as well as several types of non- monotonic reasoning, deserved a good deal of attention by logicians, philosophers, computer scientists, and probabilists, no attempt at formalizing the concept of conjecture appeared before [10] was published. In the framework of an ortholattice, conjectures were defined in [10] as those elements non-incompatible with a given set of (nonincompatible) premises reflecting the available information. That is, conjectures are those elements in the ortholattice that are “possible", once a résumé of the information given by the premises is known. This is the basic definition of which consequences (or safe, necessary conjectures), hypotheses (or explicative contingent conjectures), and speculations (or lucubrative, speculative contingent conjectures) are particular cases. It should also be pointed out that neither the set of hypotheses, nor that of speculations, can be taken as bodies of information. Processes to obtain consequences perform deductive reasoning, or deduction. Those to obtain hypotheses perform abductive reasoning, or abduction, and those to obtain speculations perform inductive reasoning, or induction, a term that is also more generally applied to obtaining either hypotheses or speculations, and then results close to the term “reasoning". Of course, in Formal Sciences and in the context of proof, the king of reasoning processes is deduction. 12.9.2 Defining the operators of conjectures only by means of consistent consequences ones (see [14]) has the drawback of placing deduction before guessing, when it can be supposed that guessing is more common and general than deduction, and this is a particular (and safe) type of the former. After the publication of some papers ([10], [12], [14], [16], [11], [17], [19]) on the subject it yet remained the doubt on the existence of conjecture’s operators obtained without consequences’ operators, and this paper liberates from such doubt by showing that to keep some properties that seem to be typical of the concept of conjecture, it suffices to only consider operators that are extensive and monotonic, but without enjoying the closure property.

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12 On an Attempt to Formalize Guessing

These operators are reached by considering (like it was done in [12]), three different ways of defining non-incompatibility by means of non-self-contradiction. Of these three ways, only one of them conducts to reach conjectures directly through logical consequences that is just the one considered in [10]. Of course, the existence of operators of conjectures not coming from extensive and monotonic operators remains an open problem.

Appendix Although basic flexible algebras are very general structures, it is desirable that they verify the principles of Non-contradiction and Excluded-middle, to ground what is represented in a ‘solid’ basement. For that goal it will be posed some definitions on the incompatibility concept of contradictory and self-contradictory elements in a BFA. In the first place, (see [2], [18]) • Two elements a, b in a BFA are said to be contradictory with respect to the negation  , if a ≤ b . • An element a in a BFA is said to be self-contradictory with respect to the negation  , if a ≤ a . The classical principles of Non-contradiction (NC) and Excluded-Middle (EM) can be defined in the way that is typical of modern logic, • NC: a·a = 0 • EM: a + a = 1 Any lattice with a strong negation (i.e. (a ) = a, for all a ∈ L) verifying these last principles, is an ortholattice. So, a Boolean Algebra verifies these principles. But, if dealing with fuzzy sets, for instance with the standard algebra of fuzzy sets ([0, 1]X , min, max, N), that is a De Morgan algebra, these principles, formulated in the previous way, do not hold. Nevertheless, if the Aristotle’s formulation of the first principle: “an element and its negation is impossible" is translated by “an element and its negation are self-contradictory", the mathematical representation of these principles changes in the form, • NC: a·a ≤ (a·a ) • EM: (a + a) ≤ ((a + a) ) With these new formulation, De Morgan algebras and functionally expressible BFA of fuzzy sets also verify those principles (see [4]), if dealing with a strong negation Nϕ for fuzzy sets, that is, Nϕ (x) = ϕ −1 (1 − ϕ (x)), with ϕ an orderautomorphism of the unit interval. In fact, it can be used as intersection any function T that verifies T (a, Nϕ (a)) ≤ ϕ −1 ( 12 ) in order to satisfy the principle of NC. In the case of EM, it is enough any function S, that satisfies ϕ −1 ( 12 ) ≤ S(a, Nϕ (a)). Notice that all t-norms are in the condition of T , and all t-conorms are in the condition of S (see [13]).

12.9 Conclusion

255

Acknowledgements The authors are in debt with the books [6], [20] and [2] for the insights they contain in reference to the development of this paper, and with Prof. Claudio Moraga (ECSC) for his kind help in the preparation of the manuscript.

References [1] Alsina, C., Frank, M.J., Schweizer, B.: Associative Functions. Triangular Norms and Copulas. World Scientific, Singapore (2006) [2] Bodiou, G.: Théorie dialectique des probabilités (englobant leurs calculs classique et quantique). Gauthier-Villars (1964) [3] Castro, J.L., Trillas, E.: Sobre preordenes y operadores de consecuencias de Tarski. Theoria 4(11), 419–425 (1989) (in Spanish) [4] García-Honrado, I., Trillas, E.: Characterizing the principles of non contradiction and excluded middle in [0,1]. Internat. J. Uncertainty Fuzz. Knowledge-Based Syst. 2, 113– 122 (2010) [5] Peirce, C.S.: Deduction, induction, and hypothesis. Popular Science Monthly 13, 470– 482 (1878) [6] Popper, K.R.: The logic of Scientific Discovery. Hutchinson & Co. Ltd., London (1959) [7] Popper, K.R.: Conjectures and Refutations. Rutledge & Kegan Paul, London (1963) [8] Pradera, A., Trillas, E., Renedo, E.: An overview on the construction of fuzzy set theories. New Mathematics and Natural Computation 1(3), 329–358 (2005) [9] Qiu, D.: A note on trillas’ chc models. Artificial Intelligence 171, 239–254 (2007) [10] Castiñeira, E., Trillas, E., Cubillo, S.: On conjectures in orthocomplemented lattices. Artificial Intelligence 117, 255–275 (2000) [11] Trillas, E., Castiñeira, E., Cubillo, S.: Averaging premises. Mathware and Soft Computing 8, 83–91 (2001) [12] Pradera, A., Trillas, E.: A reflection on rationality, guessing and measuring. In: Proceedings IPMU, Annecy, pp. 777–784 (2002) [13] Trillas, E., Alsina, C., Pradera, A.: Searching for the roots of non-contradiction and excluded-middle. International Journal of General Systems 31(5), 499–513 (2002) [14] Monserrat, M., Trillas, E., Mas, M.: Conjecturing from consequences. International Journal of General Systems 38, 567–578 (2009) [15] Trillas, E., Alsina, C., Pradera, A.: On a class of Fuzzy Set Theories. In: Proc. FUZZIEEE 2007, London, pp. 1–5 (2007) [16] Trillas, E., Pradera, A., Álvarez, A.: On the reducibility of Hypotheses and Consequences. Information Sciences 179(23), 3957–3963 (2009) [17] Trillas, E., García-Honrado, I., Pradera, A.: Consequences and conjectures in preordered sets. Information Sciences 180(19), 3573–3588 (2010) [18] Trillas, E.: Non Contradiction, Excluded Middle, and Fuzzy Sets. In: Di Gesù, V., Pal, S.K., Petrosino, A. (eds.) WILF 2009. LNCS (LNAI), vol. 5571, pp. 1–11. Springer, Heidelberg (2009) [19] Vaucheret, C., Angel, F.P., Trillas, E.: Additional comments on conjectures, hypotheses, and consequences in orthocomplemented lattices. In: Fuzzy Logic in Knowledge-Based Systems, Decision and Control, pp. 107–114 (2001) [20] Watanabe, S.: Knowing and guessing. A Quantitative Study of Inference and Information. John Wiley and sons, New York (1969) [21] Whewell, W.: Novum Organon Renovatum: Being The Second Part of The Philosophy of The Inductive Sciences. John Parker and Son, London (1858)

13 Syntactic Ambiguity Amidst Contextual Clarity Jeremy Bradley

13.1

Introduction

This contribution deals with hidden linguistic ambiguity – ambiguity that goes unnoticed by human speakers but can nevertheless cause problems in certain situations. Its primary impetus was the author’s current work in the field of text simplification for aphasic readers who have lost some, but not all, of their ability to comprehend language. By means of text simplification, it is possible to make texts more accessible to people with this disorder, thereby improving their quality of life by enabling them to participate in the information society to a greater degree without the aid of a therapist.

13.2

Linguistic Ambiguity

Human speech is, by default, loaded with ambiguity on many different levels. Words have multiple meanings and sentence structures can be parsed in various grammatically legitimate fashions. And even if expressions are lexically and grammatically clear, it is not always clear to us what someone who is making a statement is trying to say. Linguistic ambiguity occurs on the following three levels: 13.2.1

Lexical Ambiguity

In lexical ambiguity, a specific word can have different meanings – e.g. a “bank” can be an institution where you can deposit money, the land alongside a river or lake, or an airplane manoeuvre. 13.2.2

Syntactic Ambiguity

In syntactic ambiguity there are multiple ways in which a sentence can be parsed – e.g. the sentence “kissing monkeys can be dangerous” can mean that monkeys absorbed in the act of kissing are dangerous (if one interprets “kissing” as an adjective attached to the noun “monkeys”) or that it is dangerous to kiss monkeys (if one interprets “kissing” as a verbal noun and “monkeys” as its object). R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 257–266. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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Fig. 13.1. Syntactic Ambiguity – Different parses for one sentence.

13.2.3

Semantic Ambiguity

In semantic ambiguity the meaning of a parsed sentence is not clear - e.g. the sentence “quitting smoking now greatly reduces your risk of cancer” can mean that if you quit smoking now, you can increase your odds of avoiding cancer or that now, contrary to what was true in the past, quitting smoking reduces the risk of cancer.

13.3

Human Indifference to Unclear Language

Even though human speech is, by default, filled with all three types of ambiguity, competent speakers of a natural language generally manage to communicate with each other without having to request clarification after every second sentence. In fact, most linguistic ambiguity is not even noticed by speakers of a language, unless they are made explicitly aware of it or are asked to clarify some particular point, possibly by a less competent speaker of the language in question. While all of these three forms of linguistic ambiguity allow a multitude of lawyers around the globe to make a living and can cause headaches for people in certain situations, none of them make human communication impossible, since context, knowledge of the world, and common sense generally make it clear which interpretation is appropriate. For example, the warning of the US Surgeon General cited above is correctly understood by most readers, including those who choose to ignore it. Syntactically ambiguous statements will generally be semantically sound, as one of the possible syntactic interpretations would lead to a semantically nonsensical result. Take, for example, the following sentences which, on a superficial level, look very similar: “The work must be done by a professional.” – “The work must be done by Friday.” The syntax of each sentence allows two interpretations: (1) The person who does the work must be “a professional”; The person who does the work must be “Friday” (2) The work must be done by the deadline “a professional”; The work must be done by the deadline of “Friday”). Since, however, we immediately recognize “a professional” as a person and “Friday” as a point in time, the first sentence will not confuse any competent speaker of the English language and the second one will only confuse someone in the process of reading Robinson Crusoe.

13.4 Vagueness in the German Language

13.3.1

259

Reproducing Human Indifference

In order to make a computer capable of interpreting a sentence of this kind correctly, semantics must be taken into consideration. This is exactly what was done when the project “Practical Simplification of English Text” [1] [2] [3] [4] [5] [6] [7] [11] [12] [10] at the University of Sussex created an algorithm intended to transform passive sentences into active ones, on a superficial level. A class of terms used to denote temporal expressions – “Friday”, “November”, “noon”, “tomorrow” – was defined. If the preposition “by” was connected to any of these terms, the sentence was interpreted as a temporal expression and not a passive sentence. No transformation was performed. This methodology is limited, however, in that temporal expressions not covered by this set of words will not be detected – e.g. “They were done by the inauguration.” Google Translate’s [9] performance in this regard, as of May 2010, indicates that their software uses a similar methodology, as is illustrated by the following examples. Original (ENG) Translation (GER) The work must Die Arbeit muss be done von einem Fachmann by an expert. durchgeführt werden. The work must Die Arbeit müssen be done bis Freitag by Friday. erfolgen. The work must Die Arbeit muss be done von der Einweihung by the inauguration. getan werden.

13.4

Comment correct translation awkward translation based on correct interpretation incorrect interpretation

Vagueness in the German Language

While German and English are genetically closely related, a result of the extremely rapid shifts the English language has gone through historically is that the two languages are quite distant from one another from a grammatical point of view. While English has very little morphology and uses word order to express subject-object relationships (i.e. in the sentence “The farmer sees the pilot”, the fact that the farmer is the person doing the seeing is clear due to the fact that “The farmer” precedes the verb), German uses different grammatical cases to mark the subject and object(s) of transitive verbs. The subject is in the nominative case, the primary object in the accusative case, and the secondary object in the dative case. This gives German more liberty when it comes to word order – the function of constituents can be derived from their grammatical case, making their position less important. German does not, however, have completely free word order as for example Hungarian does (where all six arrangements of Subject, Object and Verb

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are allowed: SVO, SOV, OSV, OVS, VSO, VOS). Simple German declarative main sentences generally follow SVO order, as is the case in English. With the exception of the main verb, which must always be in the second position of declarative main sentences (leading to the German word order being described as “V2 word order”), constituents can be moved around to stress certain elements more than others, without changing the basic meaning of the sentence. The sentence “Der Bauer sieht den Piloten im Fenster” consists of the following constituents: “der Bauer” (the farmer – nominative), “sieht” (sees), “den Piloten” (the pilot – accusative), “im Fenster” (in the window). Regardless of how these are arranged, the sentence will always mean “The farmer sees the pilot in the window”. V2 word ordering thus allows four different arrangements. Sentence Der Bauer sieht den Piloten im Fenster Den Piloten sieht der Bauer im Fenster Im Fenster sieht der Bauer den Piloten Im Fenster sieht den Piloten der Bauer

Word Order SVO (standard) SOV VSO VOS

V2 word order only affects declarative main sentences, however. Subordinate clauses in German demand that the main finite verb be in the final position – the standard word order in these is thus SOV, but only the verb’s position is fixed. Subject and object can switch their positions. Sentence Word Order Ich glaube, dass der Bauer den Piloten sieht. SOV (standard) Ich glaube, dass den Piloten der Bauer sieht. OSV Both sentences would be translated into English as “I believe that the farmer sees the pilot”. Consequently, in spite of the restrictions affecting German word order, all six arrangements of subject, object and verb are possible in German, in principle. Practically, there are some limitations. The nominative and accusative cases only actually differ for masculine nominals. Feminine and neutral nominals are identical in the nominative and accusative cases. Nominative der Mann die Frau das Auto er sie es

Accusative den Mann die Frau das Auto ihn sie es

English Translation man woman car he she it

As a result, syntactically unambiguous switching of a sentence’s subject and object is only possible if one of these two constituents is grammatically masculine.

13.4 Vagueness in the German Language

261

However, even when it is grammatically ambiguous, this is still done if the constituents’ roles are clear from the context. Take for example the following two sentences: “Anna schreibt ihre Diplomarbeit. Die Einleitung hat sie schon geschrieben.” – literally “Anna writes her thesis. The introduction has she already written.”. Both sentences lack a grammatically masculine constituent, making it impossible to derive their constituents’ functions from their appearance. The first sentence is a standard SVO sentence – semantics would not allow any other interpretation. The second sentence is, however, a non-standard OVS sentence. This can be inferred from the fact that the interpretation reading this sentence as SVO would lead to – “The introduction writes Anna” – is not semantically sound. The syntax is ambiguous, but the context is clear. In sentences lacking a male subject or object, constituents cannot be switched around if the context does not make it clear which constituent is the subject and which is the object. Such sentences are referred to as “reversible”. Take, for example, the sentence “Die Lehrerin sieht das Mädchen” (The teacher sees the girl) – here it would not be legitimate to switch subject and object around (The girl sees the teacher), as the resulting sentence would be interpreted differently – a teacher is as likely to see a girl as a girl is likely to see a teacher. If neither syntax nor context are clear in German, word order becomes as relevant as it is in English. 13.4.1

Reproducing German Indifference to German Ambiguity

Making software sensitive to semantic context is not a trivial matter. One fairly easy way would be to first categorize nouns by their animacy, and then determine in what arrangements verbs can appear, with respect to animacy. The first step, represented by the table below, is relatively straightforward. Word Frau Mädchen Katze Buch Einleitung Raumschiff Anna

Animacy + + + +

English Translation woman girl cat book introduction spaceship Anna

Then, verbs are classified regarding the arrangements in which they can appear. In addition to being used in an intransitive context, there are four possibilities here: animate subject & inanimate object, inanimate subject & animate object, animate subject & animate subject, inanimate subject & inanimate object.

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Verb Intr. A-I schreiben + + sehen + + stören + + interessieren sein + -

I-A + + -

A-A + + + -

I-I English Translation + to write to see + to bother to interest to be

It does not seem very likely for the verb “to write” to have an animate object. Its subject could, however, be either animate or inanimate:“The woman wrote a letter”, “This drive writes DVD-R discs”. On the other hand, it is unlikely for the verb “to see” to have an inanimate subject. The object, however, can be either animate or inanimate: “The woman saw the man”, “The woman saw a car”. Using this information, a deductive process can be undertaken when a syntactically unclear sentence is encountered, e.g. the sentence mentioned above, “Die Einleitung hat Anna schon geschrieben.” When analyzing this sentence, the software will first determine that the first constituent – “Die Einleitung” – is inanimate. This is not sufficient information to determine if this is the sentence’s subject or object, as this verb’s subject and object could both be inanimate. When determining that “Anna” is animate, however, the software can make the assumption that this constituent is the sentence’s subject – within the boundaries of the logic established here, no other interpretation would be valid. Often, looking at one constituent will suffice iin such cases. Take, for example, the sentence “Das Buch interessiert Anna nicht” – “The Book does not interest Anna”. As soon as the software determines that the first constituent, “das Buch”, is inanimate, it can be identified as the verb’s subject, as “interessieren” will not generally have an inanimate object. The weakness of this approach is that it deals in absolutes. Animacy is only one rather simplified binary attribute for classifying a word’s tendency to occur as a subject or an object. Let us consider, for example, the verb “to kick”. One could say that this verb’s subject must be animate and that its object is more likely to be inanimate. However, not everything that is animate is equally likely to be this sentence’s subject – an athlete is more likely to be kicking things than a teacher is. A teacher, in turn, is more likely to be kicking things than a monkey is. A lion will be even less likely to engage in this activity. A ball is less likely than any of these to play an active role in kicking and it is the most likely to be kicked. A hierarchy could be established in which different nouns are ranked by their likeliness to be the subject or object of the activity represented by a verb. Creating such models, and an appropriate inference mechanism, would of course be difficult. Considering the enormous amount of data necessary for this approach to be feasible at all, an automated procedure would be needed to gather the data. It would require an extensive corpus in which it would determine – based on only unambiguous sentences – which words occurred in which position relative to which verb how often. Semantic nets could be used here as well – i.e. if it is determined that a “bucket” is more likely to be kicked than it is to kick, this knowledge could to some degree be transferred to semantically close words – “pail”, “pot”, “trough”.

13.5 Aphasia

13.5

263

Aphasia

As has already been illustrated, competent users of a language can generally deal with ambiguity in language, as ambiguity does not generally occur on all levels of language at once. Syntactically ambiguous statements are semantically sound; semantically reversible sentences use simple word order, making their syntactic interpretation easier. Thus, most of us manage – but only most of us. Aphasia – from Greek meaning “without language” [15] – is an acquired language disorder (or a collective term for a number of disorders) that occurs after strokes and cranial injuries, after language acquisition is complete. All language modalities – speech production, speech comprehension, reading, and writing – are affected, on all levels of language – phonology, morphology, syntax, and semantics. It is important to note that people suffering from aphasia generally have retained their intellectual capabilities – they just have a hard time dealing with language. Aphasia creates new ambiguity and vagueness, of several kinds. Both in language production and comprehension, it is typical for aphasic speakers to get words mixed up that either denote similar concepts or that are phonetically similar. An Englishspeaking aphasia patient might mix up the words “map” and “mat” due to their superficial similarity or the words “sheep” and “goat” due to the semantic relation between the concepts they represent. Correctly parsing morphology is also difficult for aphasic speakers – meaning that German-speaking patients of aphasia can, in many cases, not be expected to distinguish the nominative and accusative cases, even where they are visibly marked. This creates problems in topicalized reversible sentences, such as “Den Bäcker hat der Ingenieur gesehen” (The engineer saw the baker), in which the object is placed in the first position. Aphasic speakers not registering the accusative case would assume that this sentence follows the SVO pattern simple German sentences generally abide to and would interpret this sentence incorrectly. [13] [14] [8] Subject/object relationships are just one example of unambiguous sentence structures that aphasia can make ambiguous. Another example would be clauses of time, used to express the temporal relationship between several activities. Both in English and in German, it is primarily conjunctions that mark the temporal relationship between the different clauses. Conjunctions are, however, problematic for many aphasic speakers; one cannot expect them to be interpreted correctly. Compare the following four sentences: “After I read the letter, I turned on the light.” “Before I read the letter, I turned on the light.” “I turned on the light after I read the letter.” “I turned on the light before I read the letter.” If one ignores the conjunctions in these sentences, it is impossible to tell the temporal relationship between the clauses. One might assume that the clause that appears first describes the action that occurred first. “[xxx] I read the letter, I turned on the light.” “I turned on the light [xxx] I read the letter.”

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This is, however, not always the case. 13.5.1

Simplification for Aphasic Readers

As a rule, aphasia patients are said to communicate better than they speak. They are known to communicate using a so-called telegraphic style, in which the constituents of sentences are strung together with little or no inflection. This is more noticeable in German than it is in English, due to the larger amount of morphology in German. “Peter reads a book” might be expressed as “Peter, read, book.” To ensure optimal understanding by aphasic readers, it is best to strive to create simple short sentences that use as little morphology as possible and that individually have as little content as possible [16]. This naturally results in a larger number of sentences, as no information should be lost in the transformational process. Some examples of transformations that can be carried out to best obtain this sort of language are presented below. Compound Sentences Compound sentences – which in German are subject to their own set of complex syntactical rules – are eliminated wherever possible. Relative clauses (“The man who is standing over there is my brother.” > “A man is standing over there. The man is my brother.”) and temporal clauses (“Before I read the letter, I turned on the light.” > “I turned on the light. And then: I read the letter”) are two examples of the multitude of different compound sentence types that can be transformed. Note how in the transformation of temporal sentences, the clauses are ordered in accordance with the actual order in which events occur in reality. Conjunctive Mood Conditional sentences – “Had you called me, I would not have come” – cannot simply be split apart in the same manner that temporal sentences can, as the relationship between the two elements would be lost in the process. However, steps can be taken to make the relationship between the two clauses more explicit. In the example given here, for example, the non-negated clause (“Had you called me”) marks an event that did not occur, the negated clause (“I would not have come”) marks one that did in fact occur. For people not capable of grasping the intricacies of words like “had” and “would”, noticing this relationship may be difficult. Thus, the sentence in question can be made more explicitly understandable if formulated like this: “You did not call me. The result: I came”. Tenses Complex tenses are replaced with more simple ones – “I had not known that.” becomes “I did not know that.” Passive Sentences Passive sentences are turned into active ones – “I was seen by a policeman” becomes “A policeman saw me.”

13.6 Other Applications of Text Simplification

265

Pre- and Postpositional Phrases Information embedded in pre- and postpositional phrases (e.g. “according to my father”) is extracted and used to form a simple declarative main sentence (e.g. “My father says:”). Word Order SVO word order is always enforced in all sentences. Vocabulary Aphasia not only affects people’s ability to comprehend grammar, but also their ability to understand individual words. By replacing rarely used or complex words with more elementary synonyms (e.g. “beverage” with “drink”), a text can also be made more comprehensible.

13.6

Other Applications of Text Simplification

The implications of software of this type go beyond the field of aphasiology: Text simplification is relevant to translation software as well, and could also be an aide for people with limited competence in a language for other reasons. The possible inclusion of this software in the upcoming project at the Vienna University of Technology aimed at making its curricula accessible to people with hearing impairments us currently being considered. This target group encompasses persons who are native “speakers” of sign language and for whom German is an acquired second language, even if they were born and raised in a German-speaking country.

References [1] Canning, Y., Tait, J., Archibald, J., Crawley, R.: Cohesive Generation of Syntactically Simplified Newspaper Text. In: Proceedings of the Third International Workshop on Text, Speech and Dialogue, Brno (2000) [2] Carroll, J., McCarthy, D.: Word Sense Disambiguation Using Automatically Acquired Verbal Preferences. Computers and the Humanities 34(1-2), 109–114 (2000) [3] Carroll, J., Minnen, G., Briscoe, T.: Can Subcategorisation Probabilities Help a Statistical Parser? In: Proceedings of the 6th ACL/SIGDAT Workshop on Very Large Corpora, Montreal (1998) [4] Carroll, J., Minnen, G., Briscoe, T.: Corpus Annotation for Parser Evaluation. In: EACL 1999 Post-Conference Workshop on Linguistically Interpreted Corpora (LINC 1999), Bergen (1999) [5] Carroll, J., Minnen, G., Canning, Y., Devlin, S., Tait, J.: Practical Simplification of English Newspaper Text to Assist Aphasic Readers. In: AAAI 1998 Workshop on Integrating Artificial Intelligence and Assistive Technology, Madison, Wisconsin (1998)

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[6] Carroll, J., Minnen, G., Pearce, D., Canning, Y., Devlin, S., Tait, J.: Simplifying Text for Language-Impaired Readers. In: EACL 1999 Post-Conference Workshop on Linguistically Interpreted Corpora (LINC 1999), Bergen (1999) [7] Devlin, S., Tait, J., Canning, Y., Carroll, J., Minnen, G., Pearce, D.: Making Accessible International Communication for People with Language Comprehension Difficulties. In: Computers Helping People with Special Needs: Proceedings of ICCHP, Karlsruhe (2000) [8] Goodglass, H., Blumstein, S., Gleason, J., Hyde, M., Green, E., Statlender, S.: The Effect of Syntactic Encoding on Sentence Comprehension in Aphasia. In: Brain and Language, 7th edn., pp. 201–209 (1979) [9] Google Inc.: Google Translate, http://translate.google.com/translate_t [10] McCarthy, D., Carroll, J., Preiss, J.: Disambiguating Noun and Verb Senses Using Automatically Acquired Selectional Preferences. In: Proceedings of the SENSEVAL-2 Workshop at ACL/EACL 2001, Toulouse (2001) [11] Minnen, G., Carroll, J., Pearce, D.: Robust, Applied Morphological Generation. In: 1st International Natural Language Generation Conference (INLG 2000), Mitzpe Ramon, Israel (2000) [12] Minnen, G., Carroll, J., Pearce, D.: Applied Morphological Processing of English. Natural Language Engineering 7(3), 207–223 (2001) [13] Stark, J., Stark, H.: Störungen der Textverarbeitung bei Aphasie. In: Blanken, G. (ed.) Einführung in Die Linguistische Aphasiologie, pp. 231–285. Hochschul Verlag, Freiburg (1991) [14] Stark, J.A., Wytek, R.: The Effect of Syntactic Encoding on Sentence Comprehension Aphasia. In: Dressler, W., Stark, J. (eds.) Linguistic Analyses of Aphasic Language, pp. 82–150. Springer, New York (1988) [15] Tesak, J.: Einführung in die Aphasiologie. Georg Thieme Verlag, Stuttgart (1997) [16] Tronbacke, B.I.: Richtlinien für Easy-Reader Material. In: International Federation of Library Associations and Institutions Professional Reports, vol. 57, The Hague (1999)

14 Can We Learn Algorithms from People Who Compute Fast: An Indirect Analysis in the Presence of Fuzzy Descriptions Olga Kosheleva and Vladik Kreinovich In the past, mathematicians actively used the ability of some people to perform calculations unusually fast. With the advent of computers, there is no longer need for human calculators – even fast ones. However, recently, it was discovered that there exist, e.g., multiplication algorithms which are much faster than standard multiplication. Because of this discovery, it is possible than even faster algorithm will be discovered. It is therefore natural to ask: did fast human calculators of the past use faster algorithms – in which case we can learn from their experience – or they simply performed all operations within a standard algorithm much faster? This question is difficult to answer directly, because the fast human calculators’ self-description of their algorithm is very fuzzy. In this paper, we use an indirect analysis to argue that fast human calculators most probably used the standard algorithm.

14.1

People Who Computed Fast: A Historical Phenomenon

In history, several people have been known for their extraordinary ability to compute fast. Before the 20 century invention of computers, their computational abilities were actively used. For example, in the 19 century, Johann Martin Zaharias Dase (1824–1861) from Vienna performed computations so much faster than everyone else that professional mathematicians hired him to help with their calculations; see, e.g., [1, 3, 4, 5]. Dase: • computed π (in his head!) by using a formula





1 1 1 + arctan + arctan , π /4 = arctan 2 5 8

(14.1)

with a record-breaking accuracy of 200 digits, during the period of under two months in 1844 – while it took several centuries for researchers to compute π with a perviously known accuracy of 140 digits; • calculated the logarithms table with 7 digit accuracy – this table included logarithms of all the numbers from 1 to a million, and • performed many other computational tasks. Karl Friedrich Gauss himself, the famous Princeps mathematicorum (“The Prince of Mathematicians”), recommended that Dase be paid by the Hamburg Academy of R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 267–275. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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14 Can We Learn Algorithms from People Who Compute Fast

Science to perform calculations – this was historically the first time when a person was paid for performing mathematical calculations.

14.2

People Who Computed Fast: How They Computed?

Calculations have been (and are) important in many practical problems. Because of this practical importance, people have therefore always been trying to speed up computations. One natural way to speed up computations is to learn from the people who can do computations fast.

14.3

People Who Computed Fast: Their Self-explanations Were Fuzzy and Unclear

In spite of numerous attempts to interview the fast calculators and to learn from them how they compute, researchers could not extract a coherent algorithm. One of the main reasons is that most of the fast calculators were idiots savants: their intellectual abilities outside computations were below average. Their explanations of their algorithms were always fuzzy and imprecise, formulated in terms of words of natural language rather than in precise mathematical terms. To us familiar with fuzzy logic and its applications this fuzziness should not be surprising. It is normal that people in general – and not necessarily idiots savants – cannot precisely describe • • • • • •

how they drive, how they operate machines, how they walk, how they translate from one language to another, how they recognize faces, how they control different situations, etc.

– this is why fuzzy control and other fuzzy techniques, techniques for translating this fuzzy description into a precise strategy, have so many practical applications; see, e.g., [6]. What may be somewhat unusual about fast computations is that while the selfdescription of fast calculators was fuzzy and imprecise, the results of the computations were always correct and precise.

14.4

With the Appearance of Computers, the Interest in Fast Human Calculators Waned

With the appearance of computers, the need for human calculators disappeared, and the interest in their skills waned. Such folks may provide a good entertainment, they

14.6 A Surprising 1960s Discovery of Fast Multiplication Algorithms

269

may be of interest to psychologists who study how we reason and how we perform menial tasks – but mathematicians were no longer interested. Yes, fast human calculators can perform calculations faster than an average human being – but electronic computers can perform the same computations much much faster. From this viewpoint, fast human calculators remained a curiosity.

14.5

Why Interest Waned: Implicit Assumptions

One of the main reasons why in the 1940s and 1950s the interest in fast human calculators waned is that it was implicitly assumed that the standard techniques of addition and multiplication are the best. For example, it was assumed that the fastest way to add the two n-digit numbers is to add them digit by digit, which requires O(n) operations with digits. It was also implicitly assumed that the fastest way to multiply two n-digit numbers is the standard way to multiply the first number by each of the digits of the second numbers, and then to add the results. Each multiplication by a digit and each addition requires O(n) steps, thus multiplication by all n digits and the addition of all n results require n · O(n) = O(n2 ) computational steps. From this viewpoint, the only difference between a normal human calculator, a fast human calculator, and an electronic computer is in the speed with which we can perform operations with digits. From this viewpoint, the only thing we can learn from fast human calculators is how to perform operations with digits faster. Once the electronic computers became faster than fast human calculators, the need to learn from the fast human calculators disappeared.

14.6

A Surprising 1960s Discovery of Fast Multiplication Algorithms

The above implicit assumption about arithmetic operations was not questioned until a surprising sequence of discoveries was made in the 1960s; see, e.g., [2]. These discoveries started with the discovery of the Fast Fourier Transform algorithm, an algorithm that enables us to compute the Fourier Transform 1 fˆ(ω ) = √ · 2π



f (t) · exp(i · ω · t) dt,

(14.2)

√ where i = −1, in time O(n · log(n)) – instead of the n · O(n) = O(n2 ) time needed for a straightforward computation of each of n values of fˆ(ω ) as an integral (i.e., in effect, a sum) over n different values f (t). The ability to compute Fourier transform fast lead to the ability to speed up the computation of the convolution of two functions: def

h(t) =



f (s) · g(t − s) ds.

(14.3)

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14 Can We Learn Algorithms from People Who Compute Fast

A straightforward computation of the convolution requires that for each of the n values h(t), we compute the integral (sum) of n different products f (s) · g(t − s) corresponding to n different values s. Thus, the straightforward computation requires O(n2 ) computational steps. Indeed, it is known that the Fourier transform of the convolution is equal to the product of Fourier transforms. Thus, to compute convolution, we can do the following: • first, we compute Fourier transforms fˆ(ω ) and g( ˆ ω ); • then, we compute the Fourier transform of h as ˆ ω ) = fˆ(ω ) · g( ˆ ω ); h(

(14.4)

ˆ ω ) and compute • finally, we apply the inverse Fourier transform to the function h( the desired convolution h(t). What is the computation time of this algorithm? • Both Fourier transform and inverse Fourier transform can be computed in time O(n · log(n)). ˆ ω ) = fˆ(ω ) · g( ˆ ω ) requires n computational • The point-by-point multiplication h( steps. Thus, the overall computation time requires O(n · log(n)) + n + O(n · log(n)) = O(n · log(n))

(14.5)

steps, which is much faster than O(n2 ). V. Strassen was the first to notice, in 1968, than this idea can lead to fast multiplication of long integers. Indeed, an integer x in a number system with base b can be represented as a sum n

x = ∑ xi · bi .

(14.6)

i=1

n

In these notations, the product z = x·y of two integers x = ∑ xi ·bi and y = ∑ni=1 yi ·bi i=1

can be represented as  z=

 

n

∑ xi · b

·

i

i=1 n



n

∑ yj · b

j

=

j=1 n

n

n

∑ ∑ xi · y j · bi · b j = ∑ ∑ xi · y j · bi+ j .

i=1 j=1

(14.7)

i=1 j=1

By combining terms at different values bk , we conclude that z=

n

∑ zk · bk ,

k=1

(14.8)

14.8 Interest in Fast Human Calculators Revived

where

zk = ∑ xi · yk−i .

271

(14.9)

i

This is convolution, and we know that convolution can be computed in time O(n · log(n)). The values zk are not exactly the digits of the desired number z, since the sum (14.9) can exceed the base b. Thus, some further computations are needed. However, even with these further computations, we can multiply two numbers in almost the same time O(n · log(n) · log(log(n))). (14.10) The corresponding algorithm, first proposed by A. Schönhage and V. Strassen in their 1971 paper [8], remains the fastest known – and it is much faster than the standard O(n2 ) algorithm.

14.7

Fast Multiplication: Open Problems

Fast algorithms drastically reduced the computation time. Fast Fourier transform is one of the main tools for signal processing. These successes has led to the need to find faster and faster algorithms. From this viewpoint, it is is desirable to look for even faster algorithms for multiplying numbers. The fact that researchers succeeded in discovering algorithms which are much faster than traditional multiplication gives us hope that even faster algorithms can be found.

14.8

Interest in Fast Human Calculators Revived

Where to look for these algorithms? One natural source is folks who did compute fast. From this viewpoint, the fact that they were unable to clearly explain what algorithm they used becomes an advantage: maybe the algorithm that they actually used is some fast multiplication algorithm? This possibility revived an interest in fast human calculators. So, the question is: • did fast human calculators use fast multiplication algorithm(s), or • they used the standard algorithm but simply performed operations with digits faster? Some of the human calculators do use special tricks to speed up their computations; see, e.g., [3], [4], [10] and a website [9] hosted by Oleg Stepanov from St. Petersburg, Russia, one of the fastest living calculators. These tricks help humans compute faster – usually because they avoid the typical human limitations of memory and of computations with many digits. But computers do not have these limitations, so the known tricks of the fast human calculators cannot help them compute faster. Are there other tricks that can help computers too?

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14.9

Direct Analysis Is Impossible

The most well-known fast human calculator, Johann Martin Zacharias Dase, died almost 150 years ago. Even when he was alive, his self-descriptions were not sufficient to find out how exactly he performed the computations. We know, from experience, that self-description of people with such phenomenal abilities are often fuzzy. For example, according to detailed self-explanations provided in [7], to different symbols, we associate different colors and/or tones and then find the result by an informal process of combining and matching these colors and/or tones. There might have been hope that our knowledge of fast multiplication algorithms can help in this understanding, but it did not work out. In other words, the direct analysis of Dase’s behavior has been impossible – and it is still impossible. We therefore need to perform an indirect analysis.

14.10

Indirect Analysis: Main Idea

A natural way to check which algorithm is used by a computational device – be it a human calculator or an electronic computer – is to find out how the computation time changes with the size n (= number of digits) of the numbers that we are multiplying. • If this computation time grows with n as n2 , then it is reasonable to conclude that the standard algorithm is used – since for this algorithm, the computation time grows as n2 . • On the other hand, if the computation time grows with n as ≈ n · log(n) (or even slower), then it is reasonable to conclude that a fast multiplication algorithm is used. – It maybe a Strassen-type algorithm, for which the computation time grows as n · log(n). – It may be a (yet unknown) faster algorithm in which case the computation time grows even slower than n · log(n).

14.11

Data That We Can Use

Interestingly, there is a data on the time that Dase needed to perform multiplication of numbers of different size. This data comes from fact that Dase’s performance was analyzed and tested by several prominent mathematicians of his time – including Gauss himself. Specifically: • • • •

Dase multiplied two 8-digit numbers in 54 seconds; he multiplied two 20-digit numbers in 6 minutes; he multiplied two 40-digit numbers in 40 minutes; and he multiplied two 100-digit numbers in 8 hours and 45 minutes.

14.12 Analysis

14.12

273

Analysis

For the standard multiplication algorithm, the number of computational steps grows with the numbers size n as n2 . Thus, for the standard multiplication algorithm, the computation time for performing the computation also grows as n2 : t(n) = C · n2 .

(14.11)

So, for this algorithm, two different number sizes n1 < n2 , we would have t(n1 ) = C · n21 and t(n2 ) = C · n22 and thus, t(n2 ) n22 = = t(n1 ) n21



n2 n1

2 .

(14.12)

For a faster algorithm, e.g., for the algorithm that requires O(n · log(n)) running time, the corresponding ratio will be smaller: t(n2 ) n2 · log(n2 ) n2 log(n2 ) = = · . t(n1 ) n1 · log(n1 )) n1 log(n1 )

(14.13)

Thus, to check whether a human calculators uses the standard algorithm or a t(n2 ) faster one it is sufficient to compare the the corresponding time ratio with the t(n1 ) 2 n2 square : n1 • If the time ratio is smaller than the square, this means that the human calculator used an algorithm which is much faster than the standard one. • On the other hand, if the time ratio is approximately the same as the square, thus means that the human calculators most probably used the standard algorithm – or at least some modification of it that does not drastically speed up the computations. • If it turns out that the time ratio is larger than the square, this would mean, in effect, that a human calculator used an algorithm which is asymptotically even slower than the standard one – this can happen, e.g., if there is an overhead needed to store values etc. According to the above data, we have: • • • •

t(8) = 0.9 minutes, t(20) = 6 minutes, t(40) = 40 minutes, and t(100) = 8 · 60 + 45 = 525 minutes.

Here, for n1 = 8 and n1 = 20, we have: 2 2 t(n2 ) t(20) 6 n2 20 = = ≈ 6.7 > = = 2.52 = 6.25. t(n1 ) t(8) 0.9 n1 8

(14.14)

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14 Can We Learn Algorithms from People Who Compute Fast

For n1 = 8 and n2 = 40, we have t(n2 ) t(40) 40 = = ≈ 44 > t(n1 ) t(8) 0.9



n2 n1



2 =

40 8

2 = 52 = 25.

Finally, for n1 = 8 and n2 = 100, we have 2

t(n2 ) t(100) 525 n2 100 2 = = ≈ 583 > = = 12.52 ≈ 156. t(n1 ) t(8) 0.9 n1 8

(14.15)

(14.16)

In all these cases, the computation time of a human calculator grows faster then n2 corresponding to the standard algorithm. The same conclusion can be made if instead of comparing each value n with the smallest value n1 , we compare these values with each other. For n1 = 20 and n2 = 40, we have 2 2 t(n2 ) t(40) 40 n2 40 = = ≈7> = = 22 = 4. (14.17) t(n1 ) t(20) 6 n1 20 For n1 = 20 and n2 = 100, we have t(n2 ) t(100) 525 = = ≈ 88 > t(n1 ) t(20) 6



n2 n1

2

=

100 20

2 = 52 = 25.

Finally, for n1 = 40 and n2 = 100, we have 2

t(n2 ) t(100) 525 n2 100 2 = = ≈ 13 > = = 2.52 = 6.25. t(n1 ) t(40) 40 n1 40

(14.18)

(14.19)

Thus, it is reasonable to conclude that the fast human calculators did not use any algorithm which is faster than the standard one.

14.13

Possible Future Work

People who perform computations fast appear once in a while. It may be a good idea to record and analyze their computation time – and maybe record their fuzzy explanations and try to make sense of them.

Acknowledgments This work was partly supported by the grants “The UTEP Master Teacher Academies", “Project BEST: Bridges for Education Students to Succeed", and “Middle School Integrated Mathematics and Science" from the Texas Higher Education Coordinating Board, and by the National Science Foundation (NSF) grant DUE-0717883 “Increasing Attractiveness of Computing: The Design and Evaluation of Introductory Computing Coursework that Elicits Creativity".

14.13 Possible Future Work

275

This work was also supported in part by the National Science Foundation grants HRD0734825 and DUE-0926721 and by Grant 1 T36 GM078000-01 from the National Institutes of Health, by Grant MSM 6198898701 from MŠMT of Czech Republic, and by Grant 5015 “Application of fuzzy logic with operators in the knowledge based systems” from the Science and Technology Centre in Ukraine (STCU), funded by European Union.

References [1] Beckmann, P.: A History of π . Barnes & Noble, New York (1991) [2] Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms. MIT Press, Cambridge (2001) [3] d’Ocagne, M.: Le Calcul Simplifié par les Procédés Mecaniques et graphiques. Gauthier-Villars, Paris (1905) [4] d’Ocagne, M.: Le Calcul Simplifié: Graphical and Mechanical Methods for Simplifying Calculation. MIT Press, Cambridge (1986) [5] Hofstadter, D.R.: Godel, Escher, Bach: an Eternal Golden Braid. Basic Books (1999) [6] Klir, G., Yuan, B.: Fuzzy sets and fuzzy logic. Prentice Hall, New Jersey (1995) [7] Luria, A.R.: The mind of a mnemonist: a little book about a vast memory. Hardard University Press, Cambridge (1987) [8] Schönhage, A., Strassen, V.: Schnelle Multiplikation graßer Zahlen. Computing 7, 281– 292 (1971) [9] Stepanov, O., http://stepanov.lk.net/ [10] Tocquet, R.: The Magic of Numbers. Fawcett Publications, Robbinsdale (1965)

15 Perceptions: A Psychobiological and Cultural Approach Clara Barroso The great recompense for exercising the power of thought is in the limitless possibilities of transferring meaning to objects and events in life that were originally acquired through intellectual analysis; from this comes the permanent and unlimited increase of meanings in human life. John Dewey

15.1

The Brain: An Organ That Communicates with Other Organs

The organic school of thought defines perception as the brain’s neural capacity to fire electrochemical impulses when it receives stimuli through sensory organs and its capability to decipher those stimuli in order to behave in an appropriate manner. Perception depends on the brain’s capacity to communicate with the sensory organs which receive stimuli from the environment, its ability to respond electrochemically, and the possibility that those responses are linked to actions or facts that have psychological meaning for an individual or group. The plasticity of the human brain allows it to associate the stimuli provided by the senses with meanings constructed though its interaction with the physical and social world. In other words, the brain carries out both functions, perceiving physical stimuli and assigning meaning to them, by applying the experience it has accumulated through interaction with the world. [3], [5] From a constructivist perspective, a subject’s ’experience’ can modify or even distort the information processed by the brain when it decodes electrochemical impulses. As we shall see, ’experience’ is usually one of the factors that associate a meaning with a stimulus and, consequently assign meaning: providing perception. The experiment which demonstrated that bats guide themselves using echolocation also demonstrates that if we inhibit the reception of information from the senses, the brain is unable to perceive its environment and act in consecuence. Deprivation from physical stimuli impedes the assignation of meaning in terms of experience; it impedes perception.

R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 277–285. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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15.2

The Functional Perspective on Perception

From a linguistic and logical point of view the sentence ’give us the bread’ is correct; but it has different meanings depending on if we use it in a church, an art restoration workshop1 or in a bakery; the information is meaningless unless it is placed in a specific physical context that we can perceive. “The acquisition of definiteness and of coherency (or constancy) of meaning is derived primarily from practical activities [. . . ] So far as we sit passive before objects, they are not distinguished out of a vague blur which swallows them all.” ([4], p. 122-123.) In this sense, assigning meaning depends on the interaction between subject and reality. It is impossible to construct linguistic meanings beyond the actions that a subject carries out with the things that are being attributed these meanings. The construction of meaning depends on both the sensory perception of objects in the world and the possibility to act in the world. Actions define the precise way in which possible meanings are attributed to each object and these actions are tied to the interaction between the subject and the object. Another source of meaning is associated with our experiences and de expectations of reality that are tied to them. J. Dewey expresses it in the following description: “If a person comes suddenly into your room and calls out ’Paper’ various alternatives are possible. If you do not understand the English language, there is simply a noise which may or may not act as a physical stimulus and irritant. But the noise is not an intellectual object; it does not have intellectual value. To say that you do not understand it, and it has no meaning are equivalents. If the cry is the usual accompaniment of the delivery of the morning paper, the sound will have the meaning, intellectual content; you will understand it. Of if you are eagerly awaiting the receipt of some important document, you may assume that the cry means an announcement of this arrival. If (in the third place) you understand the English language but no context suggest itself from from your habits and expectations, the word has meaning but not the whole event” ([4], p. 116-117.) As a consequence, the semantic meaning attributed to a sentence depends on both its linguistic validity, the information provided by the environment and the expectations that our experience generates of the world that surrounds us. In other words, the brain must be able to contextualise the information in order to apply a specific meaning to the linguistic sentence. Therefore, interaction with our physical surroundings initially mediates the meaning that is assigned to a linguistic sentence; but we shall see that meaning is also linked to interaction with our socio-cultural environment. 1

Translator’s note: “Gold leaf”, the thin foil used to gild artwork, is known as pan de oro (literally “golden bread”) in Spanish; hence the possible confusion of asking for “bread” in an art restoration workshop.

15.3 The Importance of Psychology in the Construction of Perceptions

279

“However ambiguous or polysemous our discourse may be, we are still able to bring our meanings into the public domain and negotiate them there. That is to say, we live publicly by public meanings and by share procedures of interpretation and negotiation.” ([1], p. 13.) All individuals belong to a group. This fact determines how young people ’learn’ to assign meanings to perceived stimuli: each individual is incorporated into the shared experience of the group. The group’s experiences in different environments inform the individual as to what actions are possible in an environment and what behaviour is considered appropriate. The experience of a group in a perceived physical environment, reinforce the appropriateness or inappropriateness of a meaning for each individual in each context and they function as activators/inhibitors of behaviour. In this way, organic stimulation is interpreted and provided with meaning in a specific social context giving rise to complex behaviours. More importantly, sharing this accumulation of meanings for physically perceived reality allows the interaction between individuals to access the cognitive representation of the perception of the other.

15.3

The Importance of Psychology in the Construction of Perceptions

Another factor involved in perception we shall refer to as ’attention’. In certain situations things can occur which are not perceived by an individual when her attention is fixed on an objective outside of the situation; that is, when the process of assigning meanings is restricted by a prior interest. Therefore, physically perceived sensory stimuli can be mediated by prior instances that establish the relevance of what penetrates our conscience, i.e. what we perceive. ’Attention’ plays an important role in determining the relevance of the information provided by our senses, establishing what information we process consciously in order to give them meaning in specific situations. [7], [9] Finally, another factor that affects perception will be referred to as ’experience’. As we gain experience in the world our brain categorises the objects that surround us (language plays a very important role in this process). Initially our world is made up of only a few vague categories. However, through experience we accumulate information about the different objects and events surrounding us which leads to more precise categories being developed. This process allows us to optimise the decisions we take about possible actions and desirable actions. Experience, understood as an individual’s interaction with the surrounding social and physical world, constitutes the collection of knowledge that the individual possesses and which can be described as the ability to give meaning to physically perceived events, depending on the context in which they occur, and act accordingly. When dealing with the world, we expect the objects that surround us to be compatible with our prior knowledge and we attribute qualities to these objects and

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events based on previous experiences with them or others that we heuristically perceive to be similar. In other words, humans employ a collection of prior meanings -experience-in many of the processes through which we assign meaning to the physically perceived world. “Symbols are themselves, as pointed out above, particular, physical, sensible existences, like any other things. They are symbols only by virtue of what they suggest and represent i.e. meanings. They stand for these meanings to any individual only when he has experience of some situation to which these meanings are actually relevant. Words can detach and preserve a meaning only when the meaning has been first involved in our own direct intercourse with things” ([4], p. 176) How this previous experience is used is not defined a priori. There are no established rules that determine the use of the accumulated experiences. However, through experience we can ’complete’ the sensual information provided by the environment, either through intuition or the translocation of information, in order to assign it precise meanings in accordance with the environment in which we find ourselves. Because we know the characteristics of the environment in which events occur, experience allows us to give meaning even in the absence of precise information. This indicates that the set of meanings that a subject uses is not defined by associations between stimuli and meanings created through logical inferences outside of the context in which the stimuli are produced.

15.4

Perception and the Construction of Meaning

As explained earlier, we always begin to construct meaning based on sensory perception, but it is the experience accumulated throughout our conscious life which allows us to assign meanings to facts and events in the real world. We call this ’perceiving the world’. In very controlled situations meaning can even be assigned to facts that are imperceptible to the senses. If we possess theoretical knowledge of their phenomenology, we can assign meaning to facts that surpass the limits of sensory perception by using artefacts which increase the limits of our sensory organs (indirect perception). In this way an astrophysicist is able to give meaning to infrared images despite being unable to visualise that wavelength. Human perception is an emerging property in that its function in the development of actions in the real world surpasses the quantity and quality of the information that we process. It is the ability to establish interaction between the functional and the organic which makes it possible to interpret sensory information and convert it into meaningful information. In the learning process humans develop the ability to assign different meanings and different degrees of association between those meanings and specific stimuli; previous experience and perceiving context play important roles in this task. In the example, our ability to assign the sentence ’give us bread’ a precise meaning within

15.5 Perception and Artificial Intelligence

281

the context of a restoration workshop will depend on if we have experience with the processes carried out there. This will constitute the domain of knowledge that we have in a given moment. “Language is acquired not in the role of spectator but through use. Being ’exposed’ to a flow of language is not nearly so important as using it in the midst of ’doing’. Learning a language [. . . ] is learning ’how to do things with words’. The child is not learning simply what to say but how, where, to whom and under what circumstances.” ([1], p. 70-71.) Information acquires the meanings that we use to decide on which actions to take through a lifelong learning process in the area of shared experience. As explained above, in this shared experience the physical environment becomes an agent of meanings.

15.5

Perception and Artificial Intelligence

As we have seen, human perception is highly complex. Perception includes physical, psychological, biographical and cultural aspects that all play important roles. All of these facets interact with each other and take part in the processes by which human beings know how to assign meanings to the world around them and to act appropriately. A great deal of effort has been made in the field of artificial intelligence to establish rules that allow a semantic representation of what occurs in a subject’s domain of knowledge when meanings are assigned to events (or facts), but without taking into consideration the importance of the context (environment) in which the event occurs. Our perception of an environment is not based exclusively on the facts and physical objects which characterise it; our previous experience also mediates our perception of what constitutes the context and, subsequently, how we assign meanings to the events that take place in it. Experience, therefore, is part of the context. “We do not approach any problem with a wholly naive or virgin mind; we approach it with certain acquired habitual modes of understanding, with a certain store of previously evolved meanings, or at least, of experiences from which meanings may be educed.” ([4], p. 106.) A robot reacts to its environment by receiving information through sensors and processing it in accordance with the ontology represented in its computer code, it then responds according to the procedural rules defined by its computer program. A robot with artificial intelligence could act even in contexts where it had incomplete information on possible actions. However, it would have serious problems if the information corresponded to its ontology, it was capable of taking decisions on actions using incomplete or imprecise information, but the robot was in an environment that had not been previously defined. In other words, the robot would be

282

15 Perceptions: A Psychobiological and Cultural Approach

unable to give adequate meaning to the information because it would lack a contextual interpretation of the data. It could be argued that in this case the ontology would be inadequately defined. However, we hope that this work will provide some insight into how we could give the robot a key which would allow it to locate the concepts that could give meaning to the information available to it, without requiring an exhaustive description of the world. Imagine a robot that had colour and shape sensors, information on vegetables and edible plants and a program able to use all of them in a recipe that used whatever vegetables it was viewing. In addition we will give it the ability to decide if it will use all or just some of the vegetables in accordance with information on the texture of the vegetables, flavour, cooking time, nutritional value, etc. How would the robot react when faced with a painting by Arcimboldo? Would the problem be resolved by creating a new artefact for it that was capable of completing visual information with information on volumes? Would we have to give it more information on different kinds of wood, wax and other elements used to create ornamental fruits and vegetables? Describing perception as the construction of meaning adequate for a specific environment could solve the robot’s dilemma. This would allow it to assess if it was in a museum, a fruit store or a home decoration shop. In other words, having access to data about the environment could optimise the quantity of information needed to take decisions on possible actions by assigning a range of values to the actions that it was considering. The robot should process information about the world and the possible actions to take in it, and also use different degrees of association between the information received and meanings. Therefore P ⇒ Q induction relations (with P being the information received and Q the meaning that can be attributed) would assume a meaning Q depending on the environment in which P is being emitted. In other words, it would be the physical environment, the real world it was viewing, which would determine a meta-level in which certain attributions of meaning would be valid. As such, we can state that the information related to ’perceiving the environment’ determines the validity of meanings that have been assigned. In any case, it should be noted that implementing the concept of ’environment’ in artificial intelligence software would be linked to the logical representation of an argument and not the representation of a formula. If the representation of the meta-level is seen in terms of an argument structure, the attribution of meanings to an environment would only be true if the premises which describe the environment are true. In terms of logical universes, we would be in an existential environment (∃xAx) not in a universal environment (∀xAx), since any assertion that we would want to be valid in any possible environment could generate inconsistent arguments. In other words, the universal quantifier would be operative in a universe in which all objects met the properties associated with it (∀xFx ⇒ Fc), in Popperian terms we would be in a numerically universal universe in which functions would be defined by whether it was applicable or not to a specific environment in which the argument structure was valid. [8]

15.6 Concept Maps and Contextual Meaning

283

Even so, it would still be impossible for the robot to complete the construction of meaning for the environment if it was unable to represent experience at the same time. It could accumulate information on possible environments, but what knowhow would it have to develop in order to represent the processes by which humans establish the validity of multiple meanings for a single stimulus or assign different stimuli to a single meaning, choosing the most appropriate for each case?

15.6

Concept Maps and Contextual Meaning

Concept maps are emerged in education as a tool to increase meaningful learning that is learning which can be incorporated into the learner’s relevant prior knowledge [10]. The concept map is a graphical representation of specialised knowledge which shows concepts (belonging to the subject matter domain) and the relationships between them. The concepts correspond to perceived regularities in events or concepts, designated by a linguistic label. [6], [2] The concepts are connected by linking words which specify the relationship between them, this relationship is graphically expressed using connecting lines which create sentences with semantic meaning. The precise meaning of a concept is determined by the map in its entirety. In the following example ’arms’ acquires its meaning through the relationships with other concepts to which is linked.

Fig. 15.1. caption missing.

In a concept map we can distinguish different areas of knowledge reflected in the map itself that can be interpreted as linguistic sentences constructed through the links established between concepts; underlying knowledge which corresponds to the graphical form in which the explicit is represented; and implicit knowledge which corresponds to the context in which the concepts that have been represented are interpreted. Therefore a concept map contains two areas which give meaning to the concepts; the linguistic meaning of the sentences which have been constructed and the interpretative context in which the sentences acquire a precise meaning. For example, the precise meaning of the concept ’information’ changes according to the other concepts it is linked to; that is, its meaning depends on whether it is linked to concepts related to computer science, journalism, education, etc. In contrast to a graph, concept maps can present non-binary links between different concepts, so a single concept can give rise to various cross-links that express the relationship between concepts. In this sense, the relationship between the

284

15 Perceptions: A Psychobiological and Cultural Approach

Fig. 15.2. caption missing.

concepts can be located in different domains of the map, allowing for meaningful sub-domains to be represented through multiple relationships between the concepts include in the map.

Fig. 15.3. caption ???????

The representation of the experience domain (see figure 15.3) is linked to decisions that are adopted according to the concepts which are used in the map, and the relationships established between them. As a result, the precise meaning of each concept represented is given by the interpretative context in which it appears.

15.7 Conclusion

285

Consequently, the concept map can be a useful tool which provides the knowhow to express the contextual meaning of a concept, representing the universe of discourse in which a concept is used.

15.7

Conclusion

Throughout our conscious life all humans develop learning processes which allow us to acquire the ability to assign different meanings to specific physical stimuli; we also acquire the ability to establish different degrees of association between possible meanings and these stimuli. How we perceive context plays an important role in establishing which of the possible meanings is appropriate for each event, and it is in this ’how’ where our mastery of knowledge resides. This goes beyond merely accumulating information; it means that we can associate meanings with stimuli in accordance with the prior experiences we have had throughout our lives. In other words, humans choose which existential universe we are in; each fact or event possesses a meaning within this universe that is associated with a certain degree of validity and we act in consequence. Sharing this existential universe allows us to interact with other humans, by assigning collective meanings, and to evaluate the appropriateness of our actions in that universe. In order to effectively represent knowledge in artificial artefacts we must consider representing the existential universes in which concepts acquire meaning, representing the universe of discourse which corresponds to the perceptions we want the artefact to use. As we move beyond semantic networks, concept map could bring us closer to a true representation of how humans construct meanings, by incorporating the existential dimension of the universe of discourse that we wish to represent.

References [1] Bruner, J.: Acts of meaning. Harvard University Press, Cambridge (1990) [2] Coffey, J.W., Hoffman, R.R., Cañas, A.J., Ford, K.M.: A Concept Map-Based Knowledge Modeling Approach to Expert Knowledge. In: Proceedings of IASTED International Conference on Information and Knowledge Sharing, pp. 212–217 (2002) [3] Crick, F.H.C.: La búsqueda científica del alma. Debate, Madrid (1994) [4] Dewey, J.: How we think. D.C. Heath & CO. Publishers, Boston (1910) [5] Maturana, H.R., Varela, F.J.: El árbol del conocimiento. Las bases biológicas del conocimiento humano. Debate, Madrid (1990) [6] Sinha, C.: Grounding, Mapping and Acts of Meaning. In: Janssen, T., Redeker, G. (eds.) Cognitive Linguistics: Foundations, Scope and Methodology, pp. 223–255. Mouton de Gruyter, Berlin (1999) [7] Simons, D., Chabris, C.: Gorilas in our midst: sustained inattencional blindness for dynamic events. Perception 28(9), 1059–1074 (1999) [8] Popper, K.R.: The Logic of Scientific Discovery. Routledge, London (1968) [9] Ramachandran, V.S.: The emerging mind (Reith Lectures 2003). Profile Books Ltd. (2004), http://www.bbc.co.uk/radio4/reith2003/lecture2.shtml [10] Novak, J.D., Godwing, D.B.: Learning how to learn. Cambridge University Press (1984)

Part V

16 Rule Based Fuzzy Cognitive Maps in Humanities, Social Sciences and Economics João Paulo Carvalho

16.1

Introduction

Decision makers, whether they are social scientists, politicians or economists, usually face serious difficulties when trying to model significant, real-world dynamic systems. Such systems are composed of a number of dynamic qualitative concepts interrelated in complex ways, usually including feedback links that propagate influences in complicated chains. Axelrod [1] work on Cognitive Maps (CMs) introduced a way to represent real-world qualitative dynamic systems, and several methods and tools have been developed to analyze the structure of CMs. However, complete, efficient and practical mechanisms to analyze and predict the evolution of data in CMs were not available for years due to several reasons. Dynamic Cognitive Maps (DCM), where Rule Based Fuzzy Cognitive Maps (RB-FCM) are included, are a qualitative approach to modeling and simulating the Dynamics of Qualitative Systems (like, for instance, Social, Economical or Political Systems) [6, 3, 4, 5, 2, 7, 8, 9, 10, 11]. RB-FCM were developed as a tool that can be used by non-engineers and/or non-mathematicians and eliminates the need for complex mathematical knowledge when modeling qualitative dynamic systems. This work presents a case study where RB-FCM were used to model an economic system where social influences and qualitative modeling were heavily accounted for. Economic models have traditionally been based on mathematics. Econometry, the quantitative science of modeling the economy, focus on creating models to help explain and predict variables of interest in economics. However the most common econometric models are usually very imprecise and are not usually valid but on a very short term. This can easily be seen on the regular predictions made to most macroeconomic indicators: most yearly predictions made by governments, economic entities or independent experts, must usually get corrected every trimester due to inaccuracies in the models used to predict their values. This is essentially due to the fact that most econometric models tend to ignore the existence of the feedback loops that make any alteration in any component of the model to potentially be propagated until that component is affected by its own previous change on a relatively short term.The more precise models that try to address this issue are usually based on differential equations [13, 20]. However, due to the dimension of these systems (very high number of variables involved), these models demand R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 289–300. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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a strong knowledge in mathematics, and a huge amount of time to be developed. However, even when these kinds of models are possible, they usually tend to ignore that economy is a social science, and therefore is subject to qualitative uncertainties due to human and social factors that are not easily captured using strict quantitative mathematic models.The use of approaches that include both the existence of feedback cycles and the intrinsic qualitative social nature associated with economy, should lead to the implementation of more accurate models. In this paper one shows that RB-FCM can be used to model socio-economical systems using as an example a model that was originally designed in the late 90’s and yet, can explain and simulate the 2008 economic world crisis.

16.2

Dynamic Cognitive Maps

The term Dynamic Cognitive Maps has been recently used to describe techniques that allow simulating the evolution of cognitive maps through time. Axelrod work on cognitive maps (CM) [1] introduced a way to represent real-world qualitative systems that could be analyzed using several methods and tools. However, those tools only provided a way to identify the most important structural elements of the CM. Complete, efficient and practical mechanisms to analyze and predict the evolution of data in CM were not available for years due to several reasons. System Dynamics tools like those developed by J. W.Forrester [12] could have provided the solution, but since in CM numerical data may be uncertain or hard to come by, and the formulation of a mathematical model may be difficult, costly or even impossible, then efforts to introduce knowledge on these systems should rely on natural language arguments in the absence of formal models. Fuzzy Cognitive Maps (FCM), as introduced by Kosko [15, 17, 16], were developed as a qualitative alternative approach to system dynamics. However, although very efficient and simple to use, FCM are causal maps (a subset of cognitive maps that only allow basic symmetric and monotonic causal relations between concepts) [9], and, in most applications, a FCM is indeed a man-trained Neural Network that is not fuzzy in a traditional sense and does not exploit usual fuzzy capabilities. They do not share the properties of other fuzzy systems and the causal maps end up being quantitative matrixes without any qualitative knowledge.Several evolutions and extensions have been added to the original FCM model, but none addresses so many FCM issues as RB-FCM. RB-FCM were developed as a tool that models and simulates real world qualitative system dynamics while trying to avoid the limitations of those approaches. The following sub sections resume some features of RB-FCM that are useful to the comprehension of this paper. 16.2.1

Rule Based Fuzzy Cognitive Maps

RB-FCM allow a representation of the dynamics of complex real-world qualitative systems with feedback, and the simulation of events and their influence in the system. They can be represented as fuzzy directed graphs with feedback, and are composed of fuzzy nodes (Concepts), and fuzzy links (Relations). RB-FCM are true

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cognitive maps since are not limited to the representation of causal relations. Concepts are fuzzy variables described by linguistic terms, and Relations are defined with fuzzy rule bases. RB-FCM are essentially iterative fuzzy rule based systems where we added fuzzy mechanisms to deal with feedback, introduced timing mechanisms [8] and new ways to deal with uncertainty propagation, and were we defined several kinds of Concept relations (Causal, Inference, Alternatives, Probabilistic, Opposition, Conjunction, etc. [2, 5]) to cope with the complexity and diversity of the dynamic qualitative systems we are trying to model. Among new contributions brought by RB-FCM, there is a new fuzzy operation - the Fuzzy Carry Accumulation [4, 7] -, which is essential to model the mechanisms of qualitative causal relations (FCR - Fuzzy Causal Relations) while maintaining the simplicity and versatility of FCM. There are two main classes of Concepts: Levels, that represent the absolute values of system entities (e.g., LInflation is Good); and Variations, that represent the change in value of a system entity in a given amount of time (e.g., VInflation increased very much). By allowing the definition of both the absolute value of a concept and its change through time, RB-FCM have the means to properly model the dynamics of a system. 16.2.2

Expressing Time in Dynamic Cognitive Maps

Time is probably the most essential factor when modeling a dynamic system. However, most DCM approaches seem to ignore this fact. In order to maintain consistency in the process of modeling the dynamics of a qualitative system, it is necessary to develop and introduce timing control mechanisms. To allow the representation of time flow, delays, and the inhibition of certain relations when they have no influence on a given instant, changes were made to the engine of RB-FCM. More details regarding RB-FCM time mechanisms can be found in works by the present author [8, 11].

16.3

A Qualitative Macro Economic Model as an Example of DCM Modeling in Socio-Economic Systems

In this section one presents a model that was developed eight years ago. The primary goal when this problem was approached was to show the capabilities and ease of use of RB-FCM to model the dynamics of qualitative real world systems. Even if the final model is rather complex and does not contains apparent flaws, it is not, and was never intended to be, a complete model, since it was not developed by economic experts (even though some were consulted). However, as it can be seen in the obtained results, the model exhibits a behavior that is able to describe the current economic crisis and the reasons that lead to it. Classic cognitive mapping techniques [1] were used as the first step to obtain the model: the concepts and relations were extracted from a short column published in Portuguese newspaper Publico in the year 2001 consisting on an economic expert analysis regarding “Short-term Tax Rate evolution in Europe” [18]. Throughout the text, the author introduced several

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concepts, supporting its theories while explaining the relations between concepts using qualitative knowledge. The “classic” CM obtained was much simpler than the one presented here, which was expected, since the analysis of the dynamics of a much more complex model - like the one we ended up obtaining - would require several months of work using traditional quantitative econometric approaches. The first model used only the most important concepts (the ones really necessary to a short term analysis): Tax Rates, Inflation, Consumption, Oil Price, and Food Cost. Even with such a few concepts, a realistic model becomes rather difficult to analyze due to the complexity of the relations that affect the involved concepts. However, since the goal was to show RB-FCM potential to deal with larger systems and longterm simulations, the model was evolved to be more realistic therefore including more concepts and much more relations. On this step, 13 concepts were added to the original 5 (Fig.16.3). At the end of this phase of the modeling process one obtained a classic Cognitive Map - basically a graph where the nodes were the Concepts and each edge represented an existing unknown relation between a pair of Concepts. 16.3.1

Concept Modelling

The next step was refining the concepts to obtain a linguistic fuzzy representation for each one. This step consisted in defining the class(es) (Variation, Level) and the linguistic terms and membership functions for each concept. In dynamic systems, variations are much more important than absolute values, therefore, most concepts are Variations, some are Levels, and a few key concepts like Inflation, Tax Rate, etc., are both Variations and Levels (the Level value of these concepts is actualized according to its Variation using a special LV relation [5]). The linguistic terms of Levels must have a direct correspondence with the real world values. Therefore we allied common sense and expert consulting (using straight questions like “what do you consider a high value for Inflation?”, and receiving answers like “around 4%”) to define their membership functions. In the particular case of Levels that depend on LV relations, it was also necessary to define the real-world meaning of a certain amount of variation (e.g., a “Small” increase on inflation is around 0.3%). Fig.16.1 shows the linguistic terms of the Level concept LInflation.

Fig. 16.1. Level “Inflation” linguistic terms. Dotted linguistic terms represent the variation degrees of Inflation {HugeDecrease, LargeDecrease, Decrease, SmallDecrease, VSDecrease, Maintain, VSIncrease,. . ., HugeIncrease}

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Variation linguistic terms usually represent qualitative terms without a direct correspondence to absolute values. E.g., VInflation has 11 linguistic values ranging from “Huge Decrease” to “Huge Increase” (Fig.16.2). Linguistic terms of Variations can usually be represented by standard sets, which simplify and accelerate the modeling process [10].

Fig. 16.2. Variation “Inflation” linguistic terms: {Huge Decrease, Large Decrease,. . .,Huge Increase}. x-scale values are normalized values. There is no relation no real world values

16.3.2

Qualitative Modeling of a Qualitative Dynamic System

The huge advantages of using Fuzzy Rule Bases (FRB) to define qualitative relations between Concepts has been largely discussed and proved [2, 3, 5, 9, 11]. The major drawback of rule-based fuzzy inference, the combinatorial explosion of the number of rules, is avoided in RB-FCM by the use of Fuzzy Causal Relations and the Fuzzy Carry Operation [4]. Another important feature of RB-FCM is the simplicity of the process of insertion and removal of Concepts and/or Relations, which also reduces the modeling complexity of FRB [4]. Therefore one has in RB-FCM an adequate tool to model qualitative relations. However, the single fact of using linguistic rule bases to model relations does not guarantee the qualitative nature of the model. Let us see the example of Inflation modeling: A pseudo-qualitative approach using FRB would try to closely map the widespread quantitative approaches: Inflation value is predicted by a weight averaged sum of several factors (Estimated Oil inflation, estimated Food price inflation, etc.). This method is highly dependent on the precision and validity of each factor real-world absolute value. In the proposed model, a novel approach where rules are independent from the real world absolute values was used. The model is based on a qualitative definition of inflation: Economics theory states that economic growth depends on inflation - without inflation there is no growth. In fact, the worst economic crisis (30’s for instance) are associated with deflation. Therefore, it is desirable and expected that all factors that affect inflation have a certain cost increase - If all factors suffer a normal increase, then the inflation will maintain its normal and desired value. Therefore, one can state the following qualitative relation for each of those n factors: “If factor n has a normal increase, then Inflation will maintain”

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This statement is part of the fuzzy rule base of a causal relation. Since fuzzy causal effects are accumulative and their effect is a variation in the value of the consequent, then if all factors that cause inflation have the normally expected increase, Inflation will not vary. If some factors increase more than expected and the others maintain their value then inflation will somehow increase. If a factor increase less than normal, or even decreases, then its effect is a decrease in inflation (note that the final variation of Inflation is given by the accumulation of all causal variation effects e.g., if some pull it down a bit, and one pulls it up a lot, in the end inflation still can maintain its normal value). It is possible to build a completely qualitative and sound causal FRB to model each factor influence of Inflation, without ever referring to absolute values. If one intends to model inflation in South America, one can maintain the rule base. All that needs to be changed are the linguistic terms of the Level Concept associated to Inflation (for instance, normal inflation would become around 8%, and so on...). Obviously some factors are more important than others (a large increase in food might cause a large increase in Inflation, but what is considered a large increase in Oil might only cause a small increase in Inflation - average Oil price varied over 100% in the last 2 years, but other factors had a slightly above average increase, and therefore inflation had mild increase instead of a sever one...). This “relative” importance is easily modeled as a causal effect in a FRB. Table16.1 represents an example of a causal FRB. One can also mention the fact that oil price variation has a delayed effect in inflation. RB-FCM provide mechanisms to model these kinds of timing issues [8, 11]. Table 16.1. FCR7 +sl Food Cost, Inflation If Food Cost Decreases VeryMuch, If Food Cost Decreases Much, If Food Cost Decreases, If Food Cost Decreases Few, If Food Cost Decreases VFew, If Food Cost Maintains, If Food Cost Increases VFew, If Food Cost Increases Few, If Food Cost Increases Normally, If Food Cost Increases M, If Food Cost Increases VM,

Inflation has a Large Decrease Inflation has a Large Decrease Inflation has a Large Decrease Inflation Decreases Inflation Decreases Inflation Decreases Inflation has a Small Decrease Inflation has a Very Small Decrease Inflation Maintains Inflation has a Small Increase Inflation Increases

This kind of qualitative approach was used throughout the model when causal relations were involved. As it was mentioned above, variations usually have a standard set of linguistic terms. These allow the predefinition of certain common fuzzy causal relations (FCR). These FCR are called macros and were used to reduce the modeling effort.The model includes other than causal relations. For instance: Oil price variation was modeled using a classic fuzzy inference rule base (FIRB) based on oil Offer/Demand (where Oil offer was decided in simulated periodic OPEP meetings);

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the Tax Rates were modeled considering that Banks were managed as a common business with profit in mind - for example, an increase in money demand would increase Tax Rates (this would be changed later (see (16.3.4)). Regarding timing considerations, the system was modeled considering a one month period between iterations.

Fig. 16.3. RB-FCM: A qualitative model of economy. Concept 17, representing a simple FISS was added later

It is obviously impossible to detail every aspect of the system modeling in this paper. Fig. 16.3 provides a graphic representation of the final RB-FCM model. The system consists of 18 concepts and around 400 fuzzy rules to express relations (most were automatically generated using macros). The system was described using RB-FCMsyntax (a dedicated language) – a complete description is available in [2]. Here are some guidelines regarding the description of relations in Fig.16.3: “FCR+” stands for a standard positive causal relation (an increase in the antecedent will cause an increase in consequent), and “FCR-” a standard negative relation (increase causes a decrease). Several “+” or “-” represent stronger effects. A “/” represents an attenuated effect. “sl” and “sr” represent biased effects (non symmetric causal relations. A “?” represents a relation which cannot be symbolically described (one must consult the FCR). A “d” represents a delay in the effect. FIR stands for Fuzzy Inference Relation. The number after FCR or FIR is the label for the complete description of the rule base.

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Simulation Results

The simulation of the original system provided rather interesting results. The evolution of the system through time was rather independent from the initial values and the external effects. After a certain period of time, which could vary from a few months to several years (depending on a conjugation of external factors like a war, or a severe cut in oil production), economy would end up collapsing: deflation, negative growth, 0% tax-rates. Fig.16.4 represents one of those cases. Initially one could think that there was a major flaw in the model (or in the RBFCM mechanisms), but after a discussion and analysis of the results with an economics expert, the culprit was found: the model approached the economic situation before the creation of entities that control Interest Rates (like the U.S. Federal Reserve, the European Central Bank). The lack of these entities was the main cause to economic instability until 1930’s. In fact, Economics was known in the 18th and 19th century as the “Dark Science”, because all theories indicated that economy was not sustainable. According to the simulation results, depression always comes after a growth period and due to an exaggerate increase in tax rates (the banks try to maximize their profit in a short period, and their greed cause an apparently avoidable crisis). Notice the similarity with the present economic crisis - one will return to this point later on. Therefore, to support the referred theory, a simple model of the European Central Bank behavior regarding interest rates was added to the model.

Fig. 16.4. Serious economic crisis: Negative growth and deflation (Predicted Growth, Predicted TaxRate, Predicted Inflation)

16.3 A Qualitative Macro Economic Model as an Example of DCM Modeling

16.3.4

297

Modeling the European Central Bank

To simulate ECB influence, a Fuzzy Inference Subsystem (FISS) - a RB-FCM block used to model the process of decision making of system entities (FISS timing mechanisms are independent of the RB-FCM) - was added to the model (Fig.16.5). This FISS ended up as a simple FRB with 48 rules (each with 2 antecedents) [2]. These rules were designed to inhibit the greedy bank behavior that was identified as the cause to the unavoidable crisis.

Fig. 16.5. FISS: ECB decision on Interest Rate variation

16.3.5

Complete Model Simulation

With the introduction of the ECB-FISS, the system behavior changed completely and serious crisis were avoided under normal circumstances (Fig.16.6). One of the most interesting results was the fact that, under normal circumstances, the economic model stabilizes around the real-world BCE predicted ideal target value for inflation (slightly below 2%) and growth (averaging slightly above 2%). Note that these values are not imposed anywhere in the model, they result from the system itself. However, the ECB and private bank behavior in the last two years was incredibly similar to the greedy behavior exhibited by the model without ECB. Tax rates - see Euribor historical data [14], Fig.16.7 - were severely increased between 2006-08 under the pretext of controlling inflation, but, as it was found later, mostly because private banks were needing to increase their tax rates to protect themselves against prior mistakes. The variation of the Euribor tax rate +0.8% spread rate in the last 10 years (120 month), is very much similar to what was predicted in the “greedy” model that was presented 6 years before. As a result of that, and, as the original model predicted, the most severe economic crisis of the last 80 years was the result of that policy. Given that this is a very long-term 10 year simulation (executed 6-8 years before the events occurrence), the results are incredibly more accurate than current models, that don’t usually attempt to predict for longer than 2 years and usually with very inaccurate results. As a proof of this, the major economic actors were still insisting on increasing tax rates less than 1 year before the crisis having inflation control in mind, and no major economic actor was even suspecting that deflation would be the real concern in less than 8 months.

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Fig. 16.6. Avoiding economic crisis trough ECB Interest Rate control

Fig. 16.7. Euribor6+0.8 for the last 120 months

16.4

Conclusions, Applications, and Future Developments

This work exemplifies how one can use DCM to model complex qualitative socioeconomic systems, avoiding the need to use extensive and time consuming differential equation models, while obtaining very interesting and encouraging results. By using true qualitative modeling techniques, one obtained results that look more realistic (plausible) than those obtained using quantitative approaches - where results

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almost never show the short-term uncertainties that are so characteristic of qualitative real-world dynamic systems. In the end, the results of the presented model, that was developed eight years before the crisis onset, are surprisingly realistic and could have been used to predict and avoid the current world economic crisis, even if one considers its necessary incompleteness.

Acknowledgments This work was partially supported by FCT (INESC-ID multiannual funding) through the PIDDAC Program funds.

References [1] Axelrod, R.: The Structure of Decision: Cognitive Maps of Political Elites. Princeton University Press (1976) [2] Carvalho, J.P.: Mapas Cognitivos Baseados em Regras Difusas: Modelacao e Simulacao da Dinamica de Sistemas Qualitativos, PhD thesis, Instituto Superior Tecnico, Universidade Tecnica de Lisboa, Portugal (2002) [3] Carvalho, J.P., José, A., Tome, J.A.: Qualitative Optimization of Fuzzy Causal Rule Bases using Fuzzy Boolean Nets. Fuzzy Sets and Systems 158(17), 1931–1946 (2007) [4] Carvalho, J.P., Tome, J.A.: Fuzzy Mechanisms for Qualitative Causal Relations. In: Seising, R. (ed.) Views on Fuzzy Sets and Systems from Different Perspectives. Philosophy and Logic, Criticisms and Applications. Studies in Fuzziness and Soft Computing, Springer, Berlin (2009) - Qualitative Systems [5] Carvalho, J.P., Tome, J.A.: Rule Based Fuzzy Cognitive Maps D Dynamics. In: Proceedings of the 19th International Conference of the North American Fuzzy Information Processing Society, NAFIPS 2000, Atlanta (2000) [6] Carvalho, J.P., Tome, J.A.: Issues on the Stability of Fuzzy Cognitive Maps and RuleBased Fuzzy Cognitive Maps. In: Proceedings of the 21st International Conference of the North American Fuzzy Information Processing Society, NAFIPS 2002, New Orleans (2002) [7] Carvalho, J.P., Tome, J.A.: Fuzzy Mechanisms For Causal Relations. In: Proceedings of the Eighth International Fuzzy Systems Association World Congress, IFSA 1999, Taiwan (1999) [8] Carvalho, J.P., Tome, J.A.: Rule Based Fuzzy Cognitive Maps - Expressing Time in Qualitative System Dynamics. In: Proceedings of the 2001 FUZZ-IEEE, Melbourne, Australia (2001) [9] Carvalho, J.P., Tome, J.A.: Rule Based Fuzzy Cognitive Maps and Fuzzy Cognitive Maps - A Comparative Study. In: Proceedings of the 18th International Conference of the North American Fuzzy Information Processing Society, NAFIP 1999, New York (1999) [10] Carvalho, J.P., Tome, J.A.: Using Interpolated Linguistic Term to Express Uncertainty in Rule Based Fuzzy Cognitive Maps. In: Proceedings of the 22nd International Conference of the North American Fuzzy Information Processing Society, NAFIPS 2003, Chicago (2003)

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[11] Carvalho, J.P., Wise, L., Murta, A., Mesquita, M.: Issues on Dynamic Cognitive Map Modelling of Purse-seine Fishing Skippers Behavior. In: WCCI 2008 - 2008 IEEE World Congress on Computational Intelligence, Hong-Kong, pp. 1503–1510 (2008) [12] Forrester, J.W.: Several papers available online at http://sysdyn.mit.edu/sd-intro/home.html [13] Halanay, A., Samuel, J.: Differential Equations, Discrete Systems and Control: Economic Models. Mathematical Modelling: Theory and Applications. Kluwer Academic (1997) [14] http://www.euribor.org/html/content/euribordata.html [15] Kosko, B.: Fuzzy Cognitive Maps. International Journal of Man-Machine Studies (1986) [16] Kosko, B.: Fuzzy Engineering. Prentice-Hall International Editions (1997) [17] Kosko, B.: Neural Networks and Fuzzy Systems: A Dynamical Systems Approach to Machine Intelligence. Prentice-Hall International Editions (1992) [18] Neves, A.: Taxas de Juro podem descer ja este mes, Publico 2001/08/10 [19] Zadeh, L.A.: Fuzzy Sets and Applications: Selected Papers. Wiley Interscience (1987) [20] Zhang, W.-B.: Differential Equations, Bifurcations and Chaos in Economics. World Scientific (2005)

17 Voting on How to Vote José Luis García–Lapresta and Ashley Piggins

17.1

Introduction

Traditionally, social choice theory has been concerned with voting rules that select an alternative from a set of social alternatives. These rules have been analyzed in detail and some have received axiomatic characterizations. Recently, a number of authors have considered the possibility that the set of social alternatives contains the voting rules themselves, and what is sought is a method of selecting between them. In other words, what is being modeled here is how to choose how to choose. The basic literature includes Koray [16] and Barberà and Jackson [1]. A central property in the literature is an axiom called self-stability. This axiom says that the voting rule selected, given everyone’s preferences, should select itself when applied to this profile of preferences. Let us give an illustration of this concept. Imagine that the voting rule itself is selected by the majority rule. When there is a majority in favor of the majority rule, then this rule is self-stable. On the other hand, if the voting rule is selected by a dictator, then the dictator may actually prefer the majority rule. Dictatorship, in this example, is not self-stable. The first decision taken by the dictator will be to abolish his dictatorship. An interesting philosophical issue arises with models of this kind. The voting rule selected will be used to make decisions on ordinary matters; taxation levels, transfer payments etc. Individuals have preferences over these things, in addition to having preferences over voting rules. How are these preferences related? According to the so-called consequentialist hypothesis, an individual’s preferences over ordinary matters will influence his or her preferences over voting rules. This hypothesis suggests, quite naturally, that an individual will favor rules that make it more likely that his or her favored alternatives are chosen. Important references here are Koray [16] and Barberà and Jackson [1]. To put the point somewhat differently, voters are only interested in ultimate outcomes. Given a specific scenario, individuals vote for the rules that are most likely to lead to their preferred outcomes. The consequentialist hypothesis has been criticised by Houy [13, 14]. Other criticisms include Karni and Safra [15] and Segal [27]. A non-consequentialist approach has been developed by Houy [13, 14] and Diss and Merlin [4]. This approach emphasizes that individuals may have normative preferences over the rules, which are formed by their beliefs about the intrinsic merits of the rules themselves, irrespective of the outcomes that these rules ultimately bring about. R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 301–321. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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In this paper we develop an approach to this problem that is grounded in this latter tradition. More specifically, we focus on a family of majority rules which are based on the difference in support for one alternative over another. This family was introduced by García-Lapresta and Llamazares [10]. Let us take the opportunity to describe the central features of our model, and explain how it relates to this aforementioned literature. First, there is a finite set of voters. This set is denoted by V and there are m voters in V . These voters have preferences over a finite set X the elements of which correspond to basic social alternatives. The preferences each voter has over these alternatives are represented by a reciprocal preference relation. A reciprocal preference relation conveys information about how intensely a voter prefers one alternative to another. This intensity information is represented by a number between 0 and 1. If a voter prefers social alternative x to social alternative y with degree 1, then we say that this voter definitely prefers x to y. If that number is 0.5 then the voter is indifferent, and if the number is 0 then the voter prefers y to x. This is raw data in our model and it is subsequently used to generate a social ranking of basic alternatives. This social ranking of basic alternatives depends on the idea of a threshold. Let us explain how this works. Suppose we add up each person’s preference for x over y. Then add up each person’s preference for y over x. This produces a pair of numbers. Next, establish a threshold by choosing a number k, where k lies in the interval [0, m). Let us explain how this threshold is used to generate a social ranking of basic alternatives. The idea is simple. Once the threshold has been set, we declare one alternative to be social preferred to another if and only if the sum total of the (x, y) preferences exceeds the sum total of the (y, x) preferences by at least k. Of course, the higher the value of k the stronger society’s preference for x over y needs to be in order for x to be declared socially preferred to y. Conversely, a low value of k means that a weak preference for x over y is all that is needed to declare x to be socially preferred to y. The critical question then is this: what value of k should be selected? Speaking somewhat loosely, we assume that in addition to these basic preferences, individuals also have preferences over the thresholds in [0, m). These preferences are revealed indirectly through trapezoidal fuzzy numbers. Why do we use this device? The reason is that there are an infinite number of potential values for k, and so it would be impossible for an individual to compare all of them. It seems plausible that an individual would say something like “around 2” or “between 5 and 7”. Notice that these are not typically precise assessments (although they could be), they are fuzzy values that reflect the vagueness that exists in an individual’s assessment of what the threshold should actually be. We assume that each individual’s assessment of what k should be is represented by a trapezoidal fuzzy number. Perhaps surprisingly, from this trapezoidal fuzzy number we can associate a reciprocal preference relation on [0, m). What this means is that individuals are revealing their preferences over thresholds by revealing their trapezoidal fuzzy numbers. This allows us to move from trapezoidal fuzzy numbers to reciprocal preferences over thresholds.

17.2 Majorities Based on the Difference in Support

303

With these preferences over thresholds in place, we can formalize the notion of a threshold being “self-selective”. Before we do so, we would like to draw attention to the following point. An interesting feature of the fact that we can convert an individual’s evaluation into a reciprocal preference relation is that the same social choice machinery applied to basic alternatives can now be applied to thresholds. This may appear to be hopelessly circular, but it is essential to our notion of stability. Just like before, the following type of statements are now meaningful: one threshold k1 is socially preferred to another threshold k2 if and only if the sum total of everyone’s preference for k1 over k2 exceeds the sum total of everyone’s preference for k2 over k1 by at least k where k is an element of [0, m). We say that a threshold is strongly self-selective in a profile if it is socially preferred to all other thresholds when applied itself to the problem of choosing the threshold. As we show in this paper, if a strongly self-selective threshold exists then it must be unique. However, we cannot guarantee that such a threshold exists. A weaker concept is self-selectivity. A threshold is self-selective in a profile if no other threshold is preferred to it when the threshold used to determine social preference is this threshold. Unlike strongly self-selective thresholds, we show that self-selective thresholds always exist. However, they are not, in general, unique. Importantly, we describe a procedure for choosing among the self-selective thresholds in this paper. The paper concludes with some illustrative examples. The paper is organized as follows. In section 17.2 we motivate the idea of a “threshold”, and also the idea of preference intensity. Furthermore, we present our basic model. Section 17.3 presents the idea of a self-selective majority (which is a phrase that means the same thing as a self-selective threshold). This section also contains some important mathematical results. Subsection 17.3.1 contains our proposal for choosing among the self-selective majorities. Subsection 17.3.2 contains illustrative examples. Section 17.4 contains concluding remarks.

17.2

Majorities Based on the Difference in Support

According to the simple majority principle, x defeats y when the number of individuals who prefer x to y is greater than the number of individuals who prefer y to x. Since simple majority requires very poor support in order to declare that one alternative is better than another, other majorities have been introduced and studied in the literature (see Fishburn [6, chapter 6], Ferejohn and Grether [5], Saari [26, pp. 122-123], and García-Lapresta and Llamazares [9], among others). In order to avoid some drawbacks of simple and absolute majorities, and indeed other voting systems, García-Lapresta and Llamazares [9] introduced and analyzed Mk majorities, a class of voting systems based on the difference in votes. Given two alternatives, x and y, for Mk , x is collectively preferred to y, when the number of individuals who prefer x to y exceeds the number of individuals who prefer y to x by at least a fixed integer k from 0 to m− 1, where m is the number of voters. It is useful to note that Mk majorities are located between simple majority and unanimity, in the

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17 Voting on How to Vote

extreme cases of k = 0 and k = m − 1, respectively. Subsequently, Mk majorities have been characterized axiomatically by Llamazares [18] and Houy [12]. A feature of simple majority, and indeed other classic voting systems, is that they require individuals to declare dichotomous preferences: they can only declare if an alternative is preferred to another, or if they are indifferent. The “informational inputs” sent by the voters are, therefore, not particularly rich. A recurring theme throughout social choice theory is that the more information the better. For this reason, we adopt a simple model of preference intensity in this paper. Here is a quote from Nobel Prize Laureate A.K. Sen [28, p. 162]: ... the method of majority decision takes no account of intensities of preference, and it is certainly arguable that what matters is not merely the number who prefer x to y and the number who prefer y to x, but also by how much each prefers one alternative to the other. It is worth noting that this idea had already been considered in the 18th Century by the Spanish mathematician J.I. Morales, who in [23] states that opinion is not something that can be quantified but rather something which has to be weighed (see English translation in McLean and Urken [22, p. 204]), or ... majority opinion ... is something which is independent of any fixed number of votes or, which is the same, it has a varying relationship with this figure (see English translation in McLean and Urken [22, p. 214]). The importance of considering intensities of preference in the design of appropriate voting systems has been advocated by Nurmi [25]. In this way, García-Lapresta and Llamazares [8] provide some axiomatic characterizations of several decision rules that aggregate preferences through different kinds of means. Additionally, in [8, Prop. 2], simple majority has been obtained as a specific case of the aforementioned decision rules. Likewise, other kinds of majorities can be obtained through operators that aggregate fuzzy preferences (on this, see Llamazares and GarcíaLapresta [20, 21] and Llamazares [17, 19]). In addition, García-Lapresta [7] generalizes simple majority by allowing individuals to show degrees of preference in a linguistic fashion. 17.2.1

k Majorities M

Consider m voters, V = {1, . . . , m}, showing the intensity of their preferences on the alternatives in X through reciprocal preference relations Rv : X × X −→ [0, 1], v = 1, . . . , m, i.e., Rv (xi , x j ) + Rv (x j , xi ) = 1 for all xi , x j ∈ X . So, voters show intensities of preference by means of numbers between 0 and 1: Rv (xi , x j ) = 0, when v prefers absolutely x j to xi ; Rv (xi , x j ) = 0.5, when v is indifferent between xi and x j ; Rv (xi , x j ) = 1, when v prefers absolutely xi to x j ; and, whatever number different to 0, 0.5 and 1, for not extreme preferences, nor for indifference, in the sense that the closer the number is to 1, the more xi is preferred to x j (see Nurmi [24] and García-Lapresta and Llamazares [8]). With R(X ) we denote the set of reciprocal preference relations on X .

17.2 Majorities Based on the Difference in Support

305

A profile is a vector R = (R1 , . . . , Rm ) containing the individual reciprocal preferences. Accordingly, the set of profiles over basic alternatives is denoted by R(X )m . An ordinary preference relation on X is an asymmetric binary relation on X : if xi P x j , then it is false that x j P xi . With P(X ) we denote the set of ordinary preference relations on X . The indifference relation associated with P is defined by xi I x j if neither xi P x j nor x j P xi . P ∈ P(X) is an interval order if (xi Px j , x j I xk and xk P xl ) ⇒ xi P xl , for all xi , x j , xk , xl ∈ X . It is well known that if P is an interval order, then P is transitive, but I is not necessarily transitive (see, for instance, García-Lapresta and Rodríguez-Palmero [11]). We now introduce the class of majorities based on the difference in support (García-Lapresta and Llamazares [10]). k majority is the mapDefinition 17.2.1. Given a threshold k ∈ [0, m), the fuzzy M k : R(X )m −→ P(X) defined by M k (R1 , . . . , Rm ) = Pk , where ping M xi Pk x j ⇔

m

∑ Rv(xi , x j ) >

v=1

m

∑ Rv (x j , xi ) + k.

v=1

k can be defined It is easy to see (García-Lapresta and Llamazares [10]) that M through the average of the individual intensities of preference: xi Pk x j ⇔

1 m k Rv (xi , x j ) > 0.5 + . ∑ m v=1 2m

The indifference relation associated with Pk is defined by   m  m   xi Ik x j ⇔  ∑ Rv (xi , x j ) − ∑ Rv (x j , xi ) ≤ k v=1  v=1 or, equivalently,

  1 m  k   xi Ik x j ⇔  ∑ Rv (xi , x j ) − 0.5 ≤ .  m v=1  2m

k majorities is denoted by The set of M   k | k ∈ [0, m) . M= M m , given by Remark 1. The limit case of M is the unanimous majority M xi Pm x j ⇔ ∀v ∈ V Rv (xi , x j ) = 1, i.e., xi Pm x j if and only if xi Pk x j for every k ∈ [0, m).

306

17.2.2

17 Voting on How to Vote

k Majorities Preferences on M

k majority should be used in further Voters have preferences on M about which M decisions. Since M is infinite, voters cannot directly compare all the pairs in M . A possibility is that voters declare their preferences on M by means of exact as k with “k = k0 ”, or by means of imprecise sentences, as M k with sessments, as M “k near k1 ”, “k between k1 and k2 ”, “k smaller than k3 ”, “k bigger than k4 ”, and so on. All these sentences can be captured by using trapezoidal fuzzy numbers1 that generalize real numbers, intervals of real numbers and triangular fuzzy numbers. Definition 17.2.2. Given four real numbers a, b, c, d such that 0 ≤ a ≤ b ≤ c ≤ d, the trapezoidal fuzzy number (a, b, c, d) is defined by its membership function u : [0, ∞) −→ [0, 1] (see Fig. 17.1), where ⎧ 0, if x < a or x > d, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x−a ⎪ ⎪ ⎪ ⎨ b − a , if a < x < b, u(x) = ⎪ ⎪ 1, if b ≤ x ≤ c, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ d − x , if c < x < d d −c

and u(a) =

0, if a < b, 1, if a = b,

u(d) =

0, if d > c, 1, if d = c.

Fig. 17.1. Trapezoidal fuzzy number (a, b, c, d).

If a = b = c = d, then (a, b, c, d) is a real number; if a = b and c = d, then (a, b, c, d) is an interval of real numbers; and if b = c, then (a, b, c, d) is a triangular fuzzy number. 1

Trapezoidal fuzzy numbers are an appropriate tool to capture the vagueness of the assessments (see, for instance, Delgado, Vila and Voxman [3]).

k Majorities 17.3 Self-selective M

307

We now induce reciprocal preference relations on M from the corresponding trapezoidal fuzzy numbers. Let uv : [0, m) −→ [0, 1] be the restriction to [0, m) of the membership function of the trapezoidal fuzzy number that represents the imprecise sentence of voter v on what should be the threshold to use in collective decisions. We can associate a reciprocal preference relation on M in the following way:    k , M k = uv (k) − uv (k ) + 1 . Rv M 2 It is easy to see that   k , M k > 0.5 ⇔ uv (k) > uv (k ). Rv M

17.3

k Majorities Self-selective M

k majority being self-selective. We now define what we mean by an M Definition 17.3.1. Let R = (R1 , . . . , Rm ) ∈ R(M )m . k ∈ M is strongly self-selective in R if M k Pk M k for every 1. M k ∈ [0, m) \ {k}. k ∈ M is self-selective in R if not M k Pk M k for any k ∈ [0, m) \ {k}. 2. M Proposition 17.3.1. Let R = (R1 , . . . , Rm ) ∈ R(M )m . k is strongly self-selective in R, then it is self-selective in R as well. • If M k and M k are strongly self-selective in R, then k = k . • If M In what follows, we assume that R = (R1 , . . . , Rm ) ∈ R(M )m is induced by trapezoidal fuzzy numbers with corresponding membership functions uv : [0, m) −→ [0, 1], v = 1, . . . , m, and u : [0, m) −→ [0, m) is defined by u(k) =

m

∑ uv(k).

v=1

k Pk M k ⇔ u(k ) > u(k ) + k, for all k, k , k ∈ [0, m). Proposition 17.3.2. M P ROOF. k Pk M k ⇔ M

m

m

v=1

v=1

∑ Rv (M k , M k ) > ∑ Rv(M k , M k ) + k

⇔

m uv (k ) − uv (k ) + 1 uv (k ) − uv (k ) + 1 >∑ +k ⇔ 2 2 v=1 v=1 m

∑ m

m

v=1

v=1

∑ uv (k ) > ∑ uv(k ) + k

⇔ u(k ) > u(k ) + k.

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Remark 2. According to Proposition 17.3.2, we have k is strongly self-selective in R if and only if u(k) > u(k ) + k for every 1. M k ∈ [0, m) \ {k}. k is self-selective in R if and only if u(k ) ≤ u(k) + k for every 2. M k ∈ [0, m). Proposition 17.3.3. The binary relation % on [0, m) defined by k % k ⇔ u(k) > u(k ) + k is an interval order. P ROOF. First of all, we prove that % is asymmetric. By way of contradiction, suppose k % k and k % k. Then, u(k) > u(k ) + k and u(k ) > u(k) + k . Consequently, u(k) > u(k) + k + k, i.e., k + k < 0 that is absurd. Let now k1 , k2 , k3 , k4 ∈ [0, m) such that k1 % k2 , k2 ∼ k3 and k3 % k4 , where ∼ is the indifference relation associated with %. Then, we have u(k1 ) > u(k2 ) + k1 , u(k3 ) ≤ u(k2 ) + k3 and u(k3 ) > u(k4 ) + k3 . Therefore, u(k1 ) > u(k3 ) − k3 + k1 > u(k4 ) + k1 , i.e., k1 % k4 . k is strongly selfRemark 3. According to Propositions 17.3.2 and 17.3.3, M selective in R if and only if k % k for every k ∈ [0, m) \ {k}, i.e., k is the strict maximum of the interval order %. Proposition 17.3.4. 1. If u is increasing in an interval I ⊆ [0, m), then (k % k and k < k ) ⇒ k % k for all k, k , k ∈ I. 2. If u is decreasing in an interval I ⊆ [0, m), then (k % k and k > k ) ⇒ k % k for all k, k , k ∈ I. 3. If u is constant in an interval I ⊆ [0, m), then neither k % k nor k % k for all k, k ∈ I. 4. If u is increasing in an interval I ⊆ [0, m), then k < k ⇒ not k % k for all k, k ∈ I. 5. If u is decreasing in an interval I ⊆ [0, m), then k > k ⇒ not k % k for all k, k ∈ I.

k Majorities 17.3 Self-selective M

309

P ROOF. 1. From k % k , we have u(k) > u(k ) + k. Since k > k and u is increasing, then we have u(k ) ≥ u(k ) and u(k) > u(k ) + k, i.e., k % k . 2. From k % k , we have u(k) > u(k ) + k. Since k < k and u is decreasing, then we have u(k ) ≥ u(k ) and u(k) > u(k ) + k, i.e., k % k . 3. Since u(k) = u(k ), we have u(k) ≤ u(k ) + k and u(k ) ≤ u(k) + k . 4. From k < k , we have u(k) ≤ u(k ) ≤ u(k ) + k. 5. From k > k , we have u(k) ≤ u(k ) ≤ u(k ) + k. k is strongly selfProposition 17.3.5. Let R = (R1 , . . . , Rm ) ∈ R(M )m . If M selective in R, then k is the strict global maximum of u. Consequently, if u has k majority is strongly self-selective in R. not strict global maximum, then none M P ROOF. Notice that if u(k) ≤ u(k ) for some k ∈ [0, m) \ {k}, then u(k) ≤ u(k ) + k. k cannot be strongly self-selective in R. Consequently, if M k is strongly Hence, M self-selective, then k is necessarily the strict global maximum of u. Proposition 17.3.6. For every R = (R1 , . . . , Rm ) ∈ R(M )m , there always exist selfselective majorities in R. P ROOF. We consider the two possible cases. 1. If k is a global maximum of u, then u(k ) ≤ u(k) ≤ u(k)+k for every k ∈ [0, m). k is self-selective in R. Therefore, M 2. If u does not have global maximums, then there exists k0 ∈ [m − 1, m) such that u is strictly increasing in [k0 , m) and u(k0 ) > 1. Then, u(k0 ) + k0 > 1 + m − 1 = m and, consequently, u(k ) ≤ m < u(k0 ) + k0 for every k ∈ [0, m). Therefore, k is self-selective in R. M 0

17.3.1

Our Proposal

Let R = (R1 , . . . , Rm ) ∈ R(M )m . Given K ⊆ [0, m) a finite union of intervals, let μ (K) be the length of the subset K. For putting in practice our decision procedure, we will use the following sets   k Pk M k = S = (k, k ) ∈ [0, m)2 | not M   = (k, k ) ∈ [0, m)2 | u(k ) ≤ u(k) + k .   k Pk M k = T = (k, k ) ∈ [0, m)2 | M   = (k, k ) ∈ [0, m)2 | u(k) > u(k ) + k .

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Clearly, T ⊆ S .   k ∈ M | ∀k ∈ [0, m) (k, k ) ∈ S = M1 = M   k ∈ M | ∀k ∈ [0, m) u(k ) ≤ u(k) + k . = M k majorities of M that are self-selective in R. Notice that M1 is the subset of M By Proposition 17.3.6, M1 = 0. / We now introduce a count that is related to one proposed by Copeland [2] in the context of preference relations:     λ (k) = μ k ∈ [0, m) | (k, k ) ∈ T − μ k ∈ [0, m) | (k, k ) ∈ /S , k minus the amount of M k Pk M k ∈ M k ∈ M such that M i.e., the amount of M k . k Pk M such that M k ∈ M1 , then we have (k, k ) ∈ S for every k ∈ [0, m) and, Notice that if M consequently,   λ (k) = μ k ∈ [0, m) | (k, k ) ∈ T . We now introduce a sequential procedure for choosing the appropriate selfselective majority. First, we take the set of thresholds that are self-selective at profile R. We know that this set is non-empty. We then construct a subset of this set which we denote by M2 . This is how this subset is constructed. Take a self-selective threshold k ∈ [0, m). Consider the set of all thresholds that are defeated by k when k itself is used as the threshold. Calculate the length of this set. Repeat this process for every other selfselective threshold. M2 contains only those self-selective thresholds whose length is maximal (i.e. larger than for any other threshold). If M2 is a singleton, then this will be our choice of threshold. If M2 is not a singleton, then we proceed to step 2. In step 2, we construct a subset of M2 which we call M3 . For each threshold in M2 consider all thresholds that it defeats whenever the threshold is set to zero. Calculate the length of this set. M3 is the set of thresholds in M2 for which this length is maximal (i.e. larger than for any other threshold in M2 ). If M3 is a singleton, then this will be our choice of threshold. If M3 is not a singleton or is empty, then we proceed to step 3 (the final step). In step 3, if M3 contains more than one element then we select the smallest threshold in M3 . Finally, if M3 is empty then we choose the smallest threshold in M2 . This procedure is described formally below. k majorities in M1 that maximize λ : Step 1. We select M2 , the subset of M   k ∈ M1 | ∀k ∈ [0, m) λ (k) ≥ λ (k ) . M2 = M k Step 2. If M2 has more than one majority, then we select M3 , the subset of M  majorities of M2 that maximize λ , where

k Majorities 17.3 Self-selective M

λ  (k) = μ i.e.,

311

    k P0 M k k ∈ [0, m) | M = μ k ∈ [0, m) | u(k) > u(k ) ,   k ∈ M2 | ∀k ∈ [0, m) λ  (k) ≥ λ  (k ) . M3 = M

k majorities of M2 for those k that are the In other words, M3 is the subset of M global maximums of u, if they exist. k ∈ M3 , then we choose the minimum of these k, Step 3. If still there are several M i.e., the smallest k that maximizes u. If M3 = 0, / then we select the minimum k such k ∈ M2 . that M Proposition  17.3.7. If Mk ∈ M k or M3 = M k . M2 = M

is

strongly

self-selective

in

R,

then

P ROOF. By hypothesis, u(k) > u(k ) + k for every k ∈ [0, m) \ {k}. Therefore, {k ∈ [0, m) | (k, k ) ∈ T } = [0, m) \ {k}, {k ∈ [0, m) | (k, k ) ∈ / S } = 0/ and, consequently, λ (k) = m.  Consequently,  Mk ∈  M2 . Since k is the strict global k or M3 = M k . maximum of u, then M2 = M Remark 4. 1. If k = 0 is the strict global maximum of u, then u(0) >u(k ) + 0 for every  k ∈ (0, m). Thus, M0 is strongly self-selective and M3 = M0 . 2. If

1, if k = k0 , u1 (k) = · · · = um (k) = 0, otherwise for some k0 ∈ [0, m), then

u(k) =

m, if k = k0 , 0, otherwise.

 Therefore, m = u(k0 ) > k0 = u(k ) + k0 for every  k ∈  [0, m) \ {k0 }. Conse quently, Mk0 is strongly self-selective and M3 = Mk0 .

m (see Remark 1), it 3. If all the individuals propose the unanimous majority M seems clear that this majority should be used in further decisions. However, m cannot be selected as an outcome of our proposal because [0, m) is not a M m ∈ closed interval and M / M . For these reasons, in this case we propose to choose the unanimous majority.

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17.3.2

Illustrative Examples

Here we present a number of illustrative examples, which describe our procedure for selecting an appropriate threshold. Example 1. Consider two individuals who show the following trapezoidal (in fact, triangular) fuzzy numbers (a1 , b1 , c1 , d1 ) = (0, 0.5, 0.5, 1) and (a2 , b2 , c2 , d2 ) = (1, 1.5, 1.5, 2) k majority should be used. The corresponding membership functions about what M uv : [0, 2) −→ [0, 1] are (see Fig. 17.2) ⎧ 2k, if 0 ≤ k < 0.5, ⎪ ⎨ u1 (k) = 2 − 2k, if 0.5 ≤ k < 1, ⎪ ⎩ 0, if 1 ≤ k < 2,

⎧ 0, if 0 ≤ k < 1, ⎪ ⎨ u2 (k) = 2k − 2, if 1 ≤ k < 1.5, ⎪ ⎩ 4 − 2k, if 1.5 ≤ k < 2.

Fig. 17.2. Functions u1 and u2 in Example 1.

Then, function u is given by (see Fig. 17.3) ⎧ 2k, if 0 ≤ k < 0.5, ⎪ ⎪ ⎪ ⎪ ⎨ 2 2 − 2k, if 0.5 ≤ k < 1, u(k) = ∑ uv (k) = ⎪ 2k − 2, if 1 ≤ k < 1.5, ⎪ v=1 ⎪ ⎪ ⎩ 4 − 2k, if 1.5 ≤ k < 2. Since u has two non strict global maximums k = 0.5, 1.5, then no k ∈ M is strongly self-selective in this profile. M

k Majorities 17.3 Self-selective M

313

Fig. 17.3. Function u in Example 1.

After some calculations we obtain the set S (see Fig. 17.4). Consequently,   k ∈ M | 1 ≤ k < 2 M1 = M 3    0.5 , and λ (0.5) = 1 , 2 when k = 0.5, i.e., M = M 2 3 0.5 should be used in further decisions. Notice that 0.5 (see Fig. 17.4). Therefore, M is the smallest global maximum of u. and λ is maximized in

1

Fig. 17.4. Sets S and T in Example 1.

Example 2. Consider three individuals who show the following trapezoidal fuzzy numbers (a1 , b1 , c1 , d1 ) = (1, 2, 2, 3), (a2 , b2 , c2 , d2 ) = (2, 3, 3, 3) and (a3 , b3 , c3 , d3 ) = k majority should be used. The corresponding membership (1, 2, 3, 3) about what M functions uv : [0, 3) −→ [0, 1] are (see Fig. 17.5 and 17.6)

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17 Voting on How to Vote

⎧ 0, if 0 ≤ k < 1, ⎪ ⎨ u1 (k) = k − 1, if 1 ≤ k < 2, ⎪ ⎩ 3 − k, if 2 ≤ k < 3,

u2 (k) =

0, if 0 ≤ k < 2, k − 2, if 2 ≤ k < 3,

⎧ 0, if 0 ≤ k < 1, ⎪ ⎨ u3 (k) = k − 1, if 1 ≤ k < 2, ⎪ ⎩ 1, if 2 ≤ k < 3.

Fig. 17.5. Functions u1 and u2 in Example 2.

Fig. 17.6. Function u3 in Example 2.

Then, function u is given by (see Fig. 17.7) ⎧ 0, if 0 ≤ k < 1, ⎪ ⎨ 3 u(k) = ∑ uv (k) = 2k − 2, if 1 ≤ k < 2, ⎪ ⎩ v=1 2, if 2 ≤ k < 3. k ∈ M is Since u has an interval of non strict global maximums [2, 3), then no M strongly self-selective in this profile.

k Majorities 17.3 Self-selective M

315

Fig. 17.7. Function u in Example 2.

After some calculations we obtain the set S (see Fig. 17.8). Consequently,   k ∈ M | 4 ≤ k < 3 M1 = M 3 and λ is maximized in

4

3,3



  2 , and λ (2) = 2 (see when k = 2, i.e., M2 = M

2 should be used in further decisions. Again the smallest Fig. 17.8). Therefore, M global maximum of u defines the selected majority. However, this is not always true, as we can see in the next example.

Fig. 17.8. Sets S and T in Example 2.

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Example 3. Consider three individuals who show the following trapezoidal fuzzy numbers (a1 , b1 , c1 , d1 ) = (0, 0, 0, 1), (a2 , b2 , c2 , d2 ) = (0, 1, 1, 2) and (a3 , b3 , c3 , d3 ) = k majority should be used. The corresponding member(0, 0, 1, 2) about what M ship functions uv : [0, 3) −→ [0, 1] are (see Fig. 17.9 and 17.10) ⎧

k, if 0 ≤ k < 1, ⎪ ⎨ 1 − k, if 0 ≤ k < 1, u2 (k) = 2 − k, if 1 ≤ k < 2, u1 (k) = ⎪ 0, if 1 ≤ k < 3, ⎩ 0, if 2 ≤ k < 3, ⎧ 1, if 0 ≤ k < 1, ⎪ ⎨ u3 (k) = 2 − k, if 1 ≤ k < 2, ⎪ ⎩ 0, if 2 ≤ k < 3.

Fig. 17.9. Functions u1 and u2 in Example 3.

Fig. 17.10. Function u3 in Example 3.

Then, function u is given by (see Fig. 17.11) ⎧ 2, if 0 ≤ k < 1, ⎪ ⎨ 3 u(k) = ∑ uv (k) = 4 − 2k, if 1 ≤ k < 2, ⎪ ⎩ v=1 0, if 2 ≤ k < 3. k ∈ M is Since u has an interval of non strict global maximums [0, 1], then no M strongly self-selective in this profile.

k Majorities 17.3 Self-selective M

317

Fig. 17.11. Function u in Example 3.

After some calculations   we obtain M1 = M and λ is maximized in [0, 3) when 1 should 1 , and λ (1) = 1.5 (see Fig. 17.12). Therefore, M k = 1, i.e., M2 = M be used in further decisions. In this case the greatest global maximum of u defines the selected majority.

Fig. 17.12. Set T in Example 3.

Example 4. Consider three individuals who show the following trapezoidal fuzzy numbers (a1 , b1 , c1 , d1 ) = (a2 , b2 , c2 , d2 ) = (1, 2, 2, 3) and (a3 , b3 , c3 , d3 ) = (2, 3, 3, 3) about k majority should be used. The corresponding membership functions what M uv : [0, 3) −→ [0, 1] are (see Fig. 17.5) ⎧

0, if 0 ≤ k < 1, ⎪ ⎨ 0, if 0 ≤ k < 2, u1 (k) = u2 (k) = k − 1, if 1 ≤ k < 2, u3 (k) = ⎪ k − 2, if 2 ≤ k < 3. ⎩ 3 − k, if 2 ≤ k < 3,

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17 Voting on How to Vote

Then, function u is given by (see Fig. 17.13) ⎧ 0, if 0 ≤ k < 1, ⎪ ⎨ 3 u(k) = ∑ uv (k) = 2k − 2, if 1 ≤ k < 2, ⎪ ⎩ v=1 4 − k, if 2 ≤ k < 3.

Fig. 17.13. Function u in Example 4.

k

Since k = 2 is the strict global maximum of u and 2 = u(2) ≤ u(k ) + 2 for every k ∈ M is strongly self-selective in this profile. ∈ [0, 3) \ {2}, then no M

Fig. 17.14. Set S in Example 4.

After some calculations we obtain the set S (see Fig. 17.14). Consequently,   k ∈ M | 4 ≤ k < 3 . M1 = M 3

17.4 Concluding Remarks

319

  In this case T = 0/ and λ (k) = 0 for every k ∈ 43 , 3 . Therefore, M2 = M1 . 2 P0 M k for every k ∈ [0, 3) \ Since k = 2 is the strict global of u, then M  maximum  2 and M 2 should be used in further decisions. {2}. Consequently, M3 = M

17.4

Concluding Remarks

All societies need to make collective decisions, and for this they need a voting rule. It is natural to ask how stable these rules are. Stable rules are more likely to persist over time. In this paper we considered a family of voting systems in which individuals declare intensities of preference through numbers in the unit interval. With these voting systems, an alternative defeats another whenever the amount of opinion obtained by the first alternative exceeds the amount of opinion obtained by the second alternative by a fixed threshold. The relevant question is what should this threshold be? In the paper we assumed that each individual’s assessment of what the threshold should be is represented by a trapezoidal fuzzy number. From these trapezoidal fuzzy numbers we associated reciprocal preference relations on [0, m). With these preferences over thresholds in place, we formalized the notion of a threshold being “self-selective”. We established some mathematical properties of self-selective thresholds, and then described a three stage procedure for selecting an appropriate threshold. Such a procedure will always select some threshold, which will subsequently be used for future decision making. An obvious extension of our model would be to axiomatically characterize such a procedure. In this paper we have simply proposed one. Axiomatic characterization is widespread in social choice theory, and it enables us to understand better the underlying properties of any procedure. Our procedure should be understood as a first step towards a more comprehensive understanding of self-selectivity for this class of voting rules.

Acknowledgments García-Lapresta gratefully acknowledges the funding support of the Spanish Ministerio de Ciencia e Innovación (Project ECO2009–07332), ERDF and Junta de Castilla y León (Consejería de Educación, Projects VA092A08 and GR99). The authors are grateful to Semih Koray and Bonifacio Llamazares for their suggestions and comments.

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References [1] Barberà, S., Jackson, M.O.: Choosing how to choose self-stable majority rules and constitutions. The Quarterly Journal of Economics 119, 1011–1048 (2004) [2] Copeland, A.H.: A ‘reasonable’ social welfare function. University of Michigan Seminar on Applications of Mathematics to the Social Sciences. University of Michigan, Ann Arbor (1951) [3] Delgado, M., Vila, M.A., Voxman, W.: On a canonical representation of fuzzy numbers. Fuzzy Sets and Systems 93, 125–135 (1998) [4] Diss, M., Merlin, V.R.: On the stability of a triplet of scoring rules. Theory and Decision 69, 289–316 (2010) [5] Ferejohn, J.A., Grether, D.M.: On a class of rational social decisions procedures. Journal of Economic Theory 8, 471–482 (1974) [6] Fishburn, P.C.: The Theory of Social Choice. Princeton University Press, Princeton (1973) [7] García-Lapresta, J.L.: A general class of simple majority decision rules based on linguistic opinions. Information Sciences 176, 352–365 (2006) [8] García-Lapresta, J.L., Llamazares, B.: Aggregation of fuzzy preferences: Some rules of the mean. Social Choice and Welfare 17, 673–690 (2000) [9] García-Lapresta, J.L., Llamazares, B.: Majority decisions based on difference of votes. Journal of Mathematical Economics 35, 463–481 (2001) [10] García-Lapresta, J.L., Llamazares, B.: Preference intensities and majority decisions based on difference of support between alternatives. Group Decision and Negotiation 19, 527–542 (2010) [11] García-Lapresta, J.L., Rodríguez-Palmero, C.: Some algebraic characterizations of preference structures. Journal of Interdisciplinary Mathematics 7, 233–254 (2004) [12] Houy, N.: Some further characterizations for the forgotten voting rules. Mathematical Social Sciences 53, 111–121 (2007) [13] Houy, N.: Dinamics of stable sets of constitutions (2007), http://wakame.econ.hit-u.ac.jp/~riron/Workshop/2006/Houy.pdf [14] Houy, N.: On the (im)possibility of a set of constitutions stable at different levels (2009), ftp://mse.univ-paris1.fr/pub/mse/cahiers2004/V04039.pdf [15] Karni, E., Safra, Z.: Individual sense of justice: a utility representation. Econometrica 70, 263–284 (2002) [16] Koray, S.: Self-selective social choice functions verify Arrow and GibbardSatterthwaite theorems. Econometrica 68, 981–995 (2000) [17] Llamazares, B.: Simple and absolute special majorities generated by OWA operators. European Journal of Operational Research 158, 707–720 (2004) [18] Llamazares, B.: The forgotten decision rules: Majority rules based on difference of votes. Mathematical Social Sciences 51, 311–326 (2006) [19] Llamazares, B.: Choosing OWA operator weights in the field of Social Choice. Information Sciences 177, 4745–4756 (2007) [20] Llamazares, B., García-Lapresta, J.L.: Voting systems generated by quasiarithmetic means and OWA operators. In: Fodor, J., De Baets, B. (eds.) Principles of Fuzzy Preference Modelling and Decision Making, pp. 195–213. Academia Press, Ghent (2003) [21] Llamazares, B., García-Lapresta, J.L.: Extension of some voting systems to the field of gradual preferences. In: Bustince, H., Herrera, F., Montero, J. (eds.) Fuzzy Sets and Their Extensions: Representation, Aggregation and Models, pp. 292–310. Springer, Berlin (2008)

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321

[22] McLean, I., Urken, A.B. (eds.): Classics of Social Choice. The University of Michigan Press, Ann Arbor (1995) [23] Morales, J.I.: Memoria Matemática sobre el Cálculo de la Opinion en las Elecciones. Imprenta Real, Madrid (1797); English version in McLean and Urken [22, pp. 197– 235]: Mathematical Memoir on the Calculations of Opinions in Elections, pp. 197–236 [24] Nurmi, H.: Approaches to collective decision making with fuzzy preference relations. Fuzzy Sets and Systems 6, 249–259 (1981) [25] Nurmi, H.: Fuzzy social choice: A selective retrospect. Soft Computing 12, 281–288 (2008) [26] Saari, D.G.: Consistency of decision processes. Annals of Operations Research 23, 103–137 (1990) [27] Segal, U.: Let’s agree that all dictatorships are equally bad. The Journal of Political Economy 108, 569–589 (2000) [28] Sen, A.K.: Collective Choice and Social Welfare. Holden-Day, San Francisco (1970)

18 Weighted Means of Subjective Evaluations Jaume Casasnovas and J. Vicente Riera

18.1

Introduction

In recent years, a number of researchers have focused on solving performance evaluation issues by using fuzzy set theory. In [1] Biswas presented a fuzzy evaluation method for applying fuzzy sets in student’s answerscripts evaluation. Biswas’s methods show a good aplication of the fuzzy set theory [24] in education. However, different fuzzy marks may be translated into the same awarded letter grade. In [8], Chen and Lee presented two metods for applying fuzzy sets in students’ answerscripts evaluation. Their methods can overcome the drawbacks of Biswas’s methods and can evaluate students’ answerscripts in a more fair manner. In [19] and [20], Wang and Chen presented a method for students’ answerscripts evaluation using fuzzy numbers. The use of fuzzy mathematics [10, 11] allows to model the subjective nature of human grad assigners. The evaluating values are associated with degrees of confidence between zero and one and a degree of optimism of the evaluator allows to calculate a crisp value in the last step. In fact, we are faced with the problem of aggregating a collection of numerical readings to obtain an average value. Such aggregation problem is becoming present in an increasing number of areas not only of mathematics or physics, but also of engineering, economical, social and other sciences [13]. In the previously quoted methods the fuzzy numbers awarded by each evaluator are not directly aggregated, but they are previously defuzzycated and then a weighted mean is often applied. Our aim is to aggregate directly the fuzzy awards, because we think that a large amount of information and characteristics is lost in the defuzzification process. In [15] a multi-dimensional Aggregation through the Extension Principle of a quite restrictive definition of fuzzy numbers is studied. However in this work, we focus on the discrete feature of the used fuzzy numbers. In order to extend an aggregation function defined on this finite chain to fuzzy numbers defined on a finite chain we need to know the properties of this discrete aggregation. In general, the operations on fuzzy numbers u, v can be approached either by the direct use of their membership function, as fuzzy subsets of R and the Zadeh’s extension principle or by the equivalent use of the α -cuts representation [11]. Nevertheless, in the discrete case, this process can yield a fuzzy subset that does not satisfy the conditions to be a discrete fuzzy number [2, 18]. In [2, 3, 4, 6, 7], we have presented a technique that allows us to obtain closed operations on the set of discrete fuzzy numbers, and moreover, we focus these operations in discrete fuzzy R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 323–345. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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numbers whose support is an arithmetic sequence and even a subset of consecutive natural numbers. Discrete fuzzy numbers, whose support is a subset of natural numbers, are used to define the cardinality of a finite fuzzy subset [9, 11, 16, 21]. So, we will consider the case in which the subjective evaluation awarded by each evaluator is a discrete fuzzy number whose support is a subset of consecutive natural numbers defined on a finite chain L of consecutive natural numbers. In [12] and [13], Mesiar et al. deal with aggregation functions in a chain L = {0, 1. . . . , n} and in [13] an aggregation among different chains is considered. Based on these ideas, in this paper we propose to built aggregation functions with discrete fuzzy numbers whose support is a subset of consecutive natural numbers defined on a finite chain L of consecutive natural numbers. Thus, if we use this aggregation to modulate an evaluation process, then we can aggregate directly the fuzzy awards (expressed as discrete fuzzy numbers) and to get a fuzzy set (a discrete fuzzy number) that results from such aggregation. Furthermore, in [23] Yager discusses the fusion of information which has an ordinal structure and it proposes to use weighted normed operators. These operators can be used to obtain the group consensus opinion. These consensus methods can be employed in educational environments. For instance, need a professional psychological help, attitudes towards sciences or arts, assessment of personal maturity, etc. The present article proposes a way to obtain the group consensus opinion based on discrete fuzzy weighted normed operators. The paper is organized as follows: Section 18.2 is devoted to the review of discrete fuzzy number definition and its main operations and lattice structures. Section 18.3 discusses the idea of subjective evaluation and we recall some well known methods based on fuzzy set theory. Section 18.4 deals with the problem of build n-dimensional aggregation function on the set of discrete fuzzy numbers. Section 18.5 proposes a method to obtain the group consensus opinion based on discrete fuzzy weighted normed operators. Moreover, several examples are proposed. Finally, in section 18.6 some conclusions are given.

18.2

Discrete Fuzzy Numbers

In this section, we recall some definitions and the main results about discrete fuzzy numbers which will be used later. By a fuzzy subset of R, we mean a function A : R → [0, 1]. For each fuzzy subset A, let Aα = {x ∈ R : A(x) ≥ α } for any α ∈ (0, 1] be its α -level set (or α -cut). By supp(A), we mean the support of A, i.e. the set {x ∈ R : A(x) > 0}. By A0 , we mean the closure of supp(A). Definition 18.2.1. [11] A fuzzy subset u of R with membership mapping u : R → [0, 1] is called fuzzy number if its support is an interval [a, b] and there exist real numbers s,t with a ≤ s ≤ t ≤ b and such that: 1. u(x) = 1 with s ≤ x ≤ t 2. u(x) ≤ u(y) with a ≤ x ≤ y ≤ s

18.2 Discrete Fuzzy Numbers

325

3. u(x) ≥ u(y) with t ≤ x ≤ y ≤ b 4. u(x) is upper semi-continuous. We will denote the set of fuzzy numbers by FN. Definition 18.2.2. [17] A fuzzy subset A of R with membership mapping A : R → [0, 1] is called discrete fuzzy number if its support is finite, i.e., there exist x1 , ..., xn ∈ R with x1 < x2 < ... < xn such that supp(A) = {x1 , ..., xn }, and there are natural numbers s,t with 1 ≤ s ≤ t ≤ n such that: 1. A(xi )=1 for any natural number i with s ≤ i ≤ t (core) 2. A(xi ) ≤ A(x j ) for each natural number i, j with 1 ≤ i ≤ j ≤ s 3. A(xi ) ≥ A(x j ) for each natural number i, j with t ≤ i ≤ j ≤ n Remark 1. A fuzzy subset A of R whose membership mapping A : R → [0, 1] satisfies the statements 2 and 3 of Definition 18.2.2 and the additional condition A(xi ) = α for any natural number i with s ≤ i ≤ t and 0 < α ≤ 1 will be called generalized discrete fuzzy number. Remark 2. If the fuzzy subset A is a discrete fuzzy number then the support of A coincides with its closure, i.e. supp(A) = A0 . From now on, we will denote the set of discrete fuzzy numbers by DFN and the abbreviation dfn will denote a discrete fuzzy number. Theorem 18.2.1. [18] (Representation of discrete fuzzy numbers) Let A be a discrete fuzzy number. Then the following statements (1)-(4) hold: 1. Aα is a nonempty finite subset of R, for any α ∈ [0, 1] 2. Aα2 ⊆ Aα1 for any α1 , α2 ∈ [0, 1] with 0 ≤ α1 ≤ α2 ≤ 1 3. For any α1 , α2 ∈ [0, 1] with 0 ≤ α1 ≤ α2 ≤ 1, if x ∈ Aα1 − Aα2 we have x < y for all y ∈ Aα2 , or x>y for all y ∈ Aα2  4. For any α0 ∈ (0, 1], there exist some real numbers α0 with 0 < α0 < α0 such  that Aα0 = Aα0 ( i.e. Aα = Aα0 for any α ∈ [α0 , α0 ]). And conversely, if for any r ∈ [0, 1], there exist Ar ⊂ R satisfying the following conditions (1)-(4): 1. Ar is a nonempty finite for any r ∈ [0, 1] 2. Ar2 ⊂ Ar1 , for any r ∈ [0, 1] with 0 ≤ r1 ≤ r2 ≤ 1 3. For any r1 , r2 ∈ [0, 1] with 0 ≤ r1 ≤ r2 ≤ 1, if x ∈ Ar1 − Ar2 we have x < y for all y ∈ Ar2 , or x>y for all y ∈ Ar2   4. For any r0 ∈ (0, 1], there exists a real number r0 with 0 < r0 < r0 such that  Ar0 = Ar0 ( i.e. Ar = Ar0 , for any r ∈ [r0 , r0 ]) then there exists a unique B ∈ DFN such that Br = Ar for any r ∈ [0, 1].

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18.2.1

Operations on Discrete Fuzzy Numbers

In general [11, 10], the operations on fuzzy numbers u, v can be approached either by the direct use of their membership function, as fuzzy subsets of R and the Zadeh’s extension principle: O(u, v)(z) = sup{u(x) ∧ v(y)|O(x, y) = z} or by the equivalent use of the α -cuts representation: O(u, v)α = O(uα , vα ) = {O(x, y)|x ∈ uα , y ∈ vα } and

O(u, v)(z) = sup{α ∈ [0, 1]|z ∈ O(u, v)α }

Nevertheless, in the discrete case, this process can yield a fuzzy subset that does not satisfy the conditions to be a discrete fuzzy number [2, 18]. For example, let A = {0.3/1, 1/3, 0.5/7} and B = {0.4/2, 1/5, 1/6, 0.8/9} be two discrete fuzzy numbers. If we use the Zadeh’s extension principle to obtain their addition, it results the fuzzy subset S = {0.3/3, 0.4/5, 0.3/6, 0.3/7, 1/8, 1/9, 0.3/10, 0.8/12, 0.5/13, 0.5/16} which does not fulfill the conditions to be a discrete fuzzy number, because the third property of Definition 18.2.2 fails. In a previous work [2, 3] we have presented an approach to a closed extended addition of discrete fuzzy numbers after associating suitable non-discrete fuzzy numbers, which can be used like a carrier to obtain the desired addition. In [3] the authors proved that a suitable carrier can be a discrete fuzzy number whose support is an arithmetic sequence and even a subset of consecutive natural numbers. 18.2.2

Addition of Discrete Fuzzy Numbers

Let A, B ∈ DFN be two discrete fuzzy numbers. If we consider them as fuzzy subsets of R we can apply the Zadeh’s extension principle to calculate their addition. But, as we will see in the next example, it is possible to obtain a fuzzy subset that does not fulfill the conditions of Definition 18.2.2 of discrete fuzzy number. So, let A = {0.3/1, 1/3, 0.5/7} and B = {0.4/2, 1/5, 1/6, 0.8/9} be two discrete fuzzy numbers. If we use the Zadeh’s extension principle to obtain their addition, it results the fuzzy subset S = {0.3/3, 0.4/5, 0.3/6, 0.3/7, 1/8, 1/9, 0.3/10, 0.8/12, 0.5/13, 0.5/16} which does not fulfill the conditions to be a discrete fuzzy number, because the third property of Definition 18.2.2 fails. In order to overcome this drawback, several authors [2, 3, 18] have proposed other methods to get a closed addition in the set DFN:

18.2 Discrete Fuzzy Numbers

327

• [18] For each pair of the discrete fuzzy numbers A, B ∈ DFN, these authors define their addition as the discrete fuzzy number, denoted by A ⊕ B, such that W

it has as α -cuts the sets (A ⊕ B)α = W

{z ∈ supp(A) + supp(B)| min(Aα + Bα ) ≤ z ≤ max(Aα + Bα )} where supp(A) + supp(B) = {x + y| x ∈ supp(A), y ∈ supp(B)} min(Aα + Bα ) = min{x + y| x ∈ Aα , y ∈ Bα } max(Aα + Bα ) = max{x + y| x ∈ Aα , y ∈ Bα }

and by membership function (A ⊕ B)(x) = sup{α ∈ [0, 1]| x ∈ (A ⊕ B)α } W

W

Moreover, these authors prove that for each A, B ∈ DFN such that A ⊕ B ∈ DFN (obtained through the Zadeh’s extension principle) then A ⊕ B = A ⊕ B. W

• [2, 3] The authors propose a method that can be seen like a generalization of the previous method and it is based on the concept of association of a fuzzy number to a discrete fuzzy number, understanding by association a mapping A : DFN → FN that fulfills determined conditions. Thus, from each association A and for each pair A, B ∈ DFN, it is defined the A-addition of A and B, denoted by A ⊕ B, as the discrete fuzzy number which fulfills the next two conditions: A

a) It has as support the set supp(A ⊕ B) = {x + y| x ∈ supp(A), y ∈ supp(B)} A

b) And its membership function is (A ⊕ B)(z) = (A(A) ⊕ A(B))(z) ∀ z ∈ supp(A ⊕ B) A

A

where A(A) ⊕ A(B) represents the addition of the fuzzy numbers A(A) and A(B) obtained using the Zadeh’s extension principle [11]. In [3], the authors also show the next result. Let Ar be the set of discrete fuzzy numbers whose support is a set of terms of an arithmetic sequence of natural numbers with r as common difference. If A, B ∈ Ar then their addition A⊕ B (obtained using the Zadeh’s extension principle) is always a discrete fuzzy number belonging to the set Ar . Hence, the addition of any pair A, B ∈ Ar (obtained from the Zadeh’s extension principle[11]) is a closed operation on Ar , which coincides with the addition methods present in [2, 3, 18]. Remark 3. [3] Note that the set A1 is the set of discrete fuzzy numbers whose support is a set of consecutive natural numbers.

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18.2.3

Maximum and Minimum of Discrete Fuzzy Numbers

In this section we recall [4] a method to obtain the maximum and the minimum of discrete fuzzy numbers since, in general, the extension principle does not yield a fuzzy subset which it fulfills the conditions to be a discrete fuzzy number. Using these operations to represent the meet and the join, we show [5] that the set of discrete fuzzy numbers whose support is a set of consecutive natural numbers is a distributive lattice. Finally, on this lattice we construct [6] a bounded distributive lattice. Let A, B be two dfn and Aα = {xα1 , · · · , xαp }, Bα = {yα1 , · · · , yαk } their α -cuts respectively. In [4], for each α ∈ [0, 1], we consider the following sets: minw (A, B)α = {z ∈ supp(A) maxw (A, B)α = {z ∈ supp(A) 





supp(B) | {min(xα1 , yα1 ) ≤ z ≤ min(xαp , yαk )} supp(B) | {max(xα1 , yα1 ) ≤ z ≤ max(xαp , yαk )}

supp(B) = {z = min(x, y)|x ∈ supp(A), y ∈ supp(B)} and where supp(A)  supp(A) supp(B) = {z = max(x, y)|x ∈ supp(A), y ∈ supp(B)}. Proposition 18.2.1. [4] There exist two unique discrete fuzzy numbers, that we will denote by minw (A, B) and maxw(A, B), such that they have the sets minw (A, B)α and maxw (A, B)α as α -cuts respectively. The following result is not true, in general, for the set of discrete fuzzy numbers. Proposition 18.2.2. [5] The triplet (A1 ,minw ,maxw ) is a distributive lattice, where A1 denotes the set of discrete fuzzy numbers whose support is a sequence of consecutive natural numbers. Remark 4. [5] Using these operations, we can define a partial order on A1 in the usual way: A & B if and only if minw(A, B) = A, or equivalently, A & B if and only if maxw (A, B) = B for any A, B ∈ A1 . Equivalently, we can also define the partial ordering in terms of α -cuts: A & B if and only if min(Aα , Bα ) = Aα A & B if and only if max(Aα , Bα ) = Bα And moreover, Theorem 18.2.2. [6] The triplet (A1L , MIN, MAX) is a bounded distributive lattice where A1L denotes the set of discrete fuzzy numbers whose support is a subset of consecutive natural numbers of the finite chain L = {0, 1, · · · , m}. 18.2.4

Discrete Fuzzy Numbers Obtained by Extending Discrete t-norms(t-conorms) Defined on a Finite Chain

In this section we present a method to build triangular operations on the bounded lattice A1L of the set of discrete fuzzy numbers whose support is a subset of consecutive natural numbers of the finite chain L = {0, 1, · · · , m}.

18.2 Discrete Fuzzy Numbers

329

Definition 18.2.3. [14]A triangular norm (briefly t-norm) on L is a binary operation T : L × L → L such that for all x, y, z ∈ L the following axioms are satisfied: 1. T (x, y) = T (y, x) (commutativity) 2. T (T (x, y), z) = T (x, T (y, z)) (associativity) 3. T (x, y) ≤ T (x , y ) whenever x ≤ x , y ≤ y (monotonicity) 4. T (x, m) = x (boundary condition) Definition 18.2.4. A triangular conorm (t-conorm for short) on L is a binary operation S : L × L → L which, for all x, y, z ∈ L satisfies (1), (2), (3) and (4 ): S(x, e) = x, as boundary condition. Let us consider a discrete t-norm(t-conorm) T (S) on the finite chain L = {0, 1, · · · , m} ⊂ N. Let DL be the subset of discrete fuzzy number DL = {A ∈ DFN such that supp(A) ⊆ L} and A, B ∈ DL . If X and Y are subsets of L, then the subset {T (x, y)|x ∈ X, y ∈ Y} ⊆ L will be denoted by T (X, Y). Analogously, S(X, Y) = {S(x, y)|x ∈ X, y ∈ Y}. So, if we consider the α -cut sets, Aα = {xα1 , ..., xαp }, Bα = {yα1 , ..., yαk }, for A and B respectively then T (Aα , Bα ) = {T (x, y)|x ∈ Aα , y ∈ Bα } and S(Aα , Bα ) = {S(x, y)|x ∈ Aα , y ∈ Bα } for each α ∈ [0, 1], where A0 and B0 denote supp(A) and supp(B) respectively. Definition 18.2.5. [6] For each α ∈ [0, 1], let us consider the sets C α = {z ∈ T (supp(A), supp(B))| min T (Aα , Bα ) ≤ z ≤ max T (Aα , Bα )} Dα = {z ∈ S(supp(A), supp(B))| minS(Aα , Bα ) ≤ z ≤ max S(Aα , Bα )} Remark 5. [6] From the monotonicity of the t-norm(t-conorm) T (S), Cα = {z ∈ T (supp(A), supp(B))|T (xα1 , yα1 ) ≤ z ≤ T (xαp , yαk )} Dα = {z ∈ S(supp(A), supp(B))|S(xα1 , yα1 ) ≤ z ≤ S(xαp , yαk )} For α = 0 then C0 = T (supp(A), supp(B)) and D0 = S(supp(A), supp(B)). Theorem 18.2.3. [6] There exists a unique discrete fuzzy number that will be denoted by T (A, B)(S (A, B)) such that T (A, B)α = Cα (S (A, B)α = Dα ) for each α ∈ [0, 1] and T (A, B)(z) = sup{α ∈ [0, 1] : z ∈ Cα }(S (A, B)(z) = sup{α ∈ [0, 1] : z ∈ Dα }) Remark 6. [6] From the previous theorem, if T (S) is a discrete t-norm(t-conorm) on L, we see that it is possible to define a binary operation on DL = {A ∈ DFN|supp(A) ⊆ L}, T (S ) : DL × DL −→ DL (A, B) '−→ T (A, B)(S (A, B)) that will be called the extension of the t-norm T (t-conorm S) to DL . Moreover T and S are commutative and associative binary operations. Also, if we restrict these operations on the subset {A ∈ A1 | supp(A) ⊆ L = {0, 1, · · · , m}} ⊆ DL we showed that T and S are increasing operations too.

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Theorem 18.2.4. [6] Let T (S) be a divisible t-norm(t-conorm) on L and let T (S ) : A1L × A1L → A1L (A, B) '−→ T (A, B)(S (A, B)) be the extension of t-norm(t-conorm) T (S) to A1L , defined according to remark 6. Then, T (S ) is a t-norm(t-conorm) on the bounded set A1L . 18.2.5

Negations on (A1L , MIN, MAX)

In this section we recall that from the unique strong negation n defined on a finite chain L = {0, · · · , m} of consecutive natural numbers it is possible to get a strong negation on the bounded lattice A1L . Proposition 18.2.3. [7] Let us consider the strong negation n on the finite chain L = {0, 1, · · · , m}. The mapping N : A1L −→ A1L A '→ N (A) is a strong negation on the bounded distributive lattice A1L = (A1L , minw , maxw ) where N (A) is the discrete fuzzy number such that has as support the sets N (A)α = [n(xαp ), n(xα1 )] for each α ∈ [0, 1] (being Aα = [xα1 , xαp ] the α -cuts of the discrete fuzzy number A).

18.3

Subjective Evaluations

In recent years, a number of researchers [1, 8, 19, 20] have focused on solving performance evaluation issues using fuzzy set theory. In [1], Biswas presented a fuzzy evaluation method based on the degree of similarity between two fuzzy sets. In [8], Chen and Lee provided two methods for applying fuzzy sets in students’ answerscripts evaluation. Recently, in [19], Wang and Chen presented a new approach for to assess students using fuzzy numbers associated with degrees of confidence of the evaluator. For example, let us review one of the methods discussed. 18.3.1

Wang and Chen’s Method

In the Wang and Chen’s method, nine satisfaction levels are considered to evaluate students’ answerscripts regarding a question of a test/examination: Extremely Good, Very Good, Good, More or less Good, Fair, More or less Bad, Bad, Very Bad and Extremely Bad. These nine satisfaction levels are represented by triangular fuzzy numbers (see table 18.1). Each satisfaction level Fi is associated with a degree of confidence αi of the evaluator. The concept of the degree of confidence associated to a satisfaction level awarded to a question is regarded as the degree of certainty of the evaluator.

18.3 Subjective Evaluations

331

The larger the degree of confidence associated with a satisfaction level awarded to the answer of a question, the higher the degree of certainty that an evaluator awarded the satisfaction level to the answer of the question. • Step 1: To calculate the αi -cut (Fi )αi of the fuzzy number Fi 0 ≤ i ≤ n then (Fi )αi = [ai , bi ], 0 ≤ i ≤ n. • Step 2: To calculate the weighted mean of the intervals (Fi )αi , 0 ≤ i ≤ n, i.e. the ⎡ n ⎤ interval: n n ∑ si (Fi )αi ⎢ ∑ si ai ∑ si bi ⎥ ⎢ ⎥ i=1 [m1 , m2 ] = = ⎢ i=1n , i=1n ⎥ n ⎣ ⎦ s s s i i i ∑ ∑ ∑ i=1

i=1

i=1

• Step 3: The total mark of the student is evaluated as follows: (1 − λ ) × m1 + λ × m2 where λ denotes the optimism index determined by the evaluator and λ ∈ [0, 1]. The triangular fuzzy numbers used by Wang and Chen like satisfaction levels are represented in the next table: Table 18.1. Satisfaction Levels Extremely Good Very Good Good More or Less Good Fair More or Less Bad Bad Very Bad Extremely Bad

= (100, 100, 100) = (90, 100, 100) = (70, 90, 100) = (50, 70, 90) = (30, 50, 70) = (10, 30, 50) = (0, 10, 30) = (0, 0, 10) = (0, 0, 0)

Example 1. Let us suppose that the answerscript includes 4 questions and the evaluator awards the fuzzy number that can be seen in Table 18.2 respectively. The interval of confidence associated to satisfaction levels of questions Q1 , Q2 , Q3 and Q4 are [65, 75], [90, 90], [25, 35] and [99.5, 100] respectively. Now, if we consider s1 = s2 = s3 = s4 = 25 then the weighted mean of the intervals is [69.875, 75]. Finally, if we get λ = 0.75 as optimism index we obtain that the total mark is 73.71875. So, Wang and Chen aggregate intervals obtained by defuzzification. From this process a new interval yields which is defuzzificated through of an optimism index. Thus, the previously quoted method the fuzzy numbers awarded by each evaluator

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18 Weighted Means of Subjective Evaluations Table 18.2. Example of Wang and Chen’s method Degrees of Confidence of Satisfaction Levels More or Less Good 0.75 Good 1 More or less Bad 0.75 Very Good 0.95

Question No. Satisfaction Levels Q1 Q2 Q3 Q4

Fig. 18.1. Interval of confidence

are not directly aggregated. The authors obtain a weighted mean of intervals based on a degree of confidence of each fuzzy number instead of a aggregation of the evaluation expressed as fuzzy numbers. Finally, the interval obtained is defuzzificated through of an optimism index. In the previous method, as well as the procedures established in [1, 8, 19, 20] the fuzzy numbers are not aggregate directly. Hence, in the defuzzification process a large amount of information and characteristics are lost. For this reason, we think that it is interesting to build aggregation functions on fuzzy subsets which lead to a fuzzy subset too. Thus, the information loss is minimized. Next, we deal with the problem of to find aggregation functions on fuzzy numbers and discrete fuzzy numbers.

18.4 n-Dimensional Aggregation Functions

18.4

333

n-Dimensional Aggregation Functions

In this section we will discuss the issue to construct n-Dimensional Aggregation on fuzzy numbers or discrete fuzzy numbers. On the one hand, we present a well known method introduced by Mayor et al. [15] that allows to build n-dimensional aggregation function on fuzzy numbers. And on the other, we give a way to obtain n-dimensional aggregation on discrete fuzzy numbers. 18.4.1

n-Dimensional Aggregation on Fuzzy Numbers

Definition 18.4.1. [15] Given a lattice (L, ≤), a function f : Ln → L is an ndimensional aggregation function if: i. f is idempotent: f (a, . . .n , a) = a for all a ∈ L ii. f is increasing with respect to the product order of Ln , i.e. if xi ≤ yi , for all i = 1, . . . , n, then f (x1 , . . . , xn ) ≤ f (y1 , . . . , yn ) Proposition 18.4.1. [15] Let u1 , . . . , un ∈ FN be fuzzy numbers with parameters (a1 , b1 , c1 , d1 ), . . . , (an , bn , cn , dn ), respectively, and f : Rn → R a continuous and strictly increasing function. The function F : ([0, 1]R )n → [0, 1]R induced through the Extension Principle by f satisfies that F(u1 , . . . , un ) = u is a fuzzy number and, moreover: Proposition 18.4.2. [15] Let u1 , . . . , un ∈ FN be fuzzy numbers with parameters (a1 , b1 , c1 , d1 ), . . . , (an , bn , cn , dn ), respectively, and f : Rn → R a continuous and strictly increasing function. Then the α -cuts of the fuzzy number u = F(u1 , . . . , un ) are: [u]α = f ([u1 ]α , . . . , [un ]α ) = {y ∈ R : y = f (x1 , . . . , xn ), where xi ∈ [ui ]α , i = 1, . . . , n} 18.4.2

n-Dimensional Aggregation on Discrete Fuzzy Numbers

A first idea to construct n-dimensional aggregation functions on the set of discrete fuzzy number would be to consider the extension principle [11] like the method presented in previous Propositions 18.4.1-18.4.2. But in general, this way fails.

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Example 2. Let A = {0.3/1, 1/2, 0.5/3} and B = {0.4/4, 1/5, 0.8/6} be two discrete fuzzy numbers. Let f : Rn → R the aggregation function f (x1 , x2 ) = ω1 x1 + ω2 x2 with ω1 + ω2 = 1. So, if we take ω1 = 14 and ω2 = 34 it results the fuzzy set ω1 A + ω2 B = {0.3/3.25, 0.4/3.5, 0.4/3.75, 0.3/4, 1/4.25, 0.5/4.5, 0.3/4.75, 0.8/5, 0.5/5.25} that does not belong to the set of discrete fuzzy numbers, because it does not fulfill the conditions of Definition 18.2.2 as we can see in the next figure:

Fig. 18.2. The fuzzy subset ω1 A + ω2 B

Now, we will see that it is possible to build weighted arithmetic means of a finite family of discrete fuzzy numbers. For this reason, let us consider a normal weighting vector w = (w1 , · · · , wn ) and A1 , · · · , An ∈ DFN. Moreover, let us consider the set [w1 · A1 ⊕ · · · ⊕ wn · An ]α = {z ∈ (supp(w1 · A1 ) + · · · + supp(wn · An )) | min(w1 · Aα1 + · · · + wn · Aαn ) ≤ z ≤ max(w1 · Aα1 + · · · + wn · Aαn )} where min(w1 · Aα1 + · · · + wn · Aαn ) = min{x1 + · · · + xn | x1 ∈ supp(w1 · A1 ), · · · , xn ∈ supp(wn · An )}

18.4 n-Dimensional Aggregation Functions

335

max(w1 · Aα1 + · · · + wn · Aαn ) = max{x1 + · · · + xn | x1 ∈ supp(w1 · A1 ), · · · , xn ∈ supp(wn · An )} for each α ∈ [0, 1]. Proposition 18.4.3. Let us consider a normal weighting vector w = (w1 , · · · , wn ) and A1 , · · · , An ∈ DFN. Let us consider the previous set [w1 · A1 ⊕ · · · ⊕ wn · An ]α . There exists a unique discrete fuzzy number W whose level sets W α are the sets [w1 · A1 ⊕ · · · ⊕ wn · An ]α for each α ∈ [0, 1]. Proof. It is similar to the demonstration of Proposition 2 in [4].

) (

From the previous Proposition 18.4.3 we will see that it is possible to define from a weighted arithmetic mean defined on Rn , f (x1 , · · · , xn ) = ∑ni=1 wi · xi (where (w1 , · · · , wn ) is a normal weighted vector) a weighted arithmetic mean on the set of discrete fuzzy numbers. So, Definition 18.4.2. Let us consider a weighted arithmetic mean defined on Rn , f (x1 , · · · , xn ) = ∑ni=1 xi · wi (where (w1 , · · · , wn ) is a normal weighted vector).The n-ary operation on DFN, W : DFN n −− → DFN (A1 , · · · , An ) '−→ W (A1 , · · · , An ) will be called discrete fuzzy weighted arithmetic mean, being W (A1 , · · · , An ) the discrete fuzzy number built using the proposed method in Proposition 18.4.3 above. Remark 7. As f (x, · · · , x) = ∑ni=1 x · wi = x · ∑ni=1 wi = x (because (w1 , · · · , wn ) is normal weighted vector) then W (A, · · · , A) = A for all normal weighted vector on Rn and A ∈ DFN. Example 3. Let A = {0.3/1, 1/2, 0.5/3} and B = {0.4/4, 1/5, 0.8/6} be the same two discrete fuzzy numbers of example above. Let f : Rn → R the aggregation function f (x1 , x2 ) = ω1 x1 + ω2 x2 with ω1 + ω2 = 1. So, if we take ω1 = 14 and ω2 = 34 it results the discrete fuzzy number W={0.3/3.25, 0.4/3.5, 0.4/3.75, 0.4/4, 1/4.25, 0.8/4.5, 0.8/4.75, 0.8/5, 0.5/5.25}.

n-Dimensional Aggregation of Discrete Fuzzy Number whose support is a finite chain of consecutive natural numbers Definition 18.4.3. [12] Let L be a chain (called scale) and let n ∈ N be fixed. A non decreasing function m : Ln → L such that min ≤ m ≤ max will be called an n-dimensional mean on L.

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Fig. 18.3. Discrete fuzzy weighted arithmetic mean of A and B

Remark 8. [12] Due to their monotonicity means on L can alternatively be characterized by their idempotency, i.e., a non-decreasing function m : Ln → L is a mean if and only if fulfills the property m(x, · · · , x) = x for all x ∈ L. Definition 18.4.4. [13] Consider n + 1 finite chains S0 , S1 , . . . , Sn . A discrete function G : ×ni=1 Si → S0 is said to be smooth if, for any a, b ∈ ×ni=1 Si , the elements G(a) and G(b) are equal or neighboring whenever there exists j ∈ {1, . . . , n} such that a j and b j are neighboring and ai = bi , for all i = j. Remark 9. Let us point that: i. If f is a smooth mean then f (Ln ) = L, where f (Ln ) = {x ∈ L|x = f (x1 , · · · , xn ) with xi ∈ L, i = 1, · · · , n}. ii. Let us consider A1L the set of discrete fuzzy numbers whose support is a subset of consecutive natural numbers of L and Aα its α -level set. If f is a ndimensional mean then max f (Aα , · · · , Aα ) = max Aα and min f (Aα , · · · , Aα ) = min Aα , where f (Aα , · · · , Aα ) = {z ∈ L|z = f (a1 , · · · , an ) with ai ∈ Aα , i = 1, · · · , n}. Recently the authors showed [5] that A1 , the set of discrete fuzzy numbers whose support is a sequence of consecutive natural numbers, is a distributive lattice. In this lattice, we considered a partial order, obtained in a usual way, from the lattice operations of this set. So, provided this partial order, we investigated [6] the extension of monotone operations defined on a discrete setting to a closed binary operation of discrete fuzzy numbers. In this same article, we also investigated different properties such as the monotonicity, commutativity and associativity. And, in [6] we deal

18.4 n-Dimensional Aggregation Functions

337

with the construction of t-norms and t-conorms on a special bounded subset of the distributive lattice A1 which was denoted by A1L . Now we wish to see that from a n-dimensional mean defined on a finite chain L of consecutive natural numbers it is possible to construct a n-dimensional mean on the bounded distributive lattice A1L . Theorem 18.4.1. Let L = {0, 1, . . . , m} be a finite chain of consecutive natural numbers and m : Ln → L a smooth n-dimensional mean. Then, the function M : [A1L ]n −−→ A1L (A1 , · · · , An ) '−→ M(A1 , · · · , An ) where M(A1 , · · · , An ) is the discrete fuzzy number with α -cut sets M(A1 , · · · , An )α = {z ∈ L| min m(Aα1 , · · · , Aαn ) ≤ z ≤ max m(Aα1 , · · · , Aαn )} is a non-decreasing function such that MIN(A1 , · · · , An ) & M(A1 , · · · , An ) & MAX(A1 , · · · , An ) Proof. Recall that [5] A & B if and only if min(Aα , Bα ) = Aα or equivalently, A & B if and only if max(Aα , Bα ) = Bα . So, if Ai & Bi for all i = 1, · · · , n then min Aαi ≤ min Bαi and max Aαi ≤ max Bαi . Now, due to the monotonicity of the n-dimensional mean m these relations min m(Aα1 , · · · , Aαn ) = m(min Aα1 , · · · , min Aαn ) max m(Aα1 , · · · , Aαn ) = m(max Aα1 , · · · , max Aαn ) hold. Then,

m(min Aα1 , · · · , min Aαn ) ≤ m(min Bα1 , · · · , min Bαn ) m(max Aα1 , · · · , max Aαn ) ≤ m(max Bα1 , · · · , max Bαn ) min m(Aα1 , · · · , Aαn ) ≤ min m(Bα1 , · · · , Bαn ) max m(Aα1 , · · · , Aαn ) ≤ max m(Bα1 , · · · , Bαn )

for all α ∈ [0, 1]. So, using these last inequalities it is obvious that min(M(A1 , · · · , An )α , M(B1 , · · · , Bn )α ) = M(A1 , · · · , An )α for all α ∈ [0, 1], then M(A1 , · · · , An ) & M(B1 , · · · , Bn ), i.e., M is a non-decreasing function. Now, we will see that MIN(A1 , · · · , An ) & M(A1 , · · · , An ) As m is a n-dimensional mean these inequalities min(x1 , · · · , xn ) ≤ m(x1 , · · · , xn ) ≤ max(x1 , · · · , xn ) hold for all x1 , · · · , xn ∈ L. As consequence,

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min(min(Aα1 , · · · , Aαn )) ≤ min(m(Aα1 , · · · , Aαn )) max(min(Aα1 , · · · , Aαn )) ≤ max(m(Aα1 , · · · , Aαn )) and then

MIN(A1 , · · · , An )α & M(A1 , · · · , An )α

for all α ∈ [0, 1], that is, MIN(A1 , · · · , An ) & M(A1 , · · · , An ) The proof of the other inequality is similar.

) (

Definition 18.4.5. Let L be a chain of consecutive natural numbers and m a smooth n-dimensional mean on L. The function M : [A1L ]n −−→ A1L (A1 , · · · , An ) '−→ M(A1 , · · · , An ) where M(A1 , · · · , An ) is the discrete fuzzy number with α -cut sets M(A1 , · · · , An )α = {z ∈ L| min m(Aα1 , · · · , Aαn ) ≤ z ≤ max m(Aα1 , · · · , Aαn )} will be called discrete fuzzy n-dimensional mean generated by m. Proposition 18.4.4. Let L be a chain of consecutive natural numbers, m a smooth n-dimensional mean on L and M : [A1L ]n −−→ A1L (A1 , · · · , An ) '−→ M(A1 , · · · , An ) the discrete fuzzy n-dimensional mean generated by m. Then, M fulfils the property M(A, · · · , A) = A for all A ∈ A1L . Proof. Straightforward.

18.5

) (

Group Consensus Opinion Based on Discrete Fuzzy Weighted Normed Operators

Consider a multi-expert decision situation where each expert rates an alternative based on a single criterion. To rate an alternative under the considered criterion it is assumed that experts’ evaluations are performed on a finite ordinal scale and expressed as discrete fuzzy numbers whose support is a subset of consecutive natural numbers of the finite chain L = {0, 1, · · · , m}. The purpose is to establish a procedure to combine the individual opinions to form a group consensus opinion based on discrete fuzzy weighted normed operators. Moreover in this method we consider the degree of importance of each expert in the aggregation procedure too.

18.5 Group Consensus Opinion

339

Yager [22, 23] suggested a general framework for evaluating weighted t-norms (t-conorms) operators defined on a finite ordinal scale H. In order to do this, the theory is based on t-conorm (t-norm) aggregation of a collection of arguments S(a1, · · · , an )(T (a1 , · · · , an )) where the a j ∈ H (which are well defined since the associativity property of triangular operations allows us to extend this to any number of arguments). The author considered aggregations of the form S((w1 , a1 ), · · · , (wn , an )) or T ((w1 , a1 ), · · · , (wn , an )) respectively. In these formulas the a j are the argument values and w j is the importance weight associated with the argument a j with a j , w j ∈ H for all j = 1, · · · , n. The proposed methods are: a) Aggregation of S((w1 , a1 ), (w2 , a2 ), · · · , (wn , an )) 1. Transform each pair (wi , ai ) into a single value bi = T (wi , ai ) where T is a t-norm on H. 2. Perform the aggregation S(b1 , · · · , bn ). b) Aggregation of T ((w1 , a1 ), (w2 , a2 ), · · · , (wn , an )) 1. Transform each pair (wi , ai ) into a single value di = S(wi , ai ) where S is a t-norm on H and wi =negation(wi) 2. Perform the aggregation T (d1 , · · · , dn ). Based on this idea, it is possible to consider two generalizations of these methods. The first one, the values ai will be considered discrete fuzzy numbers belonging to the set A1L . The second, besides the values ai also we consider the weight associated with the argument ai as discrete fuzzy number of the set A1L . Indeed: Let us consider the finite chain L = {0, 1, · · · , m} of consecutive natural numbers and T, S a divisible t-norm and a divisible t-conorm on L respectively. Moreover, let T , S be its extensions according to Theorem 18.2.4 and A1 , A2 , · · · , An ∈ A1L . 18.5.1

Discrete Fuzzy Weighted Triangular Conorm Operators

First Method: The problem is to aggregate values from the set A1L , possibly weighted, with weights {wi }{i=1···n} belonging to the ordinal scale L. Therefore, First step: Transform each pair (wi , Ai ) into a single discrete fuzzy number belonging to A1L , Bi = T (wi , Ai ), whose α -cut sets are T (wi , Ai )α = {z ∈ L|T (wi , min Aαi ) ≤ z ≤ T (wi , max Aαi )} for each α ∈ [0, 1]. Second step: Perform the aggregation S (B1 , · · · , Bn ). This operator will be called Discrete Fuzzy Weighted Triangular Conorm. The value S (B1 , · · · , Bn ) is a discrete fuzzy number belonging to A1L whose α -cut sets are S (B1 , · · · , Bn )α = {z ∈ L|S(min Bα1 , · · · , min Bαn ) ≤ z ≤ S(max Bα1 , · · · , max Bαn )} for each α ∈ [0, 1]. Remark 10. If wi = m for all i = 1, · · · , n then Bi = T (m, Ai ) = Ai for all i = 1, · · · , n. Thus, S((m, A1 ), (m, A2 ), · · · , (m, An )) = S(A1 , · · · , An )

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Second Method: The problem is to aggregate values from the set A1L , possibly weighted, with weights {Wi }{i=1···n} belonging to A1L as well. Therefore, First step: Transform each pair (Wi , Ai ) into a single discrete fuzzy number belonging to A1L , Bi = T (Wi , Ai ), whose α -cut sets are T (Wi , Ai )α = {z ∈ L|T (minWiα , min Aαi ) ≤ z ≤ T (maxWiα , max Aαi )} for each α ∈ [0, 1]. Second step: Perform the aggregation S (B1 , · · · , Bn ). The value S (B1 , · · · , Bn ) is a discrete fuzzy number belonging to A1L whose α -cut sets are S (B1 , · · · , Bn )α = {z ∈ L|S(min Bα1 , · · · , min Bαn ) ≤ z ≤ S(max Bα1 , · · · , max Bαn )} for each α ∈ [0, 1]. Remark 11. If wi = M for all i = 1, · · · , n (where M represents the maximum of A1L ) then Bi = T (M, Ai ) = Ai for all i = 1, · · · , n. Thus, S((M, A1 ), · · · , (M, An )) = S(A1 , · · · , An ). Example 4. Consider the ordinal scale L = {N,V L, L, M, H,V H, P} where the letters refer to the linguistic terms none, very low, low, medium, high, very high and perfect and they are listed in an increasing order: N ≺ V L ≺ L ≺ M ≺ H ≺ V H ≺ P. It is obvious that we can consider a bijective application between this ordinal scale L and the finite chain L = {0, 1, 2, 3, 4, 5, 6} of natural numbers which keep the order. Furthermore, each normal convex fuzzy subset defined on the ordinal scale L can be considered like a discrete fuzzy number belonging to A1L , and viceversa. Suppose that three experts give their opinion about a commercial decision and these opinions are expressed by three discrete fuzzy numbers O1 , O2 , O3 ∈ A1L . Moreover each expert has a degree of knowledge w1 , w2 and w3 respectively. The purpose is to combine the individual opinions to form a group consensus opinion based on discrete fuzzy weighted normed operators. Let us consider O1 = {0.3/1, 0.4/2, 0.7/3, 1/4, 0.8/5, 0.6/6}, O2 = {0.2/0, 0.4/1, 1/2, 0.4/3, 0.2/4} and O3 = {0.5/2, 0.6/3, 0.7/4, 1/5, 0.7/6} belonging to A1L . Moreover, let us consider the Łukasiewicz t-norm TŁ (x, y) = max(0, x + y − 6) and the Łukasiewicz t-conorm SŁ (x, y) = min(6, x + y). First method: In this case the weights considered are w1 = 2, w2 = 3 and w3 = 2 belonging to L for O1 , O2 and O3 respectively. If we apply the proposed procedure: 1. Transform each pair (wi , Oi ) into a single discrete fuzzy number belonging to A1L , Bi = T (wi , Ai ) where i = 1, 2, 3. So, we obtain B1 = {1/0, 0.8/1, 0.6/2}, B2 = {1/0, 0.2/1} and B3 = {0.7/0, 1/1, 0.7/2}. 2. Perform the aggregation S (B1 , B2 , B3 ). Finally, it results S (B1 , B2 , B3 ) = {0.7/0, 1/1, 0.8/2, 0.7/3, 0.6/4, 0.2/5}. Now, we express this discrete fuzzy number like a normal convex fuzzy subset of the ordinal scale L = {N,V L, L, M, H,V H, P}. Finally the group consensus opinion is the normal convex fuzzy subset

18.5 Group Consensus Opinion

341

GCO = {0.7/N, 1/VL, 0.8/L, 0.7/M, 0.6/H, 0.2/VH}. So, the commercial decision in this case is very low. For instance, if the commercial decision is to sell a block of shares from a company the common decision is not to sell. Second method: In this case the weights considered are W1 = {0.6/1, 0.8/2, 1/3, 0.7/4}, W2 = {0.4/2, 0.6/3, 1/4, 0.8/5} and W3 = {0.4/3, 0.6/4, 1/5, 0.8/6} belonging to A1L for O1 , O2 and O3 respectively. If we apply the proposed procedure: 1. Transform each pair (Wi , Oi ) into a single discrete fuzzy number belonging to A1L , Bi = T (Wi , Ai ) where i = 1, 2, 3. So, we obtain B1 = {0.8/0, 1/1, 0.8/2, 0.7/3, 0.6/4}, B2 = {1/0, 0.8/1, 0.4/2, 0.2/3} and B3 = {0.5/0, 0.6/1, 0.6/2, 0.7/3, 1/4, 0.8/5, 0.7/6}. 2. Perform the aggregation S (B1 , B2 , B3 ). Finally, it results S (B1 , B2 , B3 ) = {0.5/0, 0.6/1, 0.6/2, 0.7/3, 0.8/4, 1/5, 0.8/6} Now, we express this discrete fuzzy number like a normal convex fuzzy subset of the ordinal scale L = {N,V L, L, M, H,V H, P}. Finally the group consensus opinion is the normal convex fuzzy subset GCO = {0.5/N, 0.6/VL, 0.6/L, 0.7/M, 0.8/H, 1/VH,0.8/P}. So, the commercial decision in this case is very high. For instance, if the commercial decision is to sell a block of shares from a company the common decision is to sell them. 18.5.2

Discrete Fuzzy Weighted Triangular Norm Operators

First Method: The problem is to aggregate values from the set A1L , possibly weighted, with weights {wi }{i=1···n} belonging to the ordinal scale L. Therefore, First step: Transform each pair (wi , Ai ) in to single discrete fuzzy number belonging to A1L , Bi = S (n(wi ), Ai ), whose α -cut sets are S (n(wi ), Ai )α = {z ∈ L|S(n(wi ), min Aαi ) ≤ z ≤ S(n(wi ), max Aαi )} for each α ∈ [0, 1]. Second step: Perform the aggregation T (B1 , · · · , Bn ). The value T (B1 , · · · , Bn ) is a discrete fuzzy number whose α -cut sets are T (B1 , · · · , Bn )α = {z ∈ L|T (min Bα1 , · · · , min Bαn ) ≤ z ≤ T (max Bα1 , · · · , max Bαn )} for each α ∈ [0, 1]. Remark 12. If wi = 0 for all i = 1, · · · , n then Bi = S (n(0), Ai ) = M for all i = 1, · · · , n (where M denoted the maximum of the bounded set A1L ). Thus, T ((0, A1 ), (0, A2 ), · · · , (0, An )) = S(M, · · · , M) = M. We also observe that if w j = m for all j = 1, · · · , n then n(w j ) = 0 and Bi = S (0, Ai ) = Ai for all j = 1, · · · , n. And then, T ((m, A1 ), · · · , (m, An )) = S(A1 , · · · , An ). Second Method: The problem is to aggregate values from the set A1L , possibly weighted, with weights {Wi }{i=1···n} belonging to A1L as well. Therefore, First step: Transform each pair (Wi , Ai ) in to single discrete fuzzy number belonging to A1L , Bi = T (N (Wi ), Ai ), whose α -cut sets are T (N (Wi ), Ai )α = {z ∈ L|T (min N (Wi )α , min Aαi ) ≤ z ≤ T (max N (Wi )α , max Aαi )} for each α ∈ [0, 1], and N (Wi ) is built according to Proposition 18.2.3.

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Second step: Perform the aggregation S (B1 , · · · , Bn ). The value T (B1 , · · · , Bn ) is a discrete fuzzy number whose α -cut sets are T (B1 , · · · , Bn )α = {z ∈ L|T (min Bα1 , · · · , min Bαn ) ≤ z ≤ T (max Bα1 , · · · , max Bαn )} for each α ∈ [0, 1]. Remark 13. If Wi = & 0 (& 0 denoted the minimum of A1L ) for all i = 1, · · · , n & 0, A1 ), · · · , (& 0, An )) = then Bi = S (n(0), Ai ) = M for all i = 1, · · · , n. So, T ((& S(M, · · · , M) = M. We also observe that if W j = M for all j = 1, · · · , n then N (W j ) = & 0 and Bi = S (& 0, Ai ) = Ai for all j = 1, · · · , n. And then, T ((M, A1 ), · · · , (M, An )) = S(A1 , · · · , An ). Example 5. Consider the same ordinal scale L = {N,V L, L, M, H,V H, P} of Example 18.5.1 above. Suppose that three experts give their opinion about an aspect of educational decision of a student (eg. need a professional psychological help, attitudes toward sciences or arts, assessment of personal maturity, etc.) and these opinions are expressed by three discrete fuzzy numbers O1 , O2 , O3 ∈ A1L . Moreover each expert has a degree of knowledge w1 , w2 and w3 respectively. The purpose is to combine the individual opinions to form a group consensus opinion based on discrete fuzzy weighted normed operators. Let us consider O1 = {0.3/1, 0.4/2, 0.7/3, 1/4, 0.8/5, 0.6/6}, O2 = {0.2/0, 0.4/1, 1/2, 0.4/3, 0.2/4} and O3 = {0.5/2, 0.6/3, 0.7/4, 1/5, 0.7/6} belonging to A1L . Moreover, let us consider the Łukasiewicz t-norm TŁ (x, y) = max(0, x + y − 6) and the Łukasiewicz t-conorm SŁ (x, y) = min(6, x + y). First method: In this case the weights considered are w1 = 2, w2 = 3 and w3 = 2 belonging to L for O1 , O2 and O3 respectively. If we apply the proposed procedure: 1. Transform each pair (wi , Oi ) into a single discrete fuzzy number belonging to A1L , Bi = S (n(wi ), Ai ) where i = 1, 2, 3. So, we obtain B1 = {0.3/5, 1/6}, B2 = {0.2/3, 0.4/4, 1/5, 0.4/6} and B3 = {1/6}. 2. Perform the aggregation T (B1 , B2 , B3 ). Finally, it results T (B1 , B2 , B3 ) = {0.2/2, 0.3/3, 0.4/4, 1/5, 0.4/6}. Now, we express this discrete fuzzy number like a normal convex fuzzy subset of the ordinal scale L = {N,V L, L, M, H,V H, P}. Finally the group consensus opinion is the normal convex fuzzy subset GCO = {0.2/L, 0.3/M, 0.4/H, 1/VH, 0.4/P}. So, the consensus common opinion about an aspect of educational decision of a student is very hight. For instance, if the educational decision of a student was the need for a professional psychological help the common decision would be to provide it. Second method: In this case the weights considered are W1 = {0.6/1, 0.8/2, 1/3, 0.7/4}, W2 = {0.4/2, 0.6/3, 1/4, 0.8/5} and W3 = {0.4/3, 0.6/4, 1/5, 0.8/6} belonging to A1L for O1 , O2 and O3 respectively. If we apply the proposed procedure:

18.6 Conclusions

343

1. Transform each pair (Wi , Oi ) into a single discrete fuzzy number belonging to A1L , Bi = S (N (Wi ), Ai ) where i = 1, 2, 3. So, B1 = {0.3/3, 0.4/4, 0.7/5, 1/6}, B2 = {0.2/1, 0.4/2, 0.4/3, 0.8/4, 1/5, 0.6/6} and B3 = {0.5/2, 0.6/3, 0.7/4, 0.8/5, 1/6}. 2. Perform the aggregation T (B1 , B2 , B3 ). Finally, it results T (B1 , B2 , B3 ) = {0.6/0, 0.7/1, 0.7/2, 0.8/3, 0.8/4, 1/5, 0.6/6}. Now, we express this discrete fuzzy number like a normal convex fuzzy subset of the ordinal scale L = {N,V L, L, M, H,V H, P}. Finally the group consensus opinion is the normal convex fuzzy subset GCO = {0.6/N, 0.7/VL, 0.7/L, 0.8/M, 0.8/H, 1/VH, 0.6/P}. Thus, the consensus common opinion about an aspect of educational decision of a student is very hight. For instance, if the educational decision of a student was the attitudes toward sciences the common decision would advise you to study a degree course of science.

18.6

Conclusions

In this chapter we have reviewed different approaches to assess students using fuzzy numbers associated with degrees of confidence of the evaluator. In these methods, the fuzzy numbers are not aggregate directly and a defuzzification process is used to get the result of the evaluation. But, in this way a large amount of information and characteristics is lost. For this reason, we think that it is interesting to build aggregation functions on fuzzy subsets which lead to a fuzzy subset too. On the other hand, it is well known that in an evaluation process a finite ordinal scale L is used to asses. So, the discrete fuzzy number whose support is a subset of consecutive natural numbers is a correct tool for modeling the subjective evaluation. Therefore, it is interesting to deal with the problem to build aggregation function in the set of fuzzy discrete numbers. So, we have proposed a theoretical framework to construct n-dimensional aggregation of discrete fuzzy number whose support is a finite chain of consecutive natural numbers. And, in particular, we discuss discrete fuzzy ndimensional mean generated by a mean defined on a finite chain. Finally, we present a method to obtain the group consensus opinion based on discrete fuzzy weighted normed operators.

Acknowledgments This work has been partially supported by the MTM2009-10962 project grant.

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References [1] Biswas, R.: An application of fuzzy sets in students’ evaluation. Fuzzy Sets and Systems 74(2), 187–194 (1995) [2] Casasnovas, J., Riera, J.V.: On the addition of discrete fuzzy numbers. Wseas Transactions on Mathematics 5(5), 549–554 (2006) [3] Casasnovas, J., Vincente Riera, J.: Discrete fuzzy numbers defined on a subset of natural numbers. In: Theoretical Advances and Applications of Fuzzy Logic and Soft Computing. Advances in Soft Computing, vol. 42, pp. 573–582. Springer, Heidelberg (2007) [4] Casasnovas, J., Vincente Riera, J.: Maximum and Mininum of Discrete Fuzzy numbers. In: Frontiers in Artificial Intelligence and Applications: Artificial Intelligence Research and Development, vol. 163, pp. 273–280. IOS Press (2007) [5] Casasnovas, J., Vincente Riera, J.: Lattice properties of discrete fuzzy numbers under extended min and max. In: Proceedings IFSA-EUSFLAT, Lisbon, pp. 647–652 (July 2009) [6] Casasnovas, J., Vincente Riera, J.: Extension of discrete t-norms and t-conorms to discrete fuzzy numbers. In: Fuzzy Sets and Systems, doi:10.1016/j.fss.2010.09.016 [7] Casasnovas, J., Riera, J.V.: Negation functions in the set of discrete fuzzy numbers. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds.) IPMU 2010. CCIS, vol. 81, pp. 392–401. Springer, Heidelberg (2010) [8] Chen, S.-M., Lee, C.-H.: New methods for students’evaluation using fuzzy sets. Fuzzy Sets and Systems 104(2), 209–218 (1999) [9] Dubois, D., Prade, H.: Fuzzy cardinality and the modeling of imprecise quantification. Fuzzy Sets and Systems 16, 199–230 (1985) [10] Dubois, D., Prade, H.: Fundamentals of Fuzzy Sets. The handbooks of Fuzzy Sets Series. Kluwer, Boston (2000) [11] Klir, G.J., Yuan, B.: Fuzzy sets and fuzzy logic (Theory and applications). Prentice-Hall (1995) [12] Kolesarova, a., Mayor, G., Mesiar, R.: Weighted ordinal means. Information Sciences 177, 3822–3830 (2007) [13] Marichal, J.-L., Mesiar, R.: Meaningful aggregation functions mapping ordinal scales into an ordinal scale: a state of the art. Aequationes Mathematicae 77, 207–236 (2009) [14] Mayor, G., Torrens, J.: Triangular norms on discrete settings. In: Klement, E.P., Mesiar, R. (eds.) Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms. Elsevier (2005) [15] Mayor, G., Soto, A.R., Suñer, J., Trillas, E.: Multi-dimensional Aggregation of Fuzzy Numbers Through the Extension Principle. In: Data mining, Rough sets and Granular computing. Studies in Fuzziness and Soft Computing, pp. 350–363. Physica-Verlag [16] Rocacher, D., Bosc, P.: The set of fuzzy rational numbers and flexible querying. Fuzzy Sets and Systems 155, 317–339 (2005) [17] Voxman, W.: Canonical representations of discrete fuzzy numbers. Fuzzy Sets and Systems 54, 457–466 (2001) [18] Wang, G., Wu, C., Zhao, C.: Representation and Operations of discrete fuzzy numbers. Southeast Asian Bulletin of Mathematics 28, 1003–1010 (2005) [19] Wang, H.-Y., Chen, S.-M.: Evaluating Students’Answerscripts Using Fuzzy Numbers Associated With Degrees of Confidence. IEEE Transactions on Fuzzy Systems 16, 403– 415 (2008)

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[20] Wang, H.-Y., Chen, S.-M.: Appraising the performance of high school teachers based on fuzzy number arithmetic operations. Soft Computing 12, 919–934 (2008) [21] Wygralak, M.: Fuzzy sets with triangular norms and their cardinality theory. Fuzzy Sets and Systems 124, 1–24 (2001) [22] Yager, R.R.: Weighted triangular norm and conorm using generating functions. International Journal of Intelligent Systems 19, 217–231 (2004) [23] Yager, R.R.: Aggregation of ordinal information. Fuzzy Optimization and Decision Making 6, 199–219 (2007) [24] Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965)

19 Some Experiences Applying Fuzzy Logic to Economics Bárbara Díaz and Antonio Morillas

19.1

Introduction

In this book chapter we summarize the principal results of some works in which we have applied different techniques based on fuzzy logic and robust statistics, such as fuzzy clustering, fuzzy inference systems as well as robust multidimensional outlier identification. Fuzzy inference systems can be applied to economic studies, with the objective of adding them to econometric tools, due to their ability to grasp and model the non-linearities in the relationship among variables. Fuzzy clustering allows elements to belong to deferent groups with different grades of membership. In that case, a membership value equal or close to one would identify “core” points in a cluster, while lower membership values in a cluster would identify boundary points, which are uniquely assigned to a cluster in classic cluster analysis, without any measure of their suitability to belong to it. During the past years, we have been applying these and other techniques to various economic studies [10], [40], [41], [11] and this compendium will probably give some ideas to other researchers in social sciences interested in applying fuzzy logic in their works.

19.2 19.2.1

Applications Prediction of Waged-Earning Employment in Spain with a Fuzzy Inference System

We predicted the growth rate of waged non-agricultural private employment in Spain from the available information about output, wages and productivity and carried out the comparison with the results obtained in a classic econometric model. The period under study was 1976-1998, wih quarterly based information. The information sources are: EPA (Active Population Survey), Industry and services salaries Survey and Quarterly National Accounting. An explanation of the evolution of employment is necessarily complex, due to the interdependencies existing among the economic variables that can help to understand its evolution. We have used a uniequational model, considering the wages somehow as an exogenous variable, R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 347–379. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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making it possible to give preeminence to the demand factors as explicative of the changes in employment. In the construction of a macroeconomic model for the growth rate of employment, we can distinguish two kinds of variables capable of explaining its changes. The first one is the output variables, which try to depict the economic increase or decrease, and the second one is the relative costs variables. The labor demand is a derivate demand, that is, an increase in the demand of goods and services implies an increase of labor demand, a fact that is introduced in models by means of a variable that captures the variations in the level of the economic activity. To study the evolution of the relative costs for the labor factor, the relationship between wages and productivity is taken into account; in the sense that if the nominal wages and the productivity increase in the same percentage, the unitary labor costs do not vary. If the salary rise is greater than the one produced in labor productivity, the unitary costs grow, while if it is less than this one, the unitary costs decline. The data we use as representative of waged employment is the series for private, non agricultural wage earners. We name it asan. (EPA, INE, provides the data). The study is on waged employees and not on occupation since one of the explaining variables is the salary. We don’t include agricultural wage earners. The reason is that this is a sector in which we can observe a process of employment decrease during the observed period of time because of a structural change in the Spanish economy. On the other hand, wage earners in the public sector have been increasing over the whole period, so we must think that, at least in Spain they have different mechanisms that the ones explained here (it seems they do not depend so much on the level of the economic activity). The variable chosen as a representative for the output is the private, non-agricultural gross added value (VABnap). Since the marginal productivity can not be observed, the mean productivity is usually calculated as output divided by occupied people. To have a better approximation of productivity in conditions of full employment, we have used the maximum historical occupation productivity, as the ratio: pr =

VABna OMna

(19.1)

where VABna is the non-agricultural gross added value, and OMna is the maximum non agricultural occupation reached until that moment. The variable asan represents the salaries are for the non-agricultural workers. To introduce wages and productivity in the model, it has been used a variable which tries to approximate the salary for a product unit in the following way: SUP = lnIsal78 − lnI pr78 = ln

Isal78 I pr78

(19.2)

It captures the evolution of labor costs. In this formula, Isal78 are the wages in form of indices with base 1978. Before the transformation into indices, this variable has had a seasonal adjustment by the method of moving averages, due to the seasonality it presented. On the other hand, I pr78 is the productivity given in form of indices with base in 1978.

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349

For our purposes, for variables asan and VABnap the basic growth series is used since it is more interesting to use the variation rate rather than the absolute values of employment and production. Since the data are quarterly based, we use the T41 (Y,t), which is approximated as1 : T41 (Y,t)

Yt+2 −Yt−2 ≈ Δ 4 lnYt+2 = (1 − L4)lnYt+2 Yt−2

(19.3)

So that we obtain the interanual growht. These new variables are called ASAN and VABNAP: asant+2 ASANt = ln(asant+2 ) − ln(asant−2) = ln (19.4) asant−2 VABNAPt = ln(VABnapt+2) − ln(VABnapt−2) = ln

VABnapt+2 VABnapt−2

(19.5)

Wages and gross added value are measured at current prices, due to the fact that the appropriate deflator for the nominal wages is the same one than that for the output. On the right side of figure 19.1 the relationship between the explaining variables SUP and VABnap with the explained variable ASAN is shown.

Fig. 19.1. Right: Actual data in an interpolation-generated surface; left: above: Relationship of the variables VABNAP and ASAN; below: Relationship of the variables SUP and ASAN. 1

Espasa and Cancelo (1994), [13], p. 69.

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19 Some Experiences Applying Fuzzy Logic to Economics

In figure 19.1 (left side, above), variables VABNAP and ASAN are plotted. The scatterplot shows a positive relationship between the two variables, as it could be expected from the theory underlying in the model. In Figure 19.1 (left side, below), on the other hand, variables SUP and ASAN are plotted. From the scatterplot, a rather negative relationship can be implied, although it is no so clear as the positive relationship between the two variables aforementioned. It can be observed that in the extreme values of variable SUP the negative relationship is clearer, but there is a central zone of indefinition. The reason for this indefinition, could maybe be explained by the fact that with high ratios of salary/productivity, there is an evident decrease in the growth rate of employment, and that with low ratios of salary/productivity, there is an increase in the growth rate of employment, but in intermediate situations some variable not included in the study plays a decisive role, as it could very well be the variation of the relative costs of other factors that could substitutive of the work factor. Another possible explanation could be that in intermediate situations there is more disparity among productivities of the different sectors, and then, different results regarding the change in employment. Finally, in figure 19.1 (right side) there is a 3D scatterplot where the relationship of the dependent variable, ASAN with variables VABNAP and SUP can be observed. For input selection we use the method of Tanaka, Sano and Watanabe (1995) [56]. This is a simplified version of Sugeno’s and Yasukawa’s Regularity Criterion (1993) [55], which needs less data to be computed. The first variable that joins the model is variable SUP. If the system had to use only one variable to adjust the waged employment series, the one chosen would be SUP. Then, the other variables are added separately to this one, to see which of them gives more information. The result is that VABNAP is the following chosen variable. At last, to these two variables, the variable t trying to model tendency is added, to see if it improves the system performance, but it does not. For this reason, in the fuzzy inference system, as in the regression model, the variable t is left out. For input space partition we used the subtractive clustering algorithm proposed by Chiu (1994)[7], which is a modification of the mountain clustering proposed by Yager and Filev (1994) [57]. We tried with different cluster radius, and the better results were provided by a radius ra = 0.65, with rb = 1.5ra . For this radius value, two clusters have been formed, which correspond to two rules. Smaller values for the radius ra gave a greater number of clusters and rules, but this procedure, although gave a better adjustment to the training data, did not improve the prediction. In figure 19.2 we can see the centers of the clusters with their scores for each variable. From this scores and the radio ra , we can build the gaussian membership functions in the antecedents of the rules. The coordinates of the clusters centers for each variable and the radius determine the initialization of the parameters for the adjectives of the linguistic variables in the antecedents of the rules. The membership functions are gaussians, with two parameters.

19.2 Applications

351

Fig. 19.2. Left: Clusters; right: Clusters, centers scores for variables SUP and VABNAP.

Then, the architecture of ANFIS, Jang (1993) [28], with five layers, has 8 nonlinear parameters and 6 linear parameters. It can be seen in figure 19.3. The number of total parameters, 14, is not very large, and the relationship between available data and parameters is 75/14, about 5. The estimation of the parameters of the consequences, once we have initialized the parameters in the antecedents with the results from the clusters, is made by the method of least squares, and afterwards, the backpropagation algorithm adjusts the parameters in the antecedents

Fig. 19.3. Neural Network.

The two rules generated are these: If VABNAP is medium and SUP is medium then ASAN = 1.1710 VABNAP + 0.1683 SUP − 0.0394.

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If VABNAP is high and SUP is low then ASAN = 0.6417 VABNAP − 0.1681 SUP + 0.0153. The two rules provide a surface of the values generated for the variable ASAN given the different values that the variables VABNAP and SUP can take. It can be observed in figure 19.4, left side. It can be seen the similarity that it shows with figure 19.1, left side. The relationship of each variable with the dependent variable, according to the FIS obtained have also been plotted separately in figures 19.4, right side. Figure 19.4, right and above suggests that the relationship between VABNAP and ASAN is not linear, but that from certain growth rate of VABNAP the incidence over the employment is greater, the slope changes.

Fig. 19.4. Left: FIS output; right above: FIS output relating VABNAP and ASAN; right below: FIS output relating SUP and ASAN.

In figure 19.4 the plot shows a relationship that is not uniform for the whole Surrey range. The relationship of SUP and ASAN reveals a try to model what we already saw in the original data and called an “indefinition zone”. The system generated has a root of the mean squared error (RMSE) of 0.012857 for the training data and 0.005771 for the prediction data. We also calculated the U-Theil coefficient, to facilitate the comparision of the errors in prediction for our

19.2 Applications

353

inference system and a classic econometric model applied to the same data. The RMSE itself shows that the predictions are better with the FIS, since, applied to the same data, they are perfectly comparable. The RSME depends on the unity of measurement of the predicted variable. Since ASAN is expressed in form of ratios, the RMSE is inherently low, due to the scale of the data. The U-Theil coefficient allows better apprehending the difference between the two methods. Its value is 0.046426. It can also be decomposed in the part of the error due to bias, which is a 9.3% of the differences between the predicted and the actual values, the part due to the differences between the variances of the observed and predicted values, which amounts a 19.8%, and those due to random factors, which in this case is a 70.8% of the differences. Comparing the results to those of the classic econometric model, the fuzzy inference system gave better results for the prediction. As it can be seen, a relatively complex problem, as the one presented here can be characterized with a low number of rules. The fact that each rule can be fulfilled to a degree provides the flexibility needed when trying to model this kind of phenomena, which present clear non-linearity. It is a non-parametric approach that provides very satisfactory results. There are some drawbacks that an economist must take into account: the need of a great deal of elements in the survey or very large time series, which are not always available. The number of parameters grows with the number of rules and can make the problem unapproachable. Other possible application of fuzzy inference systems is the use of the IFN (Maimon and Last, 2000) approach [37]. In this approach, the rules are extracted based on the mutual information and fuzzy logic is added so the resulting rules are simplified and expressed in linguistical terms. We applied it to the study of the economic profit value of the agricultural businesses in Andalusia. 19.2.2

Data Mining and Fuzzy Logic: An Application to the Study of the Economic Profit Value of the Agricultural Businesses in Andalusia

We studied the accounting characteristics that define the economic profit value of the businesses of the agri-food chain in Andalusia, analysing agrarian production, transformation industries, wholesale distribution and retail commerce. We work on an available set of ratios in the database of the Institute of Statistics of Andalusia, included in the Central of Balances of Enterprise Activity in Andalusia. The methodological aspects that are used in the application are advanced statistic techniques for data exploratory analysis, robust estimation, fuzzy neural networks and new knowledge discovery methods in large databases2 . We make also some contributions to the study of the businesses profit value by means of the analysis of accounting ratios (Foster, 1986) [18] as are the novel use of the previously indicated methods and the frontier regions focus, unpublished in this type of studies. The conclusions, given as fuzzy rules (as a theoretic model)) extracted form the database by the “computational theory of perceptions” (Zadeh, 2001 [58]; Last, Klein and Kandel, 2001 [33]), 2

The statistical source used in this work, the Central of Balances of Enterprise Activity in Andalucía (IEA, 1998, [24]) has information about more than 40.000 firms.

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seem fully congruent with the assessments of the financial analysis. If the assets rotation is low, there are not high profit values. High rotation of the assets with an acceptable situation of liquidity, characterize the most profitable businesses. With these two factors the 95% of the total mutual information is explained. In cases of rotations around the average, high margins play a compensatory role to give rise to high profit values. These results fit well with the agrarian businesses connected with forced cultivations behaviour (Almería, Huelva), which have high rotation of assets, and where the greater profit values are found, according to the information supplied by the Central of Balances of Enterprise Activity. The exploratory analysis has begun with the analysis of lost data. We carried out some tests on the pattern of lost data (correlation among present/ lost fictitious variables, contrast of averages and Little´s MCAR test) to verify if is or not ignorable (Rubin, 1976) [53], [34]. The result is the same in the three tests, resulting not ignorable, and so, due to this reason, with the added circumstance of very anomalous data and high correlation with other ratios, we decided to eliminate the three affected variables. We carried out a grouping of the ratios in patterns (principal components) with two goals: to avoid redundancy in the information and to work with linearly independent variables. We must also indicate that the own imprecise conception of each component selected and their composition (linear combination of ratios), makes it even more suitable for the application of fuzzy logic. We will undertake the characterization of the most profitable businesses using methods that are not dependent of all these theoretical hypothesis (normality, linear dependencies), as are neuronal networks and fuzzy logic. We have proceeded to the locating, scale and co-variation parameters robust multivariante estimation, using the Minimum Volume Ellipsoid (MVE) estimator3 . With the results of this estimation, the process of multivariate detection of anomalous values and its subsequent elimination has been optimised. It has been also obtained a matrix of robust correlations from which, with principal components technique, the accounting ratios patterns have been extracted. They are subsequently used in the characterization of the businesses with greater profit value. Such evaluation is carried out by means of knowledge discovery techniques and subsequent creation of fuzzy rules, that express knowledge by means of interpretable perceptions in terms of economic language. We must underline that all it is carried out based on a thick frontier approximation4. Once the exploratory study has been carried out, as well as the application of robust techniques to avoid the statistical problems found, and the determination of the principal components, inputs of the application, we are going to expose the adopted focus for the characterization of the businesses with high and low profit value. Based on principal components we will try to find out the rules that define the characterization in the context of what is known as data mining. We use the factors that were previously obtained in principal components, usually called accounting ratios patterns: L IQUIDITY, M ARGINS, ROTATIONS and 3 4

Rousseeuw and Leroy in [49], Rousseeuw and van Zomeren in [50], Rousseeuw and Hubert in [51], Rousseeuw and van Driessen [52]. Berger and Humphrey in [5], [6], Bauer et al. in [3].

19.2 Applications

355

I NDEBTEDNESS5 . The target is the economic profit value (ROA). We have used the data of those businesses that, based on the application of the thick frontier notion, result to be of low or high profit value6. The method is based on the IFN(Info-FuzzyNetwork)7 algorithm developed by Maimon and Last (2000) [37], in which behavior rules are induced with the aid of a of a neural network. A weight is assigned and indicates the contained information in each association among input variables and the target one (probabilistic rules), measure that is known as Mutual Information in the Information Theory. The difference between mutual information and correlation coefficient resembles the difference between entropy and variance: Mutual information is a metric-free measure, while a correlation coefficient measures a degree of functional (e.g., linear) dependency between values of random variables. So, we do not make hypothesis about the probability distributions of the variables, it is a model free technique. Besides, it can be applied to variables of different nature. In our case, all variables are continuous but the target one, that is a categorical one with only two possible values: high and low profit value. Finally, 1481 cases have been used, corresponding to the first and third quartile. We have randomly selected 59 observations (approximately a 5%), for the validation of the model. In table 19.1 the variables introduced in the study, as well as their nature and use in IFN can be observed. In this method, the interactions among input attributes and target (classification) one are modeled by means of an information-theoretic connectionist network. It consists of a root node, a variable number of hidden layers that coincides with the number of variables that finally enter the system as explanatory ones, and a target layer. In such a network, an input attribute is selected to enter the model if it maximizes the total significant decrease in the conditional entropy. This is called conditional mutual information In the first of the hidden layers, pertaining to the first variable that enters the system, the number of neurons is determined by the number of intervals in which it is partitioned, in agreement with the intervals that contribute to the maximization of the mutual information8, if the variable is continuous, or the different number of 5 6

7

8

These four components explain a 85,3% of the variance. The use of the notion of thick frontier results in an stratification by sectors of the sample, keeping for each sector those businesses that are below the first quartile and above the third one. So, we avoid the bias that could have place given the sectorial differences in profits. We are grateful to Mark Last, professor at the Information Systems Engineering Department, University of the Negev, (Israel), and creator of IFN software, his comments and suggestions, that have allowed the improvement and enlargement of a preliminary version of this work. If an input attribute is continuous, it is partitioned into intervals by means of a discretization of the attribute procedure, included in the algorithm, that in a recursive way looks for partition thresholds that maximize the contribution to the target attribute mutual information and that are automatically changed when a new attribute enters the system. To see more details about the network construction, see Maimon and Last (2000) [37], or Last, Klein, and Kandel (2001) [33].

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19 Some Experiences Applying Fuzzy Logic to Economics

values that it takes otherwise. A node in this layer is split if it provides a significant decrease in the conditional entropy of the target attribute9 . The nodes that are split will be associated in the following layer to so many neurons as intervals or different values there are for the following input variable for the system10 . The nodes of a new hidden layer are defined for a Cartesian product of split nodes of the previous hidden layer and values of the new input attribute. The construction of the network finishes when it does not exist any input candidate variable that decreases significantly (by defect in a 0´1%) the conditional entropy of the target attribute. Finally, the connection is carried out among each non split node or of the last significant (hidden) layer, called terminal ones, with nodes pertaining to the target layer. In the target layer, the number of neurons is the number of different possible values for the target attribute. Table 19.1. Attributes. Definition Attribute L IQUIDITY

M ARGINS

T URNOVER I NDEBTEDNESS

ROA (Return on assets ratio)

Nature

Use in IFN

Principal component that Continuous Input includes current ratio, candidate quick ratio, solvency ratio, cash ratio and coverage ratio Principal Component that Continuous Input includes profit margin, candidate net operating margin gross operating margin, rate of return on fixed assets, gross operating surplus ratio Asset turnover ratio Continuous Input candidate Principal Component that Continuous Input includes debt to equity ratio candidate and equity turnover Return on assets ratio Dichotomic Target

In our application, a network with six layers has been created, four of them hidden, for variables ROTATIONS , L IQUIDITY, M ARGINS and I NDEBTEDNESS . The 9

10

The significance of decrease in the conditional entropy at a hidden node (which is equal to the conditional mutual information of a candidate input attribute and a target attribute, given a hidden node) is measured by a likelihood-ratio test. The null hypothesis (H0) is that a candidate input attribute and a target attribute are conditionally independent, given a hidden node (implying that their conditional mutual information is cero). If H0 holds, the test statistic is distributed as a with degrees of freedom equal to the number of independent proportions estimated from the tuples associated with the hidden node. Anyway, a new node is created if there is at least one tuple associated to it.

19.2 Applications

357

network has 26 neurons, the number of unsplit and terminal nodes is 17 and in the target layer there are two neurons, as can be observed in figure 19.5. In table 19.2 the lower limits of the different intervals created for the variables11, as well as their code are given. Table 19.2. Lower limits of the intervals for each attribute.

Variable ROTATIONS L IQUIDITY M ARGINS I NDEBTEDNESS

Codes 0 -1´8064 -2´1014 -10´1484 -3´2001

1 2 3 4 -0´8362 -0´5884 -0´1292 -0´6631 -0´4467 -0´1936 6´1965 -0´1708 1´464

Fig. 19.5. Neural network for accounting ratios patterns.

11

These intervals are automatically created by the described algorithm in footnote number 17.

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19 Some Experiences Applying Fuzzy Logic to Economics

In the second hidden layer, pertaining to the variable L IQUIDITY it can be observed that nodes two and three are only associated to four nodes. The reason is that there are not cases in the database with the combination of values represented by the corresponding nodes (a node in a hidden layer represents a combination of values of input attributes, and each tuple can be associated with one and only one node in every hidden layer, according to its own values of the input attributes). From table 19.3 it can be deducted that the two first variables selected by the model, ROTATIONS and L IQUIDITY contribute in a 95% to the total mutual information (0.708/0.749). This means that these two variables are the most determinant ones for the expected value of the economic profit value of the business. The variable I NDEBTEDNESS is the one that less contributes to information, with a decrease in the conditional entropy of the target variable of 2´3%. Table 19.3. Selection order of input attributes. Iteration 0 1 2 3

Attribute

Conditional Conditional M.I. Mutual information entropy (acumulated) ROTATIONS 0.677 0.323 0.323 L IQUIDITY 0.292 0.385 0.708 M ARGINS 0.257 0.035 0.743 I NDEBTEDNESS 0.251 0.006 0.749

Each connection among a terminal node and a target node represents a probabilistic rule between a conjunction of input attribute-values and a target value. A node z represents an input attribute values conjunction. Given that the network represents a disjunction of these conjunctions, each conjunction is associated with one and only one node z ∈ F . So, the summation over all terminal nodes covers all the possible values of the input attributes. Subsequently, a weight is assigned to each rule of association between a terminal node and a target value V j .12 They express the mutual information among hidden nodes (that represent a conjunction of values of the input variables) and the target nodes. A connection weight is positive if the conditional probability of a value of the target variable given the node is greater than its unconditional probability and negative otherwise. A weight close to zero indicates that target attribute is independent of the node value. The sign of the weight determines if the relation is direct or inverse. We have obtained a total of 28 rules with associated weights different from zero, pertaining to the associations or probabilistic rules that are created with the combination of 17 terminal nodes (unsplit and of the last hidden layer ones) to the two target nodes. In table 19.4 we give the rules with greater weights associated: 12

ij

P(V )/z

The weigths are calculated: wz = P(Vi j ; z) · log Viijj where P(Vi j ); z) is an estimated joint probability of the target value Vi j and the node z, P(Vi j /z) is an estimated conditional (a posteriori) probability of the target value Vi j given the node z and P(Vi j ) is an estimated unconditional (a priori) probability of the target value Vi j .

19.2 Applications

359 Table 19.4. Rules with higher weights. 

Rule 18 If ROTATIONS is greater than −0 1292   and L IQUIDITY is between −0 1936 and 6 1965 then ROA is high   Rule 2 If ROTATIONS is between −1 8064 and −0 8362 then ROA is low  Rule 15 If ROTATIONS is greater than −0 1292   and L IQUIDITY is between −2 1014 and −0 6631 then ROA is low   Rule 27 If ROTATIONS is between −0 5884 and −0 1292   and L IQUIDITY is between −0 1936 and 6 1965  and M ARGINS is greater than −0 1708   and I NDEBTEDNESS is between −3 2001 and 1 464 then ROA is high  Rule 16 If ROTATIONS is greater than −0 1292   and L IQUIDITY is between −0 4467 and −0 1936 then ROA is high   Rule 9 If ROTATIONS is between −0 5884 and −0 12921292   and L IQUIDITY is between −2 1014 and −0 6631 then ROA is low



Weight = 0 2707 

Weight = 0 1974 

Weight = 0 0833 

Weight = 0 0767



Weight = 0 0586 

Weight = 0 0438

Given the values for each input variable, the value for the target variable is obtained selecting a value j∗ that maximizes the conditional (a posteriori) probability of the target variable i in the node z, (P(Vi j /z)); that is, j∗ = argmax j P(V j /z) so that this will be the predicted value of the target variable i at node z. The predictive capacity of the model generated with the formation of these rules has been verified with a 5% of the randomly chosen data that were not used in the modelling stage13 . In the data set used to train the network14, with 1422 cases, there were 1320, (92´8%) correctly classified, and in the validation set were cross validation15 has been carried out, there were 57 from 59 cases correctly classified (96´6%). Therefore, the differences in profit values are in the model well characterized by the concepts, rather inexact, in which we have included the patterns of ratios collected in the principal components analysis. It has been verified also that the model is a good predictor for observations that are not present in the sample. We 13

14

15

To estimate prediction accuracy, we have performed a validation procedure. We have ramdonly partition the data set in two sets, one for training and the other one for validation. We have made successive simulations, leaving away one tuple each time that afterward is classified (cross-validation). The prediction accuracy in training data is an optimistic estimation of the true value of the accuracy. Anyway, the model usefulness can be seen comparing the prediction accuracy in the training data (error ratio= 0,072) with the a priori error ratio (without using any model to predict, but the estimation of the target value by the majority number of cases, 0,499 here), or studying the number of well classified tuples. In the validation set the error ratio is 0,034.

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19 Some Experiences Applying Fuzzy Logic to Economics

want to emphasize that the percentage of successes in the two groups (businesses more and less profitable) is quite balanced. Since the rules thrown by the system are expressed in crisp terms, their representation power remains limited, because we are generally interested in obtaining knowledge in linguistic terms, closer to human way of reasoning. The number of rules extracted, a total of 28, if we except those of nil weight, is too much high in comparison to what an agent generally thinks of as a plan to take decisions. The method used, based on the Computational Theory of Perceptions, permits the reduction of rules in a smaller system in three steps: firstly making them fuzzy, subsequently reducing its number by means of conflicts resolution and, finally, combining rules of the reduced assembly16. To fuzzify the rules, we provide for each attribute a set of linguistic terms. We have proposed the adjectives low, medium and high for the input variables17 . In table 19.5 the fuzzy triangular numbers associated to each linguistic term are shown. Table 19.5. Linguistic terms. Attribute L IQUIDITY

Term Prototype Minimum Maximum low -2.10 -2.10 -0.24 medium -0.24 -0.61 0.36 high 9.22 -0.24 9.22 M ARGINS low -10.15 -10.15 -0.10 medium -0.10 -0.47 0.41 high 4.43 -0.10 4.43 ROTATIONS low -1.81 -1.81 -0.10 medium -0.10 -0.71 0.47 high 6.73 -0.10 6.73 I NDEBTEDNESS low -3.20 -3.20 -0.11 medium -0.11 -0.58 0.43 high 5.44 -0.11 5.44

Given that the rules obtained include associations among input values sets (for the different variables) and all the possible values of the target variable, it is possible 16

17

We use Mamdani implications. IF the crisp rule weight is positive, the relation between the antecedent and the consequent is direct and if it is negative, is an inverse  relation. The  · membership of each crisp rule, when fuzzyfing it, is: μR = w · ∏N i=1 max j μAi j (Vi ) maxk {μTk (o)}. Where w is the weight of the corresponding crisp rule, N is the number of simple conditions in the rule, V j is the crisp value of the simple condition i in the crisp rule (middle point in the interval), o is the crisp value of the target of the rule ((middle point in the target interval), μAi j (Vi ) is the membership of the simple condition i to term j and μTk (o) is the membership of the target value o in term k. We have chosen as reference values the extremes of the attributes, the mean and first and third quartile. So, the concepts of low, medium and high are context-dependent, because we have businesses with profit value under the first quartile, between the first and the third one, and above the third one.

19.2 Applications

361

that several of them have the same antecedent and different consequent. Besides, several rules can differ in their numerical values, but be identical in their linguistic terms. So, the 28 fuzzy rules obtained can be inconsistent. For this reason, to solve this conflict, the degree of each rule is calculated adding the membership degrees of all identical fuzzy rules and choosing of each group in conflict the value at the target with greater ownership. On the other hand, rules with the same consequent can be merged by means of the connective “or”, using a fuzzy union operator (generally the maximum). The reduced set of rules, once they have been fuzzified and merged can be seen in table 19.6. Table 19.6. Fuzzy rules. Rule 1 If ROTATIONS is low then ROA is low Rule 2 If ROTATIONS is medium or high and L IQUIDITY is low then ROA is low Rule 3 If ROTATIONS is high and L IQUIDITY is medium or high then ROA is high Rule 4 If ROTATIONS is medium and L IQUIDITY is medium or high and M ARGINS is low then ROA is low Rule 5 If ROTATIONS is medium and L IQUIDITY is medium and M ARGINS is high then ROA is high Rule 6 If ROTATIONS is medium and L IQUIDITY is high and M ARGINS is high and I NDEBTEDNESS is medium then ROA is high

Grade = 0.1278 Grade = 0.0156 Grade = 0.0156 Grade = 0.0036 Grade = 0.0009 Grade = 0.0003 Grade = 0.0006

The number of rules has been reduced drastically. It seems clear, according to the information that appears in the rules, that the component that better characterizes the differences among the businesses with high profit value and the less profitable, would be the rotations of assets. The businesses with low rotation of assets are characterized, in any case, by having low economic profit value. On the other hand, the presence of a rotations of assets medium or high, does not guarantee the existence of high profit value, since, as is observed in the rules two and three, when the liquidity is low, the businesses have low profit value, while the combination of high rotations with medium or high liquidity is indicative of high profit value. Finally, according to the remaining rules, businesses with medium rotations accompanied by medium or high liquidity, need to have high margins and low level of indebtedness to form part of the group of the most profitable businesses. As alternative focus to classical multivariate analysis for the study of business profit value by means of accounting ratios we have used a method that does not depend for its application on theoretical hypothesis that clearly are not given in our data. The approximation by means of the thick frontier permits one more clear characterization of the businesses with greater profit value, at the same time that protects of possible biases coming from sectorial and/or provincial aggregation. The meaning of the extracted components to decrease the dimension of the problem and to merge ratios situated under a same pattern, gives as a result some new

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19 Some Experiences Applying Fuzzy Logic to Economics

variables (principal components) characterized by its conceptual vagueness and by the imprecision derived from the information that they incorporate (a specific percentage of explained variance and a lineal combination of original variables). Its processing as fuzzy variables, therefore, seems not only fully justified, but, even, more adequate. In the relations among accounting ratios, that are clearly inexact, the behavior rules definition by means of linguistic propositions permits a conceptual interpretation of the reality that is not possible to reach with other techniques. With only six fuzzy rules we can establish the discrimination among businesses of high or low economic profit value, showing the importance of having a high rotation of the assets to reach a high level of profit value. This is a characteristic of the forced cultivations of Huelva and Almería, that have the most profitable businesses, according to the database. We must say that the use of neural networks, in conjunction with the Computational Theory of Perceptions (CTP), offers a really interesting alternative to the formation of the rules from the observation or the intuition, and permits an adequate “extraction of knowledge” of a large quantity of data. The neural network neuronal has grasped the most important relations among the variables and the TCP has interpreted them in approximate form of knowledge. In the application carried out, the fitness of the classification found, with a 92´8 % of successes for the training data and a 96´6% in the validation process can be considered highly satisfactory. Fuzzy clustering techniques are also good tools that can be applied to many problems in economics. Industrial clustering is a main topic in Input-Output analysis and Regional Economics. The groups of industries obtained by traditional clustering techniques have a mayor problem: none of them can simultaneously belong to more than a cluster, which would be desirable since it would make the results closer to reality, not fixing crisp boundaries among groups. Furthermore, never before in the literature about this subject multivariate outliers have been taken into account, being as they are a crucial problem in the clustering process, as it is well known. 19.2.3

Key Sectors, Fuzzy Industrial Clustering and Multivariate Outliers

In fuzzy clustering, each cluster is a fuzzy set. It has its origins in Ruspini (1969) [54] who pointed out several advantages of using fuzzy clustering. That is, a membership value equal or close to one would identify “core” points in a cluster, while lower membership values in a cluster would identify boundary points. “Bridge” points, in Nagy’s (1968) [43] terminology, may be classified within this framework as “undetermined points with a degree of indeterminacy proportional to their similarity to core points”. Bridges or strays between sets originated problems of misclassification in classic cluster analysis. Hathaway et al. (1989) [22] shows that there is a dual relational problem of object fuzzy c means for the special case in which the relational data coincide with the Euclidean distances among their elements. Their method is called RFCM (relational fuzzy c means). Later, Hathaway and Bezdek (1994) [23] introduce a non Euclidean relational fuzzy clustering (NERFCM). Their objective is to modify RFCM to be

19.2 Applications

363

effective with arbitrary relational data that show dissimilarities among objects. We used the NERFCM algorithm (Hathaway and Bezdek, 1994) [23].18 The objective function to minimize is: k

m ∑Nj=1 ∑Nl=1 um i j uil r jl

i=1

N 2 ∑t=1 um it

min ∑

(19.6)

where N is the number of cases (sectors), m > 1 is a parameter controlling the amount of fuzziness in the partition, k is the number of clusters, ui j is the membership of case j to cluster i, with ui j ∈ [0, 1] and ∑ki=1 ui j ∈ [0, 1]. Memberships cannot be negative and each sector has a constant total membership, distributed over the different clusters. By convention, this total membership is normalized to one, and r jl measures the relationship between cases j and l. Before applying the NERFCM algorithm, several preliminary steps are necessary, however. The first involves preprocessing of the variables, while the second provides the relational matrix on which the NERFCM will operate. Step three is to comply with the need for an initialization of the membership values for the NERFCM objective function. This is done by using a robust crisp clustering method. These tree steps are exposed in more detail in the next paragraphs. With respect to the first step, recall that in cluster analysis the presence of multicolineality among variables causes that the affected ones are more represented in the similarity measure. For this reason, and given the multidimensional nature of our approach, we will use as variables the ones resulting from a previous principal component analysis applied to our database. This allows avoiding the multicolineality problem when obtaining the similarity matrix, and also provides a better conceptual identification of the resulting clusters, reducing the number of variables for the classification task. Secondly, since NERFC operates on relational data, for the calculus of the distance matrix among sectors involving the principal components in step 1 (which are quantitative variables) and the technology variable (which is an ordinal variable) we use the daisy method proposed by Kaufman and Rousseeuw (1990) [29]. This method computes all pairwise dissimilarities among objects, allowing the use of different types of variables: nominal, binary, ordinal or interval-scaled. Finally, the NERFC algorithm needs an initialization matrix for the memberships that it will subsequently optimize. For this, we have used the results provided by the crisp partitioning around medoids (PAM) clustering method in Kaufman and Rousseeuw (1990) [29], which was used in Rey and Mattheis (2000) [46].19 This is a robust version of the well-known k means method of MacQueen (1967), [36] and is less sensitive to the presence of outliers. To select the best number of clusters k 18 19

The fanny algorithm of Kaufman and Rousseeuw (1990) [29], used in Dridi and Hewings (2003) [12], has serious problems dealing with certain types of data. The results from the PAM method are used to initialize the membership matrix assigning for each element j a membership equal to one for the cluster where it belongs to and a membership of zero to the rest of the clusters. It is thus a binary matrix with zeros and ones.

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19 Some Experiences Applying Fuzzy Logic to Economics

for the PAM method we use the silhouette width method, proposed in Rousseeuw (1987) [48]. The variables we use, their statistical sources and respective definitions, are shown in table 19.7. We have distinguished between three concepts or blocks. The economic integration block contains 4 variables. It captures both the intensity and the quality of the sectors’ interdependence. The variables are: the multiplier derived from the relative influence graph,20 its dispersion as measured by the variation coefficient, the cohesion grade21 and a topologic integration index,22 obtained for each sector. The last two variables express qualitative aspects related with the relative position of a sector in the exchange structure. The economic weights block (with 4 variables) summarises the relative participation in the national output, in exports and in wage rents and employment generation. The economic potential block (with 5 variables) includes a dynamic vision of the former variables through the observed trends plus the technological innovation capacity along with the information given by Eurostat (1998) [14], [15] and the high technology indicators of the Spanish Statistics National Institute [27]. From another (statistics) point of view, in a multivariate approach to find the key sectors from an Input-Output table, most of the sectors with difficulties to be classified could be outliers. These can deeply misrepresent the grouping and could be causing the low number of clusters observed in previous works in this field. Being indeed a serious problem it has been neglected, to our knowledge, in industrial clustering analysis or key sectors contexts. The presence of multidimensional outliers has even worst effects than the presence of one-dimensional ones23 . They cause a distortion not only in the location and dispersion measures, increasing the variance, but also in the orientation ones (decreasing the correlations)24.

20

21 22

23

24

This multiplier comes from the relative influence graph associated to the distribution coefficients matrix and can be interpreted in terms of elasticities (Lantner, 1974, 2001 [31], [32]; Mougeot et al., 1977 [42]; Morillas, 1983 [38]; and De Mesnard, 2001 [9]). Its expression is: xx ˆ −1 Δ x = (I − D)−1 eˆyˆ−1 Δ y. Since it is weighted by the export rate from outside the graph structure (eˆ ê = final demand coefficient), it could be said that it is very close to the net multiplier mentioned by Oosterhaven (2002, 2004) [44], [45]. The interindustry matrix has been deprived of self-consumption, to reinforce the interdependencies analysis. The cohesion concept in an economic model was first introduced by Rossier (1980) [47] and applied for the first time to input-output analysis by Morillas (1983) [38]. See Morillas (1983) [39]. Extending the global indicator proposed in that work, the inte1 gration index for a given sector is defined as Ri = 2n−1 ∑nj=1i= j ( e1i j + e1ji ) where e ji is an element of the distances matrix. An excellent exposition of this problem and alternative solutions can be seen in Barnett and Lewis (1994, chapter 7) [2], now in its third edition, that has become a classic in the subject. See Kempthorne and Mendel (1990) [30] for related comments on the declaration of a bivariate outlier.

19.2 Applications

365 Table 19.7. List of variables.

Variable Definition Economic integration block M ULTIPLIC Multiplier

Comments

Addition by columns of Ghosh’s inverse matrix, Augustinovics’ multiplier. C VINVERS Inverse of the multi- Each sector is characterized by the inverse of the pliers’ variation dispersion, represented by its column variation coefficient coefficient. C OHESIO Cohesion Grade Number of times that a sectoris a path (from graphs theory.) Ri Total productive A topological offer-demand integration index for each integration Index sector (see note 22). Economic weights block P RODUC Output Output at basic price by product from the Spanish symmetric table. E XPORT Exports The source is the Spanish symmetric table for 1995, INE (2000.) E MPLEO Full-time Since this data is only available by industry (use table), equivalent we have applied the correction coefficient that employment presents the compensation of employees by products. S ALARIOS Mean salaries It is estimated with corrected paid employment (also with use table) and data of compensation of employees (from the symmetric table.) Economic potential block T ENPROD Output trend It is the real output variation rate for each sector Output trend sector for the period 1995-2000 (source: [26]). T ENEMPL Employment It is the full-time equivalent employment variation rate trend for each sector respect to the initial period, from 1995 to 2000. We used the implicit sectoral deflator of the Gross Valued Added at basic prices (VABpb in the National Accounting of the Spanish National Institute of Statistics, [26]). T ENSALAR Mean salary It is the real mean salary variation rate for each sector trend for the period 1995-2000. We used the consumption prices index (IPC) variation during that period (14,2) to deflate salaries for 2000. T ENEXPO Exportation It is the real exportation variation rate for each sector trend for the period 1995-2000. The procedure used to deflate is the one used for output. T ECNO Technological Sectors classification attending to their technological innovation capacity level (high, medium or low technology.)

Especially in multivariate analysis the presence of anomalous data has a deep impact on the results, since in this kind of analysis co-variation or correlation is a key idea. Unfortunately, the sample covariance matrixes are too sensitive to the presence of outliers. In cluster analysis, the presence of outliers may cause completely twisted

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19 Some Experiences Applying Fuzzy Logic to Economics

groupings or associations among objects. When an outlier is far enough the other data look like a tight cluster by comparison and “depending on the subject matter and the task at hand, one might want to put the outlier aside for further investigation and run the clustering algorithm again on the remaining data” (Rousseeuw, 1987, p. 61) [48]. Furthermore, as it has been shown in other sciences, outliers can be important themselves because sometimes they will reveal special data having an exceptional behaviour from a multidimensional point of view. So, data with very good and diverse performances should usually belong to them. In this sense, any sector that simultaneously have anomalously high values in the variables being considered in the multidimensional analysis, can be identified as an important (key) sector from one or several special points of view (exports, interrelationships, growing, technology and so on). Of course, the sectors that are not interesting at all may be within the set of outliers also. They would be away from the majority of the data as well, although for a completely different and absolutely opposite reason. They will not have good performances in any of the variables, and will not be considered part of the key sectors group. After the outliers have been identified, we study independently, applying a fuzzy clustering analysis, the set of outliers and the remaining sectors. Its application among these last ones will allow their classification avoiding the kind of problems mentioned and making possible to find out, let us say, a “second order” or “secondary sectors” of the economy (Feser and Bergman, 2000) [16]. This strategy will be applied in this paper, to the study of the key sectors of the Spanish economy. Both in univariate and multivariate outlier identification discordancy tests25 , based in the knowledge of the distribution where the data come from, are widely used. They are meant to examine whether a particular object, say xi , is statistically unreasonable under the assumed distribution and identified as a discordant outlier. For their application, in multivariate outliers identification an appropriate sub-ordering principle needs to be applied, since the idea of extremeness arises from an “ordering of the data”. Reduced sub-ordering (See Barnett, 1976 [1] for a categorization of these principles) is almost the only principle employed for outliers study. Any multivariate observation x of dimension p is transformed to a scalar quantity, appropriately chosen, over which the test is carried out. In many works, the reduction measure used is a distance measure of the points to the location of the data (could be the zero vector, the true mean or the sample mean of the data) which considers a weighting matrix inversely related to the scatter or to the population variability (the variance-covariance matrix or its sample equivalent). The idea of multivariate distance plays a key role in outliers identification. For determining the number of outliers, Rousseeuuw and van Zomeren (1990) [50] propose the use of a discordancy test with what they call “robust distance”. It is the Mahalanobis distances of all the points respect to the robust MCD (Minimum 25

Most of the methods proposed for discordant tests are based on the multidimensional normality hypothesis of the data or on other specific multivariate distribution (exponential, Pareto).

19.2 Applications

367

Covariance Determinat) localization estimator26 . That way x¯ and Cov(x) are substituted by the MCD robust estimators of location and covariance, so that they are not affected by the presence of outliers nor the masking effect, and being these estimators affine equivariant, the Mahalanobis distance is then affine invariant, which means the shape of the data determines the distances between the points (Hardin and Rocke (2004), [20]. Under multivariate normality hypothesis, these robust distances are asymptotically distributed as a χ 2 with p (number of variables) degrees of freedom27. Rousseeuw and Van Zomeren (1990), [50], propose the consideration of the values over the 97.5% quartile in the χ 2 distribution as outliers. Other authors propose, for the outlier identification, an adaptive procedure searching for outliers specifically in the tails of the distribution, beginning at a certain chisq-quantile (see Filzmoser et al., 2005) [17]. Nevertheless, neither the χ 2 nor the F distribution of the robust Mahalanobis distances should be applied when the hypothesis of normality does not hold. As Barnett and Lewis (1994, p. 461) [2] points out, “the very presence of outliers may be masked by particularly strong real effects or by breakdowns in the conventional assumptions, such as normality, homoscedasticity, additivity and so on”. In a recent Ph. D. thesis (Dang, 2005) [8] is emphasized that normality for the data can in many occasions “not realistically be assumed”, and opts for a non parametric approach to select outliers in order to avoid this problem. Discordancy tests under normality hypothesis are not adequate for identifying the outliers in an input-output analysis context, because non-normality is a usual behaviour in the variables that are present in it. Given the deviation of the data from the normality hypothesis and the singularity of this study, we propose several simultaneous criteria for identifying the adequate group of outliers: • A plot showing, for each additional outlier, the relative decrease in the Mahalanobis distance from the robust location estimation in order to know which groups of points to check. In the axe of abscises one can see the number of outliers, from 1 to the maximum number of outliers allowed28 , [n−[(n + p + 1)/2]], with n being the number of sectors and p the number of variables. The numbers of outliers considered as good choices will be those showing acceleration in the percent decrease of the distance. We will call this graph Relative distance variation plot from now on. 26

27

28

As Rousseeuw and van Driessen (1999, p. 212) [52] point out, “Although it is quite easy to detect a single outlier by means of the Mahalanobis distances, this approach no longer suffices for multiple outliers because of the masking effect, by which multiple outliers do not necessarily have large Mahalanobis distances. It is better to use distances based on robust estimators of multivariate location and scatter”. Some authors claim that an F distribution seems to be more appropriate. See Hardin and Rocke (2005) [21] for a complete description of the distributions of robust distances coming from normally distributed data. The highest possible breakdown value for MCD is (n + p + 1)/2, see Lopuhaä and Rouseeuuw (1991), [35].

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19 Some Experiences Applying Fuzzy Logic to Economics

• A plot showing the variations in the condition number of the covariance matrix29 removing one by one each sector (in descending distance order). The relationship between the correlations and the condition number of the covariance matrix is such that when the correlation is higher so will be also the condition number (Greene (1993), [19], p. 269; Belsley et al (1980), [4]). For each additional outlier, one sees the relative change in the condition number of the resulting covariance matrix. The numbers of outliers considered as good choices will be those showing acceleration in the percentage increase of the condition number. We will call this graph Relative Condition Number Variation Plot from now on. • A two groups non-parametric analysis of the variance (Kruskal-Wallis test) applied sequentially by selecting different sets of outliers, with size from 1 to [n − [(n + p + 1)/2]], to determine the sectors that must be taken aside to get the highest number of significant variables and explaining the most of the variance in the non-outliers cloud of points . We will show subsequently that the joint use of the Relative distance variation plot, the Relative condition number variation plot and the Kruskal-Wallis test can be considered a set of tools useful when deciding the number of outliers if a multivariate normal distribution is not appropriate. After estimating for the data base the robust location and covariance parameters by the Fast MCD method, the robust Mahalanobis distances have been obtained for each one of the sectors. Regarding only the 26 possible outliers30 , these distances are shown in table 19.8. As we can see, sector number 37 (Construction) is a very extreme value in the cloud of data points. Even sectors numbers 61 (Public administration and defence; compulsory social security) and 26 (Basic metals) are very far from the rest of the sectors. It is not adequate to use the χ 2 nor the F distribution looking if they are beyond a given percentile (usually 0.975) because the variables are not normally distributed but, obviously, they can be considered as outliers. Actually, the distance of the sector number 26 is almost three times greater than the following in the distance ranking (51) and the rest of sectors seem to be a tight cluster by comparison. Anyway, it is not clear they are the only ones, because their extremely high values could be masking others important relative separations from the core of the point cloud. In order to improve the approach to select the adequate group of outliers that should be considered in our study, we will apply the three criteria that have been mentioned before: relative distance variations plot, condition number variations plot and Kruskal-Wallis test. The left side plot in figure 19.6 shows the relative variations in distance of each potential outlier from the robust location estimator of the cloud of data points31 . The right one shows the relative changes in the condition number of the covariance matrix when the corresponding set of outliers is taken away from the calculus, step by step. 29 30 31

It is defined as the square root of the ratio between the highest and lowest eigenvalue. As we have said before, the maximum number of outliers allowed is: [n − [(n + p + 1)/2]] = 26. The 3 highest distances are not included in the figure, because they would change the scale and it would be impossible to see any other change in the graphs.

19.2 Applications

369 Table 19.8. Robust Mahalanobis distances.

Sector Name 37 Construction

61 26 51 41 6 62 33 5 65 64

13 2 2

Public administration and defense; social security Basic metals Real estate services Hotel and restaurant services Electricity supply

Distance Sector Name Distance 32745.15 66 Non-market recreational, 305.77 cultural and sporting services 9440.24 63 Non-market health and 298.38 social work services 3978.10 40 Retail trade 242.74 1493.12 11 Prepared animal feeds 226.02 1197.00 48 Financial intermediation 210.46 888.36

7

Non-market education services Motor vehicles

781.17

49

733.69

47

Coke, refined petroleum products and nuclear fuel Membership organization services n.e.c. Non-market sewage and refuse disposal services, sanitation and similar services Tobacco products Crude petroleum and natural gas

446.83

20

435.28

25

434.94

36

416.57 381.43

3 54

Gas, steam and hot water supply Insurance and pension funding Telecommunication services Manufacture of chemicals and chemical products Other non metallic mineral products Recycling

200.48

Metal ores Research and development services

80.58 75.20

193.10 123.42 116.74 110.06 94.11

Fig. 19.6. Identifying outliers by using robust distance and condition number.

We can see in the distance plot that, even without the three further sectors, there are two quite important changes in the relative distances, with 8 and 20 outliers respectively. In the right side plot it is evident that a group with 20 outliers produces the most significant change in the condition number of the covariance matrix.

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Finally, fixing the p-level to select a variable in 0.10, we have found that the best two-groups non-parametric ANOVA (see table 19.9) for separating medians with a larger number of variables being statistically significant is the one containing 20 outliers in a group. Only 4 of the 12 variables used are actually not relevant. They are salaries, employment and its respective trends.32 Table 19.9. Kruskal-Wallis test for two groups; a Variables with p-level < 0.10. b Grouping variable: Out20. Multiplier Diffusion Cohesion Integration Production Export Prod. Trend χ 2 -quantil 3.761 3.240 5.087 5.091 3.707 8.381 2.757 d.of freedom 1 1 1 1 1 1 1 p-level .052 .072 .024 .024 .054 .004 .097

Export Trend 11.274 1 .001

According to the three criteria, there are much more than three outliers, as we had suspected before. It seems quite reasonable to retain the twenty sectors with the largest robust Mahalanobis distances as the right group of outliers. The relational fuzzy clustering NERFCM (Hathaway and Bezdeck, 1994) [23] for identifying key sectors was proposed to classify the sectors in an I-O table in Díaz, Moniche and Morillas (2006) [11]33 . The NERFCM clustering technique is used in this context to allow each sector to belong to any of the clusters formed with different degrees of membership, considering the clusters as fuzzy sets. NERFCM operates on relational data, which is its strength over other fuzzy clustering algorithms since with the use of an appropriate distance measure it allows the simultaneous use of variables with different measure scales such as ordinal and nominal variables along with numerical ones. In this direction, the distance measure used is daisy, from Kaufman and Rousseeuw (1990) [29], given the presence of an ordinal variable among the variables considered for the study. The results of the fuzzy clustering are given in a matrix containing the membership values of each sector to each cluster34 . The fuzzy clustering analysis with the NERFCM method of the 20 outliers gives six clusters as a result. The sectors are classified in each one of them according to their highest membership value, as can be seen in table 19.10. 32 33 34

After the principal components analysis was done, the variable production trend, with the lowest significance level, appears as irrelevant. This paper provides also an overview of different approaches for identifying key sectors. NERFCM is an iterative algorithm. For initialization of the memberships a robust crisp clustering result is provided. This is performed with a Partition Around Medoids (PAM) clustering (Kaufman and Rousseeuw, 1990) [29], so that initially the memberships are either 0 if the sector does not belong to a cluster or 1 if it belongs to it. The appropriate number of clusters has been decided based on the silhouette width (Rousseeuw (1987), [48]). These memberships will be updated within the NERFCM algorithm.

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Table 19.10. Outliers classification in six clusters by NERFCM clustering. Clusters Sectors Name Highest m. v. 1 2 Crude petroleum and natural gas 0.72228 13 Tobacco products 0.70478 62 Non-market education services 0.58723 64 Non-market sewage and refuse disposal services 0.91204 65 sanitation 0.81406 66 Membership organization services n.e.c. 0.91461 Non-market recreational, cultural and sporting services 2 5 Coke, refined petroleum products and nuclear fuel 0.58291 11 Prepared animal feeds and other food products 0.86800 26 Basic metals 0.53029 3 6 Electricity supply 0.91655 40 Retail trade 0.75204 51 Real estate services 0.82762 4 7 Gas, steam and hot water supply 0.87241 48 Financial intermediation 0.55317 49 Insurance and pension funding 0.67113 61 Public administration and defense; social security 0.79703 63 Non-market health and social work services 0.54777 5 33 Motor vehicles 0.99994 6 37 Construction 0.94861 41 Hotel and restaurant services 0.45704

The average silhouette width is 0.6435, claming for a good structured grouping and a lightly fuzzy situation in making the clusters. Both plots can be seen in Figure 19.7. The silhouette plot is clearly defined and some of the groups have a high individual silhouette, claming for the necessity they are an isolated group of sectors. On the other hand, the fact that these clusters are well defined can be seen in the right plot of the mentioned figure. The two components in the plot are explaining 80.3% of the point variability. The principal component 1 can mainly be identifying a strong local base behavior (production, multiplier and interrelationships) and the principal component 2 is in relation with good export base performances (export and its trend). In figure 19.7, the six clusters are perfectly summarized. The cluster 6 is the best in component 1 (local base) and cluster 5 in component 2 (export base). The cluster 2 has a relative good performance in both components and the cluster 3 has a good level in component 1. 35

The silhouette width has values between −1 and 1. If it equals 1, the “within” dissimilarity is much smaller than the “between” dissimilarity (minimum of the distances to the elements of other clusters), so the sector has been assigned to an appropriate cluster. If it equals 0 it is not clear whether the sector should be assigned to that cluster since the “within” and “between” dissimilarity are equal. If the silhouette width equals −1 the sector is misclassified. The overall average silhouette width is the average of the silhouette width for all sectors in the whole dataset.

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Fig. 19.7. Silhouette and clusters for outliers group.

The sectors in clusters 1 and 4 are below the mean in both of the components (See table 19.5). They seem to be irrelevant as key sectors. In fact, cluster 1 has all their indicators below the global mean for the 66 sectors in the table. The highest one is the production indicator and it only is a 29% of the global mean for this variable. They are basically non-market services plus tobacco and imported energies products, such as crude petroleum and natural gas. They are sectors with the lowest multipliers, cohesion and integration indexes. Their economic weights are not important at all, and they basically are oriented to final demand. Those are the features that are characteristic of this group. We can affirm they are outliers in an opposite sense to the kind of sectors, important sectors, we are looking for. On the other hand, sectors in cluster 4 have almost all indicators below their respective global means, except in the production and the multiplier variability with values a 46% and a 3% over the mean, respectively. The only reason for that is the presence of sector 61 (Public administration and defense) which is the most important in the production indicator. Anyway, these figures are not comparable at all with the very important percents over the mean the other clusters exhibit, as it is shown in table 19.11. So, cluster 4 can be considered as no relevant in order to select potential key sectors. Table 19.11. Centroids for the six clusters (Global average in each indicator=100). Indicator Cluster 1 Cluster 2 Cluster 3 Cluster 4 Cluster 5 Cluster 6 Technology 0 0 0 0 388 0 Multiplier 17 274 143 94 275 747 Diffusion 0 0 0 0 388 0 Cohesion 0 0 0 0 388 0 Integration 14 104 121 73 112 122 Production 29 148 258 146 250 581 Export 1 266 2 8 1468 0 Export trend 10 156 1 5 450 0

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On the contrary, cluster 5 (Motor vehicles) is by far the best in exports, having a good local base and a median technology while sectors in cluster 6 (Construction and Hotel and restaurant services) have the best performance in local base component: their production, multiplier, cohesion, integration and diffusion effects are the highest ones. Each one in their respective performance, they three are the most important sectors in the Spanish economy. It is necessary to point out that sector 41 (Hotel and restaurant services) is the only one that does not have a membership value to its group over 0.5, standing only for 0.45704. Actually, its second higher membership value (0.2016) shows that it could belong to group 3. It means that, to some degree higher than those of the any other outlier sector, it can fit different economic performances. This sector has a good performance in the cohesion and integration indicators, and this is the main characteristic of cluster 3, since their values are almost the same that those in cluster 6, the best in this subject (see table 19.5). Finally, clusters 2 and 3 can be considered as having an interesting level in some indicators, above the average, even if they are very far from those of clusters 5 and 6. In particular, cluster 2 has a good export, multiplier and diffusion indicators, and joint to cluster 5 as the only two groups of sectors above the means in both of the components. So, sectors 33, 5, 11 and 26 are important sectors because they have good behavior both in export and in production and integration. Finally, cluster 3 has a high production, and as we just mentioned above, a very good cohesion and integration indicators. Looking at these results, the hierarchy of the most important sectors in the Spanish economy and their principal performances would be the one exposed in table 19.12: Table 19.12. The Spanish economy key sectors. Sector Name 37 Construction 41 Hotel and restaurant services 33 6 40 51 5 11 26

Motor vehicles Electricity supply Retail trade Real estate services Coke, refined petroleum products and nuclear fuel Prepared animal feeds and Other food products Basic metals

Performances Production Integration Diffusion Exports Integration Production Export Diffusion

After analyzing the outliers as key sectors, we have made a fuzzy clustering for the non-outliers set of sectors, in order to find new groups of them and characterized their behaviors. The clustering of the remaining sectors gave three more groups. After these results have been analyzed, the crucial idea to remark is the selection obtained by the outlier detection procedure seems to have identified the most important

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sectors from a multivariate point of view, because none of the three new groups are as interesting as the ones mentioned before. It means that the method followed for multivariate outlier detection can be considered as a new way for identifying the key sectors. Actually, in figure 19.8 it can be seen that they have the best performances in most of the indicators, being the cluster number 5 the best in export and its trend, with a second technological level, and number 6 the most important in production, multiplier and diffusion. If we make an exception on technology in cluster 9, only other two groups of multivariate outliers that have been commented, clusters numbers 2 and 3, seem to have important sectors. So, it is obvious that the key sectors in the Spanish economy are those in table 19.12, being identified as multivariate outliers.

Fig. 19.8. Indicators of the nine clusters.

The centroids of the three new clusters (numbers 7, 8 and 9) are not good enough to claim for containing also important sectors, since they are very far from the values shown in the previously selected key sectors. Actually, only one of the three remaining clusters (number 9 in figure 19.8) may be interesting in some way, even if its production centroid is a 44% under the average, because it has a high technological level (except for sector 44, Water transport services) and the trend of its exports is slightly above the reference line. In the rest of the sectors, their indicators are on the average, in the case of the cluster 8, or even below it, as it happens in cluster 7. Only the cluster 9 could be of some interest.

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The principal reason to consider of interest the group of sectors number 9 is because, from a political economics point of view, they should be taken into account in order to expand the exports and technological level of the Spanish economy. We can say they could be a second level or potential key sectors. The sectors in this group can be seen in table 19.13. Table 19.13. Potential key sectors in the Spanish economy. Sector 20 28 29 30 31 32 34 44 47 53 54

Name Highest m. v. Manufacture of chemicals and chemical prod. 0.42022 Machinery and equipment n.e.c. 0.47575 Office machinery and computers 0.59121 Electrical machinery and apparatus n.e.c. 0.51974 Radio, television and communication equipment 0.53927 Medical, precision and optical instruments; 0.55859 Other transport equipment 0.55622 Water transport services 0.34463 Telecommunication services 0.47736 Computer and related services 0.56163 Research and development services 0.48957

It is necessary to point out that membership values are now smaller than in the previous outliers clustering and the average silhouette width is 0.46 in this case, giving information about a less structured grouping and a fuzzier situation. Actually, some of them are below 0.5 and the mentioned sector number 44 has serious problems to be assigned to this group of sectors. That means the cloud of points of the non-outliers group of sectors is much more homogeneous. For this reason, a crisp grouping is more difficult to get and, perhaps, less reasonable also to try it. With no doubt, it can be said that, once the outliers have been taken apart, there is nothing very special in the remaining sectors to be considered as key sectors. So, the key sectors can be identified as the multivariate outliers in the clouds of data we are working with. Maybe the most relevant and evident conclusion is that the presence of multivariate outliers has an enormous importance in the analysis. In relation with the study in Díaz, Moniche and Morillas (2006) [11], there has been a threefold increase in the number of clusters (9 versus 3) and, much more relevant, the key sectors identified are also quite different. Five of the nine sectors that have been considered as key in this study had serious problems to be classified in the mentioned article, having membership values under 0.5 in all of the groups. They were distributed into the three mega-clusters obtained there as follow: sector 37 and sector 41, were in cluster 1, the so called “Local base” group; sector 11 and sector 26, were in cluster 2, the group with the irrelevant sectors, which hold curiously all the sectors that can be identified as non-key outliers in this study; finally, sector number 33, was in cluster 3, the so called “Exporting base” group, to which practically belong all the called

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potential or second level key sectors in this paper. Nevertheless, this cluster was there considered the most interesting group. So, everything seems to be mixed there. The key sectors selection shown in table 19.5 is much clearer and seems even more realistic. Actually, we can affirm these nine sectors are the core of the Spanish economy. Obviously, the presence of outliers was distorting the analysis in that work, and undoubtedly something similar can happen in any multivariate study of key sectors or industrial clustering, given the effect anomalous data have in the correlations matrix, base of all the studies. In particular, in previous industrial clustering studies, it is evident the reduced number of groups and the sectors belonging to them were conditioned by the very probable existence of outliers. Detecting multidimensional outliers and avoiding its effects is absolutely necessary when it comes to identifying the industrial clusters or the key sectors in an economy. Furthermore, most of the more important ones will definitely be among them. For these two reasons it has been proposed in this article the identification of outliers as an alternative method to get the industrial clustering or the key sectors in an economy. In order to identify them, a three criteria method has been proposed. The first one, having as input the robust Mahalanobis distance, shows the highest relative jumps in this measure, where important breaks from the main cloud of points take place. The second one, by simulating in an iterative way the extraction of different number of outliers, from 1 to [n − [(n + p + 1)/2]], informs about how the condition number of the non-outliers matrix is relatively changing. So, it looks for a cloud of points highly correlated and puts aside estrange multivariate points. Finally, a nonparametric test of the variance, the Kruskal-Wallis test, is used to select the number of outliers that throws the higher number of variables in the analysis explaining the most of the variance in the non-outlier cloud of points. The proper identification of key sectors is important due to the policy implications of their designation. In the study of the Spanish Economy, the new results give much more information about the sectors strengths and weaknesses, even using the same variables and procedures, when the outliers have been taken into account. The results reached there suggest that the sectors previously identified as key sectors were not correct, because they did not contain some of the very important sectors that have the role of authentic motors of the Spanish Economy.

19.3

Conclusions

Fuzzy inference systems (FIS) allows to characterize a relatively complex problem with a low number of rules. The fact that each rule can be fulfilled to a degree provides the flexibility needed when trying to model this kind of phenomena, which present clear non-linearity. It is a non-parametric approach that provides very satisfactory results. We have pointed out some drawbacks that an economist must take into account: the need of a great deal of elements in the survey or very large time series, which are not always available. The number of parameters grows with the

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number of rules and can make the problem unapproachable. With the application of the IFN method, we have used a method that does not depend for its application on theoretical hypothesis about the distribution of the data. In this case, mutual information is the measure that allows the extraction of the crisp rules. In the algorithm, fuzzy logic is added to summarize the rules and give a more compact and understandable set of rules. When carrying out a cluster analysis, the presence of multivariate outliers has an enormous importance, given the effect anomalous data have in the correlations matrix, base of multivariate analyses. We have proposed a method for detecting multidimensional outliers and considered the identification of outliers as an alternative method to get the industrial clustering or the key sectors in an economy. Fuzzy clustering is then applied, and the results are much more informative than the results from a classic clustering method.

References [1] Barnett, V.: The ordering of multivariate data (with discussion). Journal of the Royal Statistical Society A 139, 318–354 (1976) [2] Barnett, V., Lewis, T.: Outliers in Statistical Data. Wiley, Chichester (1994) [3] Bauer, P.W., Berger, A.N., Humphrey, D.B.: Efficiency and productivity growth in U.S. banking. In: Fried, H.O., Lowell, C.A.K., Schmidt, S.S. (eds.) The measurement of productive efficiency: echniques and Applications, pp. 386–413. Oxford University Press (1993) [4] Belsley, D.A., Kuh, E., Welsch, R.E.: Regression Diagnostic: Identifying Influential Data and Sources of Collinearity. Wiley, New York (1980) [5] Berger, A.N., Humphrey, D.B.: The dominance of inefficiencies over scale and product mix economies in banking. Journal of Monetary Economics 20, 501–520 (1991) [6] Berger, A.N., Humphrey, D.B.: Measurement and efficiency issues in commercial banking. In: Griliches, Z. (ed.) Output measurement in the service sectors. University of Chicago Press (1992) [7] Chiu, S.L.: Fuzzy model identification based on cluster estimation. Journal of Intelligent and Fuzzy Systems 2(3), 267–278 (1994) [8] Dang, X.: Nonparametric multivariate outlier detection methods with applications, PhD. Thesis, University of Dallas at Texas (2005) [9] De Mesnard, L.: On Boolean topological methods of structural analysis. In: Lahr, M.L., Dietzenbacher, E. (eds.) Input-Output Analysis: Frontiers and Extensions. Palgrave Macmillan, Basingstoke (2001) [10] Diaz, B.: Sistemas de inferencia difusos. Una aplicación al estudio del empleo asalriado en España. Phd Thesis (2000) [11] Diaz, B., Moniche, L., Morillas, A.: A fuzzy clustering approach to the key sectors of the Spanish economy. Economic Systems Research 18(3), 299–318 (2006) [12] Dridi, C., Hewings, G.J.D.: Sectors associations and similarities in input-output systems: an application of dual scaling and fuzzy logic to Canada and the United States. Annals of Regional Science 37, 629–656 (2003) [13] Espasa, A., Cancelo, J.R.: El cálculo del crecimiento de variables económicas a partir de modelos cuantitativos. Boletín Trimestral de Coyuntura 54, 65–84 (1994); INE [14] EUROSTAT: Ressources humaines en haute technologie. La mesure de l’emploi dans les secteurs manufacturiers de haute technologie; une perspective européenne. Serie Statistiques en bref. Recherche et développement, Monograph, vol. 8 (January 1998)

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[15] EUROSTAT: Employment in high technology manufacturing sectors at the regional level, Doc. Eurostat/A4/REDIS/103 (1998) [16] Feser, E.J., Bergman, E.M.: National industry cluster templates: a framework for applied regional cluster analysis. Regional Studies 34, 1–19 (2000) [17] Filzmoser, P., Garrett, R.G., Reimann, C.: Multivariate outlier detection in exploration geochemistry. Computers and Geosciences 31, 579–587 (2005) [18] Foster, G.: Financial statement analysis. Prentice Hall, New Jersey (1986) [19] Greene, W.H.: Econometric Analysis. Macmillan, New York (1993) [20] Hardin, J., Rocke, D.M.: Outlier detection in the multiple cluster setting using the minimum covariance determinant estimator. Computational Statistics and Data Analysis 44, 625–638 (2004) [21] Hardin, J., Rocke, D.M.: The distribution of robust distances. Journal of Computational and Graphical Statistics 14, 1–19 (2005) [22] Hathaway, R.J., Davenport, J.W., Bezdek, J.C.: Relational duals of the c-means algorithms. Pattern Recognition 22, 205–212 (1989), http://irsr.sdsu.edu/%7Eserge/node7.html [23] Hathaway, R.J., Bezdek, J.C.: NERF c-means: Non- Euclidean Relational Fuzzy Clustering. Pattern Recognition 27, 429–437 (1994) [24] Central de Balances de Actividad empresarial en Andalucía. Instituto de Estadística de Andalucía, http://www.juntadeandalucia.es/institutodeestadistica/cenbalBD/ [25] INE, Tablas input-output de la Economía Española de 1995 (2000), www.ine.es [26] Contabilidad Nacional. Base (1995), www.ine.es [27] INEbase en la metodología expuesta en Investigación y desarrollo tecnológico. Indicadores de alta tecnología, http://www.ine.es/inebase/cgi/um?M=%2Ft14%2Fp197\&O=inebase\&N=\&L= [28] Jang, J.-S.R.: ANFIS: Adaptive Network-based Fuzzy Inference System. IEEE Transactions on Systems, Man and Cibernetics 23(3), 665–685 (1993) [29] Kaufman, L., Rousseeuw, P.J.: Finding Groups in Data. Wiley, New York (1990) [30] Kempthorne, P.J., Mendel, M.B.: Comments on Rousseeuw and Van Zomeren. Journal of the American Statistical Association 85, 647–648 (1990) [31] Lantner, R.: Théorie de la Dominance Economique. Dunod, Paris (1974) [32] Lantner, R.: Influence graph theory applied to structural analysis. In: Lahr, M.L., Dietzenbacher, E. (eds.) Input-Output Analysis: Frontiers and Extensions. Palgrave Macmillan, Basingstoke (2001) [33] Last, M., Klein, Y., Kandel, A.: Knowledge Discovery in Time Series Databases. IEEE Transactions on Systems, Man and Cybernetics, Part B 31(1), 160–169 (2001) [34] Little, R.J.A., Rubin, D.B.: Statistical analysis with missing data. John Wiley and Sons, New York (1987) [35] Lopuhaä, H.P., Rousseeuw, P.J.: Breakdown points of affine equivariant estimators of multivariate location and covariance matrices. The Annals of Statistics 19, 229–248 (1991) [36] MacQueen, J.B.: Some methods for classification and analysis of multivariate observations. In: Proceedings of 5th Berkeley Symposium on Mathematical Statistics and Probability, pp. 281–297. University of California Press, Berkeley (1967) [37] Maimon, O., Last, M.: Knowledge Discovery and Data Mining – The Info-Fuzzy Network (IFN) Methodolog. Kluwer Academic Publishers, Netherlands (2000) [38] Morillas, A.: La teoría de grafos en el análisis input-output. Secretariado de Publicaciones, Universidad de Málaga, Malaga (1983) [39] Morillas, A.: Indicadores topológicos de las características estructurales de una tabla input-output. Aplicación a la economía andaluza, Investigaciones Económicas 20, 103–118 (1983)

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[40] Morillas, A., Díaz, B.: Minería de datos y lógica difusa. Una aplicación al estudio de la rentabilidad económica de las empresas agroalimentarias en Andalucía. Revista Estadística Española, INE 46(157), 409–430 (2004) [41] Morillas, A., Díaz, B.: Key sectors, industrial clustering and multivariate outliers. Economic Systems Research 20(1), 57–73 (2008) [42] Mougeot, M., Duru, G., Auray, J.-P.: La Structure Productive Francaise. Económica, Paris (1977) [43] Nagy, G.: State of the art in pattern recognition. Proceedings IEEE 56, 836–882 (1968) [44] Oosterhaven, J., Stelder, D.: Net multipliers avoid exaggerating impacts: with a bi-regional illustration for the Dutch transportation sector. Journal of Regional Science 42, 533–543 (2002) [45] Oosterhaven, J.: On the definition of key sectors and the stability of net versus gross multipliers, SOM Research Report 04C01, Research Institute SOM, University of Groningen, The Netherlands (2004), http://som.rug.nl [46] Rey, S., Mattheis, D.: Identifying Regional Industrial Clusters in California. Volumes I-III, Reports prepared for the California Employment Development Department, San Diego State University (2000) [47] Rossier, E.: Economie Structural, Económica, Paris (1980) [48] Rousseeuw, P.J.: Silhouettes: A Graphical Aid to the Interpretation and Validation of Cluster Analysis. Journal of Computational Applied Mathematics 20, 53–65 (1987) [49] Rousseeuw, P.J., Leroy, A.M.: Robust regression and outliers detection. Wiley, New York (1987) [50] Rousseeuw, P.J., van Zomeren, B.C.: Unmasking multivariate outliers and leverage points. Journal of the American Statistical Association 85, 633–639 (1990) [51] Rousseeuw, P.J., Hubert, M.: Recent developments in PROGRESS. In: Dodge, Y. (ed.) L1-Statistical Procedures and Related Topics. IMS Lecture Notes, vol. 31, pp. 201–214 (1997) [52] Rousseeuw, P.J., van Driessen, K.: A fast algorithm for the minimum covariance determinant estimator. Technometrics 41, 212–223 (1999) [53] Rubin, D.B.: Inference and missing data. Biometrika 63, 581–592 (1976) [54] Ruspini, E.: A new approach to clustering. Information and Control 15, 22–32 (1969) [55] Sugeno, M., Yasukawa, T.: Fuzzy-Logic Based Approach to qualitative modelling. IEEE Transactions on Fuzzy Systems 1(1), 7–31 (1993) [56] Tanaka, K., Sano, M., Watanabe, H.: Modeling and control of carbon monoxide concentration using a neuro-fuzzy technique. IEEE Transactions on Fuzzy Systems 3(3), 271–279 (1995) [57] Yager, R.R., Filev, D.P.: Approximate clustering via the mountain method. IEEE Transactions on Systems, Man and Cibernetics 24, 209–219 (1994) [58] Zadeh, L.A.: A New Direction in AI - Toward a Computational Theory of Perceptions. AAAI Magazine, 73–84 (2001)

Part VI

20 Fuzzy Formal Ontology Kazem Sadegh-Zadeh

20.1

Introduction

Ontology originates with ancient Greek philosophers, especially Aristotle. A major issue he dealt with in his Metaphysics, is being as being. According to him, “There is a science which investigates being as being and the attributes which belong to this in virtue of its own nature. Now this is not the same as any of the so-called special sciences; for none of these others deals generally with being as being. They cut off a part of being and investigate the attributes of this part” (Metaphysics, Book IV 1003a 20-25). Such an inquiry Aristotle explicitly calls “the science of being as being” that he expounds in several Books of his Metaphysics. The endeavor he refers to originated with Greek philosophers preceding him, such as Heraclitus, Parmenides, and Plato. Medieval philosophers have termed it metaphysica generalis, while metaphysica specialis in their view dealt with other Aristotelean issues, such as the “first philosophy” that is a “science of the first principles and causes” (ibid., Book I 982b 9), logical foundations of philosophy, and other sciences (ibid., Book IV). The term “ontology” is just another name for metaphysica generalis coined in the early seventeenth century, i.e., “the science of being as being”.1 Until the early twentieth century, ontology was what Aristotle called “the science of being as being” concerned with the question “what does exist?” or “what is there?”. It inquired into the meaning and application of, and the philosophical problems caused by, the terms “being” and “existence” and the metaphysics of what ontologists referred to as being existent. During the twentieth century, however, the meaning of the term was extended to also include two additional fields of research. Thus, the following three areas are distinguished today: 1

The oldest record of the hybrid term “ontology” is the phrase “ontologia” found in Jacobus Lorhardus’ Ogdoas scholastica (1606 [14]) and in Rudolphus Goclenius’ Lexicon Philosophicum (1613 [9]). Jakob Lorhard or Jacobus Lorhardus (1561-1609) was born in Münsingen in South Germany and studied at the German University of Tübingen. As of 1602, he was a teacher and preacher in St. Gallen, Switzerland. His Ogdoas scholastica was concerned with “Grammatices (Latinae, Graeca), Logices, Rhetorices, Astronomices, Ethices, Physices, Metaphysices, seu Ontologia”. Rudolf Göckel or Rudolphus Goclenius (1547-1628) was a professor of physics, logic, and mathematics at the German University of Marburg. He first learned about the term “ontology” from Lorhard himself. Ontology gained currency first of all by the German philosopher Christian Wolff (1679-1754), [21].

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• Pure Ontology • Applied Ontology • Formal Ontology. Pure or philosophical ontology represents ’the good old ontology’ mentioned above. Applied ontology, also called domain ontology, is concerned (i) with the question of what entities exist in a particular domain, for example, in the domain of a scientific branch such as physics or gynecology, or even in a more special domain of a scientific theory such as the theory of active immunity; and (ii) with the ordinary taxonomy of those entities. Ordinary taxonomy is classification based on the classical, proper subsethood relation ⊂ between classes, expressed by the binary predicate “is_a”, such as, for example, pneumonia is_a respiratory disease, i.e., the class of pneumonia patients is a subset of those who have a respiratory disease. Once pure ontology has determined what kind of entities exist in general, and applied ontology has determined what entities exist in particular domains such as physics, anatomy, internal medicine, and others, inquiries can be made into whether there are any formal relationships between any of these entities. For example, is the left ventricle of the heart a part of the heart? Yes. Is the human heart a part of the human body? Yes. Then the left ventricle is a part of the human body. Such formal analyses are the subject matter of formal ontology that constructs axiomatic frameworks by means of formal logic to study formal relationships between all types of ontological categories and all types of entities existing in specific domains. The present paper is concerned with formal ontology, which is being increasingly applied in medical information sciences. We shall briefly describe the traditional, ordinary or classical formal ontology and its medical application in the next section. In Section 20.3, first steps will be taken toward constructing a fuzzy formal ontology that seems to be more appropriate for use in medicine.

20.2

Ordinary Formal Ontology

Like pure and applied ontology, formal ontology has been based on classical, bivalent logic and set theory. Therefore, we refer to it as a classical or ordinary formal ontology. The relationships that it describes between entities, are all represented by bivalent concepts such as “and”, “or”, “not”, “if-then”, and others. An example is shown below. As an alternative to this, we shall construct a fuzzy formal ontology in the next section. For a fuzzy pure ontology, see [17]. In order for a medical expert system such as an anatomical, pathological, or surgical system to process information about topographical, spatial, and functional relationships between body parts, or about relationships between pathological processes, its empirical knowledge base is enriched by some non-empirical, formal knowledge that is axiomatically-artificially constructed. A prominent example of such knowledge is mereology or partology (from the Greek term “meros” and Latin term “pars”, meaning part). Mereology is an axiomatic framework for the analysis of part-whole relationships. Its axioms and postulates are formulated in the firstorder language to enable predicate-logical inferences. A few examples are given

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below in Table 20.1 in order to refer back to them in the next section.2 In the table, the binary predicates P, PP, O and D, and the unary predicate PT are used. The first one is the primitive of the framework, and all other ones are defined thereby. They read:3 Pxy ≡ x is_a_part_of y e.g., the lens is_a_part_of the eyeball, PPxy ≡ x is_a_proper_part_of y the lens is_a_proper_part_of the eyeball, Oxy ≡ x overlaps y the eyeball overlaps the lens, Dxy ≡ x is_discrete_from y the lens is discrete from the iris, PT x ≡ x is_a_point.

Table 20.1. Some mereological axioms and definitions. Axioms: 1.

∀xPxx

2.

∀x∀y(Pxy ∧ Pyx) → x = y

3.

∀x∀y∀z(Pxy ∧ Pyz) → Pxz

Every object is a part of itself (i.e., parthood is reflexive), if x is a part of y and y is a part of x, then x and y are identical (i.e., parthood is antisymmetric), parthood is transitive,

Definitions: 4.

5. 6. 7.

∀x∀y(PPxy ↔ Pxy ∧ ¬(x = y)) an object is a proper part of another one iff it is a part thereof and different therefrom, ∀x∀y(Oxy ↔ ∃z(Pzx ∧ Pzy)) two objects overlap iff they have a part in common, ∀x∀y(Dxy ↔ ¬Oxy) an object is discrete from another one iff they do not overlap, ∀x(PT x ↔ ∀y(Pyx → y = x)) an object is a point iff it is identical. with anything of which it is a part.

According to Axioms 1-3 in Table 20.1, the relation of parthood is a partial ordering of the universe of discourse since it is reflexive, transitive, and antisymmetric. 2 3

The phrase “iff” is the connective “if and only if”. Edmund Husserl (1859-1938), an Austrian-born German philosopher and the founder of phenomenology, introduced in his Logical Investigations (Husserl, 2001), first published in 1900/01, the term “formal ontology” to analyze parts and wholes. Stanislaw Lesniewski, a Polish logician and mathematician (1886-1939), developed a formal theory of part-whole relationships from 1916 onwards and coined the term “mereology” in 1927 ([16]). The theory was improved by his student Alfred Tarski (1927). For details on mereology, see [3], [8], [15], [19].

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For instance, a human heart is a part of itself (Axiom 1). It is also a part of the circulatory system. Since the latter one is a part of the human body, a human heart is a part of the human body (Axiom 3). It is even a proper part of the human body because it is not identical therewith (sentence 4). The mereological axioms in Table 20.1 imply a large number of theorems. A few examples are listed in Table 20.2. For instance, a chromosome is a proper part of the cell nucleus because it is a part thereof and not identical therewith (sentence 4 in Table 20.1). Likewise, the cell nucleus is a proper part of the cell. Thus, a chromosome is a proper part of the cell (Theorem 10 in Table 20.2). But it is not a proper part of itself (Theorem 8), whereas it overlaps itself (Theorem 11). According to Theorems 11-12, the relation Overlap is a quasi-ordering of the universe of discourse. For more details on mereology and its application in medicine, see ([2], [10], [13], [18]). Table 20.2. Some mereological theorems. Theorems: 8. ∀x¬PPxx 9. ∀x∀y(PPxy ∧ Pyx → ¬PPyx) 10. ∀x∀y∀z(PPxy ∧ PPyz → PPxz) 11. ∀xOxx 12. ∀x∀y(Oxy → Oyx) 13. ∀x∀y(PPxy → Oxy) 14. ∀x∀y(Oxy ∧ Pyz → Oxz)

Proper parthood is irreflexive, proper parthood is asymmetric, proper parthood is transitive, overlap is reflexive, overlap is symmetric, a proper part of an object overlaps it, if x overlaps a part of z, then it also overlaps z.

Mereotopology is an extension of mereology by some additional, topological concepts dealing with spatial objects and relations such as boundaries, surfaces, interiors, neighborhood, connectedness, and others. As a simple example, consider the notion of an interior part, for instance, an interior part of an organ. When x is_a_part_of an object y, written Pxy, such that it does not touch y’s boundaries, then it is_an_interior_part_of y, written IPxy. A few axioms, definitions, and theorems listed in Table 20.3 may illustrate how such mereotopological concepts are formally characterized and how they relate to others. In the table, the following predicates are used, the first of which is a primitive and defines our second topological predicate E, i.e., “encloses”. They read: Cxy ≡ x is_connected_to y e.g., the retina is connected to the optic nerve, Exy ≡ x encloses y the pericardium encloses the myocardium.

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The remaining predicates are mereological ones. We take the predicates P, PP, and O from Table 20.1: Pxy PPxy Oxy IPxy

≡ ≡ ≡ ≡

x is_a_part_of y x is_a_proper_part_of y x overlaps y x is_an_interior_part_of y

For the sake of readability, we omit the quantifiers in Table 20.3. All sentences are tacitly universally quantified, i.e., universal closures. The phrase “iff” is the connective “if and only if”.4 Table 20.3. Some mereotopological axioms, definitions and theorems. Axioms: 15. Cxx 16. Cxy → Cyx

Connection is reflexive, Connection is symmetric.

Definitions: 17. 18.

Exy ↔ (Czx → Czy)

x encloses y iff whenever something connects to it, then it also connects to y, IPxy ↔ (Pxy ∧ (Czx → Ozy)) x is an interior part of y iff it is a part of y and everything that connects to it overlaps y,

Axioms: 19.

(Exy ↔ Ezy) ↔ x = z

20. 21.

Exx Exy ∧ Eyz → Exz

two objects are identical iff whenever one of them encloses a third object, the other one encloses it too, enclosure is reflexive, enclosure is transitive,

Theorems: 22. 23. 24. 25. 26. 27. 4

Exy ∧ Eyx → x = y Pxy → Eyx Oxy → Cxy IPxy → Pxy IPxy ∧ Pyz → IPxz Pxy ∧ IPyz → IPxz

enclosure is antisymmetric, an object encloses its parts, two overlapping objects are connected, IP is a particular kind of parthood, left monotonicity, right monotonicity,

Mereotopology comes from the philosophy of the British mathematician Alfred North Whitehead (1861-1947), especially from his work Process and Reality (1929) in which he enriched mereology with some topological notions such as connection and contiguity ([5], [6], [19], [7]).

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Sentence 17 defines enclosure by connection. Sentence 18 defines interior parthood by parthood, connection, and overlap. The latter one has been defined in Definition 5 in Table 20.1. Theorems 23-24 are bridges between mereology and topology (“mereotopology”). As was pointed out above, the goal of such axiomatizations and studies is, first, to implicitly or explicitly define the respective concepts and, second, to provide the necessary formal-ontological knowledge for use in medical decision-support systems. For details on mereotopology, see ([19], [4], [7]).

20.3

Fuzzy Formal Ontology

The mereology and mereotopology discussed above are the ordinary, classical, or crisp ones because their underlying logic is the classical predicate logic and set theory. In spite of the obvious fuzziness of the entire world surrounding us, formal ontologists still adhere to this classical approach. Medical entities, however, are vague and therefore a fuzzy approach is necessary to cope with them. For instance, we do not know where exactly the boundaries between two or more cells in a tissue, or between a cell and the extracellular space, lie; how to draw a sharp demarcation line between two different diseases in an individual; when exactly a pathogenetic process in the organism begins; how long exactly the incubation period of an infectious disease is; and so on. Consequently, the crisp mereology and mereotopology considered above may not be very relevant in medicine, because the parts of vague entities are also vague, as are the relations between them. For these reasons, it is advisable to explicitly replace in medicine and other fields the crisp mereological primitive “is_a_part_of” with the more general, fuzzy term “is_a_vague_part_of”. This vagueness may be mapped to [0, 1] to yield degrees of parthood. Thereby we would obtain a concept of fuzzy parthood that could be supplemented by additional fuzzy-mereological concepts to construct a fuzzy mereology. An analogous fuzzifying of the topological notions already mentioned would also pave the way toward fuzzy mereotopology. We shall here take only the first steps in order to stimulate further research. Thus we will cover in the next section the concept of fuzzy subsethood upon which our fuzzy mereology in Section 20.3.2 will be based. 20.3.1

Fuzzy Subsethood

Introduced below are some basic concepts of our fuzzy mereology such as “fuzzy parthood”, “fuzzy overlap”, and others. The elementary, non-mereological notion that we shall need as our basic concept is the fuzzy set-theoretical relation of subsethood. If Ω is a universe of discourse, e.g., Ω = {x1 , x2 , ..., xn } with n ≥ 1, a fuzzy set A in Ω will be represented as a set of pairs in the following way: {(x1 , μA (x1 )), (x2 , μA (x2 )), . . . , (xn , μA (xn ))} , where a μA (xi ) is the memebrship degree of the object xi ∈ Ω in fuzzy set A. Initially a concept of fuzzy subsethood was presented by the father of fuzzy logic himself

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([22], p. 340). It says that a fuzzy set A is a subset of a fuzzy set B iff the membership degree of every object xi in set A is less than or equals its membership degree in set B, that is: A ⊆ B iff μA (xi ) ≤ μB (xi ) for all xi in the universe of discourse, denoted Ω . For example, if in the universe of discourse Ω = {x1 , x2 , x3 , x4 , x5 } we have the following two fuzzy sets: A = {(x1 , 0.3), (x2 , 0.6), (x3 , 0.5), (x4 , 0), (x5 , 0)} , B = {(x1 , 0.4), (x2 , 0.6), (x3 , 0.8), (x4 , 0), (x5 , 1)} , then A ⊆ B. This concept of containment is used in fuzzy research and practice still today. However, Bart Kosko showed in 1992 that this concept is a crisp one and thus deviates from the spirit of fuzzy set theory and logic ([12], p. 278). It is a crisp one because it allows only for definite subsethood or definite non-subsethood between two fuzzy sets. There is no intermediate degree, no fuzziness of subsethood. To address this, Kosko introduced the concept of fuzzy subsethood that will be presented below. To begin, we define the count of a fuzzy set A, denoted c(A). The count of a fuzzy set: A = {(x1 , μA (x1 )), (x2 , μA (x2 )), . . . , (xn , μA (xn ))} is simply the sum of its membership degrees. That is: c(A) = ∑ μA (xi ) = μA (x1 ) + · · · + μA (xn ). For instance, for the above fuzzy set B = (x1 , 0.4), (x2 , 0.6), (x3 , 0.8), (x4 , 0), (x5 , 1) we have c(B) = 0.4 + 0.6 + 0.8 + 0 + 1 = 2.8. In order to obtain a concept of fuzzy subsethood, we first introduce a notion of fuzzy supersethood from which subsethood is easily gained by 1 − supersethood. To this end, the dominance of the membership degree: max(0, μA (xi ) − μB(xi )) of all members xi of set A over set B is measured by:

∑ max(0, μA(xi ) − μB(xi )) and then normalized by dividing it through the count of set A:

∑ max(0, μA(xi ) − μB(xi ))/c(A).

(20.1)

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The component 0 in this formula guarantees that only overhangs in set A are gathered and used to yield a measure of dominance of set A over set B. The normalized measure of dominance in (20.1) is just the degree of supersethood of set A over set B and lies between 0 and 1 inclusive. Let Ω = {x1 , x2 , . . .} be the universe of discourse and let A, B, . . . be any fuzzy sets in Ω . If we abbreviate the phrase “degree of supersethood of A over B” to “supersethood(A, B)”, we may define this function, supersethood, in the following way: Definition 20.3.1 (Fuzzy supersethood).   ∑ max 0, μA (xi ) − μB(xi ) supersethood(A, B) = · c(A) For example, if in the universe of discourse Ω = {x1 , x2 , x3 , x4 , x5 } our fuzzy sets are: A B C D ∅

= {(x1 , 0.3), (x2 , 0.6), (x3 , 0.5), (x4 , 0), (x5 , 0)} , = {(x1 , 0.4), (x2 , 0.6), (x3 , 0.8), (x4 , 0), (x5 , 1)} , = {(x1 , 1), (x2 , 1), (x3 , 1), (x4 , 1), (x5 , 1)} , = {(x1 , 1), (x2 , 1), (x3 , 0), (x4 , 1), (x5 , 0)} , = {(x1 , 0), (x2 , 0), (x3 , 0), (x4 , 0), (x5 , 0)} ,

then we have: supersethood(A, A) = (0 + 0 + 0 + 0 + 0)/1.4 = 0 supersethood(A, B) = (0 + 0 + 0 + 0 + 0)/1.4 = 0 supersethood(B, A) = (0.1 + 0 + 0.3 + 0 + 1)/2.8 = 0.5 supersethood(C, D) = (0 + 0 + 1 + 0 + 1)/5 = 0.4 supersethood(D,C) = (0 + 0 + 0 + 0 + 0)/5 = 0 supersethood(A, ∅) = (0.3 + 0.6 + 0.5 + 0 + 0)/1.4 = 1 supersethood(∅, A) = (0 + ... + 0)/0 = 0. Now, fuzzy subsethood is defined as a dual of the fuzzy supersethood from Definition 20.3.1 above, and is written subsethood(A, B) = r to say that to the extent r, fuzzy set A is a subset of fuzzy set B: Definition 20.3.2 (Fuzzy subsethood). subsethood(A, B) = 1 −supersethood(A, B). Regarding the five fuzzy sets above, we have: subsethood(A, A) = 1 − 0 = 1 subsethood(A, B) = 1 − 0 = 1 subsethood(B, A) = 1 − 0.5 = 0.5 subsethood(C, D) = 1 − 0.4 = 0.6 subsethood(D,C) = 1 − 0 = 1 subsethood(A, ∅) = 1 − 1 = 0 subsethood(∅, A) = 1 − 0 = 1.

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This concept of fuzzy subsethood is powerful enough to include both Zadeh’s initial concept as well as the classical, crisp subsethood relation ⊆ between classical sets. See, for example, the latter three subsethood degrees. They show that: If X ⊆ Y in classical sense, then subsethood(X,Y) = 1 if X ⊆ Y in Zadeh’s sense, then subsethood(X,Y) = 1. The above procedure of determining the degree of subsethood indirectly by first determining the degree of supersethood is a roundabout way. Fortunately, there is a simple theorem proven by Bart Kosko and referred to as the Subsethood Theorem that enables a straightforward calculation thus ([12], p. 287): Theorem 20.3.1 (Subsethood Theorem). subsethood(A, B) = c(A ∩ B)/c(A). By applying this theorem to the first two examples above we will show that this relationship works and may be used instead of Definition 20.3.2: A = {(x1 , 0.3), (x2 , 0.6), (x3 , 0.5), (x4 , 0), (x5 , 0)} , B = {(x1 , 0.4), (x2 , 0.6), (x3 , 0.8), (x4 , 0), (x5 , 1)} , A ∩ B = B ∩ A = {(x1 , 0.3), (x2 , 0.6), (x3 , 0.5), (x4 , 0), (x5 , 0)} c(A ∩ B) = c(B ∩ A) = 1.4 c(A) = 1 c(B) = 2.8 subsethood(A, B) = c(A ∩ B)/c(A) = 1.4/1.4 = 1 subsethood(B, A) = c(B ∩ A)/c(B) = 1.4/2.8. = 0.5. Thus fuzzy subsethood in terms of subsethood(A, B) is a binary function such that given a universe of discourse Ω , it maps the binary Cartesian product of Ω ’s fuzzy powerset, i.e., F(2Ω ) × F(2Ω ), to unit interval. That is, subsethood: F(2Ω ) × F(2Ω ) → [0, 1]. 20.3.2

Fuzzy Mereology

With the powerful concept of fuzzy subsethood in hand, we shall in what follows briefly outline a tentative fuzzy-logical approach to mereology. To this end, we shall introduce the following basic concepts: fuzzy parthood, fuzzy proper parthood, fuzzy overlap, and fuzzy discreteness. We shall also demonstrate some fuzzy-mereological theorems to show that they preserve classical-mereological relationships. The classical, crisp mereology and mereotopology discussed above are based on what we may call the definiteness postulate. According to this postulate, parts as well as wholes are clear-cut objects with a definite constitution and sharp boundaries. This definiteness allows categorical judgments about whether an object x is a part of another object y or not, is connected to that object or not, overlaps it or not, and so on. An example is “the right thumb is a part of the right hand”. Such a part-whole judgment requires that both the right thumb as well as the right hand

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be definite entities. However, it is often the case that neither the object y, the whole, nor the object x, the part, can be delimited by clear-cut, declarative statements. In an injured tissue it is often impossible to say whether at the margin of a wound, y, a particular bunch of cells, x, is or is not a part of the wound. It is only to a particular extent a part thereof because the wound does not have a clear-cut boundary. The definiteness postulate above precludes even the possibility of such vague objects. Otherwise put, the relations of parthood, connection, overlap, and others dealt with in classical mereology and mereotopology are Aristotelean, crisp relations that hold between clear-cut entities. Questions of the type “is x a part of y?”, “does x connect to y?”, and similar ones can only be answered with a yes or no by categorical statements of the type “x is a part of y”, “a is not a part of b”, and so on. It is impossible to state, for instance, that “x is a weak part of y” or “x strongly connects to y”. In order to make such statements when bivalent yes-no answers are not applicable, we must abandon the definiteness postulate of classical mereology and mereotopology. We propose an alternative approach, using a few mereological relations as our examples. We begin with the introduction of a concept of fuzzy parthood that does not presuppose the definiteness postulate. Among many other virtues, fuzzy parthood gives us the ability to access the entire corpus of fuzzy theory, thereby rendering fuzzy mereology useful in all practical domains whose objects and relations are vague, e.g., medical practice and research. We will conceive fuzzy parthood as a binary function of the structure “object x is a part of object y to the extent z” or: The degree of parthood of x in y is z symbolized by: parthood(x, y) = z where z is a real number in the unit interval [0, 1]. The relata x and y, i.e., the part and the whole, will be represented as fuzzy sets. For example, the clear-cut right_hand above may in a particular context be the fuzzy set displayed below. For the sake of readability, we will use the following abbreviations: r ≡ right r_t ≡ right thumb r_ f ≡ right forefinger r_m ≡ right middle finger r_r ≡ right ring finger r_s ≡ right small finger. For our purposes, the human right hand will be represented as an anatomically incomplete hand consisting of fingers only. However, larger sets can be made to represent the right hand as complete as “it is in reality” consisting of muscles, bones, fingers, etc., simply by continuing the fuzzy set accordingly: r_hand = {(r_t, 1), (r_ f , 1), (r_m, 1), (r_r, 1), (r_s, 1)}.

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Nevertheless, our example right hand is obviously a hand in anatomy atlases and contains each of the five fingers to the extent 1. However, consider the right hand of my neighbor Oliver who worked in a sawmill until recently: Oliver’s r_hand = {(r_t, 0), (r_ f , 0.3), (r_m, 0.5), (r_r, 0.8), (r_s, 1)}. Now if someone asks “is the right thumb a part of the right hand?”, the answer will be “yes, insofar as you mean the right hand in anatomy atlases. But if you mean Oliver’s right hand, no”. More precisely, in anatomy atlases the right thumb is a part of the right hand to the extent 1. But regarding Oliver’s right hand, which has no thumb, it is a part thereof to the extent 0. Likewise, in anatomy atlases the right forefinger is a part of the right hand to the extent 1. But regarding Oliver’s right hand, it is a part thereof only to the extent 0.3; and so on. The intuitive consideration above is based on our understanding that (i) parts and wholes are vague objects and are therefore best represented as fuzzy sets; (ii) parthood is a matter of degree; and (iii) this degree may be determined by defining fuzzy parthood by the degree of the fuzzy subsethood of the part in the whole. Before doing so, we must realize that the relation of fuzzy subsethood holds only between fuzzy sets of equal length because they must be fuzzy sets in the same universe of discourse Ω = {x1 , . . . , xn }. Two fuzzy sets which are either of unequal length or do not stem from the same universe of discourse, cannot be compared with one another. So, we face a problem: Since a part is usually “smaller” than the whole, the fuzzy set representing a part will be shorter than the fuzzy set that represents the whole. Consider, for example, a question of the form “is the right forefinger a part of the right hand?” that is based on data of the following type: {(r_ f , 1)} = a right forefinger {(r_t, 1), (r_ f , 1), (r_m, 1), (r_r, 1), (r_s, 1)} = right hand

(20.2)

such that the question reads: Is {(r_ f , 1)} a part of {(r_t, 1), (r_ f , 1), (r_m, 1), (r_r, 1), (r_s, 1)} ? In order to answer such a question with base entities of different size, we introduce the following method of representing mereological entities as fuzzy sets of equal length. We will divide a fuzzy set such as the right hand (20.2) above into two segments: {head | body} such that it starts with a head separated by a stroke “|" from the body that succeeds it. For example, the right hand in (2) above may be restructured in the following or in any other way: {(r_t, 1)} | {(r_ f , 1), (r_m, 1), (r_r, 1), (r_s, 1)} .

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What the head of such a restructured fuzzy set contains, depends on the subject x of the mereological question “is x a part of y?” that is posed. For instance, if it is asked “is a right forefinger a part of the right hand”, the subject of the question, referred to as the query subject, is a right forefinger: {(r_ f , 1)} Thus, its object y, referred to as the query object, is the restructured fuzzy set: {(r_ f , 1)} | {(r_t, 1), (r_m, 1), (r_r, 1), (r_s, 1)} . whose segment (r_ f , 1) is taken to be its head and the remainder of the set to be its body. By so doing we need not examine the entirety of a whole to determine whether the query subject {(r_ f , 1)} is a part thereof or not. We look only at the head of the restructured fuzzy set to examine whether, and to what extent, the query subject matches it. In our present example, the query subject, i.e., (r_ f ,1), and the head of the restructured fuzzy set match completely. So, the answer to the query is “a right forefinger is a part of the right hand to the extent 1”. This extent, 1, we obtain by determining the degree of fuzzy subsethood of the query subject, i.e., {(r_ f , 1)}, in the head of the restructured fuzzy set that represents the right hand. See Definition 20.3.3 below. Asking the same question of the following restructured fuzzy set: {(r_ f , 0.3)} | {(r_t, 0), (r_m, 0.5), (r_r, 0.8), (r_s, 1)} = Oliver’s right hand the answer is “a right forefinger is a part of Oliver’s right hand only to the extent 0.3”. We obtain this extent, 0.3, in the same fashion as above by determining the degree of fuzzy subsethood of the query subject {(r_ f , 1} in the head of the restructured fuzzy set that represents Oliver’s right hand. It is of course possible that when inquiring into whether an entity x is a part of an entity y, the entity x is not an elementary one such as “a right forefinger”. It may also be a compound such as “a right forefinger and a right middle finger” to raise the question whether “a right forefinger and a right middle finger are a part of the right hand”. In this case our query subject and restructured fuzzy sets are: {(r_ f , 1), (r_m, 1)}

= a right forefinger and a right middle finger {(r_ f , 1), (r_m, 1)} | {(r_t, 1), (r_r, 1), (r_s, 1)} = a right hand in anatomy atlases, {(r_ f , 0, 3), (r_m, 0, 5)} | {(r_t, 0), (r_r, 0, 8), (r_s, 1)} = Oliver’s right hand. Regarding the fuzzy parthood of the compound query subject {(r_ f , 1), (r_m, 1)} in each of the two right hands above we obtain the following results: • {(r_ f , 1), (r_m, 1)} is a part of the right hand in anatomy atlases to the extent 1, • {(r_ f , 1), (r_m, 1)} is a part of Oliver’s right hand to the extent 0.4.

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It is worth noting that a restructured fuzzy set, represented as {head | body}, may either have an empty head or an empty body in the following form: {∅ | body} {head | ∅} depending on the extent of overlap between the query subject and the query object. In any event, it can easily be determined whether and to what extent the query subject is a fuzzy subset of the head of the query object. This is exactly our concept of fuzzy parthood introduced in Definition 20.3.3 below. To arrive at that definition, we denote the head of a restructured fuzzy set A by head(A). For example, if A is Oliver’s restructured right hand {(r_ f , 0.3), (r_m, 0.5)} | {(r_t, 0), (r_r, 0.8), (r_s, 1)}, then we have head(A) = {(r_ f , 0.3), (r_m, 0.5)}. We must now establish how to decide the length of a fuzzy set in a mereological discourse. For example, it does not make sense to ask whether a hand is a part of the forefinger because a hand represented as a fuzzy set, is much longer than a forefinger represented as a fuzzy set, and renders a fuzzy set-theoretical comparison unfeasible. However, this can be prevented with the following three auxiliary notions: First, two universes of discourse, Ω1 and Ω2 , are said to be related if one of them is a subset of the other one, i.e., if either Ω1 ⊆ Ω2 or Ω2 ⊆ Ω1 . Otherwise, they are unrelated. For example, {thumb, forefinger, middle_finger} and {thumb, middle_finger} are related, while {thumb, forefinger, middle_finger} and {eye, ear} or {ring_finger, small_finger} are unrelated. Second, two fuzzy sets A and B are said to be co-local if their universes of discourse, ΩA and ΩB , are related, i.e., (i) A is a fuzzy set in Ω A ; (ii) B is a fuzzy set in ΩB ; and (iii) ΩA and Ω B are related. For example, {(r_t, 1), (r_ f , 1), (r_m, 1)} and {(r_t, 0), (r_ f , 0.3)} are co-local, while {(r_t, 1), (r_ f , 1), (r_m, 1)} and {(eye, 1), (ear, 0.7)} are not co-local. Third, two fuzzy sets A and B are of equal length if they have the same number of members. For instance, {(r_t, 1), (r_ f , 1), (r_m, 1)} and {(r_t, 0), (r_ f , 0.3), (r_m, 0.5)} are of equal length, while {(r_t, 1), (r_ f , 1), (r_m, 1)} is longer than {(r_t, 0), (r_ f , 0.3)}. The phrase “the length of the fuzzy set A” is written “length(A)”. Definition 20.3.3 (Fuzzy parthood). If X and Y are co-local fuzzy sets then: ⎧ ⎪ if head(Y ) = ∅ ⎨0 parthood(X,Y) = or length(X) > length(Y ) ⎪ ⎩ subsethood(X, head(Y)) otherwise As an example, we will now calculate the degree of parthood of the fuzzy set: A = {(a, 1), (b, 0.4), (c, 0.6)}

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in the following fuzzy structures: A = {(a, 1), (b, 0.4), (c, 0.6)} = {(a, 1), (b, 0.4), (c, 0.6)} |∅ B = {(x, 0.9), (a, 0.6), (y, 1), (b, 0.2), (z, 0.5), (c, 1)} = {(a, 0.6), (b, 0.2), (c, 1)} | {(x, 0.9), (y, 1), (z, 0.5)} C = {(x, 1), (a, 1), (y, 0), (b, 0.4), (z, 1), (c, 0.6)} = {(a, 1), (b, 0.4), (c, 0.6)} | {(x, 1), (y, 0), (z, 1)} D = {(x, 1), (a, 0), (y, 0), (b, 0), (z, 1), (c, 0)} = {(a, 0), (b, 0), (c, 0)} | {(x, 1), (y, 0), (z, 1)} = ∅| {(x, 1), (y, 0), (z, 1)} E = {(a, 0), (b, 0), (c, 0)} = {(a, 0), (b, 0), (c, 0)} |∅ = ∅|∅ F = {(a, 1), (b, 0.4), (c, 0.6)} = {(a, 1), (b, 0.4), (c, 0.6)} |∅ By employing the Subsethood Theorem (20.3.1) we obtain the following degrees of parthood: parthood(A, A) = 1 parthood(A, B) = subsethood(A, head(B)) = 0.6. + 0.2 + 0.6/1 + 0.4 + 0.6 = 1.4/2 = 0.7 parthood(A,C) = 1 parthood(A, D) = 0 parthood(A, E) = parthood(A, ∅) = 0 parthood(A, F) = parthood(A, A) = 1. Since fuzzy parthood is defined by fuzzy subsethood, its degree lies in the unit interval [0, 1]. The last examples show that an object is a part of itself to the extent 1 and a part of the empty object ∅ to the extent 0. Something may be partially or completely a part of something else. The question arises whether it is a proper part thereof. We define the degree of fuzzy proper parthood of an object A in another object B, written p-parthood(A, B), by the notion of fuzzy parthood thus: Definition 20.3.4 (Fuzzy proper parthood). If X and Y are co-local fuzzy sets with length(X) ≤ length(Y ), then: ⎧ ⎪ if head(Y ) = ∅ or X = Y ⎨0 p-parthood(X,Y ) = ⎪ ⎩ parthood(X,Y ) otherwise. For instance, the degree of proper parthood of the above example fuzzy set A = {(a, 1), (b, 0.4), (c, 0.6)} in fuzzy sets A through F are:

20.3 Fuzzy Formal Ontology

397

p-parthood(A, A) = 0 p-parthood(A, B) = 0, 7 p-parthood(A,C) = 0 p-parthood(A, D) = 0 p-parthood(A, E) = p-parthood(A, ∅) = 0 p-parthood(A, F) = p-parthood(A, A) = ∅. Without fuzzifying the entities, we may give some intuitive examples to show that the above approach also includes the classical concepts of parthood and proper parthood: parthood(r_thumb, r_thumb) = 1 parthood(r_thumb, r_hand) = 1 parthood(r_thumb, r_kidney) = 0 parthood(r_thumb, ∅) =0 parthood(∅, r_hand) =1 parthood(∅, r_thumb) =1

p-parthood(r_thumb, r_thumb) = 0 p-parthood(r_thumb, r_hand) = 1 p-parthood(r_thumb, r_kidney) = 0 p-parthood(r_thumb, ∅) = 0 p-parthood(∅, r_hand) = 1 p-parthood(∅, r_thumb) = 1.

Parthood and proper parthood are binary functions which map the fuzzy powerset of the Cartesian product of two categories of entities, Ω1 and Ω2 , to the unit interval [0, 1]. That is: parthood : F(2Ω1 ) × F(2Ω2 ) → [0, 1] p-parthood : F(2Ω1 ) × F(2Ω2 ) → [0, 1]. We will now continue our project of constructing a fuzzy mereology by introducing the concepts of fuzzy overlap and fuzzy discreteness: overlap(X,Y ) = r discrete(X,Y ) = r

reads : X overlaps Y to the extent r, X is discrete from Y to the extent r.

To apply these concepts, we need the following, auxiliary notion of the degree of intersection of two fuzzy sets: intersection(X,Y ) = r

reads : the degree of intersection of X and Y is r,

Definition 20.3.5 (Degree of fuzzy intersection). If X and Y are two fuzzy sets in the universe of discourse Ω , then intersection(X,Y ) = c(X ∩Y )/c(X ∪Y ).

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For example, if our entities are the following fuzzy sets in the universe Ω = {x, y, z}: A = {(x, 0.3), (y, 0.8), (z, 0.4)} , B = {(x, 1), (y, 0.5), (z, 0.3)} , then we have: intersection(A, B) = (0.3+0.5+0.3)/(1+0.8+0.4) = 1.1/2.2 = 0.5. Definition 20.3.6 (Degree of fuzzy overlap). If X and Y are co-local fuzzy sets, then: ⎧   ⎪ ⎨intersection X, head(Y ) if lenght(X) ≤ lenght(Y ) overlap(X,Y ) = ⎪   ⎩ intersection head(X),Y otherwise. For instance, the above-mentioned two co-local fuzzy sets A and B overlap to the extent 0.5. A more instructive example may be: A = {(r_ f , 1), (r_m, 1)}

≡ a right forefinger and a right middle finger B = {(r_t, 0), (r_ f , 0.3), (r_r, 0.8), (r_m, 0.5), (r_s, 1)} ≡ Oliver’s right hand. These two fuzzy sets of unequal length overlap to the following extent: overlap(A, B) = (0.3 + 0.5)/1 + 1 = 0.4 where head(B) = {(r_ f , 0.3), (r_m, 0.5)} because B = {(r_ f , 0.3), (r_m, 0.5)} | {(r_t, 0), (r_r, 0.8), (r_s, 1)} . Definition 20.3.7 (Degree of fuzzy discreteness). If X and Y are co-local fuzzy sets, then discrete(X,Y ) = 1 − overlap(X,Y). For example, the last fuzzy sets A and B mentioned above are discrete from one another to the extent 1 − 0.4 = 0.6. But the same set A is totally discrete from the following fuzzy set: C = {(r_ f , 0), (r_m, 0)} that is, discrete(A,C) = 1. So far we have looked at the following concepts: degrees of parthood, proper parthood, intersection, overlap, and discreteness. All of them have been defined by means of purely fuzzy set-theoretical notions. Thus, none of them is a primitive. Table 20.4 below displays some theorems as corollaries of the definitions above. They express important fuzzy-mereological relationships and show that classicalmereological relationships are only limiting cases thereof, i.e., bivalent instances of them with limiting values 0 and 1. For example, the first theorem corresponds to the axiom of reflexivity of classical parthood, the second one corresponds to its axiom of symmetry, etc.

20.4 Conclusion

399 Table 20.4. Some fuzzy-mereological theorems.

Axioms: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

20.4

parthood(A, A) = 1 parthood(A, B) = paarthood(B, A) = 1 → A = B parthood(A, B) > 0 ∧ parthood(B,C) > 0 → parthood(A,C) > 0 parthood(A, B) > 0 → overlap(A, B) > 0 parthood(A, B) > 0 → parthood(A, B) = overlap(A, B) A = B → p-parthood(A, B) = parthood(A, B) p-parthood(A, A) = 0 p-parthood(A, B) = r ∧ r > 0 → parthood(A, B) = r p-parthood(A, B) > 0 → p-parthood(A, B) = parthood(A, B) p-parthood(A, B) > 0 ∧ p-parthood(B,C) > 0 → p-parthood(A,C) > 0 overlap(A, A) = 1 overlap(A, B) = overlap(B, A) overlap(A, B) > 0 ∧ parthood(B,C) > 0 → overlap(A,C) > 0 overlap(A, B) > 0 → ∃C [parthood(C, A) > 0 ∧ parthood(C, B) > 0] discrete(A, B) = 1 → overlap(A, B) = 0

Conclusion

As an example of formal-ontological research, a miniature system of classical mereology and mereotopology has been briefly outlined. Such systems are currently applied in medical knowledge-based systems. Since they are not applicable whenever there are only partial mereological and mereotopological relationships between two objects, the core of a fuzzy mereology has been put forward as an alternative to stimulate further research.

References [1] Aristotle: The Metaphysics. Translated by Lawson-Tancred, H. Penguin Group, New York (1999) [2] Bittner, T., Donnelly, M.: Logical properties of foundational relations in bio-ontologies. Artificial Intelligence in Medicine 39, 197–216 (2007) [3] Carnap, R.: Introduction to Symbolic Logic and Its Applications. Dover Publications, New York (1954) [4] Casati, R., Varzi, A.C.: Parts and Places: The Structures of Spatial Representation. MIT Press, Cambridge (1999) [5] Clarke, B.L.: A calculus of individuals based on "connection”. Notre Dame Journal of Formal Logic 22, 204–218 (1981) [6] Clarke, B.L.: Individuals and points. Notre Dame Journal of Formal Logic 26, 61–75 (1985) [7] Cohn, A.G., Varzi, A.C.: Mereotopological connection. Journal of Philosophical Logic 32, 357–390 (2003)

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[8] Donnelly, M., Bittner, T., Rosse, C.: A formal theory of spatial representation and reasoning in biomedical ontologies. Artificial Intelligence in Medicine 36, 1–27 (2006) [9] Goclenius, R.: Lexicon Philosophicum, quo tanquam clave Philosophiae fores aperiuntur. Musculus, Frankfurt am Main (1613) [10] Hovda, P.: What is classical mereology? Journal of Philosophical Logic 38, 55–82 (2009) [11] Husserl, E.: Logical Investigations. Two volumes. Routledge, London (2001); (First published in German. Max Niemeyer, Halle (1900-1901) [12] Kosko, B.: Neural Networks and Fuzzy Systems. A Dynamical Systems Approach to Machine Intelligence. Prentice Hall, Englewood Cliffs (1992) [13] Koslicki, K.: The Structure of Objects. Oxford University Press, Oxford (2008) [14] Lorhardus, J.: Ogdoas scholastica. Georgium Straub, Sangalli (St. Gallen) (1606) [15] Pontow, C., Rainer Schubert, R.: A mathematical analysis of theories of parthood. Data and Knowledge Engineering 59, 107–138 (2006) [16] Rickey, V.F., Srzednicki, J.T. (eds.): Lesniewski’s Systems: Ontology and Mereology. Springer, Heidelberg (1986) [17] Sadegh-Zadeh, K.: Handbook of Analytic Philosophy of Medicine. Springer, Dordrecht (2011) [18] Schulz, S., Hahn, U.: Part-whole representation and reasoning in formal biomedical ontologies. Artificial Intelligence in Medicine 34, 179–200 (2005) [19] Simons, P.M.: Parts: A Study in Ontology. Oxford University Press, New York (2000) [20] Whitehead, A.N.: Process and Reality. Macmillan, New York (1929) [21] Wolff, C.: Philosophia Prima sive Ontologia. Libraria Rengeriana, Frankfurt (1730) [22] Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965)

21 Computational Representation of Medical Concepts: A Semiotic and Fuzzy Logic Approach Mila Kwiatkowska, Krzysztof Michalik, and Krzysztof Kielan

21.1

Introduction

In terms of human history, computer-based information and communication systems are new phenomena. The development of such systems, whether they are designed to perform word processing or to support clinical decision making, is still a very young discipline. Yet the use of computers to support various aspects of human activities is growing exponentially and, today, computers affect almost every aspect of human life. Among the many disciplines rapidly utilizing computational modeling and computer-based technologies, the fields of medicine and biology are two of the fastest growing. In medicine, computer-based processing allows healthcare providers to perform previously impossible diagnostic tasks, for example, the analysis of cross-sectional slices of the brain using computed tomography or magnetic resonance imaging. The advances in medical imaging allow humans to study live organisms using multiple scales: from the entire organisms, through specific organs, to individual cells. Recently, medicine and biology have been coupled with computing science into interdisciplinary fields of medical informatics, biomedical informatics and bioinformatics. These newly developed fields concern not only the management and use of biomedical data, but also, and even more importantly, the computational representation of biomedical knowledge. Building computational model to represent biomedical systems poses several challenges. We focus on three categories of problems. The first category is related to the characteristics of the modeling process - mapping between the realworld biomedical systems and the computational models and their implementations. The second category is related to the characteristics of the modeled systems - the biomedical systems. The third category is related to the characteristics of the models themselves - computational models and their implementations. 1. Characteristics of modeling process. Computational modeling uses constructs from three separate domains: (1) medical domain (conceptual models in problem domain), (2) computational domain (conceptual models in solution domain), and (3) implementation domain (physical models of the systems corresponding R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 401–420. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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to the conceptual models). These three domains represent fundamentally different areas of study, with different methodologies and different goals. Medicine studies the functioning of biological organisms and, as a practical discipline, provides treatment of patients. The computational domain studies and creates abstract, mathematically-based constructs, which, at the same time, must be parsimonious and computable. The implementation domain studies human-created artifacts such as software and hardware. The implementation domain is characterized by an engineering approach, which means that the artifacts must be functional and must be constructed within the constraints of limited time and limited resources. Ideally, the computational model of the system should be independent from the physical aspects of the implementation. However, in the case of the software systems it is often difficult to separate the computational domain and the implementation domain. Some aspects of the systems, for example, the time- and security-related functions, depend on the physical implementation of the system. The main problems of the modeling within three dissimilar domains are (1) explicit specification of the assumptions underlying each domain and (2) validation of the computational and implementational models against the real-world medical domain. This is especially true for the highly complex biological systems and their computational models. As stressed by Hiroaki Kitano, an understanding of complex biological systems requires a combination of computational, experimental, and observational approaches [17]. Thus, the computational models must incorporate explicit specification of the design assumptions and a mechanism for validation of the model. 2. Characteristics of the modeled systems. Biomedical systems are characterized by varied spatial and temporal scales and high complexity. The first problem is the modeling of wide-ranging spatial and temporal scales. As noted by Hunter and Nielsen, “The wealth of data now available from 50 years of research in molecular and cellular biology, coupled with recent developments in computational modeling, presents a wonderful opportunity to understand the physiology of the human body across the relevant 109 (from nanometers to meters) range of spatial scales and 1015 (from microseconds to a human lifetime) range of temporal scales.” [14] Thus, the modeling process should explicitly represent various scales and, as we postulate, the modeling process should extend into multi-level models of the functioning of human organisms within their natural and social environments. The second problem is the modeling of highly complex systems using an emergence principle and a symbiotic approach. The emergence principle is characterized by Andrzej Wierzbicki as “the emergence of new properties of a system with increased level of complexity, qualitatively different than and irreducible to the properties of its parts.” [36] The symbiotic characteristic of a biological system is described by Hiroaki Kitano as a layer of complexity: “Unlike complex systems of simple elements, in which functions emerge from the properties of the networks they form rather than from any specific element, functions in biological systems rely on a combination of the network and the specific elements involved.” [17] Thus, the computational

21.1 Introduction

403

model should provide explicit representation for the quantitative, as well as, qualitative properties of the specific system components and the configuration of components. 3. Characteristics of the models. The very nature of the computational models poses additional challenges. First, the models represent the real-world biomedical systems; therefore, the external validity of the computational models and their implementations hinges upon the construction of valid mapping from the problem domain (biomedicine) to the solution domain (computational models and their implementations) and from the solution domain (implementations of the models) to the problem domain. Second, computational models must be computer-readable and constructed in such a way that the machine will be able not only to perform calculations on data, but also to infer information from data and use the information for the reasoning about concepts. Third, computational models, in their original form, or after suitable translation, should be comprehensible to humans, so they will be able to validate the machine-produced results and follow the reasoning process and explanations. Considering the three categories of modeling problems, we stress the importance of building explicit models within each of the three domains: medical, computational, and implementational. We argue that without a clearly specified modeling process and without explicit models for each of the three domains, large computerbased systems are often developed using a mixture of often inadequate methods and intuitions. The results are often disastrous. The now classical example of a major software disaster is the Therac-25, a linear accelerator used for radiotherapy in the US and Canada between June 1985 and January 1987 [8]. Therac-25 administered massive overdoses of radiation to at least six patients, causing the deaths of three of them. The main problem was the modeling errors in the system’s software and hardware. One of the fundamental errors was the reuse of the old Therac-20 software system for the new Therac-25 hardware. The old software system was functioning correctly with the old hardware; however, the new series of Therac machines had undergone a major change in their construction: the physical backup safety system had been removed from the hardware and the system was constructed under the assumption that the software system would solely handle the safety mechanism. As a result of the erroneous designing assumption, some patients received doses of up to 20 000 rads instead of the typical 200 rads of therapeutic dose (a whole-body dose of 500 rads is lethal in 50% of cases). The main problem was summarized by Coriera in the following way: “software modeled to one machine’s environment was used in the second context in which that model was not valid.” [8] The explicit model (specification) of the domain environment was missing. To avoid similar disasters, we advocate the use of an explicit meta-modeling framework which makes a clear distinction between the reality being modeled, the conceptual model of the reality and the implementation of the conceptual model. The goal of this semioticallybased approach to modeling is to separate the three domains and to explicitly specify the mappings between them. Thus, we view computational representation as a specification (or a set of specifications) of a system. The conceptual specification is

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a description of the domain concepts (system in the problem domain) mapped to a specific conceptual model and expressed in a particular language. The implementation specification is a mapping from the conceptual specification into the particular implementation of the system. To generalize our idea, we define specification as a sequence of n layers. A specification at the layer i = 1,. . . , n contains text (or multiple texts) expressed in a particular language (or languages) and referring to a particular model (or models) of the specific domain. We understand language as a conventional code system. The term language encompasses natural languages, as well as formal languages, graphical, and other multimedia systems. Each layer corresponds to a specific level of abstraction. For example, specification at the level of “problem domain” describes the medical concept using the language and models from medicine. Specification at the level of “solution domain” describes the computational concepts using the language and models from computing science. Specification at the level of the “implementation domain” describes the implementation (artifacts) concepts using the language and models from the physical world of constructing programs and hardware systems. Specification also includes the mapping between the layers. Thus, we define a specification Si at level i as a tuple (21.1): Si =< Ti , Li ,Vi , Mi >

(21.1)

where Ti is the set of specification texts for layer i, Li is a set of languages used at layer i, Vi is a set of models used at layer i, Mi is a set of mapping functions from layer i − 1 to layer i. Set Mi can be empty. This chapter is organized around three key ideas. First, we define the central terms: the notion of a concept as a fundamental unit of knowledge, the principles of the computational representation of concepts, and the characteristics of the medical concepts, specifically their historical and cultural changeability, their social and cultural ambiguity, and their varied levels of precision. Second, we describe a conceptual framework based on fuzzy logic to represent the inherent imprecision of concepts and semiotics to represent the interpretative aspects of the medical concepts used in the diagnostic process. Third, we provide the example of computational representation of medical concepts. The chapter is structured as follows. Section 21.2 defines the terms: definition of a concept, characteristics of medical concepts, and definition of computational representation. Section 21.3 presents the fuzzy-semiotic framework and provides the example of clinical depression. The final section, Section 21.4, provides the conclusions.

21.2

Representation of Medical Concepts

In this section, we focus on the understanding of the term “concept” and its computational representation. We specifically focus on medical concepts and their characteristics and, fthen, we direct our discussion towards practical utilization of computational representation in the diagnosis and treatment of mental disorders. In

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405

subsection 21.2.1, we describe the concept and its role as a principle of classification. In subsection 21.2.2, we discuss concept representation in a broader context of knowledge representation. In subsection 21.2.3, we describe two specific characteristics of the medical concepts: changeability and imprecision. 21.2.1

Concepts

Concepts are essential to our understanding of the world, to meaningful communication and to the creation of human knowledge. As a fundamental construct in any scientific endeavor, the term “concept”, its meaning, and its many classifications have been a subjects of studies in various disciplines, most prominently, philosophy, cognitive science, and mathematics. In philosophy, concepts were studied using many perspectives. Two extreme points are represented by nominalism and conceptualism. The nominalists understand concepts as general words or uses of words. The traditional conceptualists understand concepts as mental representations (ideas). In cognitive science, a concept is a fundamental construct; it is a cognitive unit of meaning. In mathematics, the formal concept analysis defines a concept as a unit of thought with its extension and intension. The discussion of the various perspectives on the notion of a concept and an attempt at unanimous definition of the term “concept” are beyond the scope of this chapter. We will interpret “concept” as a principle of classification, since this specific approach to concept analysis has been broadly used in the medical diagnostic systems. The notion of a concept has been studied from the classification perspective as a class or as a category. A class can be defined by description of the members and the similarity (typicality) of the members belonging to the same class. The notions of similarity and typicality have been studied in the context of several disciplines. In cognitive psychology, the term typicality is used to describe members within the same category. [6], [27] Typicality is used as “a measure of how well a category member represents that category” [8]. The key questions in cognitive psychology are “How do people classify objects into categories?” and “How do people create categories?” Answering these questions is essential for the creation of computational models. The findings from cognitive psychology about human categorization and classification have a significant bearing on the representational methods. Studies in cognitive psychology have demonstrated that category learning and classification in the real world are different from the creation and classification of mathematical categories. [26] Cognitive psychology uses three approaches to the mental representation of natural categories: a classical approach, a prototype approach, and an exemplar approach. [22], [20] In the classical approach, objects are grouped based on their properties. Objects are either a member of a category or not, and all objects have equal membership in a category. A category can be represented by a set of rules, which can be evaluated as true (object belongs to the category) or false (object is not in the category). In the prototype approach, a category is based on prototype, which exists as an ideal member of a category, and the other members of the same category may share some of the features with the ideal member. [27] In this approach, the members are more or less typical of the category; in other words, they belong to the category

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to a certain degree. The prototype of the category usually represents the central tendency of the category and could be defined as the “average” of all the members of the category. In the exemplar approach, all exemplars of a category are stored in memory, and a new instance is classified based on its similarity to all prior exemplars. [21] This representation requires specification of a similarity measure, as well as storage and retrieval of multiple exemplars. In the exemplar-based representation, the category is defined by all exemplars belonging to a given category. Computational representation of natural categories presents several challenges: 1. Typicality gradient. The members of a natural category may not be equally representative for a category; thus, the members may vary in their typicality each member of a natural class has a typicality gradient. 2. Family resemblance. The natural categories do not have to share all attributes; a natural category may have some attributes which are common to many members, and some attributes which may be shared by only a few members. [37] This characteristic of natural categories is the basis for the distinction between monothetic classification and polythetic classification. In monothetic classification, categories are defined by objects possessing necessary and sufficient attributes. In polythetic classification, each member may possess some, but not all of the attributes. A notion of family resemblance has been introduced by Wittgenstein as a measure of how many members of a category share attributes. 3. Hierarchical organization of categories. Natural categories have a hierarchical organization, which often has three levels: the superordinate categories, the basic-level categories, and the subordinate categories. [28] The superordinate categories share only few attributes, while the subordinate categories share many attributes. 4. Goal-oriented categories. Some natural categories are created not based on the similarity of their features or family resemblance. Instead, the objects are classified together to satisfy specific goals. [4] These goal-oriented categories are formed based on the degree to which they satisfy the goal. 21.2.2

Computational Representation

In this subsection, we discuss the computational representation in a broader context of knowledge representation. We use the term “computational representation” to emphasize the role of computational models in biomedicine; however, many AI researchers use the term “knowledge representation” to signify “computational knowledge representation.” The studies of knowledge representation have a long history in philosophy, linguistics, logic, and computing science. We will limit our discussion to the five fundamental principles of knowledge representation presented initially by Randall Davis, Howard Schrobe, and Peter Szolovits in their influential paper

21.2 Representation of Medical Concepts

407

“What is a knowledge representation?” [10] and later discussed by John Sowa in his book on “Knowledge Representation.” [35] These principles can be summarized as the following five characteristics of knowledge representation: 1. A knowledge representation is most fundamentally a surrogate. Computational models use signs to represent the concepts or physical objects from the application domain (problem domain). These computational representations (surrogates) are used to simulate, analyze, and reason about the external real-world systems. 2. A knowledge representation is a set of ontological commitments. Computational models represent one of many possible ways of thinking about the external world. Computational models determine the categories (concepts) and the relationships between the categories. 3. A knowledge representation is a fragmentary theory of intelligent reasoning. Computational models are constructed based on some underlying theories on intelligent reasoning. Thus, the models adopt particular specific approaches from various fields, for example, mathematical logic (e.g., deduction), psychology (e.g., human behaviors), biology (e.g., connectionism), statistics (e.g. probability theory), and economics (e.g., utility theory). 4. A knowledge representation is a medium for pragmatically efficient computation. Computational models are constructed for the purpose of computation. Their construction is limited by the available computer technologies and human resources. 5. A knowledge representation is a medium of human expression. Computational models should facilitate communication between the people working in the three domains: problem domain, computational domain, and implementational domain. 21.2.3

Medical Concepts Characteristics

Medical concepts reflect the rapidly expanding and evolving nature of medical knowledge. They are characterized by a high level of changeability and imprecision. This is especially evident in the definition and classification of mental disorders. Changeability of Concepts We view human knowledge, in the spirit of Peirce’s pragmatism, as always incomplete and as always requiring constant inter-subjective communication and argumentation for its formation. Thus, all concepts, and specially medical concepts, are undergoing constant evolutionary changes and, in some critical moments in the history of science, are the subjects of major revolutionary changes. We emphasize

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here the unavoidable problem in modeling concepts: their historical evolution. As noted in 1935 by Ludwik Fleck (1896-1961), Polish bacteriologist and philosopher, in his book Genesis and Development of a Scientific Fact: “Concepts are not spontaneously created but are determined by their ancestors.” [11] Further, Fleck claims that the concept of disease is not static and it often fluctuates: “As history shows, it is feasible to introduce completely different classifications of diseases. Furthermore, it is possible to dispense with the concept of a disease entity altogether, and to speak only of various symptoms and states, of various patients and incidents. This latter point of view is by no means impracticable because, after all, the various forms and stages as well as the various patients and constitutions must always be treated differently. It is evident that the formation of the concept “disease entity” involves synthesis as well as analysis , and that the current concept does not constitute the logically or essentially only possible solution”. [11] On the other hand, as Fleck acknowledges, there is a practical importance in the “naming” of disorders. Since each concept has its genesis and development, computational representation should address the changeable characteristic of medical concepts. Imprecision of Concepts Vagueness, inexactness and imprecision, as well as imperfection of information in general, have been studied for many years in the context of computer-supported decision making, knowledge engineering, and artificial intelligence. Although imprecision is intrinsic to information and knowledge in the real world, oftentimes, the models of reality created for computational purposes are oversimplified. A simplified representation of reality is often necessary in order to design and develop feasible information systems. On the other hand, such simplified models may create a false assurance that they are themselves complete and precise and that they reflect a complete and precise reality. This caution is especially important in complex disciplines such as medicine. Often, in medical care, the decisions are made based on subjective, uncertain, multidimensional, and imprecise information. Thus, the computerized models, in order to represent real life data, information, and knowledge used in diagnosis, prognosis, and treatment, must represent various forms of imprecision and must provide reasoning methods which tolerate imprecision. As it was emphasized by Zadeh imperfections must be studied and accounted for in the models of reality. [39], [40] Imprecision has been defined using various approaches and classifications. However, there is no unanimous definition. We describe imprecision as a concept with the following characteristics: 1. Imprecision is distinct from incompleteness (absence of value), inaccuracy (value is not close to the “true” value), inconsistency (dissimilar values from several sources), and uncertainty (probability or belief that the value is the “right” value). 2. Imprecision is highly contextual and interpretative, i.e. imprecision is a quality of specific value used in a reference to a concept in a specific context. Often,

21.3 A Framework for Modeling Changeability and Imprecision

409

imprecise values are sufficient, since their precision may be not possible, may be impractical, expensive, or unnecessary. 3. Imprecision is not a binary concept. Each concept, its representation, and its interpretation have certain degree of imprecision, which can be ordered from the lowest level to the highest level. We define two aspects of imprecision: qualitative and quantitative. Qualitative imprecision is a result of a vagueness in the concept (e.g., loss of self-confidence) and the lack of precise measures of the concept (e.g., measures of appetite). Quantitative imprecision is a result of a lack of precision in a measurement (e.g., loss of weight).

21.3

A Framework for Modeling Changeability and Imprecision of Medical Concepts

In this section, we present a conceptual framework for modeling changeability and imprecision of medical concepts. This framework is based on semiotics and fuzzy logic. The semiotic approach provides models for context-dependent representation and interpretation of concepts. The fuzzy-logic approach provides explicit representation for inherent imprecision of medical concepts. 21.3.1

Semiotic Approach

Originally, the term ’semiotics’ (from a Greek word for sign “semeion”) was introduced in the second century by the famous physician and philosopher Galen (129199), who classified semiotics (the contemporary symptomatology) as a branch of medicine. 19 The use of term semiotics to describe the study of signs was developed by the Swiss linguist Ferdinand de Saussure (1857-1913) and the American logician and philosopher Charles Sanders Peirce (1839-1914). Contemporarily, semiotics is a discipline which can be broadly defined as the study of signs. Since signs, meaning-making, and representations are all present in every part of human life, the semiotic approach has been used in almost all disciplines, from mathematics through literary studies and library studies to information sciences. [31], [7], [34], [32] A semiotic paradigm is, on one hand, characterized by its universality and transdisciplinary nature, but, on the other hand, it is associated with different traditions and with a variety of empirical methodologies. Our intention is not to present the field of semiotics; rather, our goal is to define the basic terminology needed to present an example of the semiotic approach to modeling of vague medical concept such as clinical depression. These examples illustrate that the meaning of a sign arises in its interpretation or even multiple possible interpretations. Thus, the notion of imprecision is not universal and absolute, but should be studied in context of the interpretations of the sign.

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Peirce defined “sign” as any entity carrying some information and used in a communication process. Peirce, and later Charles Morris, divided semiotics into three categories [7]: syntax (the study of relations between signs), semantics (the study of relations between signs and the referred objects), and pragmatics (the study of relations between the signs and the agents who use the signs to refer to objects in the world). This triadic distinction is represented by a Peirce’s semiotic triangle: the representamen (the form which the sign takes), an interpretant (the sense made of the sign), and an object (an object to which the sign refers). The notion of “interpretant”, is represented by a set of pragmatic modifiers: agents (e.g., patients, health professionals, medical sensors, computer systems), perspectives (e.g., health care costs, accessibility, ethics), biases (e.g., specific subgroups of agents), and views (e.g., variations in the diagnostic criteria used by individual experts or clinics). Peirce’s semiotic triangle is illustrated in Figure 21.1.

Fig. 21.1. Peircean semiotic triangle.

21.3.2

Fuzzy-Logic Approach

The first step towards creation of a formal mathematical representation of vagueness was the work of Jan Łukasiewicz, who in the 1920’s introduced multi-valued logic. Łukasiewicz extended the traditional two-valued logic (values: true and false) to a closed real interval [0,1] representing the possibility that a given value is true or false. While in traditional logic and set theory, an element either belongs to a set or not, in fuzzy set, an element may belong to a set “partially” with some degree of membership. At the same time, Emil Post introduced similar ideas in logics which are more general than two-valued logic. In 1937, Max Black published the paper, “Vagueness: an exercise in logical analysis”, in which he introduced “vague sets” and operations. In 1965, Lotfi Zadeh published a paper “Fuzzy sets” [39]. Zadeh introduced the term “fuzzy set”, extended the fuzzy set theory, and created fuzzy logic as a new field of study. Fuzzy set theory has been used for the representation of imprecision in a wide-range of systems. Fuzzy logic is well suited for humanand computer-readable representation of imprecise measurements. Moreover, fuzzy logic is well suited for representation of medical concepts and for modeling of medical decision making [30], [33]. The application of fuzzy logic in medicine started

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in the early 70’s. One of the first medical expert systems to apply fuzzy logic was CADIAG-2 [1], [2], which become later MedFrame/CADIAG-IV [29]. Since the 1970s, fuzzy logic has been utilized in numerous medical applications: from processing of biomedical signals and images [9] through epidemiological studies of sleep disorders [24] to control of fluid resuscitation of patients in an intensive care [5]. Fuzzy logic approach has been used in the systems supporting the diagnosis of mental disorders, for example, for prediction of suicidal attempts [23], diagnosis of Tourette’s syndrome [38], and treatment of depressive episodes [16]. One of the key concepts in fuzzy logic is the linguistic variable (fuzzy variable). A linguistic variable may be qualitative, for example self reproach, decreased energy, depressed moods, or quantitative, for example, loss of weight, waking in the morning. A linguistic variable is associated with terms. A set of terms describes the possible states of the variable. A linguistic variable is a quintuple: L =< X, T (X),U, G, M >, where X is the name of the variable, T (X) is the set of terms for X, U is the universe of discourse (the set of all possible values of a linguistic variable), G is the set of grammar rules to generate T (X), and M is the set of semantic rules M(X). 21.3.3

Computational Representation of Depression

Depression is a term used to cover a wide range of states, from feeling sad or helpless, through minor depression to major depression (MD). There are many approaches to the definitions, classifications, diagnostic criteria, and assessment of depression. These diverse approaches reflect the fact that depression has a complex etiology and presents itself with a variety of symptoms, which differ in different patients. For example, the modeling of the assessment process involves three aspects: conceptualization (what to measure), operationalization (how to measure), and utilization (how the measure is used). These three aspects are mapped to the semiotic triangle, shown in Figure 21.2.

Fig. 21.2. Peircean semiotic triangle for modeling depression.

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The syndrome of depression, S, is defined by a triplet S =< O, M,U >, where O represents a set of objects, M a set of measures, and U a set of utilization parameters. Depression can be represented as a set of objects O = depressive episode, symptom, intensity, frequency, duration, the possible measures can be represented by a set diagnostic criteria and assessment instruments. The set of utilization U contains clinical guidelines for the assessment. Conceptualization of Depression Several conceptual approaches to depression exist, and many authors view depression as a syndrome rather than a single diagnostic entity. In this paper, we focus on major characteristics: (1) depression can be defined by a set of presenting symptoms, which display specific severity, frequency, and duration, (2) depression can be viewed as a dimensional concept in which symptoms may be grouped or clustered into specific dimensions, and (3) depression may be conceptualized as a state or trait. The symptomatic approach to depression identifies more than 10 symptoms, which have varied definitions and which are used in different ways by the diagnostic criteria. The symptoms are generally grouped into three classes: affective (crying, sadness, apathy), cognitive (thoughts of hopelessness, helplessness, suicide, worthlessness, guilt), and somatic (sleep disturbance, changes in energy level, changes in appetite, and elimination). Not all symptoms are present at the same time, and the severity of symptoms differs. Moreover, the symptoms may vary in their “directions”. For example, two subtypes of depression are distinguished: depression with vegetative symptoms (e.g., appetite loss, weight loss, insomnia) and depression with reverse vegetative symptoms (e.g., appetite increase, weight gain, hypersomnia). The second subtype, according to the epidemiological studies, is characteristic of one-fourth to one-third or of all people with major depression, and it is more common among women. Operationalization of Depression In our discussion, we refer to two general diagnostic criteria: WHO’s International Classification of Disease (ICD-10 originating in Europe) and the Diagnostic and Statistical Manual of Mental Diseases Fourth Edition (DSM-IV originating in the United States). Both criteria are based on a symptomatic approach and have many similarities and differences [12]. They differ in the set of symptoms; however, both have eight symptoms in common: depressed mood, loss of interest, decrease in energy or increased fatigue, sleep disturbance, appetite disturbance, recurrent thoughts of death, inability to concentrate or indecisiveness, psychomotor agitation or retardation. The ICD-10 and DSM-IV criteria have a significant overlap in diagnosis, yet in some cases the patient may meet the diagnostic criteria in one system but not in the other. Both diagnostic criteria have been modified several times and remain subjects of ongoing discussions. These modifications clearly indicate the difficulties in defining such a heterogeneous syndrome. Current versions of Diagnostic

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and Statistical Manual of Mental Disorders (DSM-IV-TR) and ICD (ICD-10) define depression a polythetic and categorical concept. However, as discussed by Kruger and Bezdjian [19] ,the clinical experience and research results demonstrate that the classification of mental disorders has three problems: comorbidity (patients meet criteria for multiple disorders), within-category heterogeneity (patients who meet criteria for the same disorder may have few or even no symptoms in common), and lack of abrupt thresholds (patients who have subthreshold symptoms may be significantly affected in their daily life). The members of the DSM-5 Task Force and Work Group have been working on the development of new diagnostic criteria and changes to DSM-IV for the last 14 years. [3] The publication of DSM version 5 (DSM-5) manual has been scheduled for May 2013. The new version will reflect new advances in the science and conceptualization of mental disorders, as well as it will incorporate patients’ perspective. Ontological Representation of Depression In recent years, ontologies have been successfully used to standardize and share data among various projects. For example, the Gene Ontology has been used for the standardization and comparison of genomic data. The classification of mental disorders has also been represented by computer-readable ontologies. For example, the DSM-V classification of depression has been represented using OWL [25]. This ontology is mainly concerned with the categorization of different types of depression, excluding related concepts such as environmental factors and life events. Recently, an outline of unifying psychosis OWL ontology has been proposed by Kola et al. [18] The OWL ontology proposed in this chapter represents the concepts associated with chronic depression. The representation includes concepts beyond the basic diagnosis based on DSM or ICD criteria. Figure 21.3 shows the overall ontological structure of the fundamental concepts such as reporting factors, symptoms, diagnostic methods, general concept of a mental disorder, contextual factors, and the concept of a patient. Symptoms are divided into three sub-classes: affective, cognitive, and somatic. We have created this ontology to exemplify an ontological approach rather than to provide a comprehensive knowledge schema. Assessment of Depression We place assessment of depression in a broader context of clinical decision making. We describe the treatment evaluation process, which typically is used in psychiatric clinics to assess clinical situations and to evaluate the effectiveness of pharmaceutical treatment. We have applied the fuzzy-semiotic framework to model the assessment of depression in the context of treatment evaluation. The evaluation protocol is based on our earlier work on a fuzzy-logic based system to support a depressive episode therapy [16]. The treatment evaluation requires at least two consultations: pre- and post-treatment. During the first consultation, the clinician evaluates the patients according to diagnostic criteria accepted by the clinic, such as DSM-IV

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or ICD-10. In this specific application, the clinician uses the Research Diagnostic Criteria (RDC) for ICD-10 to assess the severity of a depressive episode as mild, moderate, or severe, and to classify it as an episode with or without psychotic symptoms and with or without somatic symptoms. During the second consultation, the clinician repeats the assessment, compares the results with the results from the first consultation, and evaluates the clinical situation as a recovery, partial improvement, lack of improvement, and deterioration. The RDC for ICD-10 have 16 items, which are rated by the clinician using the Hamilton Rating Scale for Depression (HRSD) and eight additional questions for the items not covered by HRSD.

Fig. 21.3. OWL class hierarchy for chronic depression.

Assessment Instruments The Hamilton Rating Scale for Depression (HRDS) is an assessment instrument, which has been the most frequently used clinical rating scale since its inception in 1960[13]. The HRDS is completed by a clinician, and it used to indicate the severity of depression in patients already diagnosed with a depressive disorder. The HRDS has 21 items; 17 items are usually used for scoring. The items are measured on a three-point scale (0, 1, 2) or a five-point scale (0, 1, 2, 3, 4). The items are based on symptoms. The items on the three-point scale are quantified as 0 = “symptom

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absent”, 1 = “slight or doubtful”, and 2 = “clearly present”. The items on the fivepoint scale are quantified in terms of increasing intensity: 0 = “symptom absent”, 1 = “doubtful or trivial”, 2 = “mild”, 3 = “moderate”, and 4 = “severe”. For example, the symptom depressed mood is quantified as 0 = “absent”, 1 = “feeling of sadness, gloomy attitude, pessimism about future”, 2 = “occasional weeping”, 3 = “frequent weeping”, 4 = “extreme symptoms”. Typically, the scores from 17 HRDS’s items are added together, and the final score ranges from 0 to 52 points. Originally, Hamilton did not specify cutoff points; however, generally the scores lower than 7 indicate an absence of depression, scores 7 to 17 indicate mild depression, scores 18 to 24 indicate moderate depression, and scores of 25 and above indicate severe depression. Generally, the HRDS has high reliability and validity. On the other hand, the literature on the HRSD includes many papers criticizing the scale on a number of grounds. One of the important issues for our explicit model of imprecision is a critical assumption about the type of a measurement. The problem is related to the quantification of the concept: is the HRDS an ordinal measurement scale or is it an interval measurement scale? Thus, although from a theoretical perspective, the HRDS is an ordinal scale, from the practical perspective, the HRDS is perceived as an interval measurement, which means that one unit on the scale represents the same magnitude of change across the entire scale.

Fig. 21.4. Example of 11-point scale for 16 items of the RDC.

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We have observed that (1) the HRDS represents an ordinal scale, and the assumption of the equity of distances between units introduces a large measurement imprecision, and (2) the range of values on three-point scale and five-point scale is not sufficient for scaling small changes in symptoms during the treatment. To address these two problems, we have introduced an 11-point rating scale - a range of values from a discrete set: 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1. The value 0 means “normal state” (not precisely defined), and the value 1 means extreme pathology (also not specified), the medium values are based on the HRDS range of scores 0-4 (1 = 0.2, 2 = 0.4, 3 = 0.6, and 4 = 0.8). Each item has the same rating range. The 16 items on the RDC for ICD-10 and an example of values for a patient are presented in Figure 21.4. The item “thoughts of death” has been highlighted, since currently this factor is included with an equal weight as other 15 items. However, the proposed DSM-V classification treats the suicidal thoughts as a separate dimension. Representation of Intensity and Frequency of Symptoms We appplied the fuzzy-logic approach to the standardized rating guidelines for the administration of the HRDS, GRID-HAMD-17 [15]. The GRID-HAMD-17 overcomes several shortfalls in the original HRDS, particularly, the high level of imprecision in measures of frequency and intensity. The GRID-HAMD-17 provides specific instructions for the evaluation of the frequency and intensity for the 12 items and intensity only for 5 items (frequency is not applicable for these items). The frequency is represented by four linguistic terms: absent or clinically insignificant, occasional, much of the time, and almost all of the time. The GRID-HAMD-17 guidelines specify the mapping between the linguistic terms and the frequency of symptoms measured in days/week. The mapping is defined as follows: absent = not occurring, occasional = less than 3 day/week, much of the time = 3-5 days/week, almost all the time = 6-7 days/week. The same definition of frequency is used for all applicable items. The intensity is represented by five linguistic terms: absent, mild, moderate, severe, and very severe. The terms have specific qualitative mappings for each item. We have used the fuzzy-logic approach for one of the items on the GRID-HAMD-17, depressed moods. We have constructed two linguistic variables: Frequency and Intensity of Depressed Moods. The variable Frequency has four terms: absent, occasional, much of the time, almost all the time. The membership functions, MFs, for the symptom frequency are shown in Fig. 21.5. They have been created based on the frequency measured by days per week. The MFs for the intensity of a depressed mood are shown in Fig. 21.6. The MFs are based on a continuous scale from 0 to 1, which corresponds to the clinician’s rates on a discrete scale from 0 to 1 with an increment of 0.1. The intensity of the symptom is rated by the clinician based on the GRID-HAMD-17 specification. For example, the depressed mood with a severe intensity is described as intense sadness; hopelessness about most aspects of life, feeling of complete helplessness or worthlessness.

21.4 Conclusions and Future Work

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Fig. 21.5. Membership functions for the frequency of symptoms.

Fig. 21.6. Membership functions for the intensity of a symptom.

21.4

Conclusions and Future Work

In this chapter, we have examined the meta-modeling framework for creating computational representation of medical concepts. We have demonstrated that (1) medical concepts are evolving and changing, (2) interpretation of medical concept is highly contextual, and (3) imprecision is an intrinsic part of medical concepts and

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their utilization in diagnosis and treatment. To address these issues, we have presented a conceptual framework for explicit modeling of changeability and imprecision. Our framework has its theoretical foundations in semiotics, fuzzy logic, and analytical approach to computational modeling. We have observed that interpretation of concepts is highly contextual and goal-oriented, which has led us to the application of a semiotic approach. Semiotics provides the modeling constructs for the description of the concept, its representation, and its interpretation. Furthermore, we required a formal framework to explicitly represent the imprecision of medical concepts. To address this problem, we have applied a fuzzy logic approach. Fuzzy logic provides representational constructs for reasoning with quantitative values, as well as qualitative terms, and produces quantifiable results. To address the multidimensionality of medical concepts, such as mental disorders, we have used an analytical approach to model individual symptoms and their features. We have applied the semiotic and fuzzy-logic approach to define a concept of “clinical depression”. We have used the classical Peircean triangle to represent the concept of depression, its measurements, and its interpretations. We have demonstrated that imprecision is an intrinsic part of the assessment of such a complex disorder as depression. Furthermore, we have presented a fuzzysemiotic framework for the explicit representation of qualitative, quantitative, contextual, and interpretative aspects of imprecision. We have used a semiotic approach to represent the concept of depression, its symptomatic representations, and the clinical utilization of the measures. We are planning to expand and further formalize the proposed framework and to build a comprehensive computational model for the medical concept of depression and its assessment in treatment evaluation and screening. We will apply this model in a clinical decision support system for the diagnosis and treatment of depression, as well as in a support system for the treatment of sleep disorders. Furthermore, we plan to utilize the proposed computational model for the analysis of patients’ data from clinics which use diverse diagnostic criteria. The explicit modeling of interpretation will allow us to compare treatment results from various clinics. The explicit modeling of imprecision will allow us to analyze and integrate patients’ data characterized by varied granularity and heterogeneity.

References [1] Adlassnig, K.-P.: A fuzzy logical model of computer-assisted medical diagnosis. Methods of Information in Medicine 19, 141–148 (1980) [2] Adlassnig, K.-P.: Fuzzy set theory in medical diagnosis. IEEE Trans. on Systems, Man, and Cybernetics SMC-16, 260–265 (1986) [3] American Psychological Association DSM-5 Development, http://www.dsm5.org/Pages/Default.aspx (accessed May 10, 2010) [4] Barsalou, L.W.: Context-independent and context-dependent information in concepts. Memory and Cognition 10(1), 82–93 (1982) [5] Bates, J.H.T., Young, M.P.: Applying fuzzy logic to medical decision making in the intensive care unit. American Journal of Respiratory and Critical Care Medicine 167, 948–952 (2003)

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[6] Bruner, J.S., Goodnow, J.J., Austin, G.A.: A Study of Thinking. Wiley, New York (1956) [7] Chandler, D.: Semiotics: The Basics. Routledge, London (2002) [8] Coriera, E.: Guide to Health Informatics, 2nd edn. Hodder Arnold, London (2003) [9] Ernest, C., Leski, J.: Entropy and energy measures of fuzziness in ECG signal processing. In: Szczepaniak, P., Kacprzyk, J. (eds.) Fuzzy Systems in Medicine, pp. 227–245. Physica-Verlag, Heidelberg (2000) [10] Davis, R., Schrobe, H., Szolovits, P.: What is a knowledge representation? AI Magazine 14(1), 17–33 (1993) [11] Ludwik, F.: Genesis and Development of a Scientific Fact. The University of Chicago Press, Chicago (1979) [12] Gruenberg, A.M., Goldstein, R.D., Pincus, H.A.: Classification of Depression: Research and Diagnostic Criteria: DSM-IV and ICD-10. In: Licincio, J., Wong, M.-L. (eds.) Biology of Depression. From Novel Insights to Therapeutic Strategies. WIILEYVCH, Weinheim (2005) [13] Hamilton, M.: A rating scale for depression. Journal of Neurology, Neurosurgery and Psychiatry 23(1), 56–61 (1960) [14] Hunter, P., Nielsen, P.: A Strategy for Integrative Computational Physiology. Physiology 20, 316–325 (2005) [15] Kalali, A., Williams, J.B.W., Kobak, K.A., et al.: The new GRID HAM-D: pilot testing and international field trials. International Journal of Neuropsychopharmacology 5, 147–148 (2002) [16] Kielan, K.: The Salomon advisory system supports a depressive episode therapy. Polish Journal of Pathology 54(3), 215–218 (2003) [17] Kitano, H.: Computational systems biology. Nature 420, 206–210 (2002), doi:10.1038/nature01254 [18] Kola, J.(Subbarao)., Harris, J., Lawrie, S., Rector, A., Goble, C., Martone, M.: Towards an ontology for psychosis. Cognitive Systems Research (2008), doi:10.1016/j.cogsys, 08.005 [19] Kruger, R.F., Bezdjian, S.: Enhancing research and treatment of mental disorders with dimensional concepts: toward DSM-V and ICD-11. World Psychiatry 8, 3–6 (2009) [20] Nosofsky, R.M.: Exemplars, prototypes, and similarity rules. In: Healy, A., Kosslyn, S., Shiffrin, R. (eds.) From Learning Theory to Connectionist Theory: Essays in Honour of William K. Estes, vol. 1. Erlbaum, Hillsdale (1992) [21] Medin, D.L., Schaffer, M.M.: Context theory of classification learning. Psychological Review 85, 207–238 (1978) [22] Minda, J.P., Smith, J.D.: The effects of category size, category structure and stimulus complexity. Journal of Experimental Psychology: Learning, Memory and Cognition 27, 755–799 (2001) [23] Modai, I., Kuperman, J., Goldberg, I., Goldish, M., Mendel, S.: Fuzzy logic detection of medically serious suicide attempt records in major psychiatric disorders. The Journal of Nervous and Mental Disease 192(10), 708–710 (2004) [24] Ohayon, M.M.: Improving decision making processes with the fuzzy logic approach in the epidemiology of sleep disorders. Journal of Psychosomatic Research 47(4), 297– 311 (1999) [25] Ontology Lookup Service (OLS). http://www.ebi.ac.uk/ontology-lookup/browse.do?ontName=DOID (accessed May 2, 2010) [26] Reed, S.K.: Cognition. Theory and Applications, 4th edn. Brooks/Cole Publishing, Pacific Grove (1996)

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[27] Rosch, E., Mervis, C.B.: Family Resemblances: Studies in the Internal Structure of Categories. Cognitive Psychology 7, 573–605 (1975) [28] Rosch, E., Mervis, C.B., Gray, W.D., Johnsen, D.M., Penny, B.-B.: Basic objects in natural categories. Cognitive Psychology 8, 382–440 (1976) [29] Rothenfluh, T.E., Bögl, K., Adlassnig, K.-P.: Representation and acquisition of knowledge for a fuzzy medical consultation system. In: Szczepaniak, P.S., Kacprzyk, J. (eds.) Fuzzy Systems in Medicine, pp. 636–651. Physica-Verlag, Heidelberg (2000) [30] Sadegh-Zadeh, K.: Fuzzy health, illness, and disease. The Journal of Medicine and Philosophy 25, 605–638 (2000) [31] Sebeok, T.A.: Signs: An introduction to semiotics. University of Toronto Press (1999) [32] Sebeok, T.A., Danesi, M.: The Forms of Meaning: Modeling Systems Theory and Semiotic Analysis. Mounton de Gruyter, Berlin (2000) [33] Seising, R.: From vagueness in medical thought to the foundations of fuzzy reasoning in medical diagnosis. Artificial Intelligence in Medicine 38, 237–256 (2006) [34] Sheng-Cheng, H.: A semiotic view of information: semiotics as a foundation of LIS research in information behavior. Proceedings of the American Society for Information Science and Technology 43(1), 66 (2006) [35] Sowa John, F.: Knowledge Representation: Logical, Philosophical, and Computational Foundations. Brooks/Cole (2000) [36] Wierzbicki, A.P.: Modelling as a way of organising knowledge. European Journal of Operational Research 176(1), 610–635 (2007) [37] Wittgenstein, L.: Philosophical Investigations. Blackwell, Oxford (1953) [38] Yin, T.-K., Chiu, N.-T.: A computer-aided diagnosis for distinguishing Tourette’s syndrome from chronic tic disorder in children by a fuzzy system with a two-step minimization approach. IEEE Transactions on Biomedical Engineering 51(7), 1286–1295 (2004) [39] Zadeh, L.A.: Fuzzy Sets. Information and Control 8(3), 338–353 (1965) [40] Zadeh, L.A.: A note on prototype theory and fuzzy sets. Cognition, 291–297 (1982)

Part VII

22 Invariance and Variance of Motives: A Model of Musical Logic and/as Fuzzy Logic Hanns-Werner Heister

22.1

Introduction: Congruities, Similarities and Analogies between Fuzzy Logic and Musical Logic

Until November 2008 my knowledge of Fuzzy Logic was vague, implicit, fuzzy. A presentation by Rudolf Seising1 in the context ’Cybernetics’ [69] supplied me with more and more un-fuzzy information, which caught my attention. Reading his book [70] has reinforced this former fascination.2 Besides the general relevance I was particularly intrigued by the many parallels to thinking within and about music. It is an inherent part of music that the object itself is fuzzy in many respects, just as a certain fuzziness in analysis, to various degrees, is necessary. Taking into account fuzziness and precision in their calculation go together according to the subject matter, I shall discuss Fuzzy Logic rather qualitatively than quantitatively, in its application as a thought process or form of thinking and of methodical approach and not in and as its mathematical calculation. 1

2

First of all I would like to thank Rudolf Seising and the European Centre for Soft Computing for the opportunity to present some ideas from my area of expertise in this context, hoping to contribute to the diversification and differentiation of thinking about and with Fuzzy Logic. In respect to the presentation in September 2009 at Mieres I would also like to thank my assistant Hanjo Polk for his help with Powerpoint, and for the translation there and the revised and expanded version for print here my son Hilmar. The discovery and development of Fuzzy Logic in the contexts of set theory and other mathematical and engineering sciences continues lines of the traditions of differential and infinitesimal calculus. These falsified long ago the naïve idea of the absolute exactness as main feature of maths – an idea which even after Gödel seems ineradicable. In the late enlightenment Lichtenberg wrote: “The big stratagem to reckon minor divergences from the truth as truth itself, whereupon is built the whole differential calculus, is withal the reason for our whitty cogitations wherein the whole would result, if we would take the deviations in a philosophical rigour.” (“Der große Kunstgriff kleine Abweichungen von der Wahrheit für die Wahrheit selbst zu halten, worauf die ganze Differential-Rechnung gebaut ist, ist auch zugleich der Grund unsrer witzigen Gedanken, wo oft das Ganze hinfallen würde, wenn wir die Abweichungen in einer philosophischen Strenge nehmen würden.”) [54].

R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 423–450. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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Seemingly precise and sharp dualistic differentiation proves to be fuzzy and inappropriate, in music as in life3 , as well in regard to the various levels of a musical piece as to the layers of the musical work process: for example the opposition of consonance/dissonance or minor and major. Even more so the conventional association of consonance=beautiful, dissonance=ugly or major=happy ad minor=sad are simply not true in this distinction - listening to Bach, Schubert or Spanish traditional folk music make this obvious. And in many idioms of contemporary music after the so called “emancipation of dissonance” (Arnold Schönberg) the binary opposition consonance/dissonance has changed into a segmented continuum (’gegliedertes Kontinuum’). The binary two-valued (’zweiwertige’) Aristotelian Logic of the either/or therefore can be transformed in a Fuzzy Logic or at least be modified by this logic.4 Instead of dualities there are therefore gradual transitions, grades, shades, not binary divisions with precise boundaries. Historically and culturally various expressions further add to this differentiation. On the other hand there are areas of obvious yes/no-distinctions, as between A and C, and also between (German) B and H (English: B flat versus B sharp). They are valid at least within a given tonal system - and in a system of pitches. 22.1.1

Musical Logic

“Musical Logic” (1788 by Johann Nicolaus Forkel) means both the “logicality of music” and the “involvement/participation of musical processes in the laws of reason”.5 Motives, themes, parts of movements, forms are supposedly related to each other: horizontally in time in a consequent and apparent manner, either as derived variation or as contrast, and vertically in (musical) space as an order of various voices simultaneously in the score. In both dimensions it should sound meaningful and beautiful, harmonical without dissolving contradictions (but with contrasts, even extreme ones), and following a guideline of consequence, development and intentional-teleological and dramaturgical finality. That’s valid even for improvisations. (Here in the artwork the traditional Aristotelian “causa finalis”, chimerical in reality outside of human/social contexts, as an inner principle of the artwork has its place.) On a high level of abstraction this logic is valid universally. But in concrete music it depends on a complicated framework of implicit or explicit principles, rules, on the conditions of genres, forms, of period norms and styles etc. (Cf. e.g. [7]) And furthermore it depends from the even more complicate interactions between artwork, realization, reception (Cf. [51]), and then diversification 3

4 5

And, of course, in “Life Sciences” from biology to “neurosciences”. And even in the so called “exact” sciences and also in technical sciences as in technic itself - the decisive starting point for Lotfi Zadeh. - Maths is in contrary to the common prejudice not a (natural) science but a human discipline (“Geisteswissenschaft”), and an extreme one (together with philosophy, in particular logic. That maths can be applied nearly universally to matters of nature, technics, society, mind is founded in its quality (the same with logic) of reflecting in a high degree of abstraction real relations, nexuses, proportions etc. Cf. drafts of a poly-valued (“mehrwertige”) logic, e.g. [47], [23], [24]; http://de.wikipedia.org/wiki/Polykontexturalitätstheorie (22.7.2009). New tendencies among others: [62].

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of the public in social strata of class, gender, generation etc., at last in individual differences too. The logic of music however reaches even further. It includes cultural and historical contexts. And it includes analogies, similarities up to almost identity to mathematical methods and procedures, e.g. shaping forms with the proportions of durations following the famous ’Fibonacci Row’ and other types of rows organizing musical measures (as Boris Blachers “Variable Metren”, variable metres with measure-sequences as e.g. 1/4, 1/4, 2/4, 3/4, 5/4, 8/4, 11/4, 7/4, 4/4 etc., even in retrograde mode), permutations on different levels of the material of music and the techniques of composition etc.6 On different levels of organisation it includes analogies and similarities to methods and procedures of Fuzzy Logic. Part of the musical logic and its realization in concrete artworks is the motivicthematic work (“motivisch-thematische Arbeit”). Musical motives, often combined to more expanded themes, form shapes (“Gestalten”) with a fairly stable core and diverse and multiple varied elaborations in the course of the art work as process. Core can be interpreted as the inner kernel, partially subconscious, quasi also the ’generative’ dimension. To the core as essence correlate as phenomenon respectively phenomena in different variants, the ’performative’ dimension. The ’contours’ of these variants for their part are a generalization and a fuzzily identity-establishing commonality of the different various forms and appearances of the kernel. The interrelations between both these entities belong to the general dialectic of essence and appearance (the Hegelian ’Wesen’ vs. ’Erscheinung’) - notabene a real dialectic, because both sides or dimensions participate in the constitution of the object, and both belong to the objects of Fuzzy Logic. There are other types of variation and development of variants in some styles and genres of music. In these the dialectic of core und contour is not fully brought to bear, because there is no stable, fixed kernel as identical starting point and reference point for the differences. This is nearly the rule in many fields of music outside of the European art music and is often in folk music - but not in Popular Music at least in the present. “The chain of variants [. . . ] is also not stamped by forms of memory, by any form-building restatement of something already performed. Following the principle of generation and formation of ’variative Gestalten’ (varying shapes) the variation takes place successively in the melody and in case of heterophony simultaneously in different parts.” ([64], pp. 473) The immediate forms of appearance of a piece of music as well as the historical formation of tradition follows a sort of Fuzzy Logic. “Passed on from generation to generation are the basic shapes (“Grundgestalten”) on one side and on the other the methods to vary them.” ([64], pp. 473) Here variative, vague Fuzzy Logic is necessary to realize identity. Identity in and as process As term covering such facts I propose a concept, which is valid also for other matters: the only seemingly paradoxical identity in and as process (’prozessierende 6

More detailed see [41].

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Identität’)7: In a given phase the identical is not invariably ’given’ and present but develops in time and history.8 This – relative – identity is the initial point and endpoint of a process and is realized as this process. The change or turn (’Umschlag’) from quantitatively measurable differences between the variants to a new, different identity as a new quality is as much an object for Fuzzy Logic as the inner differences. The respective identical is not invariably preset but develops in time and history. It is starting point and endpoint of a process and is realized as this process. This concept can be applied to very different matters and facts, from big systems or the unity of individuals and Egos as subjects up to “subjects” (sogetti, themes) of fugues. Necessarily there are blurs not only on the edges of the motives. Thought patterns and procedures of Fuzzy Logic serve as methodical instrument that expands and enhances conventional thought patterns of music analysis, beginning with the segmentation and emphasis of significant motives from the score and its flow, but also regarding the concrete definition of “musical logic”: for fundamental aspects like the interrelations of motives, not only within a single piece of work, the quotation and/or the allusion, the unaware quote, the reminiscence, in general the perception and remembering of motif complexes and more. I will demonstrate this with some concrete examples. (I must leave aside the psycho-logic – causes, motivations, motives, the question of affects, emotions –, the social and sociological questions of interaction between musical process and public, the influence of situations on realization and reception etc.) Inversely music (and similarly the fine arts) appear as part and concrete elaboration of the aesthetic and sensual perception, which form their own laws, logics, layers and types of cognition9 , and therewith nothing less than a paradigm of Fuzzy Logic. 22.1.2

Filtering: From Chaos to Cosmos

Filtering is a fundamental process in the realm of artistic mimesis, in respect to the evolution of single signs as to the combination of these signs.10 It could be discussed 7 8

9

10

The term and the concept to the first time exposed in [40]. Cf.: “As the same thing in us is living and dead, waking and sleeping, young and old. For these things having changed around are those, and conversely those having changed around are these. (DK22B88)” Daniel W. Graham http://www.iep.utm.edu/heraclit/ (31.12.2010) as English translation of Heraklit, Fragment 88, [14]: “Und es ist immer ein und dasselbe was in uns wohnt: Lebendes und Totes und das Wache und das Schlafende und Jung und Alt.” There are probably not more than three basic forms of the reflection and ideational appropriation of the world: Arts, religion, science. Other candidates I don’t see. Religion, being an inverted and distorted reflection of reality, should eventually be classified as a derivative form and type, counted and included here in contradistinction to its truth content because of its widespread occurrence even today. That must of course be discussed - but not here. As filtering we can even describe the evolution of language in respect to the basic elements: The phonemes, defined by opposite/binary qualities of distinctive features, filter the fluid and continuous phonetical raw material of the phones. For filtering processes on a higher level, in particular words, cf. e.g. [42], p. 13.

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if perhaps selection would be a better term11 , especially with its relation to historical processes - certainly without implications of biologism. Anyway, ’filtering’ may fit for historical reasons in the context of discovering Fuzzy Logic; and it covers a wide span from natural (water through gravel aggregate bed filter), industrial (chemical, technical etc.), trivial as coffee filter, social (“infiltration” by “foreigners” from outside the own ethnos, nation and so on - and their filtering out as potential “terrorists” or similar), and also mathematically abstract (“Peano-Sieve”). Musical materials, meaning sounds, colors, forms have an aesthetically higher intrinsic value as material in scientific reasoning, deviating from everyday experiences. Apart from that the filtering runs analogue to processes and methods of abstraction and terminology as used for scientific reasoning. Here one speaks of reality as the total set, of which subsets are extracted through filtering, there one sees the musical world of sound with all its cognitive components divided, segmented and gridded in various manners. Filtering has further implications that also show correlations to music-external processes. One essential aspect is the reduction of entropy. Chaos becomes cosmos, but as a “moveable/fluid order” (as nature is for Goethe), especially in music as fluid equilibrium and dynamic, dialectical homeostasis of stability an variability. The opening of Haydn’s oratorio Die Schöpfung (The Creation) is prototypical for musical evolvement of cosmos out of chaos: an unshapely, harmonically diffuse sound arrangement without melodic contours gradually crystallizes to clear shape and forms. Wagner radicalizes this topos in the opening of his musical drama Rheingold, when for several minutes the orchestra reverberates only with a single Eflat-major chord. The constitution of motives and themes as filtering, as elaborating a raw material here is itself part of the artwork as process. In a more general sense, especially since the Viennese Classic, the theme often evolves gradually from single motives. Once the motif or theme is established, it exists as a fixed, shaped – and developing – form (Gestalt). In this development of variation there are various degrees and ways of similarities between primary shape and reproduction. Often enough the relation of motif contours (a term of the Viennese music theoretician and Beethoven specialist Rudolf Réti): instead of congruity or strict step-by-step derivation contexts, a similarity of motives and themes concerning contours, outlines, course, basic structures, edges, is sufficient.12 The metamorphosis of motives expresses itself in concrete appearances like a literal quote, paraphrase, unconscious reminiscence, allusion, parody, caricature and others. In this case we can speak of a specific “fuzzification”. This occurs in the movement of an artwork, the artwork as a whole, the oeuvre of a composer, in relations of different composers, in the process of the different forms of existence (’Existenzformen’) of an artwork. 11 12

[17] e.g. see ’selection’ as a basic principle also for processes and developments in the brain. This corresponds also with features of the neuronal processes in perception and cognition - for ’edges’ and the like are responsible special clusters and mechanisms in the brain, especially in visual perception.

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We can also detect parallels to the constitution of the Ich, the human subject. The multiform, or polymorphism, which evolves from the unshapeliness, and undergoes continuous metamorphoses reminds of Freuds term of the “polymorph-perverse”, from which the concentrated, mature “genital sexuality” originates, not destroying earlier states but incorporating them. Another link: the boundaries of the Ich during this ontogenesis are at first quite fuzzy. Subject-object relations are formed gradually in a process of interaction. Once developed, the achievement of the Ich becomes pleasure and danger at the same time in a reminiscence of its origins, the regression to the “oceanic feeling” (Freud). 22.1.3

De-fuzzyfication as Filtering - Tone versus Clangour, Chord versus Cluster as Turn towards a New Quality

Filtering is also a basic category for music and its evolution. Even from a structural and historical perspective the constantly renewed filtering of (especially acoustic) elements of reality continues in producing music. This process creates general music material - sounds, sequences, tonal systems or sustained sounds or rhythmic systems, the combination of pitched sequences and rhythms with melodies and more. Not only the motive, but already the tone is filtered from the fullness of the real clangours, noises, sounds, and is embedded in a system of at least partially ’enkaptic’ structure. In general we must note that musically not the physical-acoustical fact and technical signal but the musical-aesthetical and psycho-social sign is essential and decisive. This applies already to the elementary level of pitches. Also an acoustical fuzzily realized tone which deviates more or less from the exact pitch, a nearly wrong tone, can in the framework of the system be judged and perceived as a right tone13 . This too is a specific form of filtering. In music, being communication and art, consciousness and intentionality play a great role. The musical cluster (not to be confused with “cluster analysis”) in relation to the chord is a good example. In the cluster the vertical arrangement of pitches is quasi ’naturally’ fuzzified – densely packed, undifferentiated or not yet differentiated. In 1919 the US-American composer Henry Cowell coined the term “tone cluster” for complexes of tones sounding big and small seconds simultaneously. Such a tonal complex should be treated like a single tone as a musical unit, to be more closely defined by its ambitus (extension in the vertical tonal space), with its variable fringes. In the 1960ies the composer Pierre Boulez pointed out the connection to the “mediation of tone and sound” and the “relatedness of vertical tonal complexes to the diagonal glissando”, which also blurs the distinct and clearly different pitches for the benefit of a segmented continuum – a seemingly realistic paradox – instead of a gliding, fluid continuum as continua normally are imagined (see fig. 22.1 and 22.2). 13

See [52] and [53]. – Many very interesting types of acoustical illusions http://deutsch.ucsd.edu/ with further links (29.6.2009). See also for the special branch “Cognitive Musicology” with relations to cognitive science in general and psychology of music [12], [66], [10].

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Fig. 22.1. Left: BACH as Cluster;

right: Interval: A C.

By filtering we get intervals, here a minor third, which can be arranged successively as ’melody’ or simultaneously as ’harmony’. The chord as filtering plus superimposing is a defined, complex unit of vertically layered discrete pitches. Normally – many exceptions granted – it is organized at the core in the form of layered thirds, as can be seen in the following example fig. 22.2.

Fig. 22.2. Chord: B D F A.

22.1.4

Formation of Motives as Filtering

A seemingly exact filtering can be found in internet censorship (or even in simple search engines within a digital text, spam-filters, etc.). An example taken from my own experience in electronic communication with the administration of my university. The mail, referring to "Staatsexamen", never reached its destination. Because: Staatsex amen (States’ Examine) Staat SEX amen14 Instead of just regarding single isolated words, the context has to be taken into account - a general problem, and in detail a complicated issue. It requires on both sides - humanoid searcher and computer engine - an natural intelligence with regard to language and logic, which might be not so common in this area. Maybe Artificial Intelligence with Fuzzy Logic could compensate this lack of natural an technical intelligence at least partially. The given example relates to music in a more substantial manner, given my focus on procedures of theme and motif. Motives are shortened elements resituated or permutated. The filtering and the metamorphoses of musical motives are, being mostly fuzzy, in general more exact – and more productive. – Motif can be defined as the smallest 14

Even imitable in English: State Sex Amine. The combination of sexuality and Christianity is spicy.

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unit bearing meaning in music, semantically and syntactically. It is comparable to the morpheme in language. Motives are filtered from the total set of the music material, the store of pitches and rhythms, specifically for a piece, for a composer, or, as in the case of B-A-C-H in a manner transcending individual circumstances. That’s done during the more or less long process of composition, in different, even temporarily disrupted phases, pre- and retro-thinking, combining, cutting, amending etc., with or without conscious decisions for the respective selection or choice. And it can be done during an improvisation, with more pressure from the uninterrupted stream of time and (therefore) less possibilities for choices. A motif mostly contains about 2 to 5 tones. The upper limit should lie around six tones - but this would need to be ascertained empirically by analyzing historical styles, genres and more. As border case, not very known and reflected, there exist 1-tone-motives. A famous one of these relatively rare cases of 1-tone-motives is to be found in Alban Berg’s opera Wozzeck - the H (engl. B) as symbol of death, played by the entire orchestra in an immense crescendo, formally-dramaturgically functioning as interlude. More popular might be A. C. Jobim’s One Note Samba: only 1 tone is enough in each of the both first sections - in the melody; the harmony chances of course. Duke Ellington needs for his C-Jam Blues 2 melody tones. A motif is, according to historical style and genre, more or less a distinct and independent element of the complete work. But it is also a building unit, element of a superordinate “set”. The theme represents this larger unit. In the process the motif undergoes changes - in language respectively phonetic-phonematic - as well as semantic, within and through the given context. I mentioned morpheme in correlation to language. However, the musical motif seems to be rather more easily and extensively variable, as by permutations of sequence, as indicated earlier, augmentation or reduction, abbreviation or extension, of pitch or duration, and so on . . . At first the BACH-motif as a whole. B in English = B flat, H in English = B. For musical reasons I will stick to the German denotation of B as H. BACH BA CH BAC HCA

Already a part, the BA, is a motif: for one thing as a descending half tone step forming a topos of lament, specifically, depending on rhythm, a “sigh motif”. In a similar way the transposition to a higher pitch, CH. Even the BAC as an extension is in itself again a motif - as for example a “question motif” in Beethoven or Wagner. A question sentence in German (and in other languages) has an ascending intonation of the voice as marker. An answer to this question could sound H-C-A, thus rounding the melodic line. Also C-A descending resp. falling is the so called “Kuckucks”- or “Rufterz”, a minor third often used with texts for calling someone, in the kernel a two-syllable name as e.g. “Ru-di”. And so on. Important: The motifs are similar in structure, mainly ’logical’ and evolutionary, but very different in meaning.

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As can be seen here, the dissection or assembling, division, permutation of sequence (anagram), all without changing the intervals of the tones, and other methods lead to semantic and syntactic variations. This variety is multiplied through combinations with rhythmic variations - not to be further discussed at this point.

22.2

Models: The B-A-C-H-Motif

Basically B-A-C-H is a motif in spite of its brevity. Historically it would best be named a "soggetto", an older expression for "theme". After J.S. Bach it becomes subject to numerous variations, be it by Robert Schumann, Max Reger or Hanns Eisler or many others. 22.2.1

BACH - Variations and “Constancy of Shape” (“Gestaltkonstanz”)

Variations of the BACH-motif are traditionally created by the method of counterpoint: basic figure, inversion, retrograde, retrograde inversion or transposition. These methods constitute a first type of similarity (I). In the thinking of dodecaphony or twelvetone music these methods are reintroduced. Counterpoint as procedure of variation and transformation Inversion: Intervals, spaces accordingly running in (vertical) opposite direction in the tonal space -what went up, now goes down, and vice versa. Retrograde (Palindrom): order of tones inverted horizontally in time, back to front.

1) BACH 2) HCAB 3) BHAsA 4) BACH

Grundgestalt Krebs Umkehrung oder (transponiert) Umkehrung: Krebsumkehrung

basic shape/form retrograde inversion or transposed inversion retrograde inversion

Basic form and retrograde in this case remain at an identical pitch. The nontransposed inversion (3) introduces new pitches (As). Transposition: 11 further tonal steps within the 12-tone scale of the well tempered octave (last transposition = 1), altogether 12 steps of transposition Here we can see a chart of possible variations. B as initial tone is given as invariant, stable. Transposition means in the shifting of the motif to different pitches. The similarity in this case includes the differences. In isolation the transposed forms are identical relative to pitch. Only with perfect/absolute pitch can the differences be detected within a framework of pitch classes. But even withous this the forms in this context are noticeably different, when the motif is repeated directly and seriatim at these different pitches.

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01

BACH

02 03 04 05 06 07 08 09 10 11 12

H BDesC C HDCis Cis ... D Dis E F Fis G Gis A

B=1

The symmetries or rather reflections (or inflection) over a horizontal or vertical axis as well as the shift of transposition parallel to the axis all preserve the “constancy of shape” (Gestaltkonstanz) to various degrees. Boundaries of this constancy in the reflections in tonal space are rotations by 90 degrees or such. As remarked the “Gestalt” of a motif or melody etc. is relatively stable at least in its kernel; on the other side it’s one of the tasks of Fuzzy Logic to analyse also quantitatively the grades and degrees of the “more/less” similarities of variants and starting point. Permutation15 01 02 03

BACH ABHC ABHC etc. (as exhaustive schedule see fig. 22.3)

In addition the procedure of permutation, meaning, as known, the rearrangement of sequence, constitutes another type of similarity, with internal grades of similarity: In general the similarity is mathematically greater than musically, in particular varying – ABHC e.g. is a simple chromatic sequence with relatively little significance as shape. What remains unchanged here is the ambitus of the motif, the border or edges/ contour as upper and lower pitch demarcation - remember the cluster I mentioned earlier. This ambitus changes in the following variations. The intervals, the spaces between tones are changed: augmented or extended. (In principle a reduction or compression is possible - but not with half tones here being the smallest interval). Nevertheless, contours remain visible. Here we have variations of the basic shape, with more and more augmented intervals. 15

The whole Viola-Concerto of Karl Amadeus Hartmann, the Konzert für Bratsche mit Klavier, begleitet von Bläsern und Schlagzeug (1954/1955) is a study in and about BACH.

22.2 Models: The B-A-C-H-Motif

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Fig. 22.3. BACH-Permutations (underlying the work: Karl Amadeus Hartmann, Konzert für Bratsche mit Klavier, begleitet von Bläsern und Schlagzeug (1954/1955) [18], p. 31)

Formula: BACH −1 + 3 − 1 1 semitone − descending + ascending

Variation of intervals BAC −1 + 3 − 1 B As C −2 + 4 − 1 B As C −2 + 4 − 2 B As Des BGC BGC BGC B G Des . . . etc.

H H B C H B A C

Contour and core pattern remain as a constant shape, at least to a certain degree, but still intact and visible. However, the similarity is reduced more and more. The sequence and cross structure as an essential aspect (see below) is maintained even when the tone intervals are augmented. The borders are once again fluid, the edges fringed – a typical marker of Fuzzy Logic. Furthermore the borderlines are fringed; there is no clearly defined set of variations – a frequent occurrence in music. These borders can be verified empirically by comparably simple experiments.

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As common in Fuzzy Logic, the degrees of fuzziness could and should be precisely quantified. Thereby we have an opulent quantity of variations. On the one hand they are precisely defined set, created through various procedures following exact rules or algorithms. So far the variations result from a simply binary logic: EITHER B before A OR A before B etcetera, as mentioned above. But: The shapes differ from the basic shape according to the degree of their similarity. Insofar the variations follow a Fuzzy Logic with fringed borders, transitions: a structured or segmented continuum once more. Precisely this potential variety is employed by J.S. Bach himself. Of the countless instances to be found in his work with just as many variations I will show only one example. In the 14th and last “Contrapunctus” in his last opus Kunst der Fuge (Art of the Fugue) he quotes himself and his name, in a way leaving a signature (bar 194f.) Furthermore the BACH-signature was implied and intended right from the start in this huge cycle (see fig. 22.4).16

Fig. 22.4. J.S. Bach, Kunst der Fuge, Contrapunctus 14. 16

Detailled [30].

22.2 Models: The B-A-C-H-Motif

22.2.2

435

BACH as Emblem in and before J.S. Bach – The Cross as Generating Substructure

With BACH the filtering of the core motif is unproblematic. However, even here we encounter problems at the fringes in the transitions, degrees of similarity - and even types and sorts of similarity, as in a subset (group) of cyclic replacements within the permutations. On a meta-level the BACH itself is a self-structured “set of “fuzzy sets”; and someone more qualified in math than me could in a second step of analysis determine the exact degree of relatedness. St. Andrew’s cross, rotated But there is a further, rather hidden generating and regulating agent of the BACHcipher. The pious Christian J. S. Bach consciously made use of it. Historically these traces would require empirical scrutiny to be securely confirmed. The structure is that of the cross. - On the one hand J.S. Bach inscribes his name into the tonal arrangement by using the BACH-motif, motivated biographically. A coincidence readily made use of by Bach, and due to his extraordinary position in music history an beyond and due to its structure this motif has lived through a spectacular career in music history and composition. On the other hand this motive via its cross structure makes a reference to religion, mentality and history. Bach delivers a profession. This rhetorical aspect, together with some wordings found in canons or cantatas, implies even an emblematic metaphor. The shape of the cross is primarily conveyed by the notation, rather for the optic than the acoustic sense. An it is the St. Andrew’s cross, a diagonal (not the orthogonal “normal” cross), used as railway crossing sign - and additionally laid down with a rotation of 90◦ (see fig. 22.5).

Fig. 22.5. BACH with cross. J.S. Bach: Duett F-Dur BWV 803, im III. Teil der “ClavierÜbung” (Druck 1739) Shape of the cross in fractions on various levels of the piece. Ulrich Siegele, quoted and extensively discussed in [30], p. 906f.

In the duet BWV 803 Bach probably inflected the Bach-cross-shape on various levels of his work. This figure or configuration is in no way to be heard. It is possible, but not confirmed, that Bach himself arranged this configuration. But geometric arrangements generally play an important role in music17 , at least, African metrical and rhythmical patterns referring to wettlework included, from Guillaume 17

Cf. e.g. [55], [56].

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Machault in the 14th century till Iannis Xenakis in the 20th and afterwords. Therefore we can readily assume Bach’s intentionality as a pious composer, giving the BACH-motif such a prominent position. (Even if this question is always discussed, mainly with the not very honourable intention to negate semantics in music, it is principally mostly of secondary importance, if motivic metamorphoses, inventions, variations and developments are made consciously or unconsciously. For it might be one of the characteristics of creativity that both psychic spheres are permeable to each other, more fluid and moveable than commonly and in non-creative persons or states of mind.) Acoustically the cross-structure, articulated as a sequence, refers to further traditions of religiously influenced music, just as the dense chromaticism, completely within the minimal space of ABHC. Sequence - Dies irae First on sequence: another line of tradition comes into play. The sequencing crossmotif is related to an old and also countlessly reproduced motif: the “Dies irae” (Day of Wrath) of the Requiem. The famous Dies irae, the sequence of the requiem, the mass on Judgement Day, has a similar chiastic structure - the cross again. The procedure of sequencing creates it. Sequence means: the same interval or minimotif is repeated at a different pitch, higher or lower. - The formula (transposed from F to C): di-es / i-rae ... (dies illa) -1 +1 -3 CH CA (H-G-A-A)

I illustrate the motive with musical examples from the Finale of Gustav Mahler’s 2nd Symphony. It begins with a scene of the Last Judgement - which, as later in the last parts of the movements is revealed, is adjourned in favour of a redemption and reconciliation. But first the “Di-es i-rae” (see fig. 22.6).

Fig. 22.6. Musical ex. Mahler, 2nd Symphony, Floros 1985, vol. III, p. 70.

Mahler variates the Dies-irae-motif in extenso: with intervallic formulas as -1 +1 -5 or -1 +1 -6 (instead of the original -1 +1 -3).18 We see, that the identical 18

Musical examples [20], p. 71.

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initial core is retained, the cauda (or tail) varied. And then, too, Mahler makes it similar or transforms it into his semantic counterpoint, the “theme of resurrection” (“Auferstehungsthema”) – formulas −2 + 2 − 4, . . .− 2 + 2 + 5, partial inversion the of the initial core +1 -1 -3; +4 +2 -2 . The similarities are as evident as fuzzy; once again a challenge for Fuzzy Logic. Retained is, as contour, the return to the beginning tone. The descending diatonic line Dies irae, bar 3s., and Resurrection, bar 5s. as sequence complete the metamorphosis of the motives. (des-C-B... As / As-G-F-Es; and inverted and expanded to a 6-tone-scale, bar 6s., As-B-c-des-es-f) – see fig. 22.7).

Fig. 22.7. Musical ex. Mahler, 2nd Symphony, Finale (Floros 1985, vol. III, p. 69.

The Dies-irae-motive remains within the frame of a diatonic, primarily 7-tone system. Bach, however, already applied the 12-tone system. The dense chromaticism BACH can be extended to a 6-tone motif, which as a lament-phrase entirely fills the tetrachord, the frame of the fourth: D d minor C H B A or ABHC C sharp D. Lament - Miserere Characteristic for the BACH-motif is the easily reached combination with text of the lament of Mi-se-re-re, equivalent to Kyrie eleison (Lord, have mercy). The semantics of tone and word are in conjunction with the doubled tone of lament, the descending minor second (of course there are also other Miserere-motives). Guillaume Dufay uses the Bach-cross-structure in his Ave Regina Celorum: Eb-D-F-E = 1 + 3 − 1 to the words in the two upper voices S “MI-SE-” and A1 “-SE-” and in the two other voices as B-A-C-H A2 “A-” and B “-RE-” (see fig. 22.8). Again this can be deduced from the descending half tone, the submotif BA or CH, as mentioned earlier. Josquin des Prez in the Agnus Dei from the Mass Missa L’homme armé super voces musicales uses the doubled Lamento-semitonestep as ABHC ascending, adapted to the keyword “(tol-)lis” (you bear). (See fig. 22.9)

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Fig. 22.8. Guillaume Dufay: Ave Regina celorum. [30], p. 322.

Fig. 22.9. Josquin des Prez: Agnus Dei from the Mass Missa L’homme armé super voces musicales [30], p. 323

The BACH-motive thus results historically as well as esthetico-’logically’ in its structure and in its semantic field from manifold, crossing and self reinforcing reasons and motives – a multiple, complex logic. – Quasi as a palimpsest the MiserereLament gets an ambiguous, double sense in the Agnus Dei of Schuberts late Es-DurMesse ((mass in E flat major) and the late Lied Der Doppelgänger (Heinrich Heine; 1828). In both cases the same 4-tone resp. 4-chord figure (in Doppelgänger in the sinister key B-minor) is functioning as an ostinato throughout the whole movement. The chronology is uncertain; therefore Schubert probably connects sacred and secular spheres not blasphemously but under a sign of sighing as a nation and a person crucified by repressive social-political circumstances of the Habsburgian restauration. (See fig. 22.10.)

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Intervallic formula −1 + 4 − 1 = B A Cis C H Ais d cis

Fig. 22.10. Franz Schubert: Der Doppelgänger (Heinrich Heine, 1828)

22.2.3

Arnold Schönberg’s BACH Variations

In the large-scale Variations for orchestra op. 31 (1928) Schönberg uses the twelve tone technique (dodecaphony) for the first time in connection with a large work of classic dimension. Starting point and foundation of pitch organisation is a specifically organised twelve tone row.

Fig. 22.11. Basic Series Schönberg: Variationen für Orchester op. 31

The tone row forms or shapes constitute an already in itself variating generative substructure. (See fig. 22.11.) Schönberg uses complex techniques of composition, which incorporate classical, especially counterpoint-devices. Therewith he attempts to link his radical innovation back to tradition as a means of justification – more clearly and timidly this can be found with his student Anton Webern. One of these means is the partial identification with Bach. The “figure” – as defined in Gestalt psychology – being a primer in our perception, here the concrete formation of motif and theme and the variation of the underlying sequential forms, becomes the “ground” or rather background. From this evolves the decisive part. As pertaining to sound Schönberg foregrounds the emblem as a quote. He almost chisels the BACH out of various sequence forms. “Soggetto cavato”, excavated theme,was the phrase used in the Renaissance for such a semantically coined motif of notes expressed in letters.

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Arnold Schönberg becomes BACH 1. A rnol D eS CH ön B E rG A ...... D eS CH ..... B E.G 2. There is even a second, more hidden appearance of the motto motif, to be found by transposition: D DeS e eS = intervallic structure the same as with B ACH −1 + 3 − 1 3. In the succession inside the name there’s a permutation: A CH B ACHB 2342 1234 BACH Even if the holy name BACH pervades the whole 12-tone-structure, only in one crucial moment during the Introduction Schönberg quotes unmistakeably the MottoMotif, similar to a cantus firmus in the bass line (see fig. 22.12):

Fig. 22.12. BACH as Emblematic Rebus-Figure inside the Variants of the Series in Schönberg: Variationen für Orchester op. 31 [9], p. 9.

There is evidently no Fuzzy Logic necessary for excavating this motto-subject, but however for finding the influences and similarities to the 12-tonerow in its structure and its variants and other motives in the course of the Variationen.

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22.3

Outlook: Generalization and Connection of Fuzzy Logic with Further Musical and Artistic Subject Fields

22.3.1

Learning Neuronal Networks and BACH-Variations in the Internet

Schönberg dissects the BACH emblem in its historical form from a pitch material, that at least in its acoustic appearance seems amorph, unshapely, chaotic: Again filtering as principle. BACH emerges as a quote and clearly discernible figure, clearly outlined within the context, which is more and more “defuzzified” in respect to this focus. Analogues processing procedures we find in neuronal networks, in the living brain as in its simulations via Artificial Intelligence.19 A short example with letters, similar to BACH see fig. 22.13.20

Fig. 22.13. A Kohonen-Network is learning. “To the left are four out of 26 input patterns. In the center the Kohonen layers are represented, at the top at the beginning of training, in the middle after 200 viewings of the alphabet and below after 700. To the right a graphic representation of the weight of the synapses of the dominant neuron during presentation of the letters. The letters on the neurons show the dominant neurons. The four examples show how the weight of the synapses changes during the experiment. At first few synaptic weights can be seen, since no learning has taken place yet. After 200 repetitions (center) a sensible distribution can be detected, but still without much precision. After 700 repetitions (bottom) the letters are distributed evenly according to their graphic attributes [. . . ]”[75], p. 106. 19 20

Cf.: [1]. Cf. [68] - Referring to music e.g.: [80], [81].

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The search for motives similar to BACH in the internet is nearly an experiment looking for mechanisms of pattern perception, implying learning processes. One will encounter analogies to the physiological “neuronal” assimilation of motives, artificial intelligence and networks. A melody-search-engine (accessible over wikipedia) reflects in a technical manner some of the problem of intramusical similarities and correlations. This search must be regarded as a special case of storage and pattern recognition as well as filtering in general. The internet search for melodies runs according to so-called keymotives. Probably the algorithms behind this are not precise - or fuzzy - enough. The program recognizes half tones and chromatic patterns, but takes them into account quite randomly (as far as I could see). Some of the results I got by this special type of fuzzy logic at least were interesting: they pointed at similarities starting with BACH which with conscious and classical research methods nearly never would have been found: César Franck, Les Djinns / Grieg, Peer Gynt (Hall of the King of the Mountains) / El Garrotín – this a remarkable discovery, since the Garrotín dance is of Asturian origin and belongs to the group of the Flamenco!21 If these similarities make sense - that is another story. Searching with the various interfaces mostly leads to the same results.22 The vital, essential aspects, the chromatic and sequenced or chiastic order, is not taken into consideration by the search filter. As often the search engine delivers a vast amount of results, but mostly only seemingly valid or helpful information. A 4-tone motif is excluded from the contour search, this requires at least five tones - feasible, but not sensible. The musically essential evaluation of the quality of significance, especially of the cross structure, is not possible within the parameters of this program. The sequence of events reflects dimly some degree of similarity, but not very precisely or logically. In total I would dare say, this is in part more a fuzzy logic than a truly multi-dimensional Fuzzy Logic. Nonetheless, at least one result was productive for music analysis, even if at the first glance seemingly estranging: the cross structure, underlying a famous thema of Beethoven. (See fig. 22.14 and 22.15).

Fig. 22.14. Beethoven, 5th symphony, 1st movement, main theme.

As model form my ideas about music and Fuzzy Logic I chose the pitch motif. The main motif of Beethoven’s 5th symphony added another model, where the 21 22

Cf. the article using mathematical methods, tendentially also Fuzzy Logic: [13]. The interesting “Query by humming” as a very different method would deserve a further research and interpretation.

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rhythmic configuration was more essential. Not without reason the search engine chose this as an example. For the sequence and cross figure plays a vital role in the Beethoven motif. Therefore it is a variation of BACH with very augmented intervals.

Fig. 22.15. Contour of the theme.

A general contour is retained by the identical diastematic movement - + - (descending / ascending / descending). BACH −1 + 3 − 1 Dies irae −1 + 1 − 3 Beethoven −4 + 2 − 3 This leeway is left by Fuzzy Logic. Surprising is how the detour of a negatively fuzzy filtering process can lead to a highly relevant connection. However, the motif undergoes considerable changes in character. Instead of Christian devotion Beethoven articulates with his “fate motif” the uprising against fate: the self-confident and active citizen. In the Second World War this motif became the acoustical signal of the BBC broadcasts to Germany. Primarily it had been found as “V” = Victory as common abbreviation both for the Flemish and Wallonian resistance in Belgium (“vrijheid”/“victoire”). Secondarily the similarity, nearly identity of this “V” as Morse character and of Beethoven’s motif was discovered. It was indeed adequate not only structurally but also semantically as symbol of resistance against Nazism. 22.3.2

Fuzzy Logic and the Existence Forms of Music

Fuzzy processes and fuzzy facts in a different musical context are reflected and quasi ’fractally’ reproduced on higher levels reproduced in the complex of forms of appearance or existence-forms of music (’Existenzformen’)23 - the sequence in time 23

Cf. e.g. [71]. He mentions only four main segments - Composer, Performer, Room, Listener. Lengthily and with at least thirteen different forms see [39].

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of different states, stadiums of conception, productive and receptive realization of a piece of music between idea and storing in memory. In all these forms of existence there are two opposed processes in regard to filtering and fuzzification. The existence form “Work as complete Idea” is the culmination and turning point in regard to precision. From here on the indistinctness increases, unconscious or conscious formation of variations and more – just to mention the problem of musical interpretation. Altogether this sequence of existence forms results in a sort of unintentional or directionless morphing - which doesn’t only exist for pictures, but also for music. In other words: The whole of the artwork as processing identity evolves and is established in and by the different forms of existence. The identity is, again following the configuration of core and contours, in itself necessarily fuzzy. But this does not mean the “anything goes”, whateverism and mere subjectivism: the objective measure of the differences is qualitatively and quantitatively definable. 22.3.3

Interdisciplinarity: Fuzzy Logic as an Underlying Generative form of Thought: Towards a General Theory of Similarity?

Till now I did not find really convincing algorithms with which to measure similarities exactly and quantitatively. To produce and to formulate it would transcend my mathematical capabilities - the indispensible ’inner interdisciplinarity in particular of social and art sciences as musicology reaches here personal limits. ’Inner interdisciplinarity’ I define as cooperation of different disciplines, in particular methods and materials, inside of a discipline, asking for connections between the three main realms of reality, i.e. nature, society, mind. To evolve and to practice this ’inner interdisciplinarity’ would, by the way, in return not be inappropriate for other sciences and humanities. One way or another: Here real interdisciplinarity is asked, especially from mathematics, informatics and computer sciences and technologies in general. In respect to music and from the view of musicology they often stay restricted to very elementary matters and questions. (From the other side much of my deliberations may appear as naïve or primitive.) Collectively and concertedly we anyway would proceed faster and better. Scientific progress demands such cooperation beyond declarations. Many problems and demands could be listed. I mention in conclusion only one, but a central one. One of the great lacunae, a gap to be bridges by common efforts, and of common relevance for different spheres of reality as of disciplinary matters, is what I would call a ’general theory of similarity’. Of course similarities of styles (by the way in music24 as in literature and other arts) are important in different fields and methods of research25 and intensively studied. Fuzzy Logic would be completely in its element here and find one of its proper purposes. But (as far as I see) there is no very clear distinction, differentiation, hierarchical and reflected system of terms (and meanings), systematisation in general/special, relations and interrelations etc. 24 25

See e.g. [16]. For searching stylistic similarities and patterns in arts and architecture cf. e.g. [74]; for filtering similarities of words in different dimensions (“Wortähnlichkeiten”) from a corpus for scopes of psychological-linguistical research e.g. [42], p. 13.

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I think of categories as reflection, representation, image, isomorphy, homomorphy, analogy, mirroring etc. ’Mimesis’ e.g. is specific for arts, one of the four essential an basic principles of art (with katharsis, aisthesis and poiesis). Art is mirroring, but as a special ’mirror’ it is as reality and not as reality: It is similar.26 Our main task in this field is twofold. First the systematic application and elaboration of Fuzzy Logic within each discipline - including the partial non-applicability. This will be long lasting and may lie at least partially in the future for disciplines more or less far from the core of genesis, development and application of Fuzzy Logic. The second task we have begun and are beginning here and now with congress and book: systematic analysis and discussion of overlaps and points of contact, of analogies and isomorphisms between Fuzzy Logic and the Logics of other matters and disciplines.

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26

’Mimesis’ is not meant as eidetic or pictorial. It is totally independent from any stilistic or idiomatic options, be it naturalistic or “abstract”-nonfigurative.

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[31] Heidenreich, S.: Form und Filter - Algorithmen der Bilderverarbeitung und Stilanalyse. Zeitenblicke 2(1) (2003), http://www.zeitenblicke.historicum.net/2003/01/heidenreich/index.html (January 4, 2010) [32] Heister, H.-W.: Natur, Geist, Gesellschaft. Musik-Anthropologie im Gefüge der Musikwissenschaft. In: Petersen, P., Rösing, H. (Hg.) 50 Jahre Musikwissenschaftliches Institut in Hamburg. Bestandsaufnahme - aktuelle Forschung - Ausblick (Hamburger Jahrbuch für Musikwissenschaft, Bd. 16), Peter Lang, pp. 65–82. Europäischer Verlag der Wissenschaften, Frankfurt am Main (1999) [33] Heister, H.-W.: Geschlechterverhältnisse als Modell. Musik-Anthropologie: Gegenstände, Themen, Forschungsperspektiven. In: Stahnke, M. (Hg.) Musik - nicht ohne Worte. Beiträge zu aktuellen Fragen aus Komposition, Musiktheorie und Musikwissenschaft (Musik und. Eine Schriftenreihe der Hochschule für Musik und Theater Hamburg, Bd. 2), pp. 13–46. Von Bockel-Verlag, Hamburg (2000) [34] Heister, H.-W.: Natura, spirito, società. Antropologia musicale e nuova musicologia. Musica/Realtà 22(65), 25–54 (Part I); 22(66), 115–150 (Part II) (2001) [35] Heister, H.-W.: Konstanz und Wandel. Zu Hartmanns konzertanten Werken zwischen 1930 und 1955. In: Dibelius, U. (Hg.) Karl Amadeus Hartmann. Komponist im Widerstreit, pp. 147–173. Bärenreiter-Verlag, Kassel (2004) [36] Heister, H.-W.: Mimetische Zeremonie - Gesamtkunstwerk und alle Sinne. Aspekte eines Konzepts. In: Mimetische Zeremonien - Musik als Spiel, Ritual, Kunst (Musik und. Eine Schriftenreihe der Hochschule für Musik und Theater Hamburg. Neue Folge, Bd. 7), pp. 143–185. Weidler Buchverlag, Berlin (2007) [37] Heister, H.-W.: Homöostase historisch: Zu Fragen der Regelung des musikalischen Tonsatz-Systems, Beitrag zum Kolloquium 1948-2008: 60 Jahre Kybernetik - eine nach wie vor junge Wissenschaft?!, Jahrestagung der Deutschen Gesellschaft für Kybernetik, Berlin (November 22, 2008) [38] Heister, H.-W.: Weltenmusik und Menschenmusik. Ein Knotenpunkt in den Traditionslinien der universistischen und mathematisch-quadrivialen Musikauffassung. In: Banse, G., Küttler, W., Roswitha, M. (Hrsg.) Die Mathematik im System der Wissenschaften (Abhandlungen der Leibniz-Sozietät der Wissenschaften, Bd. 24), pp. 193– 216. trafo Wissenschaftsverlag, Berlin (2009) [39] Heister, H.-W.: Kybernetik, Fuzzy Logic und die Existenzformen der Musik. Ein vorläufiger Entwurf. In: Institut für Interkulturelle Innovationsforschung (Hg.) Innovation aus Tradition - Festschrift Hermann Rauhe zum 80, pp. 93–115. Mainz, Geburtstag (2010) [40] Heister, H.-W.: Encodierung und Decodierung im Musikprozeß, Vortrag beim Symposion Musik und Psychoanalyse hören voneinander, Institut für Musiktherapie, Hochschule für Musik und Theater, Hamburg (to appear, 2011) (in print) [41] Heister, H.-W.: Klang und Kosmos. Mathematik, Musikdenken, Musik (work in progress) [42] Heister, J., et al.: dlexDB - eine lexikalische Datenbank für die psychologische und linguistische Forschung. Psychologische Rundschau 62(1), 10–20 [43] Hochstein, S.: “Runnin” und “Empire State of Mind”: Zwei Modelle aus den Subgenres Independent-und Mainstream-Hip-Hop, Hausarbeit für die Erste Staatsprüfung im Fach Musik, Hochschule für Musik und Theater Hamburg (Juni 2010) [44] Hörz, H., Liebscher, H., Löther, R., Wollgast, S. (Hrsg.): Philosophie und Naturwissenschaften. Wörterbuch zu den philosophischen Fragen der Naturwissenschaften. Dietz-Verlag, Berlin (1983)

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[45] Hörz, H., Liebscher, H., Löther, R., Wollgast, S. (Hrsg.): Philosophie und Naturwissenschaften. Wörterbuch zu den philosophischen Fragen der Naturwissenschaften, Dietz (1983), http://www.alphagalileo.org/VieewItem.aspx?ItemId=56202\&CultureCode=en [46] Hüther, G.: Die Macht der inneren Bilder. Wie Visionen das Gehirn, den Menschen und die Welt verändern. Vandenhoeck & Ruprecht, Göttingen (2006) [47] Kaehr, R., Ditterich, J.: Einübung in eine andere Lektüre: Diagramm einer Rekonstruktion der Güntherschen Theorie der Negativsprachen, Philosophisches Jahrbuch, 86. Jhg, 385–408 (1979), http://www.vordenker.de/ggphilosophy/kaehr_einuebung.pdf (July 23, 2009) [48] Klaus, G., Liebscher, H. (Hrsg.): Wörterbuch der Kybernetik. Dietz-Verla, Berlin (1976) [49] Kosko, B.: Die Zukunft ist fuzzy. Unscharfe Logik verändert die Welt. Piper, München und Zürich (1999) ; English: Kosko, B.: The Fuzzy Future: From Society and Science to Heaven in a Chip. Random House/Harmony Books (1999) [50] Kühn, C.: Kompositionsgeschichte in kommentierten. Beispielen, Kassel, et al. Bärenreiter-Studienbücher Musik, vol. 9 (1998) [51] Leman, M.: Expressing Coherence of Musical Perception in Formal Logic. In: [3], pp. 185–198 [52] León, T., Liern, V.: Obtaining the compatibility between musicians using soft computing. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds.) IPMU 2010. Communications in Computer and Information Science, vol. 81, pp. 75–84. Springer, Heidelberg (2010) [53] León, T., Carrión, V.L.: Mathematics and Soft Computing in Music (in this volume) [54] Lichtenberg, G.C.: Sudelbuch A. Erstes bis fünftes Heft (1765-1770) [55] Mazzola, G.: The Topos Geometry of Musical Logic. In: [3], pp. 199–214 [56] Mazzola, G., von Daniel Muzzulini, u.M., von Georg Rainer Hofmann, m.e.B.: Geometrie der Töne: Elemente der mathematischen Musiktheorie. Birkhäuser, Basel (1990) [57] Messing, J.: Allgemeine Theorie des menschlichen Bewusstseins. Weidler Buchverlag, Berlin (1999) [58] Moles, A.A.: Informationstheorie und ästhetische Wahrnehmung. Dumont Verlag, Köln (1971) [59] Morris, D.: Biologie der Kunst. Ein Beitrag zur Untersuchung bildnerischer Verhaltensweisen bei Menschenaffen und zur Grundlagenforschung der Kunst. Karl RauchVerlag, Düsseldorf (1963); English: Morris, D.: The Biology of Art. Methuen, London (1962) [60] Müller, G.K., Christa: Geheimnisse der Pflanzenwelt. Manuscriptum, Waltrop und Leipzig (2003) [61] Neuhoff, H.: Mensch - Musik - Maschine. Das digitale Credo und seine Probleme. Eine Antwort auf Reinhard Kopiez. Musica 1, 27–32 (1996) [62] Nicolas, F.: Questions of Logic: Writing, Dialectics and Musical Strategies. In: [3], pp. 89–112 [63] Oberhoff, B. (Hg.): Das Unbewußte in der Musik. Imago Psychosozial-Verlag, Gießen (2002) [64] Oesch, H., von Max Haas, U.M., Haller, P.: Außreuropäische Musik (Teil 2), Neues Handbuch der Musikwissenschaft, 12 Bde., Hrg. von Carl Dahlhaus (+). Fortgeführt von Hermann Danuser, Bd. 9. Laaber-Verlag, Laaber (1987) [65] Pias, C.: Hollerith ’gefiederter’ Kristalle. Kunst, Wissenschaft und Computer in Zeiten der Kybernetik. In: Hagner, M., Hörl, E. (Hrsg.) Die Transformation des Humanen. Beiträge zur Kulturgeschichte der Kybernetik, pp. 72–106. Frankfurt am Main, Suhrkamp (2008)

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[66] Risset, J.-C.: Computing Musical Sound. In: [3], pp. 215–232 [67] Schacter, D.L.: Wir sind Erinnerung. Gedächtnis und Persönlichkeit. Rowohlt, Reinbek (2001) [68] Schwamm, D.: Fuzzy Logic und neuronale Netze (March 1994), http://www.henrys.de/daniel/ index.php?cmd=texte_fuzzy-logic-neuronale-netzwerke.htm (June 17,2010) [69] Seising, R.: Die Fuzzifizierung der Systeme. Die Entstehung der Fuzzy Set Theorie und ihrer ersten Anwendungen - Ihre Entwicklung bis in die 70er Jahre des 20. Jahrhunderts. Franz Steiner Verlag, Stuttgart (2005); (English: Seising, R.: The Fuzzification of Systems: The Genesis of Fuzzy Set Theory and Its Initial Applications - Developments Up to the 1970s. Springer-Verlag, Berlin (2007) [70] Seising, R.: Information - ein Begriff im Fluss. 60 Jahre ’Mathematische Theorie der Kommunikation’, Beitrag zum Kolloquium 1948 -2008: 60 Jahre Kybernetik - eine nach wie vor junge Wissenschaft?! Vortrag im Rahmen der Jahrestagung der Deutschen Gesellschaft für Kybernetik, Berlin, November 22 (2008) [71] Serra, X.:The Musical Communication Chain and its Modeling. In: [3], pp. 243-256 [72] Serres, M., Farouki, N. (Hrsg.): Thesaurus der exakten Wissenschaften. Zweitausendeins Verlag, Frankfurt a. M (2000); (Serres, M., Farouki, N.: Le Trésor. Dictionnaire des Sciences. Flammarion, Paris (1997) [73] Solms, M., Turnbull, O.: Das Gehirn und die innere Welt. Neurowissenschaft und Psychoanalyse. Patmos-Verlag, Düsseldorf (2002); (English: Solms, M., Turnbull, O.: The Brain and the Inner World. Karnac/Other Press, London (2002) [74] Sondereguer, C.: Sistemas compositivos Amerindos. Morfoproporcionalidad / El concepto arquitectonico-escultorico en Amerindia. Praxis de un pensmiento morfoespacial, urbanistico y simbolico, Buenos Aires 2010 (Corregidor) (2010) [75] Spitzer, M.: Geist im Netz. Spektrum Akademischer Verlag, Heidelberg (1996) [76] Spitzer, M.: Musik im Kopf. Hören, Musizieren, Verstehen und Erleben im neuronalen Netzwerk. Schattauer, Stuttgart 5. Auflage (2005) [77] Toiviainen, P.: Musikalische Wahrnehmung und Kognition im Computermodell. In: Bruhn, H., Kopiez, R., Lehmann, A.C. (Hrgs.) Musikpsychologie. Das neue Handbuch, Rowohlt, Reinbek (2006) [78] Vogler, H.: Polyphonie: Komplexität der Komposition versus Kapazität der rezeptiven Verarbeitung. Staatsexamensarbeit Hochschule für Musik und Theater, Hamburg (2010) [79] Völz, H.: Computer und Kunst. Urania, Leipzig, Jena und Berlin (1st edn., 1988, 2nd edn.,1990) [80] Weyde, T.: Lern- und wissensbasierte Analyse von Rhythmen - zur Entwicklung eines Neuro-Fuzzy-Systems für Erkennung und Vergleich rhythmischer Muster und Strukturen. Dissertation, Universität Osnabrück (2002) [81] Weyde, T., Dalinghaus, K.: Recognition of Musical Rhythm Patterns Based on a NeuroFuzzy-System. In: Dagli, C.H. (Hg.) Proceedings of the 11th Conference on Artificial Neural Networks in Engineering (ANNIE 2001), St. Louis, Missouri (2001) [82] Wiener, N.: Kybernetik. Regelung und Nachrichtenübertragung in Lebewesen und Maschine. Rowohlt, Reinbek (1968); English: Wiener, N.: Cybernetics or Control and Communications in the Animal and the Machine. MIT Press, Cambridge, Massachusetts (1948)

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[83] Wieser, W.: Organismen, Strukturen, Maschinen. Zu einer Lehre vom Organismus. Fischer Bücherei, Frankfurt am Main (1959) [84] Wolf, G.(Hrsg.): BI-Lexikon Neurobiologie. Bibliographisches Institut der DDR, Leipzig (1988) [85] Zadeh, L.A., Polak, E. (eds.): System Theory. McGraw-Hill, New York (1969) [86] Zadeh, L.A.: From Computing with Numbers to Computing with Words - from Manipulation of Measurements to Manipulation of Perceptions. IEEE Transactions of Circuits and Systems 45(1), 105–119 (1999) [87] Zadeh, L.A.: Fuzzy Sets. Information and Control 8, 338–353 (1965) [88] Zadeh, L.A.: Linguistic Cybernetics. In: Proceedings of the International Symposium on Systems Sciences and Cybernetics, Oxford University, pp. 1607–1617 (1972)

23 Mathematics and Soft Computing in Music Teresa León and Vicente Liern

23.1

Introduction

Nowadays, emphasizing the strong relationship between music and mathematics seems unnecessary. Mathematics is the fundamental tool for dealing with the physical processes that explain music but it is also in the very essence of this art. How to choose the musical notes, the tonalities, the tempos and even some methods of composition is pure mathematics. In the sixth century B.C. the Pythagoreans completed and transmitted the Chaldean practice of selecting musical notes from the proportions between tight strings. They created a link between music and mathematics which has still not been broken. An example of this relationship is the use, sometimes intuitive, of the golden ratio in the sonatas of Mozart, Beethoven’s Fifth Symphony or more recently in pieces by Bartok, Messiaen and Stockhausen. Besides, mathematicians throughout time have made music their object of study and now, both in musical and mathematical journals and in the Internet, many documents can be found in which mathematics is used in a practical way in the creation or analysis of musical pieces. One question is to explore the common ground between these disciplines and quite another is the use that musicians make of the models and the solutions provided by mathematics. For example, the musical notes, the first elements which music works with, are defined for each tuning system as very specific frequencies, but the instrumentalist knows that a small change in these values does not have serious consequences. In fact, sometimes consensus is only reached if the entire orchestra alters the theoretical pitches. Does this mean that musicians must restrict their use of mathematics to the theoretical aspects? In our opinion, what happens is that musicians implicitly handle very complex mathematical processes involving some uncertainty in the concepts and this is better explained in terms of fuzzy logic (see [12]). Another example: why do two different orchestras and two directors offer such different versions of the same work? Our answer to this question is that a musical score is a very fuzzy system. Composers invent sound structures, they are creators. If they develop such musical ideas and notate them in a musical score, the expert re-creators (instrumentalists) interpret the notation and transform it into sound. The creator and the instrumentalist are not usually the same person and different types of uncertainty may be present at many stages of this process. One example of a fuzzy approach to music and art is provided by J.S. Bach, who did not prescribe the tempo, tuning or even the instrumentation of some of his compositions in the corresponding scores. R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 451–465. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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23 Mathematics and Soft Computing in Music

Fig. 23.1. Excerpt from “El jove jardí de Joanot” by Llorenç Barber.

El jove jardí de Joanot, a piece for dolçaina and drums by the Valencian composer Llorenç Barber (1948-), is an ideal example of the unconventional notation practices and performance expectations found in contemporary pieces. Figure 1 displays several bars from the piece in which the composer uses non-traditional notation. For instance, we can read the instruction “NOCTURNO FANTASIOSO E IRREGULAR, QUASI LAMENTUOSO”. Finally, we would like to cite a paper by J. Haluska [14] where a “type of uncertainty related to the creativity and psyche of the interpreter and listener of the composition” is mentioned. “Perhaps, the idea is best visible on Indian ragas when the same raga is played in different pitch systems depending on the mood, year season, occasion, place, etc.” If we perform a search in a database entering the keywords “music” and “fuzzy” we do not obtain so many retrievals as could be expected. This may be surprising due to the essentially subjective characteristics of music. We do not intend to make an exhaustive review of the literature but to offer some insights into the use of soft computing techniques in music. In our search we find out that, in comparison to other areas, fuzzy logic has seldom been used in the artistic and creative fields. Only a few applications related to creative activities, such as musical composition and sound synthesis, have been reported in the literature. Some papers are related to the creative manipulation and transformation of digital sound. In [7], two applications of fuzzy logic in music are presented: a fuzzy logic-based mapping strategy for audiovisual composition and an audio synthesis

23.1 Introduction

453

technique based on sound particles and fuzzy logic. An extension of an existing performance system, capable of generating expressive musical performances based on case-based reasoning techniques (called SaxEx) is presented in [2] where the authors model the degree of different expressive parameters by means of fuzzy sets. Elsea [10] also applies concepts of fuzzy logic to problems in music analysis and composition. Several digital audio processing applications based on fuzzy logic have been proposed. Some of them are mainly focused on technical questions such as digital signal restoration [8] or the design of adaptive filters in acoustic echo cancelation [6]. One of the most cited references describes an algorithm for the classification, search, and retrieval of audio files [21]. This paper presents a general paradigm and specific techniques for analyzing audio signals in a way that facilitates content-based retrieval. A recent paper [5] presents a soft computing procedure which automatically generates sequences of songs. Starting from a given seed song, a sequence is generated while listening to the music because the user can express his or her dislike for the song being played by pressing a skip button. Some other references related to audio retrieval and soft computing techniques can be found in the literature (see for instance [16] and [18]), and other interesting references are reviewed in [22] and [7]. After this brief review of the literature, let us focus on our main concern. We are interested in a conceptual problem in music theory: What do we mean by a welltuned note or passage? And we also deal with the apparently conflictive coexistence of multiple tuning systems in an orchestra. In 1948, N. A. Garbuzov published a paper entitled “The zonal nature of the human aural perception”, where twelve bars of the aria from the suite in D by J. S. Bach played by three famous violinists: Oistrakh, Elman and Cimbalist were analyzed (see [15]). This article showed that, to an accuracy of five cents, most of the notes played by those violinists did not belong to the tuning system in which the musicians thought they were tuned, the twelve-tone equal temperament. Some of these notes correspond to other tuning systems but the rest of them did not belong to any system. However, when hearing the passage, the feeling was not only pleasant, but even persons endowed with a very sensitive ear rated their performance as welltuned. Thus, a question arises that we believe deserves to be analyzed in greater detail. We could think that the result of Garbuzov’s experience was forced as it was performed with violins which have no frets and where the pitch depends largely on the instrumentalist. Certainly, such an experiment could not have been carried out with fixed tuning instruments. But what happens with most wind or fretless string instruments? Or maybe we should reformulate our question: What do we mean when we accept that a note or a passage is finely-tuned? This question has been and continues to be a matter for discussion among many researchers in Musical Acoustics. Actually in the late 20th century we can observe a revival of interest in tone systems among musicians and in the industry in connection with the development of computer music and the production of electronic musical instruments with computer control.

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23 Mathematics and Soft Computing in Music

Different criteria have been used to select the sounds that music uses. A set containing these sounds (musical notes) is called a tuning system. Most of them have been obtained through mathematical arguments (see [3], [4], [11], [17]). The numerical nature of these systems facilitates their transmission and the manufacture of instruments. However, the harshness of the mathematical arguments relegated these tuning systems to theoretical studies while in practice musicians tuned in a more flexible way. Because of this, many musicians feel that the mathematical arguments that justify tuning systems are impractical. Modelling the notes as fuzzy sets and extending the concept of tuning systems allows us to connect theory and practice, and understand how musicians work in real-life. The notes offered by a musician during a performance should be compatible with the theoretical ones, but not necessarily equal. A numerical experiment conducted with the help of a saxophonist illustrates our approach and also highlights the need for considering the sequential uncertainty previously studied by Garbuzov.

23.2

Some Concepts and Notation

The word “tone” is used with different meanings in music. In [1] we can read that a tone is “a sound of definite pitch and duration, as distinct from noise and from less definite phenomena, such as the violin portamento.” In this dictionary we find that notes are “the signs with which music is written on a staff. In British usage the term also means the sound indicated by a note”. A pure tone can be defined as the sound of only one frequency, such as that given by an electronic signal generator. The fundamental frequency of a tone has the greatest amplitude. The other frequencies are called overtones or harmonics and they determine the quality of the sound. Loudness is a physiological sensation. It mainly depends on sound pressure but also on the spectrum of harmonics and physical duration. Although timbre and loudness are very important, we are focusing on pitch. Pitch is a psychological concept depending on the frequency of the tone. A higher frequency is perceived as a higher pitch. In music only a small choice of possible sounds is used and a tuning system is the system used to define which tones to use when playing music; these tones are the tuned notes. We will identify each note with the frequency of its fundamental harmonic (the frequency that chromatic tuners measure). The usual way to relate two frequencies is through their ratio; this number is called the interval. Actually, some authors (see [13]) identify the note with its relative frequency to the frequency of a fundamental, fixed tone (conventionally, such a tone is usually taken as A=440 Hz) It is well known that in the middle zone of the audible field, the “pitch sensation” changes somewhat according to the logarithm of the frequency, so the distance between two sounds whose frequencies are f1 and f2 can be estimated by means of the expression    f1   d( f1 , f2 ) = 1200 × log2 . (23.1) f2 

23.2 Some Concepts and Notation

455

where the logarithm in base 2 and the factor 1200 have been used in order to express d in cents. Let us define the well-known concept of an octave mathematically: given two sounds with frequencies f1 and f2 , we say that f2 is one octave higher than f1 if f2 is twice f1 . Two notes one octave apart from each other have the same letter-names. This naming corresponds to the fact that notes which are one octave apart sound like the same note produced at different pitches and not like entirely different notes. Based on this idea, we can define in R+ (the subset of all the frequencies of all the sounds) a binary equivalence relation, denoted by R, as follows: f1 R f2 if and only if ∃n ∈ Z such that f1 = 2n × f2 . Therefore, instead of dealing with R+ , we can analyze the quotient set R+ /R, which for a given fixed note f0 (diapason) can be identified with the interval [ f0 , 2 f0 [. For the sake of simplicity, we can assume that f0 = 1 and work in the interval [1, 2[. Octave equivalence has been an assumption in tonal music; however, the terminology used in atonal theory is much more specific. Let us offer an outline of the main tuning systems used in Western music. For an overview of the topic, including historical aspects we recommend interested readers the book by Benson [3]. The Pythagorean system is so named because it was actually discussed by Pythagoras, who in the sixth century B.C. already recognized the simple arithmetical relationship involved in the intervals of octaves, fifths, and fourths. He and his followers believed that numbers were the ruling principle of the universe, and that musical harmonies were a basic expression of the mathematical laws of the universe. Pythagorean tuning was widely used in Medieval and Renaissance times. All tuning is based on the interval of the pure fifth: the notes of the scale are determined by stacking perfect fifths without alterations. The Just Intonation (Zarlinean version) can be viewed as an adaptation of the Pythagorean system to diminish the thirds; it can be obtained by replacing some fifths of the Pythagorean system 3:2, by syntonic fifths 40:27 (see [20]). For these two tuning systems the circle of fifths is not closed, hence to establish an appropriate number of notes in an octave, some additional criteria are necessary. In order to avoid this question and also to permit transposing, the temperaments were introduced. If every element in the tuning system is a rational number, we say that it is a tuned system, whereas if some element is an irrational number then the system is a temperament. The most-used temperaments are the equal cyclic temperaments that divide the octave into equal parts. Hölder’s temperament divides the octave into 53 parts, providing a good approximation to the Pythagorean system. The Twelve-Tone Equal Temperament 12-TET is today’s standard on virtually all western instruments. This temperament divides the octave into twelve equal half steps. Tuning systems based on a unique interval (like the Pythagorean) admit a direct mathematical construction. However, the definition of systems generated by more than one interval requires specifying when and how many times each interval appears. Next, we give a general definition of a tuning system (see [19])

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23 Mathematics and Soft Computing in Music

Definition 23.2.1. (tuning system) Let Λ = {λi }ki=1 ⊂ [0, 1[ be a family of functions  k F = {hi : Z → Z}ki=1 . We call the tuning system generated by the intervals 2λi i=1 and F the set

' ( ) SΛF =

k

2cn : cn = ∑ λi hi (n) − i=1

k

∑ λihi (n)

,n ∈ Z

i=1

where +x, is the integer part of x (which is added to gain octave equivalence). The advantage of expressing the tuned notes as 2cn is that if our reference note is 20 , in accordance with (1), the exponent cn provides the pitch sensation. Let us mention that the family of integer-valued functions F marks the “interval locations”. For those systems generated by one interval (for instance the Pythagorean) they are not really necessary, while they are for the other systems. For instance, in the Just Intonation h1 (n) and h2 (n) indicate the position of the fifths and the thirds considered as tuned. Table 23.2 displays some examples of tuning systems. S

F

Pythagorean 12-TET Zarlinean (Just Intonation)

Neidhart’s temperament (1/2 & 1/6 comma)

Fig. 23.2. Table 1. Examples of generators of some tuning systems.

Although we only analyze Pythagorean, Zarlinean and 12-TET systems, the study of other tuning systems would be similar.

23.3

Notes as Fuzzy Sets

If we take the note A = 440Hz (diapason) as our fixed note, then a note offered with a frequency of 442Hz would be out of tune from the point of view of Boolean logic. However, anybody that hears it would consider it to be in tune.

23.3 Notes as Fuzzy Sets

457

While not trying to delve into psychoacoustic issues, we need to make some brief remarks about hearing sensitivity. According to J. Piles (1982) (see [20]), there is no unanimity among musicologists about aural perception. Roughly speaking, we could distinguish between two great tendencies: those who, following the work of Hermann von Helmholtz (1821 - 1894), consider that a privileged and educated ear can distinguish a difference of two cents, and those who fix the minimum distance of perception at 5 or 6 cents. For instance, Haluska states that the accuracy of an instrumentalist is not better than 5 or 6 cents and that this accuracy is between 10 and 20 cents for non-trained listeners. Nonetheless, this threshold of the human aural perception depends on many factors: the sensitivity of the ear, the listener’s age, education, practice and mood, the intensity and duration of the sounds, etcetera. As the human ear is not “perfect”, a musical note should be understood as a band of frequencies around a central frequency and it is appropriate to express it as a fuzzy number. Therefore the modal interval corresponding to the pitch sensation of a tone with frequency f should be expressed as [log2 ( f ) − ε , log2 ( f ) + ε ], where ε = 3/1200 (for instrumentalists) or ε ∈ [5/1200, 10/1200] for non-trained listeners (see [20], [14]). Accordingly, we define the band of unison as: [ f 2−ε , f 2ε ], where ε > 0, and where 1200ε expresses, in cents, the accuracy of the human ear to the perception of the unison. Next, let us focus on its support. If the number of notes per octave is q, the octave can be divided into q intervals of widths 1200/q cents. So, if we represent it as a segment, the (crisp) central pitch would be in the middle, and the extremes would be obtained by adding and subtracting 1200/(2 × q) cents. In fact, chromatic tuners assign q = 12 divisions per octave, suggesting that a tolerance of δ = 50/1200 = 1/24 is appropriate. Therefore, the support of the pitch sensation should be expressed as [log2 ( f ) − δ , log2 ( f ) + δ ], where δ = 1/(2 × q). Therefore, we can express the interval of the note f as: [ f 2−δ , f 2δ ]. Notice that the quantity Δ = 1200δ expresses, in cents, the tolerance that we 1 admit for every note, and for q = 12, we have Δ = 1200 2×12 = 50 cents. These arguments justify the expression of a musical note as a trapezoidal fuzzy number with peak [ f 2−ε , f 2ε ] and support [ f 2−δ , f 2δ ]. For notational purposes let us recall the definition of a LR-fuzzy number (see [9]). Definition 23.3.1. (LR-fuzzy number) M˜ is said to be a LR-fuzzy number, M˜ = (mL , mR , α L , α R )LR if its membership function has the following form: ⎧  L  m −x ⎪ x < mL ⎪ ⎨L α L ,

μM˜ =

 1,  ⎪ ⎪ ⎩R x−mR , αR

mL ≤ x ≤ mR x > mR

where L and R are reference functions, i.e. L, R : [0, +∞[→ [0, 1] are strictly decreasing in suppM˜ = {x : μM˜ (x) > 0} and upper semi-continuous functions such that L(0) = R(0) = 1. If suppM˜ is a bounded set, L and R are defined on [0, 1] and satisfy L(1) = R(1) = 0.

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23 Mathematics and Soft Computing in Music

Moreover, if L and R are linear functions, the fuzzy number is called trapezoidal, and is defined by four real numbers, A˜ = (aL , aR , α L , α R ). As notes are expressed as powers of two, it is not only more practical to express the fuzzy musical notes using their exponent but it also makes more sense, because as we have already mentioned, the exponent reflects the pitch sensation. Therefore we represent the pitch sensation of a note 2t˜ as the trapezoidal fuzzy number t˜ = (t − ε ,t + ε , δ − ε , δ − ε ). Now that notes are modelled as fuzzy numbers, the concept of fuzzy tuning system arises naturally: Definition 23.3.2. (fuzzy tuning system) Let δ ∈ [0, 1], Λ = {λi }ki=1 ⊂ [0, 1[, and a family of functions F = hi : Z → Z ki=1 . A fuzzy tuning system generated by the inter k vals 2λi i=1 and F is the set:

S˜ΛF (δ )

=

 2 : c˜n = c˜n

k

'

k

(

∑ λi hi (n) − ∑ λi hi (n)

i=1

 ,δ

) n∈Z

i=1

In [19] the compatibility between two fuzzy notes is defined as the Zadeh consistency index between their pitch sensations. Figure 23.3 illustrates the definition of α -compatibility. Definition 23.3.3. (compatibility) Let 2t˜ and 2s˜ be two musical notes, where t˜ = (t − ε ,t + ε , δ − ε , δ − ε ) and s˜ = (s − ε , s + ε , δ − ε , δ − ε ). The degree of compatibility between 2t˜ and 2s˜ is defined as comp[2t˜, 2s˜] = maxx μs∩ ˜ t˜(x), and we say that 2t˜ and 2s˜ are α -compatible, α ∈ [0, 1], if comp[2t˜, 2s˜] ≥ α .

Fig. 23.3. Graph illustrating the concept of α -compatibility between two notes.

23.4 A Numerical Experiment and Sequential Uncertainty

459

By a direct calculus we can obtain the formula which allows us to calculate the compatibility between two notes. ⎧ 1 i f |t − s| < 2ε ⎪ ⎨ ε comp[2t˜, 2s˜] = 1 − |t−s|−2 i f 2 ε ≤ |t − s| < 2δ 2(δ −ε ) ⎪ ⎩ 0 i f |t − s| ≥ 2δ Moreover, the concept of compatibility is also extended for fuzzy tuning systems. The definition of compatibility between two tuning systems reflects both the idea of proximity between their notes and also whether their configuration is similar. Definition 23.3.4. (compatibility between two tuning systems) Let S˜q (δ ) and T˜q (δ ) be two tuning systems with q notes. We say that S˜q (δ ) and T˜q (δ ) are α -compatible, if for each 2s˜i ∈ S˜q (δ ) there is a unique 2t˜i ∈ T˜q (δ ) such that comp[2s˜i , 2t˜j ] ≥ α . The quantity α is the degree of interchangeability between S˜q (δ ) and T˜q (δ ) and the uniqueness required in the definition guarantees that these systems have a similar distribution in the cycle of fifths. Note that the α -compatibility does not define a binary relation of equivalence in the set of tuning systems because the transitive property is not verified. Example Let us analyze the compatibility between the 12-TET and the Pythagorean system with 21 notes. Our data are the exact (crisp) frequencies, displayed in Table 23.4. The following pairs of notes are said to be enharmonic: (C# , Db ), (D# , E b ), (F # , Gb ), (G# , Ab )and(A# , Bb ) because although they have different names they sound the same in the 12-TET. For a better visualization, instead of the exact compatibilities between the notes, we show their graphical representation in Figure 23.5. We have set δ = 50/1200 and ε = 3/1200 (suitable for trained listeners). The minimum compatibility between the notes is equal to 0.84, however the systems are not 0.84-compatible because the uniqueness property does not hold. Nonetheless, if we consider the 12-TET and the Pythagorean system with 12 notes, C, C# , D, E b , E, F, F # , G, G# , A, Bb , B, they are α -compatible for α ≤ 0.84.

23.4

A Numerical Experiment and Sequential Uncertainty

The purpose of the experiment described in this section is to study the different variations of a note that usually occur in a wind instrument (a baritone saxophone) where the pitch may be subject to the interpretation of the performer or the characteristics of the instrument. In order to set one of the parameters of the experiment, the same saxophonist performed five interpretations of the excerpt represented in Figure 23.6. The recordings took place on the same day without changing the location of the recording or its

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23 Mathematics and Soft Computing in Music

physical characteristics such as temperature and humidity. The measurements were R The saxophone brand name is Studio. We made with the free software Audacity-. considered two possible conceptual frameworks: “static tuning”, in which each note is treated separately, and "dynamic tuning" where notes are studied in their context. In this section we will describe our results for the second approach, which seems more relevant for this study. Note C B# Db # C D Eb D# b F E F E# b G # F G Ab # G A Bb A# b C B

Pythagorean 260,74074 264,29809 274,68983 278,4375 293,33333 309,02606 313,24219 325,55832 330 347,65432 352,39746 366,25311 371,25 391,11111 412,03475 417,65625 440 463,5391 469,86328 488,33748 495

12-TET 261,6265 261,6265 277,1826 277,1826 293,6648 311,127 311,127 329,6275 329,6275 349,2282 349,2282 369,9944 369,9944 391,9954 415,3047 415,3047 440 466,1638 466,1638 493,8833 493,8833

Fig. 23.4. Table 2. Exact frequencies of the notes.

Firstly we obtained the compatibility between the notes recorded and the notes tuned in the fuzzy 12-TET. We fixed δ = 50/1200 and ε = 6/1200. Figure 23.6 is a graphic representation of the compatibility values. We can observe that the worst compatibilities with the theoretical notes occur for the notes D#4, C#4, G4, D#4. A first conclusion is that the saxophonist should make an effort to improve his interpretation of these notes. However, our analysis should be completed by taking “sequential uncertainty” into account. We have already mentioned the musicologist N. A. Garbuzov in the introduction. According to J. Haluska [15], “. . . (he) revolutionized the study of musical intervals suggesting a concept of musical “zones” in the 1940s. This theory can be characterized in the present scientific language as an information granulation in the sense of Zadeh”. Table 23.9 was obtained by Garbuzov from hundreds of measurements. We

23.4 A Numerical Experiment and Sequential Uncertainty

461

are taking it as a reference, although bearing in mind that it should be recomputed to take into account the higher precision of the present measurement instruments.

Fig. 23.5. Compatibility between the 12-TET and the Pythagorean tuning system.

For the 12-TET, two notes which differ by a semitone are exactly 100 cents apart. However, according to Garbuzov’s experiments if two notes differ between 48 and 124 cents and they are played consecutively, the human ear perceives them to be a semitone apart.

Fig. 23.6. Score for the excerpt interpreted by the musician.

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23 Mathematics and Soft Computing in Music

Note

Rep1

Rep2

Rep3

Rep4

Rep5

C3 # F3 B3 F3

#

139.7 185.12 247.36 174.34

138.77 184.19 249.17 175.52

139.3 185.38 247.83 175

137.75 184.84 247.48 174.8

137.12 183.14 244.55 174.84

A3 # D3 B3 D# 4

218.26 156.49 247.41 323.38

218.38 156.64 248.65 324.19

218.86 157.27 246.99 323.33

218.73 156.61 245.28 321.43

217.41 155.93 242.06 322.31

C4 G4 D# 4 A# 3

#

284.48 401.2 322.08 234.48

284.75 401.1 322.03 234.14

284.57 401 320.03 234.11

283.43 399.31 319.9 233.56

284.49 399.79 320.01 233.56

A3 F3 E3 D# 3

219.11 175.46 164.3 156.13

219.19 175.8 164.46 156.11

219.12 175.21 164.61 155.65

219.06 174.97 165.05 156.42

218.75 174.82 165.15 156.74

Fig. 23.7. Table 3. Exact (crisp) frequencies of the notes offered.

When these two notes are played simultaneously (for instance two instrumentalists are playing together) and they differ between 66 and 130 cents, they are perceived to be a semitone apart. Clearly, for our experiment we should only take into account the second column because our saxophonist is playing “a solo”. Column 1 in Table 5 contains the “low compatibility notes”, the second column the corresponding Garbuzov zones, Column 3 the distance in semitones from an offered note to its previous one and Colums 4-8 the distances in cents from an offered note to its previous one.s which differ by a semitone are exactly 100 cents apart. However, according to Garbuzov’s experiments if two notes differ between 48 and 124 cents and they are played consecutively, the human ear perceives them to be a semitone apart. When these two notes are played simultaneously (for instance two instrumentalists are playing together) and they differ between 66 and 130 cents, they are perceived to be a semitone apart. We can see that the distances are out of the Garbuzov zone for only two notes. In the other cases, the saxophonist and those listening to his interpretation would probably perceive them as correct.

23.4 A Numerical Experiment and Sequential Uncertainty

463

Fig. 23.8. Compatibility between the theoretical notes and those offered by the musician.

Granule Unison (octave) Minor second Major second Minor third Major third Fourth Tritone Fifth Minor sixth Major sixth Minor seventh Major seventh

Melodic (-12, 12) (48, 124) (160, 230) (272, 330) (372, 430) (472, 530) (566,630) (672,730) (766, 830) (866, 930) (966, 1024) (1066, 1136)

Harmonic (-30,30) (66, 130) (166, 230) (266, 330) (372, 430) (466, 524) (566, 630) (672, 730) (766, 830) (866, 924) (966, 1024) (1066, 1136)

Fig. 23.9. Table 4. Garbuzov zones in cents: sequential and simultaneous uncertainty (source [15]).

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23 Mathematics and Soft Computing in Music

Note #

D4 # C4 G4 D#4 A# 3

Zone

semitones

Rep1

Rep2

Rep3

Rep4

Rep5

2 1 3 2 2.5

463.59 221.88 595.19 380.28 549.55

459.27 224.57 593.12 380.12 551.79

466.27 221.07 593.78

468.09 217.815 593.42

495.7 216.09 589.04

390.47 541.22

383.866 544.593

385.35 545.19

[372,430] [160,230] [566,630] [372,430] [472,530]

Fig. 23.10. Table 5. Low compatibility notes (5 repetitions).

23.5

Conclusions

Defining tuning systems as comprised of fuzzy notes allows us to include the daily reality of musicians and their theoretical instruction in a mathematical structure. We can justify that the adjustments the musicians make to play together constitute a method for increasing the compatibility level among systems. Complex tuners indicating precisely the difference between the note offered and the desired pitch could suggest to musicians that they should aspire to achieving “perfect tuning”; however, getting a high degree of compatibility or similarity with the score is a more achievable and reasonable goal. On the other hand, knowing the compatibility between notes allows musicians to improve their performance by choosing between different tuning positions, increasing lip pressure, etcetera. The numerical example that we have presented causes us to reflect: it is not only important to consider compatibility with the theoretical notes (allowing the coexistence of different instruments in an orchestra), but also that a new concept of sequential compatibility should be considered to better explain instrumentalists’ performances. In addition, we should not forget “simultaneous uncertainty”. We intend to define the concepts of sequential compatibility and simultaneous compatibility in order to aggregate them with the compatibility between the theoretical notes and the notes offered. The weights of these quantities in the aggregation should depend on whether a musician is playing a solo, a duet or playing with the orchestra.

Acknowledgments The authors acknowledge the kind collaboration of Julio Rus-Monge in making the recordings used in our numerical experiment and would also like to thank the Science and Innovation Department of the Spanish government for the financial support of research projects TIN2008-06872-C04-02 and TIN2009-14392-C02-01.

23.5 Conclusions

465

References [1] Apel, W.: Harvard Dictionary of Music: Second Edition, Revised and Enlarged, 2nd edn. The Belknap Press of Harvard University Press, Cambridge, Massachussets (1994) [2] Arcos, J.L., De Mantaras, R.L.: An interactive case-based reasoning approach for generating expressive music. Applied Intelligence 14, 115–129 (2001) [3] Benson, D.: Music: a Mathematical Offering. Cambridge University Press, Cambridge (2006), http://www.maths.abdn.ac.uk/~bensondj/html/music.pdf [4] Borup, H.: A History of String Intonation, http://www.hasseborup.com/ahistoryofintonationfinal1.pdf [5] Bosteels, K., Kerre, E.E.: A fuzzy framework for defining dynamic playlist generation heuristics. Fuzzy Sets and Systems 160, 3342–3358 (2009) [6] Christina, B.: A robust fuzzy logic-based step-gain control for adaptive filters in acoustic echo cancellation. IEEE Transactions on Speech and Audio Processing 9(2), 162– 167 (2001) [7] Cádiz, R.F.: Fuzzy logic in the arts: applications in audiovisual composition and sound synthesis. In: Proceedings of NAFIPS 2005 - 2005 Annual Meeting of the North American Fuzzy Information Processing Society, pp. 551–556. IEEE/IET Electronic Library, IEL (2005) [8] Civanlar, M.R., Joel Trussel, H.: Digital restoration using fuzzy sets. IEEE Transactions on Acoustics, Speech and Signal Processing ASSF-34(4), 919–936 (1986) [9] Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York (1980) [10] Elsea, P.: Fuzzy logic and musical decisions (1995), http://arts.ucsc.edu/EMS/Music/research/FuzzyLogicTutor/FuzzyTut.html [11] Goldáraz Gaínza, J.J.: Afinación y temperamento en la música occidental. Alianza Editorial, Madrid (1992) [12] Hall, R.W., Josíc, K.: The mathematics of musical instruments. The American Mathematical Monthly 108, 347–357 (2001) [13] Haluska, J.: Equal temperament and Pythagorean tuning: a geometrical interpretation in the plane. Fuzzy Sets and Systems 114, 261–269 (2000) [14] Jan, H.: Uncertainty measures of well tempered systems. International Journal of General Systems 31, 73–96 (2002) [15] Haluska, J.: The Mathematical Theory of Tone Systems. Marcel Dekker, Inc., Bratislava (2005) [16] Kiranyaz, S., Qureshi, A.F., Gabbouj, M.: A Generic Audio Classification and Segmentation Approach for Multimedia Indexing and Retrieval. IEEE Transactions on Audio, Speech, and Language Processing 14(3), 1062–1081 (2006) [17] Lattard, J.: Gammes et tempéraments musicaux. Masson Éditions, Paris (1988) [18] Lesaffre, M., Leman, M., Martens, J.-P.: A User-Oriented Approach to Music Information Retrieval. Dagstuhl Seminar Proceedings 06171 (2006), http://drops.dagstuhl.de/opus/volltexte/2006/650 [19] Vicente, L.: Fuzzy tuning systems: the mathematics of the musicians. Fuzzy Sets and Systems 150, 35–52 (2005) [20] Piles Estellés, J.: Intervalos y gamas. Ediciones Piles, Valencia (1982) [21] Wold, E., Blum, T., Keislar, D., Wheaton, J.: Content-Based Classification, Search, and Retrieval of Audio. IEEE MultiMedia Archive 3(3), 27–36 (1996) [22] Yilmaz, A.E., Telatar, Z.: Potential applications of fuzzy logic in music. In: Proceedings of the 18th IEEE International Conference on Fuzzy Systems, pp. 670–675. IEEE Press, Piscataway (2009)

24 Music and Similarity Based Reasoning Josep LLuís Arcos

24.1

Introduction

Whenever that a musician plays a musical piece, the result is never a literal interpretation of the score. These performance deviations are intentional and constitute the essence of the musical communication. Deviations are usually thought of as conveying expressiveness. Two main purposes of musical expression are generally recognized: the clarification of the the musical structure and the transmission of affective content [18, 9, 5]. The challenge of the computer music field when modeling expressiveness is to grasp the performers “touch”, i.e., the musical knowledge applied when performing a score. The research interest on music expressiveness comes from different motivations: to understand or model music expressiveness; to identify the expressive resources that characterize an instrument, musical genre, or performer; or to build synthesis systems able to play expressively. One possible approach to tackle the problem is to try to make explicit this knowledge, mainly designing rule-based systems, using musical experts. One of the first expert systems developed was a rule-based system able to propose tempo and articulation transformations to Bach’s fugues [8]. The rules of the system were designed from two experts. The transformations proposed by the system coincided with well-known commented editions but the main limitation of the system was the lack of generality. Another successful research was the system developed by the KTH group from Stockholm. Their effort on the director musicies system [4] incorporated rules applied to tempo, dynamic, and articulation transformations to MIDI files. Also related to knowledge-based systems, is the long-term research led by G. Widmer [20] that applied machine learning techniques to acquire rules from a large amount of high-quality classical piano performances. See [13] for a more complete survey on the design of systems using explicit knowledge. The main drawback when using rule-based systems is the difficulty to find rules general enough to capture the diversity present in different performances of the same piece, or even the variety within the same performance. An additional problem with this approach is the interdependency of the rules. For instance, rules applied to change dynamics affect those related to tempo. An alternative approach, much closer to the human observation-imitation process, is to directly work with the knowledge implicitly stored in musical recordings and let the system imitate R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 467–478. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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these performances. This alternative approach, also called lazy learning, focus on locally approximating a complex target function when a new problem is presented to the system. An example of this alternative approach is Case-Based Reasoning. Case-Based Reasoning (CBR) [1, 14] is a similarity based methodology that exploits prior experiences (called cases) when solving new problems by identifying relevantly similar cases and adapting them to fit new needs. CBR is appropriate for problems where either (1) many examples of solved problems are available or (2) a main part of the knowledge involved in the solution of problems is implicit, difficult to verbalize or generalize. As we will present in this chapter, music expressiveness presents the two characteristics described above. CBR inference consists of four steps: case retrieval, which determines which cases address problems most similar to the current problem; case adaptation, which forms a new solution by adapting/combining solutions of the retrieved cases; case revision, which evaluates and adjusts the adapted solution; and case retention, in which learning is performed by storing the result as a new case for future use. In this chapter we will describe how the CBR methodology has been successfully used to design different computer systems for expressive performance. First, we will present the SaxEx system that is able to generate high-quality human-like melody performances of jazz ballads based on examples of human performances. Next, we will present the TempoExpress system that is able to perform expressiveness-aware tempo transformations of saxophone jazz recordings. Finally, we will show how a similarity-based approach can be used for identifying professional violin performers in commercial recordings.

24.2

SaxEx

The goal of the SaxEx system [2, 3] was to generate expressive melodies by means of performing transformations on non-expressive recordings. That is, to add expressiveness according to user’s preferences. SaxEx was applied in the context of saxophone jazz ballads. Specifically, SaxEx manages five different expressive parameters: dynamics, tempo, vibrato, articulation, and note attacks. SaxEx was designed with the claim that the best way for a musician to elicit her expertise is to directly provide examples, i.e. playing her instrument. Then, SaxEx was designed as a CBR system where the cases hold the implicit knowledge collected in human recordings. To achieve its task, SaxEx uses a case memory containing human performances (ballad recordings), the scores associated to these performances (containing both melodic and harmonic information), and musical knowledge used to define similarity among melodic fragments. The information about the expressive performances contained in the examples of the case memory is represented by a sequence of affective regions. Affective regions group (sub)-sequences of notes with common affective expressiveness. Specifically, an affective region holds knowledge describing the following affective dimensions: tender-aggressive, sad-joyful, and calm-restless (see Figure 24.1).

24.2 SaxEx

469

Agressive

Sad

s

tles

Res

Joyful

m

Cal

Tender

Fig. 24.1. Affective dimensions.

These affective dimensions are described using five ordered qualitative values expressed by linguistic labels as follows: the middle label represents no predominance (for instance, neither tender nor aggressive), lower and upper labels represent, respectively predominance in one direction (for example, absolutely calm is described with the lowest label). For instance, a jazz ballad can start very tender and calm and continue very tender but more restless. Such different nuances are represented in SaxEx by means of different affective regions. Additionally, expressive performances (recordings) are analyzed using spectral techniques using using SMS [19] and annotated using fuzzy techniques. The advantage of using fuzzy techniques is that we can abstract numerical features provided by a low level analysis by a collection of fuzzy labels with associated membership degrees. This approach facilitates the reasoning process without losing information. A new problem for SaxEx is a musical phrase, described by its musical score and a non-expressive recording, together with a description of the desired affective expressiveness for the output. Affective information can be partially specified, i.e. the user may only provide values for some of the three affective dimensions. Values of desired affective expressiveness will guide the search in the memory of cases. 24.2.1

Melodic Similarity

The two main inference processes in SaxEx are the retrieval and the adaptation steps. The goal of the retrieval step is to select, from the case memory, a set of similar notes to the current problem (input musical phrase). To do that, the similarity criterion is the key factor. SaxEx uses Narmour’s Implication-Realization (IR) model [17] for similarity assessment. IR model is based on a perception and cognitive approach to analyze the structure of a musical piece. Specifically, IR models the patterns of expectations generated in people when listening a melody. It follows the approach introduced by Meyer [15] that applies the principles of Gestalt Theory to melody perception.

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Gestalt theory states that perceptual elements are grouped together to form a single perceived whole (called ‘gestalt’). This grouping follows some principles: proximity (two elements are perceived as a whole when they are perceptually close), similarity (two elements are perceived as a whole when they have similar perceptual features, e.g. color in visual perception), and good continuation (two elements are perceived as a whole if one is a ‘natural’ continuation of the other). Narmour claims that similar principles hold for the perception of melodic sequences. In IR, these principles take the form of implications and involve two main principles: registral direction (PRD) and intervallic difference (PID). The PRD principle states that small intervals create the expectation of a following interval in the same registral direction (for instance, a small upward interval generates an expectation of another upward interval), and large intervals create the expectation of a change in the registral direction (for instance, a large upward interval generates an expectation of a downward interval). The PID principle states that a small (five semitones or less) interval creates an expectation of a following similarly-sized interval (plus or minus two semitones), and a large interval (seven semitones or more) creates an expectation of a following smaller interval. Based on these two principles IR model proposes some basic melodic patterns that are used as an approximation of the local structure of a melody (see Figure 24.2). Moreover, their characterization in terms of melodic intervals and relative durations is used by SaxEx as similarity assessment.

Fig. 24.2. Eight of the basic structures of the IR model

24.2.2

Applying Expressive Transformations

After the retrieval of the notes more similar to the current problem, the role of the adaptation step is to determine the expressive transformations to be applied to each note. That is, for each note in the problem melody, a value for each of the five expressive parameters (dynamics, tempo, vibrato, articulation, and note attacks) must be determined. To determine expressive transformations, the first step is to inspect the solutions given in the cases retrieved (the values chosen in each similar note for each expressive parameter). Notice that these values are represented as fuzzy labels with associated membership degrees. Because these values are never exactly the same, SaxEx has a collection of adaptation criteria that can be selected by the user. For instance, the majority criterion will choose only the values belonging to the linguistic fuzzy label applied in the majority of cases. Next, these values are combined by applying a fuzzy aggregation operator. Figure 24.3 illustrates an example with two similar notes whose fuzzy values for loudness are, respectively, 75dB and 82dB. The system first computes the

24.3 TempoExpress

471

0.9 0.6

82

75

Loudness

COA

Fig. 24.3. Fuzzy combination and defuzzification of loudness value.

maximum degree of membership of each one of these two values with respect to the five linguistic values characterizing loudness. Next, it computes a combined fuzzy membership function, the fuzzy disjunction, based on these two values. Finally, a defuzzificaton process, computing the center of area, gives a precise value of loudness to be applied to the initially inexpressive note. Expressive examples generated by SaxEx contained some interesting results such as: crescendos placed in ascending melodic progressions, accentuated legato in sad performances, staccato in joyful ones, and “swingy” rhythmic patterns. Readers are encouraged to judge SaxEx results at http://www.iiia.csic.es/Projects/music/Saxex.html

24.3

TempoExpress

TempoExpress [6] is A CBR system for realizing tempo transformations of audio recordings at the musical level, taking into account the expressive characteristics of a performance. Existing high-quality audio algorithms are only focused on maintaining sound quality of audio recordings, rather than maintaining the musical quality of the audio. In contrast, TempoExpress aims at preserving the musical quality of monophonic recordings when their tempo is changed. Because expressiveness is a vital part of performed music, an important issue to study is the effect of tempo on expressiveness. An study conducted by H. Honing demonstrated that listeners are able to determine, based only on expressive aspects of the performances, whether audio-recordings of jazz and classical performances are original or uniformly time stretched recordings [7]. The goal of TempoExpress is to transform audio recordings in a way that (ideally) listeners should not be able to notice whether tempo has been scaled up or down. TempoExpress performs tempo transformations by using a case-based reasoning approach. The design of the system was similar to the SaxEx system: a case memory storing symbolic descriptions of human performances, the scores associated to these performances, and musical knowledge used to define similarity among a new problem and the cases. Human performances are analyzed using audio analysis

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techniques [12] and a symbolic description of each performance is automatically calculated. 24.3.1

Similarity Assessment

Analogously to the SaxEx system, IR model is used to calculate melodic similarities. However, the structure of the case memory in TempoExpress is more complex than in SaxEx. First at all, in TempoExpress an input problem is now an expressive performance. Moreover, available performances may become part of the problem description or a solution because source and target tempos vary. Thus, the pairs of problem-solution are generated dynamically according to the characteristics of a new problem. First, performances with tempos very different from the source and target tempos are filtered out. Next, IR similarity is applied as a second retrieval filter. Finally, a pool of cases are constructed as pairs of problem performances (those with tempo close to the source tempo) and solution performances (those with tempo close to the target tempo). The first consequence of dealing with input expressive performances is that TempoExpress must include a similarity measure among performances. It is common to define musical expressiveness as the discrepancy between the musical piece as it is performed and as it is notated. This implies that a precise description of the notes that were performed is not very useful in itself. Rather, the relation between score and performance is crucial. TempoExpress uses a representation of expressiveness that describes the musical behavior of the performer as performance events. The performance events form a sequence that maps the performance to the score. For example, the occurrence of a note that is present in the score, but has no counterpart in the performance, will be represented by a deletion event (since this note was effectively deleted in the process of performing the score). A key aspect of performance events is that they may refer to particular notes in the notated score, the performance, or both. Based on this characteristic the taxonomy of performance events is summarized in Figure 24.4. Similarity among performances is calculated using edit-distance techniques [11]. The edit-distance is defined as the minimal cost of a sequence of editions needed to transform a source sequence into a target sequence, given a predefined set of edit operations (classically deletion, insertion, and replacement of notes). The cost of a particular edit-operation is defined through a cost function for that operation, that computes the cost of applying that operation to the notes of the source and target sequences. Case adaptation is performed by a set of defined adaptation rules. Adaptation rules capture the perceptual similarity of two performances. See [6] for a detailed description of the adaptation process.

24.3 TempoExpress

473

Reference

Performance Reference

Score Reference

Dynamics

Deletion

Correspondence

Insertion

Pitch deviation Duration deviation Onset deviation

Ornamentation Nr. of notes Melodic direction

Consolidation Nr. of notes

Transformation

Fragmentation Nr. of notes

Fig. 24.4. Taxonomy of performance events.

24.3.2

Evaluating the Quality of Tempo Transformations

To evaluate TempoExpress four different jazz standards were recorded at various tempos (about 12 different tempos per song). In total 170 interpretations of 14 musical phrases were recorded (corresponding to 4.256 played notes). The methodology conducted was to compare results from TempoExpress with results from a uniform time strecthing algorithm. The quality of a tempo transformation was calculated as the distance to a human performance. The key aspect of this approach was that results are very sensitive to the distance metric used to compare performances. It is conceivable that certain small quantitative differences between performances are perceptually very significant, whereas other, larger, quantitative differences are hardly noticeable by the human ear. To overcome this problem, the distance measure used for comparing performances was constructed by human similarity judgments. Specifically, a web based survey was set up to gather information about human judgments of performance similarity. Survey questions were presenting to subjects a target performance A (the nominal performance, without expressive deviations) of a short musical fragment, and two different performances B and C of the same score fragment. The task of subjects was to indicate which of the two alternative performances was perceived as most similar to the target performance. Thus, subjects were asked with questions of the form “is more similar A to B or more similar to C ?”. A total of 92 subjects responded to the survey, answering on average 8.12 questions (listeners were asked to answer at least 12 questions, but were allowed to interrupt the survey). Only those questions answered by at least ten subjects and with significant agreement between the answers were considered. A leave-one-out setup was used to evaluate TempoExpress. Each of the 14 phrases in the case base was segmented into 3 to 6 motif-like segments generating 6.364 transformation problems from all pairwise combinations of performances for each segment. Experiments demonstrated that TempoExpress clearly behaves

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better than a Uniform Time Stretch when the target tempo is slower than the source tempo. When the target tempo is higher than the source tempo, TempoExpress behaves similarly as UTS (the improvement is not statistically significant). Readers are encouraged to judge TempoExpress results at http://www.iiia.csic.es/Projects/music/TempoExpress.html

24.4

Identifying Violin Performers

The last system presented has the goal of identifying violinists from their playing style. Violinists’ identification is performed by using descriptors that are automatically extracted from commercial audio recordings by means of state-of-the-art feature extraction tools [16]. The system was tested with Sonatas and Partitas for solo violin from J.S. Bach [10]. Sonatas and Partitas for solo Violin by J.S. Bach is a six work collection (three Sonatas and three Partitas) that almost every violinist plays during her artistic life. Moreover, several players have multiple recordings of this piece. Thus, commercial recordings of the best known violin performers may be obtained easily. The problem is challenging due to two main issues. First, since we consider audio excerpts from quite different sources, we assume a high heterogeneity in the recording conditions. Second, as state-of-the-art audio transcription and feature extraction tools are not 100% precise, we assume a partial accuracy in both the melodic transcription and the extraction of audio features (for example in note durations). Taking into account these constraints, the system is based on the acquisition of trend models that characterize violin performers through capturing only their general expressive footprint. To achieve this task, first a higher-level abstraction of the automatic transcription is generated, based on the melodic contour, and recordings are segmented based on the Implication/Realization (IR) model [17]. Next, a trend model is constructed by a set of discrete frequency distributions for a given audio descriptor. Each of these frequency distributions represents the way a given IR pattern is played against a certain audio descriptor. 24.4.1

Acquiring Trend Models

To generate trend models for a particular performer and audio descriptor, we use values extracted from the notes identified in each IR segment. From these values, a qualitative transformation is first performed in the following way: each value is compared to the mean value of the segment and is transformed into a qualitative value where + means ‘the descriptor value is higher than the mean’, and − means ‘the descriptor value is lower than the mean’. In the current approach, since we are segmenting the melodies in groups of three notes and using two qualitative values, eight (23 ) different combinations may arise. Next, a histogram per IR pattern with these eight qualitative combinations is constructed by calculating the percentage of occurrence of each combination. Figure 24.5 shows an example of the trend models for duration and energy descriptors.

24.4 Identifying Violin Performers

P

D

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ID

IP

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IR

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duration energy Fig. 24.5. Example of trend models for duration and energy descriptors.

24.4.2

Identifying the Performer in New Recordings

A nearest neighbor classifier is used to predict the performer of new recordings. Trend models acquired in the training stage are used as class patterns, i.e. each trained performer is considered a different solution class. When a new recording is presented to the system, its trend model is created and this trend model is compared with the previously acquired models. The system generates a ranked list of performer candidates where distances determine the order, being first position the most likely performer relative to the results of the training phase. The distance di j between two trend models i and j (i.e. the distance between two performances), is defined as the weighted sum of distances between the frequency distributions of IR patterns: di j =

∑ wnij Δdist(ni , n j )

(24.1)

n∈N

where N is the set of the different IR patterns considered; dist(ni , n j ) measures the distance between two frequency distributions (see equation (24.3) below); and wnij are the weights assigned to each IR pattern. Weights have been introduced for balancing the importance of the IR patterns with respect to the number of times they appear. Frequent patterns are considered more informative due to the fact that they come from more representative samples. Weights are defined as the mean of cardinalities of respective histograms for a given pattern n: wnij = (Nin + N nj )/2

(24.2)

Mean value is used instead of just one of the cardinalities to assure a symmetric distance measure in which wnij is equal to wnji . Cardinalities could be different because recognized notes can vary from a performance to another, even though the score is supposed to be the same. Finally, distance between two frequency distributions is calculated by measuring the absolute distances between the respective patterns: dist(s, r) =

∑ |sk − rk |

(24.3)

k∈K

where s and r are two frequency distributions for the same IR pattern; and K is the set of all possible values they can take (in our case |K| = 8). Since duration and energy are analyzed in the system, their corresponding distances are aggregated.

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24.4.3

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System Performance

We analyzed music recordings from 23 different professional performers with a variety of performing styles (see Figure 24.6). Some of them are from the beginning of the last century while others are modern performers still active. Violinists like B. Brooks, R. Podger, and C. Tetzlaff play in a Modern Baroque style; whereas violinists like S. Luca, S. Mintz, and J. Ross are well-know for their use of détaché. We also included two recordings that are clearly different from the others: G. Fischbach and T. Anisimova. G. Fischbach plays with a sustained articulation. T. Anisimova is a cellist and, then, the performance is clearly very different from the others. Different experiments were conducted to to assess the capabilities of trend models approach. When comparing two movements from the same piece, the correct performer was mostly identified in the first half of the list, i.e. at most in the 12th position. The correct performer is predicted, in the worst case, 34.8% of times as the first candidate, clearly outperforming the random classifier (whose success rate is 4.3%). Additionally, using the four top candidates the accuracy reaches the 50% of success. When comparing two movements from different pieces, a most difficult scenario, the 90% of identification accuracy was overcame at position 15. Selecting only seven performers, the accuracy reaches the 60% of success.

Fig. 24.6. Hierarchical clustering for the Sixth movement of Partita No. 1

In order to better understand what are capturing the different trend models, we calculated the distances di j between all of them (see equation 24.1) and applied a hierarchical clustering algorithm (using a complete linkage method). Figure 24.6 shows the dendrogram representation of the hierarchical clustering for the Sixth movement of Partita No. 1. It is interesting to remark that some of the different performing styles are captured by the trend models. For instance, the most different

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477

recordings (G. Fischbach and T. Anisimova) are clearly far from the rest; violinists playing in a Modern Baroque style (B. Brooks, R. Podger, and C. Tetzlaff) are clustered together; violinists using détaché (S. Luca, S. Mintz, and J. Ross) appear also close to each other; and the usage of expressive resources such as portamento, vibrato, or ritardandi presents a relationship with the result of the clustering.

Acknowledgments This work was partially funded by projects NEXT-CBR (TIN2009-13692-C03-01), IL4LTS (CSIC-200450E557) and by the Generalitat de Catalunya under the grant 2009-SGR-1434.

References [1] Aamodt, A., Plaza, E.: Case-Based Reasoning: Foundational Issues, Methodological Variations, and System Approaches. Artificial Intelligence Communications 7(1), 39– 59 (1994) [2] Arcos, J.L., de Mántaras, R.L.: An interactive case-based reasoning approach for generating expressive music. Applied Intelligence 14(1), 115–129 (2001) [3] Arcos, J.L., de Mántaras, R.L.: Combining fuzzy and case-based reasoning to generate human-like music performances. In: Technologies for Constructing Intelligent Systems: Tasks, pp. 21–31. Physica-Verlag GmbH (2002) [4] Bresin, R.: Articulation rules for automatic music performance. In: Proceedings of the International Computer Music Conference - ICMC 2001, San Francisco, pp. 294–297 (2001) [5] Gabrielsson, A.: Expressive Intention and Performance. In: Steinberg, R. (ed.) Music and the Mind Machine, pp. 35–47. Springer, Heidelberg (1995) [6] Grachten, M., Arcos, J.L., de Mántaras, R.L.: A case based approach to expressivityaware tempo transformation. Machine Learning 65(2-3), 411–437 (2006) [7] Honing, H.: Is expressive timing relational invariant under tempo transformation? Psychology of Music 35(2), 276–285 (2007) [8] Johnson, M.L.: An Expert System for the Articulation of Bach Fugue Melodies. In: Baggi, D.L. (ed.) Readings in Computer-Generated Music, pp. 41–51. IEEE Computes Society Press (1992) [9] Juslin, P.N.: Communicating emotion in music performance: a review and a theoretical framework. In: Juslin, P.N., Sloboda, J.A. (eds.) Music and Emotion: Theory and Research, pp. 309–337. Oxford University Press (2001) [10] Lester, J.: Bach’s Works for Solo Violin: Style, Structure, Performance. Oxford University Press (1999) [11] Levenshtein, V.I.: Binary codes capable of correcting deletions, insertions and reversals. Soviet Physics Doklady 10, 707–710 (1966) [12] Maestre, E., Gómez, E.: Automatic characterization of dynamics and articulation of expressive monophonic recordings. In: AES 118th Convention (2005) [13] de Mántaras, R.L., Acros, J.L.: Arcos: AI and music, form composition to expressive performance. AI Magazine 23(3), 43–57 (2002) [14] de Mántaras, R.L., McSherry, D., Bridge, D., David, L., Smyth, B., Susan, C., Faltings, B., Maher, M.L., Cox, M.T., Forbus, K., Keane, M., Aamodt, A., Watson, I.: Retrieval, Reuse, Revision, and Retention in CBR. Knowledge Engineering Review 20(3) (2005)

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[15] Meyer, L.: Emotion and Meaning in Music. University of Chicago Press, Chicago (1956) [16] Molina-Solana, M., Arcos, J.L., Gómez, E.: Identifying Violin Performers by their Expressive Trends. Intelligent Data Analysis (2010) [17] Narmour, E.: The Analysis and cognition of basic melodic structures: the implicationrealization model. University of Chicago Press, Chicago (1990) [18] Palmer, C.: Anatomy of a performance: Sources of musical expression. Music Perception 13(3), 433–453 (1996) [19] Serra, X.: Musical Sound Modeling with Sinusoids plus Noise. In: Roads, C., Pope, S.T., Picialli, A., De Poli, G. (eds.) Musical Signal Processing, pp. 91–122. Swets and Zeitlinger Publishers (1997) [20] Widmer, G.: Discovering simple rules in complex data: A meta-learning algorithm and some surprising musical discoveries. Artificial Intelligence 146(2), 129–148 (2003)

25 T-Norms, T-Conorms, Aggregation Operators and Gaudí’s Columns Amadeo Monreal

25.1

Introduction

Antoni Gaudí i Cornet (1852-1926) is one of the most famous modernist architects and put this art movement to its highest standards [11]. His astonishing shapes, many of them inspired by nature, are fascinating and his work is worldwide famous by this. One of the most interesting features in Gaudí’s architecture is that these shapes are not in general generated arbitrarily, as could be thought in a first sight, but following precise geometric patterns. He was a master using conic section curves as well as catenary curves. Quadric surfaces can also be found in many of his masterpieces. To give an example, hyperboloids and hyperbolic paraboloids are the elements with which the vault of the Expiatory Temple of the Sagrada Família is built. Another way in which Gaudí uses geometry is generating twinnings, these being the intersection of different solid geometric figures. The columns of the Sagrada Família are based on the intersection of two salomonic columns and hence they are nice examples of twinnings (figure 25.1). Some capitals and pinnacles are also generated under the same law (figure 25.2). The way they are generated was studied and formalized in descriptive terms by M. Burry [4], [5], [6] and deserves a special attention. This paper is mainly devoted to give a mathematical formalization for these geometric objects as well as to some generalizations based on the use of tnorms, t-conorms and aggregation operators. If a Jordan curve (namely the ’basis curve’) on a plane π rotates around a point P of the bounded region at constant angular speed and simultaneously ascends along the axis r perpendicular to π and passing through P, we obtain a salomonic column. These columns were used mainly in the baroque period. Gaudí stars from this idea to generate the main columns of the Sagrada Família. He rotates the basis curve anticlockwise, thus generating a salomonic column, and clockwise, generating a second one, and then he takes the intersection of both columns. If we continue the column indefinitely, at a certain height h, the sections of both salomonic columns will coincide again and the column will be periodic, but Gaudí’s columns are more elaborated and built in pieces. By design, when the column reaches the height h/2, the sections of both salomonic columns are in its maximum gap of difference. At this level, the intersection of both copies of the basis curve is taken as the new basis curve to generate a second span of the column of height h/4, from h/2 R. Seising & V. Sanz (Eds.): Soft Comput. in Humanit. and Soc. Sci., STUDFUZZ 273, pp. 479–497. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com 

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Fig. 25.1. Examples of columns and capitals in the Temple of the Sagrada Família.

25.1 Introduction

481

Fig. 25.2. Examples of columns and capitals in the Temple of the Sagrada Família.

to 3h/4, until both copies of the new basis curve are at their maximum gap. From the intersection of these curves as new basis curve, a third span of height h/8, from 3h/4 to 7h/8, is generated, and so on. At the limit, as the number of pieces tends to infinity (and the total height tends to h) the obtained section would be the inscribed circle in the original basis curve (see figure 25.3).

Fig. 25.3. A column using a t-norm and its basis curve.

The rotating and ascending speed depends on h and, more interesting, on the symmetry of the original basis curve as we will see in section 25.2. For any section S of the column, the distance of a point in the contour of S and the axis is the minimum of the distances of the corresponding sections of both salomonic ones.

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25 T-Norms, T-Conorms, Aggregation Operators and Gaudí’s Columns

Taking into account that Minimum is a t-norm and an aggregation operator, this paper generalizes the generation of double rotation columns in the context of tnorms, t-conorms and aggregation operators.

25.2

Gaudí’s Columns

Though Gaudí starts with specific basis curves, the generations of a Gaudí’s column can be done from any closed plane curve. For the sake of simplicity, we will restrict our study to Jordan curves bounding a star domain. Definition 25.2.1. A Jordan curve is a (continuous) curve γ : [a, b] → R2 such that 1. It is closed ( γ (a) = γ (b)). 2. γ : (a, b) → R2 is injective. In other words, γ is homeomorphic to a circle. Definition 25.2.2. A set S in R2 is called a star domain if there exists a point P ∈ S such that for all Q ∈ S the line segment from P to Q is in S. A point such as P will be called a centre of the domain. N.B.: In the sequel we will assume that the bounded region defined by the Jordan curve is a star domain. Let us consider a centre P inside a Jordan curve γ and select an Euclidean coordinate reference system in R3 with γ in the plane XY and the origin in P. The curve γ can be described in polar coordinates as ρ = ρ (α ) (figure 25.4).

Fig. 25.4. An arbitrary Jordan curve.

When γ rotates anticlockwise a certain angle α0 , its polar equation becomes

ρ1 (α0 , α ) = ρ (α − α0 ).

(25.1)

25.2 Gaudí’s Columns

483

whereas if it rotates clockwise the same angle α0 , its polar equation is

ρ2 (α0 , α ) = ρ (α + α0 ).

(25.2)

Finally, the polar equation of the intersection of the two copies of the original Jordan curve γ due to this double rotation is

ρdr (α0 , α ) = Min (ρ1 (α0 , α ), ρ2 (α0 , α ))

(25.3)

see Figure 25.5.

Fig. 25.5. Double rotation of the Jordan curve and polar equation of the intersection of both copies.

When γ is used to generate a Gaudí’s column, α0 must depend linearly on the height z along the column: if αi is the initial angle applied to the double rotation of γ at the initial height zi and α f is the final angle corresponding to the final height z f , then

α0 (z) =

αi · (z f − z) + α f · (z − zi ) z f − zi

(25.4)

Thus, the salomonic column SC1 generated by γ rotating anticlockwise and ascending trough the OZ axis has parametric equation

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25 T-Norms, T-Conorms, Aggregation Operators and Gaudí’s Columns

SC1 (α , z) = (ρ1 (α0 (z), α ) · cosα , ρ1 (α0 (z), α )sinα , z)

(25.5)

while the salomonic column SC2 generated by γ rotating clockwise has parametric equation SC2 (α , z) = (ρ2 (α0 (z), α ) · cosα , ρ2 (α0 (z), α )sinα , z)

(25.6)

A piece of a Gaudí’s column is the intersection of both columns. Definition 25.2.3. Let γ be a Jordan curve, P a centre inside γ and consider an Euclidean coordinate system with γ in the plane XY and origin P. Let ρ = ρ (α ) be the equation of γ in polar coordinates. A piece of a Gaudí’s column GC is the (solid limited by the) surface defined by the parametric equation GC(α , z) = (ρdr (α0 (z), α ) · cosα , ρdr (α0 (z), α )sinα , z)

(25.7)

according with formulas (25.1) to (25.6). Let us investigate the rotations of different pieces or spans in relation with the group of symmetry [13] of the basis curve. Definition 25.2.4. Let A be a subset of R2 . The group of symmetry GA of A is the set of all the isometries of R2 that maintain A fixed. So f ∈ GA if and only if f (A) = A. Proposition 25.2.1. If A is bounded and GA = {Id}, then there exists a point P such that GA consists of only rotations around the fixed point P and axial symmetries with axes passing trough P. Proposition 25.2.2. The only finite groups of symmetry of plane figures are cyclic groups Cn and dihedral groups Dn . Cn consists of a rotation around a point of angle 2π /n and its multiples and has cardinality n. Dn consists of the same rotations as Cn plus axial symmetries and has cardinality 2n. Let us consider that the first span of a Gaudí’s column starts, at its basis, with a Jordan curve γ1 (i.e., γ1 is the initial basis curve of the column) with finite group of symmetry Gγ1 and let αmin = 2π /n be the smallest rotation angle of this group different from 0 around a fixed point P for some natural number n. If Gγ1 = {Id}, then n = 1 and αmin = 2π . If we rotate γ1 αmin /2 (anticlockwise) and −αmin /2 (clockwise) around P, both copies will coincide. So the maximal gap will be when γ1 is rotated ±αmin /4. This must be the double rotation reached by γ1 at the top of the first piece and the intersection of these two copies at this level is γ2 , the basis curve for the second piece of the column. γ2 has duplicated the group of symmetry of γ1 , therefore, the maximal gap for γ2 is obtained with a double rotation of an angle ±αmin /8, at the top of the second piece of the column, and so on.

25.3 Generalizing Gaudí’s Columns Using T-Norms and T-Conorms

485

According with formulas (25.4) to (25.7), if the group of symmetry of the initial basis curve γ1 is Cn or Dn , the first piece of the Gaudí’s column has the values αi = 0, π α f = 2n , zi = 0 and z f = h2 . Then, α0 (z) = πnhz and formulas 1,2 and 7 become

and

ρ1 (α0 (z), α ) = ρ (α −

πz ) nh

(25.8)

ρ2 (α0 (z), α ) = ρ (α +

πz ), nh

(25.9)

  πz   πz GC(α , z) = ρdr , α · cosα , ρdr ( , α ) · sinα , z nh nh

(25.10)

respectively, with α ∈ [0, 2π ] and z ∈ [0, h2 ]. For the second piece of the column, the new curve is γ2 with polar equation ρ =  π π ρdr 2n , α and the values are αi = 0, α f = 4n , zi = h2 and z f = 3h 4 , and so on. The shafts of the columns of the Temple of the Sagrada Família are formulated this way using foliated stars like that of the figure 25.3 as initial basis curve with different number of points depending on its position and its hierarchy in the temple, while the capitals like that of the figure 25.2 use a rectangle as initial basis curve but, in this case, proceeding top-to-bottom.

25.3

Generalizing Gaudí’s Columns Using T-Norms and T-Conorms

In this section we will use the same notation of the previous one. Once we have provided a mathematical formulation for the Gaudí’s double rotation, we dispose of a mathematical device that allows us to design new objects apart of those of the Sagrada Família simply generalizing the diverse elements or parameters appearing in the parametric functions. A first and easy generalization can be obtained changing the Jordan curve used as basis curve. A second way of generalization consists on letting the size of the basis curve vary along the axis according to some outline function. In the figure 25.6 we see some examples of both types of generalization. Another way of generalization, a bit more subtle, points to the binary operation applied, at each section of the column, to the two rotated copies of the basis curve γ ; the Minimum in the case of Gaudí. In fuzzy logic, the logical conjunction and disjunction are modeled by a t-norm and a t-conorm respectively. Since the Minimum is a t-norm, the Gaudí’s column, as we have formulated above, can be interpreted as the conjunction of SC1 and SC2 . The question that arises in this context is: What happens if the Minimum is replaced by another conjunction, i.e.

486

25 T-Norms, T-Conorms, Aggregation Operators and Gaudí’s Columns

another t-norm? In a similar way, we could define a new column as the union of the two salomonic columns by using the Maximum instead the Minimum. Replacing the t-conorm Maximum by another disjunction (i.e. a different t-conorm) we have another generalization of Gaudi’s column.

Fig. 25.6. A first set of generalizations.

Let us first recall some results on t-norms and t-conorms [1], [10], [12]. Definition 25.3.1. A continuous t-norm is a continuous map T : [0, 1] × [0, 1] → [0, 1] satisfying, for all x, y, x , y ∈ [0, 1], 1. T

(x, T (y, z)) = T (T (x, y), z). If x ≤ x , then T (x, y) ≤ T (x , y) 2. If y ≤ y , then T (x, y) ≤ T (x, y ) 3. T (x, 1) = T (1, x) = x. Example 1. 1. The Minimum t-norm Min defined for all x, y ∈ [0, 1] by Min(x, y). 2. The Product t-norm TΠ defined for all x, y ∈ [0, 1] by TΠ (x, y) = xy.

25.3 Generalizing Gaudí’s Columns Using T-Norms and T-Conorms

487

3. The Łukasiewicz t-norm TŁ defined for all x, y ∈ [0, 1] by TŁ (x, y) = Max(0, x + y − 1) Proposition 25.3.1. Let t : [0, 1] → [0, ∞] be a continuous decreasing map with t(1) = 0. The map T : [0, 1] × [0, 1] → [0, 1] defined for all x, y ∈ [0, 1] by

t −1 (x) if t(x) ≤ t(0) [−1] [−1] T (x, y) = t (T (x) + T (y)) where t (x) = 0 otherwise is a continuous t-norm. A t-norm that can be obtained in this way is called Archimedean and t an additive generator of T . Proposition 25.3.2. Let {Ti }i∈I be a set of continuous Archimedean t-norms and {(ai , bi )}i∈I a family of pairwise disjoint (non-empty) open subintervals of [0, 1] (I a finite denumerable set). Then the map T : [0, 1] × [0, 1] → [0, 1] defined as  

y−ai i ai + (bi − ai ) · Ti bx−a , if x, y ∈ (ai , bi ) i −ai bi −ai T (x, y) = Min(x, y) otherwise is a continuous t-norm called the ordinal sum of {Ti }i∈I , {(ai , bi )}i∈I . Definition 25.3.2. A continuous t-conorm is a continuous map S : [0, 1] × [0, 1] → [0, 1] satisfying, for all x, y, z, x , y ∈ [0, 1], 1. S(x, S(y, z)) = S(S(x, y), z). If x ≤ x , then S(x, y) ≤ S(x , y) 2. If y ≤ y , then S(x, y) ≤ S(x, y ) 3. S(x, 0) = S(0, x) = x. Example 2. 1. The Maximum t-conorm Max defined for all x, y ∈ [0, 1] by Max(x, y). 2. The Product t-conorm SΠ defined for all x, y ∈ [0, 1] by SΠ (x, y) = x + y − xy. 3. The Łukasiewicz t-conorm SŁ defined for all x, y ∈ [0, 1] by SŁ (x, y) = Min(1, x + y). Proposition 25.3.3. Let s : [0, 1] → [0, ∞] be a continuous increasing map with t(0) = 0. The map S : [0, 1] × [0, 1] → [0, 1] defined for all x, y ∈ [0, 1] by

s−1 (x) if s(x) ≤ s(1) [−1] [−1] S(x, y) = s (s(x) + s(y)) where s (x) = 1 otherwise is a continuous t-conorm. A t-conorm that can be obtained in this way is called Archimedean and t an additive generator of S.

488

25 T-Norms, T-Conorms, Aggregation Operators and Gaudí’s Columns

Proposition 25.3.4. Let {Si }i∈I be a set of continuous Archimedean t-conorms and {(ai , bi )}i∈I a family of pairwise disjoint (non-empty) open subintervals of [0, 1] (I a finite denumerable set). Then the map S : [0, 1] × [0, 1] → [0, 1] defined as  

y−ai i ai + (bi − ai) · Si bx−a , if x, y ∈ (ai , bi ) i −ai bi −ai S(x, y) = Max(x, y) otherwise is a continuous t-conorm called the ordinal sum of {Si }i∈I , {(ai , bi )}i∈I . With this background, we can generalize the Gaudí’s columns using these binary operators. In our case, in which our purpose is to do Geometry and to design objects rather than to face logical issues, we must take into account the scale of the object and, thus, we will need to fix in every case which length will be understood as the “unit” for the t-norms and t-conorms, i.e., the “unit” will be an additional parameter for these operator. For example, if T is a usual t-norm valued in [0, 1] and ρre f is the length of reference, i.e., the unit, we will use the “parameterized t-norm” T& defined as

x y & T (ρre f ; x, y) = ρre f · T , ρre f ρre f In some cases, in order to obtain a richer variety of designs, well will also accept values for ρ (α ) beyond ρre f . Definition 25.3.3. Let T be a continuous t-norm, γ a Jordan curve, P a centre in the region of the plane bounded by γ and consider an Euclidian coordinate system with γ in the plane XY an origin P. Let ρ = ρ (α ) be the equation of γ in polar coordinates. a) A double rotation induced by T is the curve with polar equation

ρdr,T (α0 , α ) = T (ρ (α + α0 ), ρ (α − α0 )) = T (ρ1 (α0 , α ), ρ2 (α0 , α )) (25.11) (recall formulas (25.1)) and (25.2)) b) A piece of a conjunctive Gaudí’s column with respect to T , GCT , is the (solid limited by the) surface defined by the parametric equation   GCT (α , z) = ρdr,T (α0 (z), α ) · cosα , ρdr,T (α0 (z), α ) · sinα , z (25.12)

25.3 Generalizing Gaudí’s Columns Using T-Norms and T-Conorms

489

Example 3. The column of figure 25.7 has been built using the basis curve of the same figure and the Product t-norm TΠ . Please notice that it is not a Gaudí’s column, since the first piece is repeated periodically. Definition 25.3.4. Let S be a continuous t-conorm, γ a Jordan curve, P a centre in the region of the plane bounded by γ and consider an Euclidian coordinate system with γ in the plane XY an origin P. Let ρ = ρ (α ) be the equation of γ in polar coordinates. a) A double rotation induced by S is the curve with polar equation

ρdr,S (α0 , α ) = S (ρ (α + α0 ), ρ (α − α0 )) = S (ρ1 (α0 , α ), ρ2 (α0 , α )) (25.13) (recall formulas (25.1)) and (25.2)) b) A piece of a disjunctive Gaudí’s column with respect to S, GCS , is the (solid limited by the) surface defined by the parametric equation   (25.14) GCS (α , z) = ρdr,S (α0 (z), α ) · cosα , ρdr,S (α0 (z), α ) · sinα , z

Fig. 25.7. A column using a t-norm and its basis curve.

Example 4. The column of figure 25.8 has been built using the Łukasiewicz t-conorm SŁ and the same basis curve of the figure 25.7. Again, the column grows periodically and hence is not a Gaudí’s one.

490

25 T-Norms, T-Conorms, Aggregation Operators and Gaudí’s Columns

Fig. 25.8. A column using a t-conorm.

25.4

Generalizing Gaudí’s Column Using Aggregation Operators

In a similar way as in the previous section, since the Minimum is an aggregation operator, the Gaudí’s column can be interpreted as the aggregation of two salomonic ones. Let us first recall the definition of an aggregation operator [2]. Definition 25.4.1. An aggregation operator is a continuous map  n A: ∞ n=1 [0, 1] → [0, 1] satisfying 1. A(0, . . . , 0) = 0, A(1, . . . , 1) = 1 2. A(x1 , . . . , xn ) ≤ A(y1 , . . . , yn ) if xi ≤ yi ∀xi , yi ∈ [0, 1]. Definition 25.4.2. An aggregation operator A is bisymmetric if and only if for all x1 , . . . , xn+m ∈ [0, 1], A(A(x1 , . . . , xn ), A(xn+1 , . . . , xn+m )) = A(x1 , . . . , xn+m ). Example 5. Let t : [0, 1] → [−∞, ∞] be a continuous strictly monotonous map. The map mt : ∞ n n=1 [0, 1] → [0, 1] defined for all x1 , . . . , xn ∈ [0, 1] by

t(x1 ) + . . . + t(xn ) mt (x1 , . . . , xn ) = t −1 n is a bisymmetric aggregation operator called the quasi-arithmetic mean generated by t. mt is continuous if and only if at most one of the elements −∞, ∞ belongs to the image of t. Example 6. 1. If t(x) = x for all x ∈ [0, 1], then mt is the arithmetic mean mt (x, y) = x+y 2 . 2. If t(x) = logx for all x ∈ [0, 1], (log(0) = −∞), then mt is the geometric mean √ mt (x, y) = xy.

25.5 A Park Güell’s Tower

491

The following definition appears naturally as a generalization of Gaudí’s columns using aggregation operators. Definition 25.4.3. Let A be an aggregation operator, γ a Jordan curve, P a centre in the region of the plane bounded by γ and consider an Euclidian coordinate system with γ in the plane XY an origin P. Let ρ = ρ (α ) be the equation of γ in polar coordinates. a) A double rotation induced by A is the curve with polar equation ρdr,A (α0 , α ) = A (ρ (α + α0 ), ρ (α − α0 )) = A (ρ1 (α0 , α ), ρ2 (α0 , α ))

(25.15)

(recall formulas (25.1)) and (25.2)) b) A piece of an aggregated Gaudí’s column with respect to A, GCA , is the (solid limited by the) surface defined by the parametric equation GCA (α , z) = ρdr,A (α0 (z), α ) · cosα , ρdr,A (α0 (z), α ) · sinα , z)

(25.16)

Example 7. The column of figure 25.9 has been built using the geometric mean and the same basis curve of the figure 25.7.

Fig. 25.9. A column using a an aggregation operator.

25.5

A Park Güell’s Tower

What is really amazing is that Gaudí in fact did use t-conorms or, in a more prudent way, t-conorms can be used to model some of its works. At the main entrance of Park Güell in Barcelona there are two buildings, one of them with a blue and white ventilation tower (figure 25.10). The way it is conceived or modeled is based on the double rotation of a basis curve, but with some differences with respect to the columns presented so far.

492

25 T-Norms, T-Conorms, Aggregation Operators and Gaudí’s Columns

Firstly, the tower is generated in only one piece, but allowing the sections to diminish as the height increases (the tower is inscribed in a hyperboloid) as in our second generalization. Also, periodically, in the ascending of the column, just when both copies of the basis curve coincide, the rotation stops for a while. But (most interesting in this paper) a t-conorm is used to calculate each section. The t-conorm is very close to the Maximum but, since the tower not has rough edges, the t-conorm must be smooth. A good candidate for a smooth t-conorm close to the Maximum is one of Frank’s family. Definition 25.5.1. Let s ∈ (0, 1) ∪ (1, ∞).The Frank t-conorm with parameter s is defined by

(sx − 1)(sy − 1) s S (x, y) = x + y − logs 1 + . (25.17) s−1

Fig. 25.10. The actual tower at the main entrance of Park Güell.

Figure 25.11 shows the model of the tower using Frank’s t-conorm for s = 0.5.

25.6 The Limit Curve

493

Fig. 25.11. The model of the tower at the main entrance of Park Güell.

25.6

The Limit Curve

In this section we will assume that “1” means the “unit” of the column, as we have stated in section 25.3. We will also restrict our study to Jordan curves satisfying the following constraint: if ρ = ρ (α ) is the polar equation of the Jordan curve, then Sup {ρ (α ), 0 ≤ α ≤ 2π } ≤ 1. As we stated in the introductory section, the section of a Gaudí’s column when the height is h is the inscribed circle to the original basis curve. In this section we investigate the limit curve of conjunctive, disjunctive and aggregated Gaudí’s columns. The results are quite simple for t-norms and tconorms, but more interesting for aggregation operators. If the Minimum is replaced by an Archimedean t-norm or t-conorm, the following lemma follows trivially. Lemma 1. Let T be a continuous Archimedean t-norm, γ a Jordan curve, ρ = ρ (α ) its equation in polar coordinates with Sup {ρ (α ), 0 ≤ α ≤ 2π } ≤ 1 ≤ β ≤ 1 and GCT the conjunctive Gaudí’s column generated from γ . The sections of the column tend to a point of the axis. Proposition 25.6.1. Let T be a continuous Archimedean t-norm, γ a Jordan curve different from the circle ρ = 1 and GCT the conjunctive Gaudí’s column generated from γ . The sections of the column tend to a point of the axis. Proof. If the group of symmetry of γ is Cn or Dn and β is some value with ρ (β ) = 1, by continuity there exists an interval T = (β − ε , β + ε ) of angular length 2ε with ρ (x) = 1 for all x ∈ T . In the first section of the second piece or span of the column, there will be 2n intervals of length 2ε with all their points x satisfying ρ (x) < 1; in the first section of the nth piece there will be 2n−1n intervals of length 2ε with all their points x satisfying ρ (x) < 1. When 2n−1 n × 2ε > 2π , all points of the section will be at distance smaller than 1 to the axes. 

494

25 T-Norms, T-Conorms, Aggregation Operators and Gaudí’s Columns

Similarly, Proposition 25.6.2. Let S be a continuous Archimedean t-conorm, γ a Jordan curve, ρ = ρ (α ) its equation in polar coordinates with Sup {ρ (α ), 0 ≤ α ≤ 2π } ≤ β < 1 and GCs the disjunctive Gaudí’s column generated from γ . The sections of the column tend to the circle with centre in the axis and radius 1. More interesting is the study of the limit curve for aggregating columns. Let γ be a Jordan curve and P a centre point in the region of the plane bounded by γ and consider an Euclidian coordinate system with γ in the plane XY and origin P. Let ρ = ρ (α ) be the equation of γ in polar coordinates. For the sake of simplicity, let us first suppose that the group of symmetry of γ consists only of the identity. Then the maximum gap between the two sections will be when α0 = π /2. The section will then be   π  π  A ρ α+ ,ρ α − . 2 2 The final section of the second piece will then be    π π  π π    π π  π π  A A ρ α+ + ,ρ α − + ,A ρ α + − ,ρ α − − . 2 4 2 4 2 4 2 4

Assuming that A is a bisymmetric aggregation operator, the last expression becomes   π π  π π  π π  π π  A ρ α+ + ,ρ α − + ,ρ α + − ,ρ α − − . 2 4 2 4 2 4 2 4 and the final section of the nth   piece of Gaudí’s column will be the aggregation of the elements ρ α + ∑ni=1 ε 2πi , where ε can take the values 1 and −1.   Lemma 2. The set An = ∑ni=1 εi 2i1 , εi = −1, 1 coincides with the set * n + −2 + 1 −2n + 3 −1 1 2n − 3 2n − 1 Bn = , ,··· n , n , , . 2n 2n 2 2 2n 2n Proof.

An = *

) n ε −1 −2n + 1 2  i + n ∑ 2i + 2 ∑ 2i , εi = 0, 1 = n 2 2 i=1 i=1 n

n

∑

) 2n−i εi , εi

= 0, 1

=

i=1

+ * n + −2n + 1 2k −2 + (2k + 1) n n + , k = 0, . . . , 2 − 1 = , k = 0, . . . , 2 − 1 = Bn . 2n 2n 2n

 Proposition 25.6.3. Let γ be a Jordan curve and P a centre point in the region of the plane bounded by γ and consider an Euclidian coordinate system with γ in the plane XY and origin P. Let ρ = ρ (α ) be the equation of γ in polar coordinates. Let the group of symmetry of γ consists only of the identity. The final section of the nth

25.6 The Limit Curve

495

piece of the aggregated Gaudí’s column with respect to a bisymmetric aggregation operator A is



2n − 1 2n − 3 A ρ α− π , ρ α − π ,... 2n 2n







1 1 2n − 3 2n − 1 ...,ρ α − n π ,ρ α + n π ,ρ α + π , ρ α + π . 2 2 2n 2n If we take the limit of the last expression when n tends to ∞, we obtain a kind of Riemann integral. In particular, if A is the quasi-arithmetic mean mt generated by t, we obtain Proposition 25.6.4. Let γ be a Jordan curve and P a centre point in the region of the plane bounded by γ and consider an Euclidian coordinate system with γ in the plane XY and origin P. Let ρ = ρ (α ) be the equation of γ in polar coordinates. Let the group of symmetry of γ consist only of the identity. The polar equation of the last section of the aggregated Gaudí’s column with respect to the quasi-arithmetic mean mt generated by t is  2π

1 −1 ρ (α ) = t t ◦ ρ (x)dx 2π 0 Note that the integral does not depend on the value of α and therefore the curve is a circle of this radius. A similar integral appeared in [8] when aggregating a non-finite family of indistinguishability operators. Proof. The limit of

       ⎞ n 2n −3 2n −3 2n −1 t ◦ ρ α − 2 2−1 n π + ρ α − 2n π + · · · + ρ α + 2n π + ρ α − 2n π ⎠ t −1 ⎝ 2n ⎛





1 2n − 1 2n − 3 · t ◦ρ α − π + ρ α − π + ... 2π 2n 2n





2n − 3 2n − 1 2π ... + ρ α + π + ρ α − π · 2n 2n 2n

= t −1



when n → ∞ is t

−1



1 2π

 2π 0

t ◦ ρ (α + x)dx .

Putting α + x = u, dx = du and the last expression becomes  α +π

1 −1 t t ◦ ρ (u)du . 2π α −π Since ρ is a periodic function of period 2π , we get the result.



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25 T-Norms, T-Conorms, Aggregation Operators and Gaudí’s Columns

If the group of symmetry of the basis curve γ is Cn or Dn we get the following similar result Proposition 25.6.5. Let γ be a Jordan curve and P a centre point in the region of the plane bounded by γ and consider an Euclidian coordinate system with γ in the plane XY and origin P. Let ρ = ρ (α ) be the equation of γ in polar coordinates. Let the group of symmetry of γ be Cn or Dn and P the centre of all rotations of the group. The polar equation of the last section of the aggregated Gaudí’s column with respect to the quasi-arithmetic mean mt generated by t is   2π  n n ρ (α ) = t −1 t ◦ ρ (x)dx 2π 0 Again, this result not depend on the value of α . For other aggregation operators, different results would be obtained. For example, if the aggregation operator is the projection to the first or the last coordinate, we obtain simply one of the salomonic columns, SC1 or SC2 . In general, non commutative aggregation operators would give more importance to one salomonic column than to the other one.

25.7

Concluding Remarks

In this paper a mathematical model for Gaudí’s columns, providing its parametric equations based on the polar equation of the basis curve and the Minimum operator, have been given and also it have been given different generalizations of it by replacing the Minimum in its formula by a t-norm, a t-conorm or an aggregation operator. Let us recall that, in a linearly ordered universe X , a map ant: X → X is called and antonymy map when it is involutive and decreasing. The antonym υ of the fuzzy set μ of X is the defined by υ (x) = μ (ant(x)) [3]. In particular, the map φ : [0, 2π ] → [0, 2π ] given by φ (α ) = 2π − α is an antonymy map. Also, as the polar equation ρ (α ) is a periodic map, we have

ρ2 (α0 , α ) = ρ (α − α0 ) = ρ (α + (2π − α0 )) = ρ2 (2π − α0 , α ) = ρ1 (φ (α0 ), α ). Thus, ρ2 can be understood as an antonymy of ρ1 with respect to the first variable. This gives another interpretation to the two columns as one can be thought as the antonym of the other. Combining two columns related by a different antonymy map would give new shapes. The results of this paper open a novel application of the most used operators in soft computing (t-norms, t-conorms and aggregation operators) to the Computer Aided Design, a branch in which these tools have been rarely used [7], [9]. The generalizations developed in this paper suggest that many of Gaudí’s ideas have not yet been completely explored.

25.7 Concluding Remarks

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Acknowledgments The author whish to thank the Junta Constructora del Temple Expiatori de la Sagrada Família for its authorization to the use of the pictures of the Temple included in this work. This work is partially supported by the project TIN2009-07235.

References [1] Alsina, C., Frank, M.J., Schweizer, B.: Associative Functions: Triangular Norms and Copulas. World Scientific (2006) [2] Calvo, T., Kolesárová, A., Komorníková, M., Mesiar, R.: Aggregation operators: Properties, classes and construction methods. In: Aggregation Operators: New Trends and Applications. Physica-Verlag (2002) [3] De Soto, A.R., Recasens, J.: Modelling a Linguistic Variable as a Hierarchical Family of Partitions induced by an Indistinguishability Operator. Fuzzy Sets and Systems 121, 57–67 (2001) [4] Giralt, M.D.: Gaudí, la búsqueda de la forma, Lunwerg (ed.) Barcelona 2002. [5] Gómez, J., Coll, J., Burry, M.C., Melero, J.C.: La Sagrada Familia. In: De Gaudíal, C.A.D. (ed.) UPC, Barcelona (1996) [6] Gómez, J., Coll, J., Burry, M.C.: Sagrada Familia s. XXI. In: Gaudíahora (ed.) UPC, Barcelona (2008) [7] Jacas, J., Monreal, A., Recasens, J.: A model for CAGD using fuzzy logic. International Journal of Approximate Reasoning 16, 289–308 (1997) [8] Jacas, J., Recasens, J.: Aggregation of T-Transitive Relations. Int J. of Intelligent Systems 18, 1193–1214 (2003) [9] Jacas, J., Monreal, A.: Gaudíen el siglo XXI. In: V International Conference Mathematics and Design, Univ. Regional de Blumenau (2007) [10] Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Kluwer, Dordrecht (2000) [11] Navarro, J.J.: Gaudí, el arquitecto de Dios. Planeta, Barcelona (2002) [12] Schweizer, B., Sklar, A.: Probabilistic metric Spaces. North-Holland (1983) [13] Yale, P.B.: Geometry and Symmetry. Dover, New York (1988)

Authors

Josep Lluís Arcos is a Research Scientist of the Artificial Intelligence Research Institute of the Spanish National Research Council (IIIA-CSIC). Dr. Arcos received an M.S. in Musical Creation and Sound Technology from Pompeu Fabra University in 1996 and a Ph.D. in Computer Science from the Universitat Politècnica de Catalunya in 1997. He is co-author of more than 100 scientific publications an co-recipient of several awards at case-based reasoning conferences and computer music conferences. Presently he is working on case-based reasoning and learning, on self-organization and self-adaptation mechanisms, and on artificial intelligence applications to music. Txetxu Ausín is Tenured Scientist and Chair of the Unit of Applied Ethics at Spanish National Research Council (CSIC), Madrid. He received his Ph.D. in Philosophy and First Prize for the year from the University of the Basque Country in 2000. His research focuses are deontic logic, bioethics, ethics of communication, and human rights. He is Editor of the web and journal about Applied Ethics DILEMATA: www.dilemata.net. Clara Barroso has a Ph.D. in Philosophy and Education Science from the University of La Laguna, Spain where is Associated Professor at the Dpt. of Logic and Moral Philosophy at the same university. She has made research stays at the Institute of Philosophy of CSIC, the Centre d’Estudis Avançats of Blanes, the Institut de Investigació en Intelligencia Artificial of CSIC and University of Edinburgh Her areas of interest are in general related to the field of Science, Technology and Society and, in particular, to the application of information technologies to teaching and learning processes and knowledge representation. Jeremy Bradley is a doctoral candidate at Vienna University of Technology, specializing in the field of computer linguistics. He holds master’s degrees in both computer science and linguistics and is currently employed as a researcher at the Institute for European and Comparative Linguistics and Literature at the University of Vienna. His main endeavours currently are the creation of text simplification software for patients of aphasia and software aiding the revitalization and preservation of endangered languages. Joao Paulo Carvalho has a PhD and MsC degrees from the Technical University of Lisbon (Instituto Superior Técnico), Portugal, where he is currently Professor at the Electrical and Computer Engineering Department. He has taught courses on Computational Intelligence, Distributed Systems, Computer Architectures and Digital Circuits since 1998. He is also a senior researcher at L2F - Spoken Language Systems Laboratory of INESC-ID Lisboa, where

500

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he has been working since 1991. His main research interests involve applying Computational Intelligence techniques to solve problems in non-computing related areas. Jaume Casasnovas was born in Palma (Mallorca, Spain) in 1951 and he passed away on July 14th, 2010. He received the B.S. degree in mathematics from the University of Barcelona, Barcelona, Spain, in 1973 and the Ph.D. degree in Computer Science, from the University of the Balearic Islands, Palma (Mallorca), Spain in 1989. In 1994, he won a position of University teacher at the University of the Balearic. He was a member of Fuzzy Logic and Information Fusion (LOBFI) Research group and a member of the research group BIOCOM (UIB) in the fields of Computational Biology and Bioinformatics. His research interests included fuzzy logic, mathematics education and bioinformatics. Barbara Diaz Diez received her BA and PhD in Economics from Malaga University, Spain. She is an Associate Professor at the Statistics and Econometrics Department, University of Malaga. She was a visiting scholar at University of Marseille and NEURINFO (NeuroInformatique et Systèmes Flous). She had a postdoctoral scholarship for two years, funded by the Spanish State Secretary of Education and Universities and the European Social Fund, to support her stay as a visiting scholar at BISC (Berkeley Initiative in Soft Computing) of the EECS Department, UC Berkeley. Her research topics of interest are Fuzzy logic, approximate reasoning, robust statistics and Input-Output Analysis. Lukas Gander graduated in computer science in 2008 at the University of Innsbruck with special subject Databases and Information Systems. Since 2009 he is working at the Department for Digitisation and Digital Preservation at the library of the University of Innsbruck. His current research area is the reconstruction of the logical structure of books based on OCR output result files of still images. Itziar García-Honrado (Gijón, Spain, 1985) obtained her bachelor degree in Mathematics from the University of Oviedo in 2007. Since then, she is carrying out her PhD Studies at the University of León and is working at the European Centre for Soft Computing as Research Assistant. She has published some papers in international journals and conferences. Her research is focused on the meaning of connectives in the context on Computing With Words, and in the analysis of models of everyday reasoning, using models based on conjectures, hypothesis and consequences. José Luis García-Lapresta is a Professor of Quantitative Methods in Economics and Business in the Dep. of Applied Economics of the University of Valladolid (Spain). He received a Ph.D. in Mathematics from the Barcelona University (Spain) in 1991. His research in mainly devoted to preference modelling, social choice and decision making. He has published more than 50 international papers in scientific journals (European Journal of Operational Research; Fuzzy Optimization and Decision Making; Fuzzy Sets and Systems; Group Decision and Negotiation; Information Sciences; International Journal of Computational Intelligent Systems; International Journal of Intelligent Systems; International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems; Journal of Mathematical Economics; Mathematical Logic Quarterly; Mathematical Social Sciences; Neural Network World; Public Choice; Review of Economic Design; Social Choice and Welfare; and Soft Computing, among others) and books (Elsevier, IEEE Press, Springer-Verlag and World Scientific, among others). He has contributed with more than 170 papers in scientific conferences in near 30 different countries, and he has made several stays in different universities of Australia, Austria, France, Ireland, Italy, The Netherlands and Turkey.

Authors

501

Hanns-Werner Heister was born 1946 in Plochingen/Neckar, Germany. Dr. phil. habil., Professor for Musicology at the Hochschule für Musik und Theater Hamburg. - Publications on methodology of musicology, music aesthetics and sociology, music history, political, popular music and new music, music and musical culture in the Nazi era, in resistance and in exile, aesthetics and history of music theatre, media and institutions of music culture, music anthropology (in particular music and human perception, origins of art), music and other arts. Among others: Das Konzert. Theorie einer Kulturform [The Concert. Theory of a Cultural Form], 2 volumes (1983), Jazz (1983); Vom allgemeingültigen Neuen. Analysen engagierter Musik: Dessau, Eisler, Ginastera, Hartmann [Of the Universal New. Analyses of engaged Music:. . . ] (2006); (photographies by Ines Gellrich) Un/Endlichkeit. Begegnungen mit György Ligeti [In/Finity. Encountering Görgy Ligeti] (2008); Hintergrund Klangkunst [Background Sound Art] (2008; publ. 2009). Co-publisher of (among others): Musik und Musikpolitik im faschistischen Deutschland [Music and Music Politics in Fascist Germany] (1984); Komponisten der Gegenwart [Contemporary Composers] (loose leaf lexicon, since 1992, so far 38 deliveries); Zwischen Aufklärung & Kulturindustrie [Between Enlightenment and Cultural Industry] (3 volumes, 1993); Musik und. Eine Schriftenreihe der Hochschule für Musik und Theater [Music and] Hamburg (since 2000). - Editor of among others: Zwischen/Töne. Musik und andere Künste [Over/tones. Music and other Arts] (series since 1995; Musik/Revolution [Music/Revolution] (3 volumes, 1996/97); “Entartete Musik” 1938 - Weimar und die Ambivalenz [“Degenerated Music” 1938 - Weimar and Ambivalence], 2 volumes (2001); Geschichte der Musik im 20. Jahrhundert [History of Music in the 20th Century], Vol. III: 1945-1975 (2005); Zur Ambivalenz der Moderne [On the Ambivalence of Modernity] (series Musik/Gesellschaft/Geschichte), 4 volumes, Vol. 1 2005; Vol. 2-4 2007. Krzysztof Kielan received his Medical Diploma from Medical University of Silesia, Poland, in 1988. He completed his clinical training in psychiatry and he received the PhD degree in medicine in 1995. He is currently a Consultant Psychiatrist at the Department of Psychiatry, Harrison House, Peaks Lane, Grimsby, UK, and an honorary senior lecturer at Hull & York Medical School. His clinical practice and research interests include: brain mapping, QEEG assessment, new ways of working within acute setting, rehabilitation of cognitive functions, ADD/ADHD and Asperger spectrum, dynamic psychotherapy, and computer-based medical decision support systems. Olga Kosheleva received her M.Sc. in Mathematics from Novosibirsk University, Russia, in 1978, and M.Sc. in Computer Science (1994) and Ph.D. in Computer Engineering (2003) from the University of Texas at El Paso. In 1978-80, she worked with the Special Astrophysical Observatory (representation and processing of uncertainty in radioastronomy). In 1983-87, she was with the Computing Center and Department of Automated Control Systems, Technological Institute for Refrigerating Industry, St. Petersburg, Russia; in 1987-89, worked as a Senior Research Associate with “Impulse”, a Consulting Firm in Applied Mathematics and Computing, St. Petersburg, Russia. Since 1990, she is with the University of Texas at El Paso. She also served as a Visiting Researcher with the Euler International Mathematical Institute, St. Petersburg, Russia (2002), and with the Catholic University of Pelotas, Brazil (2003-04). Main current interest: applications of information technology, especially applications to education in mathematics, science, and engineering. Published more than 60 journal papers and more than 80 papers in refereed conference proceedings. Served on National Science Foundation panels.

502

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Vladik Kreinovich received his M.Sc. in Mathematics and Computer Science from St. Petersburg University, Russia, in 1974, and Ph.D. from the Institute of Mathematics, Soviet Academy of Sciences, Novosibirsk, in 1979. In 1975-80, he worked with the Soviet Academy of Sciences, in particular, in 1978-80, with the Special Astrophysical Observatory (representation and processing of uncertainty in radioastronomy). In 1982-89, he worked on error estimation and intelligent information processing for the National Institute for Electrical Measuring Instruments, Russia. In 1989, he was a Visiting Scholar at Stanford University. Since 1990, he is with the Department of Computer Science, University of Texas at El Paso. Also, served as an invited professor in Paris (University of Paris VI), Hong Kong, St. Petersburg, Russia, and Brazil. Main interests: representation and processing of uncertainty, especially interval computations and intelligent control. Published 3 books, 6 edited books, and more than 700 papers. Member of the editorial board of the international journal “Reliable Computing” (formerly, “Interval Computations”), and several other journals. Co-maintainer of the international website on interval computations http://www.cs.utep.edu/interval-comp. Honors: President-Elect, North American Fuzzy Information Processing Society; Foreign Member of the Russian Academy of Metrological Sciences; recipient of the 2003 El Paso Energy Foundation Faculty Achievement Award for Research awarded by the University of Texas at El Paso, and a co-recipient of the 2005 Star Award from the University of Texas System. Mila Kwiatkowska received the MA degree in Interdisciplinary Studies of Polish Philology and Informatics from the University of Wroclaw in Poland, the MSc degree in Computing Science from the University of Alberta in Canada, and the PhD degree in Computing Science from Simon Fraser University in Canada. She is currently an Assistant Professor in the Computing Science Department at Thompson Rivers University (TRU) in Kamloops, Canada. Her research interests include medical data mining, medical decision systems, clinical prediction rules, knowledge representation, fuzzy logic, semiotics, and case-based reasoning. She is author or co-author of over 30 peer-reviewed publications. She teaches courses in database systems, data mining, web-based systems, and biomedical informatics. Teresa León received her degree in Mathematics in 1987 and is a member of the Department of Statistics and Operations Research of the University of Valencia since 1988. She began to work in Fuzzy Logic in 1998; she has published several theoretical and applied papers related with fuzzy optimization (multi-objective programming, portfolio selection and human resources management), fuzzy data envelopment analysis, fuzzy random variables and aggregation operators. Recently, she has become interested in the study of Music and Soft Computing. Vicente Liern is a mathematician (he received his degree in Mathematics in 1987) and also a musician: he plays the trombone. Currently, he is a professor in the Departmento de Matemática para la Economía y la Empresa of the University of Valencia. He began to work in Fuzzy Logic in 1998 and since then he has published several theoretical and applied papers related with fuzzy optimization: multi-objective programming, portfolio selection, human resources management and the fuzzy p-median problem. He has always been interested in the relationship between Music and Mathematics and he has published a number of papers about this subject. Gaspar Mayor was born in Palma (Mallorca, Spain) in 1946. He received a B.S. degree in Mathematics from the Universitat de Barcelona in 1971, and a Ph. D. degree in Mathematics from the Universitat de les Illes Balears in 1984. He is currently a Professor at the Department

Authors

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of Mathematics and Computer Science of the University of the Balearic Islands in Palma, where he is the Head of the LOBFI Research Group, which is mainly engaged in research of fuzzy logic and fusion of information. His research interests include aggregation operators, multidimensional aggregation problems, fuzzy connectives and functional equations. Krzysztof Michalik is a Professor in the Department of Knowledge Engineering, University of Economics, Katowice, Poland. His research interests include fuzzy logic, knowledge management, and medical expert systems. He is a founder of an AI software company, AITECH. His hybrid expert system, SPHYNX, has been utilized in many medical and financial applications. Antonio Morillas holds a Ph.D degree in Economics from the University of Malaga, where he carries out teaching and research activities as a Full Professor at the Applied Economics (Statistics and econometrics) Department. He is author of numerous papers in scientific books and reviews, with the application of quantitative techniques, such as graph theory or fuzzy logic, to the economic analysis, especially in the Regional Economics area, Input-Output Analysis and the Environment. In 1992 he was a visiting scholar at U.C. Berkeley for a year, at the Institut of Urban and Regional Development, and he became a member of the Berkeley Iniciative in Soft Computing (BISC) from that date. Amadeo Monreal Pujadas got his Bachelor of Sciences (Mathematics) from the Universitat Autònoma de Barcelona (1989), PhD in Mathematics from the Universitat Politècnica de Catalunya (UPC) (2001), with the thesis: Modelling of curves and surfaces with applications to the Computer Aided Geometric Design and to the Architecture; Current job: Lecturer from the Technical School of Architecture of Barcelona, UPC, where he teaches common mathematics and modelling of curves and surfaces. His research is devoted to developing a mathematical grammar based on mathematical formulations enabling families or types of geometric objects as a basis for computer aided design. Regarding this subject, he has published several articles and participated in several conferences and seminars. He collaborates, through an agreement, with the technical office that manages the construction of the Sagrada Família’s temple in Barcelona, for the implementation of programs for generating conics and quadrics, and for solving diverse geometric problems that frequently arise in the process of realization of the project. Takehiko (Take) Nakama received two Ph.D.s from The Johns Hopkins University in Baltimore, Maryland, USA. He completed his first Ph.D. program in 2003 by conducting quantitative neurophysiological research at The Johns Hopkins Krieger Mind/Brain Institute. In 2009, he completed his second Ph.D. program in the Department of Applied Mathematics and Statistics by conducting research on stochastic processes that mathematically characterize stochastic search and optimization algorithms. In 2010, he completed the master’s program at the European Center for Soft Computing and joined Enrique Ruspini’s research unit as a postdoctoral researcher. His research interests include measure-theoretic probability, stochastic processes (Markov chains in particular), statistics for Hilbert-space-valued data (including fuzzy data), stochastic optimization, analysis of algorithms, system identification, mathematical foundations of fuzzy logic, and reasoning under uncertainty. José A. Olivas, born in 1964 in Lugo (Spain), received his M.S. degree in Philosophy in 1990 (University of Santiago de Compostela, Spain), Master on Knowledge Engineering of the Department of Artificial Intelligence, Polytechnic University of Madrid in 1992, and his Ph.D. in Computer Science in 2000 (University of Castilla-La Mancha, Spain). In 2001 was

504

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Postdoc Visiting Scholar at Lotfi A. Zadeh’s BISC (Berkeley Initiative in Soft Computing), University of California-Berkeley, USA. His current main research interests are in the field of Soft Computing for Information Retrieval and Knowledge Engineering applications. He received the Environment Research Award 2002 from the Madrid Council (Spain) for his PhD. Thesis. Lorenzo Peña is a Spanish logician and lawyer. He received his Ph.D. degree in philosophy from Liege University in 1979. He is research professor at the CSIC, the Spanish leading academic institution devoted to scientific research. He has worked on systems of paraconsistent gradualistic logic ant their applications to the field of legal and deontic reasoning. Ashley Piggins is a Lecturer in Economics at the J.E. Cairnes School of Business and Economics, National University of Ireland Galway. He received a Ph.D. In Economics at the University of Bristol in 1999. His research is mainly devoted to social choice theory and issues on the boundary of economics and philosophy. He has published in Social Choice and Welfare, Economic Theory, Journal of Mathematical Economics and Journal of Logic and Computation among others. He is an Associate Editor of Economics Bulletin and was chair of the Public Economic Theory conference in 2009 which was held in Galway, Ireland. In 2008, Ashley was appointed a visiting Professor of Economics at the Université de Cergy-Pontoise in Paris. He has given research seminars at the Université de Paris 1 Panthéon-Sorbonne, University of Bath, University of Birmingham, University of Bristol, Trinity College Dublin, Queen’s University Belfast, Université de Caen, Universidad de Murcia, Universität Osnabrück, University of St. Andrews, University of East Anglia and NUI Galway among others. Cristina Puente Águeda, born in 1978 in Madrid (Spain), received her M.S. degree in Computer Engineering in 2001 (ICAI-Pontificia Comillas University, Madrid, Spain). She is Phd. student (ICAI-Pontificia Comillas University, Madrid, Spain). Principal Employment and Affiliations: From 2005: Associate Professor of the Department of Telecommunications, SEK (IE) University, Segovia, Spain. From 2006: Professor of the Department of Computer Science, ICAI -Pontificia Comillas University, Madrid, Spain. From 2007: Professor of the Department of Mathematics, ICAI -Pontificia Comillas University, Madrid, Spain. Ulrich Reffle graduated in 2006 in computational linguistics at the “Center for information and language processing” at the University of Munich. Since then he has worked at this institute as a researcher in the group of Prof. Klaus U. Schulz. His main fields of interest include finite state technology, enhancement and post-correction of OCR, and the processing of historical language. Juan Vicente Riera was born in Palma (Mallorca, Spain) in 1969. He received the B.S. degree in Mathematics from the University of Salamanca, Spain, in 1994. Currently, he is a Secondary school teacher and Associate Professor in the Department of Mathematics and Computer Science, University of Balearic Islands, Palma (Mallorca), Spain, and a member of Fuzzy Logic and Information Fusion (LOBFI) Research group. He is working on the research of fuzzy logic, including aggregation operators, fuzzy connectives, discrete fuzzy numbers, fuzzy multisets and functional equations. Christoph Ringlstetter finished his Ph.D. in Computational Linguistics in 2006. From 2006 – 2008, he was a postdoc at the Alberta Ingenuity Center for Machine Learning (AICML), University of Alberta, Canada. In 2008 Christoph Ringlstetter joined EC IMPACT project

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on mass-digitization of historical books. Research interests are corpus linguistics, document post-processing, information retrieval in noisy environments and semantic search. He is now with the Center of Information and Language Processing, University of Munich. Kazem Sadegh-Zadeh, born in 1942, is an analytic philosopher of medicine. He studied medicine and philosophy at the German universities of Münster, Berlin, and Göttingen with Internship and residency 1967-1971, assistant professor and lecturer 1972-1982, full professor of philosophy of medicine at the University of Münster 1982-2004. SadeghZadeh was born in Tabriz, Iran, and is a German citizen. He is the founder of the analytic philosophy of medicine. He has been working in this area since 1970. He is the founding editor of the international journals Metamed, founded in 1977 (current title: Theoretical Medicine and Bioethics, published by Springer Verlag) and Artificial Intelligence in Medicine, founded in 1989 (published by Elsevier). His work includes: Handbook of Analytic Philosophy of Medicine (2011), “The Prototype Resemblance Theory of Disease” (2008), theory of fuzzy biopolymers (2000), and theory of the Machina Sapiens (2000). From: htt p : //en.wikipedia.org/wiki/KazemS adegh − Zadeh Veronica Sanz earned a Ph.D. in Philosophy at the University Complutense of Madrid (Spain) in February 2011 with a dissertation titled “Contextual Values in Science and Technology: the case of Computer Technologies”. During 2003-2004 she received a scholarship from University of California and University Complutense to attend during two semesters the Graduate Program of the Office for the History of Science and Technology (History Department) at University of California at Berkeley. From 2005 to 2007 she has been an Assistant Researcher at the Department of Science, Technology and Society of the National Council of Scientific Research (CSIC), Spain. In 2008 she was during 5 months a Fellow Researcher at the Institute of Advances Studies on Science, Technology and Society (IAS-STS) in Graz (Austria). From December 2008 to October 2010 has been Visiting Graduate Student at the European Centre for Soft Computing in Mieres (Asturias). At the moment she is a Postdoctoral Researcher at the Science, Technology and Society Center (STSC) in the University of California at Berkeley. Her main research areas are Philosophy of Technology, Science and Technology Studies and Feminist Technoscience Sudies, with a particular focus on Information and Communication Technologies and Artificial Intelligence. Sven Schlarb holds a PhD in Humanities Computer Science from the University of Cologne. Before joining the Austrian National Library, where he is participating in the EU-funded projects PLANETS and IMPACT, he worked as a software engineer in Cologne and Madrid and as support consultant at SAP in Madrid. Klaus U. Schulz finished his Ph.D. in Mathemetics in 1987. After a visiting professorship at the University of Niteroi he was appointed professor of Computational Linguistics at the University of Munich (LMU) in 1991. He is a technical director of the Centrum für Informationsund Sprachverarbeitung (CIS) of the LMU. Recent research interests are concentrated on text correction, document analysis, information retrieval and semantic technologies. Rudolf Seising, born 1961 in Duisburg, Germany, received an MS degree in mathematics from the Ruhr University of Bochum in 1986, a Ph.D. degree in philosophy of science from the Ludwig Maximilians University (LMU) of Munich in 1995, and a postdoctoral lecture qualification (PD) in history of science from the LMU of Munich in 2004 for his book on the history of the theory of fuzzy sets (published in English in 2007: The Fuzzification of Systems, Springer-Verlag and in German in 2005: Die Fuzzifizierung der Systeme,

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Steiner-Verlag). He has been scientific assistant for computer sciences at the University of the Armed Forces in Munich from 1988 to 1995 and scientific assistant for history of sciences at the same university from 1995 to 2002. From 2002 to 2008 he was scientific assistant in the Core unit for medical statistics and informatics at the University Vienna Medical School, which in 2004 became the Medical University of Vienna. Since 2005 he is also college lecturer in the faculty of History and Arts, institute of history of sciences, at the LMU of Munich. From April to September 2008 was acting as a professor (substitute) for the history of science at the Friedrich-Schiller-University Jena and from October 2009 to March 2010 at the LMU of Munich. He has been visiting scholar at the University of California at Berkeley in 2000, 2001 and 2002 and at the University of Turku, Finland in 2008. From January to September 2009 and again since April 2010 he is Visiting Researcher at the European Centre for Soft Computing in Mieres, Spain. Rudolf Seising is Chairman of the IFSA Special Interest Group on History of Fuzzy Sets - IFSA SIG History and of the EUSFLAT Working Group “Philosophical Foundations”, and he is the editor of the book Views on Fuzzy Sets and Systems from Different Perspectives. Philosophy and Logic, Criticisims and Applications, (Springer-Verlag 2009). Alejandro Sobrino Cerdeiriña is Ph. D. in Philosophy from the University of Santiago de Compostela, Spain. Currently is Associated Professor at the Dpt. of Logic and Moral Philosophy at the same university, where he teaches ’Formal Grammars’ and ’Logic Programming’. His areas of interest are: vagueness, fuzzy logic and (vague) natural language processing. Settimo Termini is Professor of Theoretical Computer Science at the University of Palermo and has directed from 2002 to 2009 the Istituto di Cibernetica “Eduardo Caianiello” of the CNR (National Research Council). Among his scientific interests, we mention: the introduction and formal development of the theory of (entropy) measures of fuzziness; an analysis in innovative terms of the notion of vague predicate as it appears and is used in Information Sciences, Cybernetics and AI. He has been interested also in the problem of the connections between scientific research and economic development of a country. Recently he has extended his interest on the conceptual foundations of Fuzzy Sets and Soft Computing in the analysis of the role that Fuzzy Set Theory can play in the “interface” between, Hard Sciences and Humanities. He is Fellow of the International Fuzzy System Association and of the Accademia Nazionale di Scienze, Lettere ed Arti of Palermo. Main books: Aspects of Vagueness, Reidel (1984), edited with E. Trillas and H. J. Skala; Imagination and Rigor, Springer (2006); Contro il declino, Codice edizioni (2007), coauthored with Pietro Greco; Memoria e progetto, Edizioni GEM (2010), edited with Pietro Greco. Enric Trillas (Barcelona, 1940) is Emeritus Researcher at the European Centre for Soft Computing (Mieres, Asturias, Spain). He got a Ph.D. on Sciences from the University of Barcelona and he became Professor at the Technical University of Catalonia in 1974. In 1988 moved to the Technical University of Madrid, where he was Professor at the Department of Artificial Intelligence until September 2006. Formerly, and among other positions, he was Vice-Rector of the Technical University of Catalonia, President of the High Council for Scientific Research (CSIC), Director General of the National Institute for Aerospace Technology (INTA), Secretary General of the National Plan for Scientific and Technological Research, and Chairman of the company Aerospace Engineering and Services (INSA). Other than several distinctions and medals, he is Fellow of the International Fuzzy Systems Association (IFSA), got the Fuzzy Pioneers Award of the European Society for Fuzzy Logic and Technologies, the Fuzzy Systems Pioneer Award of the IEEE Computational Intelligence Society, the Spanish National Prize on Computer Sciences in 2007 and the Kampé de Fériet

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Award in 2008. He has published over two hundred fifty papers in Journals, Conference’s Proceedings, and editor’s books,as well as several books. He serves in the Editorial Board of many International Journals. His current research interests are, Fundamentals of Fuzzy Set Theories, and Fuzzy Logic, Methods of reasoning: conjectures, hypotheses and consequences. Raphael Unterweger, born on 10.10.1978 in Feldkirch, Austria. 1994-1999 Higher Technical School for communication engineering in Rankweil. Presenzdienst in 2000. 2001-2007 Technical computer science at the Leopold-Franzens University in Innsbruck. Since 2005 in the Department for digitisation and digital preservation at the University Innsbruck Library.

Abstracts

On Some “family resemblances” of Fuzzy Set Theory and Human Sciences Settimo Termini The aim of this paper is to underline the importance of detecting similarities or at least, ’family resemblances’ among different fields of investigation. As a matter of fact, the attention will be focused mainly on fuzzy sets and a few features of human sciences; however, I hope that the arguments provided and the general context outlined will show that the problem of picking up (dis)similarities among different disciplines is of a more general interest. Usually strong dichotomies guide out attempts at understanding the paths along which scientific research proceed; i.e., soft versus hard sciences, humanities versus the sciences of nature, Naturwissenschaften versus Geisteswissenschaften, Kultur versus Zivilization, applied sciences and technology versus fundamental, basic (or, as has become recently fashionable to denote it, “curiosity driven”) research. However, the similarity or dissimilarity of different fields of investigation is – to quote Lotfi Zadeh – “a matter of degree”. This is particularly evident in the huge, composite, rich and chaotic field of the investigations having to do with the treatment of information, uncertainty, partial and revisable knowledge (and their application to different problems). The specific points treated in this paper can be then seen as case studies of a more general crucial question. A question which could be important in affording also the problems posed by interdisciplinarity. The specific point of the interaction between fuzzy sets and human sciences can be seen as an episode of a larger question. There is a long history, in fact, regarding the mutual relationship existing between the (so-called) humanities and the (so-called) hard sciences, that has produced the so-called question of the two Cultures. At the end of the paper possible epistemological similarities between the development of Fuzzy Set theory and new emerging disciplines, like Trust Theory, will be briefly discussed. Warren Weaver’s “Science and Complexity” Revisited Rudolf Seising The mathematician Warren Weaver was an important science administrator during and after World War II. As the director of natural science of the Rockefeller Foundation he was significantly involved in changing the leading sciences from physics to life sciences. In his 1949 article “Science and Complexity” Weaver associates this change with the location of a “great middle region” of scientific problems of organized complexity” between the “problems of simplicity” that physical sciences are concerned with and the “problems of disorganized

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complexity” that can be solved by probability theory and statistics. Weaver stated that “something more is needed than the mathematics of averages.” To solve such problems he pinned his hope on the power of digital computers and on interdisciplinary collaborating “mixed teams”. These quotations sound very similar to statements of Lotfi A. Zadeh’s, when he founded his theory of “Fuzzy sets”. In this contribution we consider the theory of Fuzzy Sets as an approach to solve Weaver’s “problems of organized complexity”. How Philosophy, Science and Technologies Studies, and Feminist Studies of Technology Can Be of Use for Soft Computing Veronica Sanz Artificial Intelligence has been one of the fields within Computer Science that has generated more interest and debates among philosophers. Later on, the most recent fields of Science and Technology Studies (STS), and Feminist Studies of Technology (FST) have also shown some interest in AI. In both cases most of the authors have been quite critical about the promises, practices and, particularly, the epistemological basis of Classical AI. The first part of the paper consists on an enumeration of the most important authors and their critiques to AI from Philosophy, STS studies and FST. Since Soft Computing entails important changes with respect to traditional AI approaches, the second part of the paper will be devoted to confront Soft Computing with the previous critiques and challenges to AI and to weight up to what extent Soft Computing could (or could not) answer differently than other AI approaches to these critiques and challenges. Explicatum and Explicandum and Soft Computing Settimo Termini The aim of this paper is twofold. First of all I want to present some old ideas revisited in the light of some of the many interesting new developments occurred in the course of these last ten years in the field of the foundations of fuzziness. Secondly I desire to present a tentative general framework in which it is possible to compare different attitudes and different approaches to the clarification of the conceptual problems arising from fuzziness and soft computing. In the paper, then, I shall use some names as banners to indicate a (crucial) problem (i.e., Carnap’s problem, von Neumann’s problem, Galileian science, Aristotelian science and so on). As it will be clear by reading the paper, the association of a name to a certain problem should not be considered as the result of a historically based profound investigation but only as a sort of slogan for a specific position and point of view. It is well known that Rudolf Carnap in the first pages of his Logical foundations of probability faced the (difficult) problem of the ways and procedures according to which a prescientific concept (which by its very nature is inexact) is trasformed into a (new) exact scientific concept. He called this transformation (the transition from the explicandum, the informal, qualitative, inexact prescientific notion to the explicatum, its scientific, quantitative, exact substitute) the procedure of explication, a procedure which, as Carnap immediately observed, presents a paradoxical aspect. While in ordinary scientific problems, he, in fact, observes, “both the datum and the solution are, under favorable conditions, formulated in exact terms ... in a problem of explication the datum, viz., the explicandum, is not given in exact terms; if it were, no explication would be necessary. Since the datum is inexact, the problem itself is not stated in exact terms; and yet we are asked to give an exact solution. This is one of the puzzling peculiarities of explication”. One of the leading ideas of the paper will be to use

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the distinction made by Carnap between “explicandum” and “explicatum” to analyze a few conceptual questions arising in Fuzzy Set Theory (and Soft Computing). Axiomatic Investigation of Fuzzy Probabilities Takehiko Nakama, Enric Trillas, and Itziar García-Honrado Various forms of probability and probabilistic concepts have been discussed in fuzzy logic since its very inception, but their mathematical foundations have yet to be fully established. In this paper, we investigate theoretical issues concerning probability functions as membership functions, probability measures for fuzzy sets, and fuzzy-valued probabilities. Fuzzy Deontics Kazem Sadegh-Zadeh A concept of fuzzy deontic set is introduced that makes it possible to represent moral and legal norms as fuzzy deontic rules. As numerical fuzzy rules, they enable the formulation of comparative norms. And as qualitative fuzzy rules, they may aid fuzzy deontic decisionmaking in ethics, law, and medicine. The emerging field of research is termed fuzzy deontics.

Soft Deontic Logic Txetxu Ausin and Lorenzo Peña Deontic logic is the theory of valid inference rules containing qualifications of prohibition, duty, or permission. Three main assumptions of standard deontic-logic approaches are here discussed and rejected: (1) the modal conception of deontic logic, according to which “licit” means “possible without breaking the rules”; (2) the gap between facts and norms, which contends that what actually happens has no bearing on duties; and (3) bivalence, which bars any situation inbetween absolute truth and downright falseness. As against such approaches, we put forward an alternative deontic logic we call soft, which, while being fuzzy (and based on a a paraconsistent gradualistic sentential calculus), binds duties to facts by espousing the implantation principle, according to which, if A is the case, then the duty (or permission) to do B − i f − A implies the duty (or permission) to do B. By embracing degrees of licitness our proposal upholds a principle of proportionality ruling out deontic leaps. A formalized axiomatic system along those lines is developed. Retrieving Crisp and Imperfect Causal Sentences in Texts: From Single Causal Sentences to Mechanisms Cristina Puente, Alejandro Sobrino and José Angel Olivas Causality is a fundamental notion in every field of science. In empirical sciences, such as physics, causality is a typical way of generating knowledge and providing explanations. Usually, causation is a kind of relationship between two entities: cause and effect. The cause provokes an effect, and the effect is derived from the cause, so there is a relationship of strong dependence between cause and effect. Causality and conditionality are closely related. One of the main topics in the field of causality is to analyze the relationship between causality and conditionality, and to determine which causal relationships can be formulated as conditional links. In this work a method has been developed to extract causal and conditional sentences from texts belonging to different genres or disciplines, using them as a database to study imperfect causality and to explore the causal relationships of a given concept by means of a causal graph. The process is divided into three major parts. The first part creates a

512

Abstracts

causal knowledge base by means of a detection and classification processes which are able to extract those sentences matching any of the causal patterns selected for this task. The second part selects those sentences related to an input concept and creates a brief summary of them, retrieving the concepts involved in the causal relationship such as the cause and effect nodes, its modifiers, linguistic edges and the type of causal relationship. The third part presents a graphical representation of the causal relationships through a causal graph, with nodes and relationships labelled with linguistic hedges that denote the intensity with which the causes or effects happen. This procedure should help to explore the role of causality in different areas such as medicine, biology, social sciences and engineering. Facing Uncertainty in Digitisation Lukas Gander, Ulrich Reffle, Sven Schlarb, Klaus Schulz and Raphael Unterweger In actual practice, digitisation and text recognition (OCR) refers to a processing chain, starting with the scanning of original assets (newspaper, book, manuscript, etc.) and the creation of digital images of the asset’s pages, which is the basis for producing digital text documents. The core process is Optical Character Recognition (OCR) which is preceded by image enhancement steps, like deskewing, denoising, etc., and is followed by post-processing steps, such as linguistic correction of OCR errors or enrichment of the OCR results, like adding layout information and identifying semantic units of a page (e.g. page number). In this paper, the focus lies on the post-processing steps. Two selected research areas of the European project IMPACT (IMProving ACcess to Text) will be outlined. Firstly, we present a technology for OCR and information retrieval on historical document collections, and discuss the potential use of fuzzy logic. Secondly, we present the Functional Extension Parser, a software that implements a fuzzy rule-based system for detecting and reconstructing some of the main features of a digitised book based on the OCR results of the digitised images. The Role of Synonymy and Antonymy in a ’Natural’ Fuzzy Prolog Alejandro Sobrino The aim of this paper is to attempt a first approach to a kind of ’natural Fuzzy Prolog’ based on the linguistic relations of synonymy and antonymy. Traditionally, Prolog was associated to the clausal logic, a disposition of the classical logic in which the goals are conjectural theorems and the answers, provided by the interpreter, are achieved using resolution and unification. Both resolution and unification are the core of a Prolog interpreter. Classical Prolog has had and still currently has interesting applications in domains as natural language processing where the problems are verbalized using crisp language and algorithmic style. But as Zadeh pointed out, natural language is essentially ill-defined or vague. Fuzzy Prolog provides tools for dealing with tasks that involve vague or imprecise statements and approximate reasoning. Traditionally, fuzzy Prolog was related with the specification of facts or rules as a matter of degree. Degrees adopted several forms: single degrees, intervals of degrees and linguistic truth-values, represented by triangular or trapezoidal numbers. Fuzzy solutions using degrees are valuable, but far from the way employed by human beings to solve daily problems. Using a naive style, this paper introduces a ’natural fuzzy Prolog’ that deals with a kind of natural resolution applying antonymy as a linguistic negation and synonymy as a way to match predicates with similar meanings.

Abstracts

513

On an Attempt to Formalize Guessing Itziar García-Honrado and Enric Trillas Guessing from a piece of information is what humans do in their reasoning processes, that is why reasoning and obtaining conjectures can be considered almost equivalent. These reasoning processes are nothing else than posing new questions which possible answers are non contradictory with the available information. This is the idea that allows to introduce different mathematical models by means of different conjecture operators, built up depending on how the concept of non-contradiction is understood. A relevant contribution of this chapter is that there can be conjecture operators not coming from Tarski’s operators of consequences, in this way untying the concept of conjecture from a previously given deductive system. The concept of a conjecture proves to include those of logical consequences, hypotheses and speculations.

Syntactic Ambiguity Amidst Contextual Clarity – Reproducing Human Indifference to Linguistic Vagueness Jeremy Bradley Even though human speech is, by default, filled with ambiguity, competent speakers of a natural language generally manage to communicate with each other without having to request clarification after every second sentence. In fact, most linguistic ambiguity is not even noticed by speakers of a language, unless they are made explicitly aware of it or are asked to clarify some particular point, possibly by a less competent speaker of the language in question. In certain situations, linguistic ambiguity can cause headaches for people, but it does not make human communication impossible. Context, knowledge of the world, and common sense generally make it clear which interpretation of an ambiguous statement is appropriate. While the word “bank” in the statement “I’ve got some money in the bank” is lexically ambiguous – a “bank” in English can be a financial institution, the land alongside a river or a lake, or an airplane manoeuvre – semantics only allow for one interpretation. People suffering from aphasia – the partial loss of language skills due to brain damage – have to deal with more linguistic ambiguity than others. Aphasia affects people’s capability to correctly interpret morphology and syntax, stripping away the much needed context competent speakers of a language need to correctly interpret human language. This paper details the handling of ambiguity in text simplification software aimed at patients of this illness. Can We Learn Algorithms from People Who Compute Fast: An Indirect Analysis in the Presence of Fuzzy Descriptions Olga Kosheleva and Vladik Kreinovitch In the past, mathematicians actively used the ability of some people to perform calculations unusually fast. With the advent of computers, there is no longer need for human calculators – even fast ones. However, recently, it was discovered that there exist, e.g., multiplication algorithms which are much faster than standard multiplication. Because of this discovery, it is possible than even faster algorithm will be discovered. It is therefore natural to ask: did fast human calculators of the past use faster algorithms – in which case we can learn from their experience – or they simply performed all operations within a standard algorithm much faster? This question is difficult to answer directly, because the fast human calculators’ selfdescription of their algorithm is very fuzzy. In this paper, we use an indirect analysis to argue that fast human calculators most probably used the standard algorithm.

514

Abstracts

Perceptions: A Psychobiologial and Cultural Approach Clara Barroso This article examines human perception in order to establish which aspects of the process could be used to inform models that represent meaning in artificial artefacts. Human perception is analysed from two approaches that we initially differentiated as organic and functional. Both are necessary and largely determine how an intelligent subject interacts with the objects that surround it and, as a consequence, how it behaves in the world. Both facilitate the emergence of meaning from what is perceived. Finally, we will establish the relevance that these considerations could have for the decision making process of artificial artefacts. Rule Based Fuzzy Cognitive Maps in Humanities, Social Sciences and Economics Joao Paulo Carvalho Decision makers, whether they are social scientists, politicians or economists, usually face serious difficulties when trying to model significant, real-world dynamic systems. Such systems are composed of a number of dynamic qualitative concepts interrelated in complex ways, usually including feedback links that propagate influences in complicated chains. Axelrod work on Cognitive Maps (CMs) introduced a way to represent real-world qualitative dynamic systems, and several methods and tools have been developed to analyze the structure of CMs. However, complete, efficient and practical mechanisms to analyze and predict the evolution of data in CMs were not available for years due to several reasons. System Dynamics tools like those developed by J. W. Forrester could have provided the solution, but since in CMs numerical data may be uncertain or hard to come by, and the formulation of a mathematical model may be difficult, costly or even impossible due to their qualitative and uncertain nature, then efforts to introduce knowledge on these systems should rely on natural language arguments in the absence of formal models. Fuzzy Cognitive Maps (FCM), as introduced by Kosko, were developed as a qualitative alternative approach to system dynamics. However, FCM are Causal Maps (a subset of Cognitive Maps that only allow basic symmetric and monotonic causal relations), and in most applications do not explore usual Fuzzy capabilities. They do not share the properties of other fuzzy systems and the causal maps usually result in quantitative matrixes without any qualitative knowledge. This talk introduces Rule Based Fuzzy Cognitive Maps (RB-FCM), a new approach to model and simulate real world qualitative dynamic systems (social, economic, political, etc.) while avoiding the limitations of the above alternatives. Voting on How to Vote José Luis García-Lapresta and Ashley Piggins All societies need to make collective decisions, and for this they need a voting rule. We ask how stable these rules are. Stable rules are more likely to persist over time. We consider a family of voting systems in which individuals declare intensities of preference through numbers in the unit interval. With these voting systems, an alternative defeats another whenever the amount of opinion obtained by the first alternative exceeds the amount of opinion obtained by the second alternative by a fixed threshold. The relevant question is what should this threshold be? We assume that each individual’s assessment of what the threshold should be is represented by a trapezoidal fuzzy number. From these trapezoidal fuzzy numbers we associate reciprocal preference relations on [0, m). With these preferences over thresholds in place, we formalize the notion of a threshold being “self-selective”. We establish some mathematical properties of self-selective thresholds, and then describe a three stage procedure for

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selecting an appropriate threshold. Such a procedure will always select a threshold which can then be used for future decision making. Weighted Means of Subjective Evaluations Juan Vicente Riera and Jaume Casasnovas In this article, we recall different student evaluation methods based on fuzzy set theory. The problem arises is the aggregation of this fuzzy information when it is presented as a fuzzy number. Such aggregation problem is becoming present in an increasing number of areas: mathematics, physic, engineering, economy, social sciences, etc. In the previously quoted methods the fuzzy numbers awarded by each evaluator are not directly aggregated. They are previously defuzzycated and then a weighted mean or other type of aggregation function is often applied. Our aim is to aggregate directly the fuzzy awards (expressed as discrete fuzzy numbers) and to get like a fuzzy set (a discrete fuzzy number) resulting from such aggregation, because we think that in the defuzzification process a large amount of information and characteristics are lost. Hence, we propose a theoretical method to build n-dimensional aggregation functions on the set of discrete fuzzy number. Moreover, we propose a method to obtain the group consensus opinion based on discrete fuzzy weighted normed operators. Some Experiences Applying Fuzzy Logic to Economics Bárbara Díaz Diez and Antonio Morillas Economy becomes a field of special interest for the application of fuzzy logic. Here we present some works carried out in this direction, highlighting their advantages and also some of the difficulties encountered. Fuzzy inference systems are very useful for Economic Modelling. The use of a rule system defines the underlying economic theory, and allows extracting inferences and predictions. We applied them to modelling and prediction of waged-earning employment in Spain, with Jang´s algorithm (ANFIS) for the period 1977-1998. As additional experiences in this direction, we have applied the IFN algorithm (Info-FuzzyNetwork) developed by Maimon and Last to the study of the profit value of the Andalusian agrarian industry. The search for key sectors in an economy has been and still is one of the more recurrent themes in Input-Output analysis, a relevant research area in the economic analysis. We proposed a multidimensional approach to classify the productive sectors of the Spanish InputOutput table. We subsequently analyzed the problems that can arise in key sector analysis and industrial clustering, due to the usual presence of outliers when using multidimensional data. Fuzzy Formal Ontology Kazem Sadegh-Zadeh Formal-ontological frameworks such as mereology and mereotopology are increasingly applied in medical knowledge-based systems. But they are based on classical logic, and therefore, not useful in fuzzy environments. An attempt has been made in the present paper to conceive an approach to fuzzy formal ontology. The result is the core of a fuzzy mereology.

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Computational Representation of Medical Concepts: A Semiotic and Fuzzy Logic Approach Mila Kwiatkowska, Krzysztof Michalik and Krzysztof Kielan Medicine and biology are among the fastest growing application areas of computer-based systems. Nonetheless, the creation of a computerized support for the health systems presents manifold challenges. One of the major problems is the modeling and interpretation of heterogeneous concepts used in medicine. The medical concepts such as, for example, specific symptoms and their etiologies, are described using terms from diverse domains - some concepts are described in terms of molecular biology and genetics, some concepts use models from chemistry and physics; yet some, for example, mental disorders, are defined in terms of particular feelings, behaviours, habits, and life events. Moreover, the computational representation of medical concepts must be (1) formally or rigorously specified to be processed by a computer, (2) human-readable to be validated by humans, and (3) sufficiently expressive to model concepts which are inherently complex, multi-dimensional, goal-oriented, and, at the same time, evolving and often imprecise. In this chapter, we present a meta-modeling framework for computational representation of medical concepts. Our framework is based on semiotics and fuzzy logic to explicitly model two important characteristics of medical concepts: changeability and imprecision. Furthermore, the framework uses a multi-layered specification linking together three domains: medical, computational, and implementational. We describe the framework using an example of mental disorders, specifically, the concept of clinical depression. To exemplify the changeable character of medical concepts, we discuss the evolution of the diagnostic criteria for depression. We discuss the computational representation for polythetic and categorical concepts and for multi-dimensional and noncategorical concepts. We demonstrate how the proposed modeling framework utilizes (1) a fuzzy-logic approach to represent the non-categorical (continuous) nature of the symptoms and (2) a semiotic approach to represent the contextual interpretation and dimensional nature of the symptoms. Invariance and Variance of Motives: A Model of Musical Logic and/as Fuzzy Logic Hanns-Werner Heister “Musical Logic” (term 1788 by Johann Nicolaus Forkel) means both the “logicality of music” and the “involvement/participation of musical processes in the laws of reason”. Motives, themes, parts of movements, forms are supposedly related to each other: horizontally in time in and vertically in (musical) space. Part of the musical logic and its realization in concrete artworks is the motivic-thematic work (“motivisch-thematische Arbeit”). Musical motives,often combined to more expanded themes, form shapes (“Gestalten”) with a fairly stable core and diverse and multiple varied elaborations in the course of the art work as process. Core can be interpreted as the inner kernel, partially subconscious, quasi also the ’generative’ dimension. To the core as essence correlate as phenomenon respectively phenomena in different variants, the ’performative’ dimension. The ’contours’ of these variants for their part are a generalization and a fuzzily identity-establishing commonality of the different various forms and appearances of the kernel. The interrelations between both these entities belong to the general dialectic of essence and appearance (the Hegelian ’Wesen’ vs. ’Erscheinung’) - notabene a real dialectic, because both sides or dimensions participate in the constitution of the object, and both belong to the objects of Fuzzy Logic. As a concept, which is valid also for other matters, I propose that of an identity in and as process (’prozessierende Identität’): In a given phase the identical is not invariably ’given’ and present but develops in time and history. This — relative — identity is the initial point and endpoint of a process and

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is realized as this process. I will demonstrate this with concrete examples, especially with the BACH-cipher, a historically long lasting and multi-dimensional motif with a remarkable cross-structure (leaving aside the psycho- and socio-logic of music – causes, motivations, functions etc.) Inversely music (and similarly the fine arts) appear as part and concrete elaboration of the aesthetic and sensual perception, which form their own laws, logics, layers and types of cognition, and therewith nothing less than a paradigm of Fuzzy Logic. Mathematics and Soft Computing in Music Teresa León and Vicente Liern Mathematics is the fundamental tool for dealing with the physical processes that explain music but it is also in the very essence of this art. Musical notes, the first elements which music works with, are defined for each tuning system as very specific frequencies; however, instrumentalists know that small changes in these values do not have serious consequences. In fact, sometimes consensus is only reached if the entire orchestra alters the theoretical pitches. The explanation for this contradiction is that musicians implicitly handle very complex mathematical processes involving some uncertainty in the concepts and this is better explained in terms of fuzzy logic. Modelling the notes as fuzzy sets and extending the concept of tuning systems lead us to a better understanding on how musicians work in real-life. The notes offered by a musician during a performance should be compatible with the theoretical ones but not necessarily equal. A numerical experiment, conducted with the help of a saxophonist, illustrates our approach and also points to the need for considering sequential uncertainty. Music and Similarity-Based Reasoning Josep Lluis Arcos Whenever that a musician plays a musical piece, the result is never a literal interpretation of the score. These performance deviations are intentional and constitute the essence of the musical communication. Deviations are usually thought of as conveying expressiveness. Two main purposes of musical expression are generally recognized: the clarification of the the musical structure and the transmission of affective content. The challenge of the computer music field when modeling expressiveness is to grasp the performers “touch”, i.e., the musical knowledge applied when performing a score. One possible approach to tackle the problem is to try to make explicit this knowledge using musical experts. An alternative approach, much closer to the human observation-imitation process, is to directly work with the knowledge implicitly stored in musical recordings and let the system imitate these performances. This alternative approach, also called lazy learning, focus on locally approximating a complex target function when a new problem is presented to the system. Exploiting the notion of local similarity, the chapter presents how the Case-Based Reasoning methodology has been successfully applied to design different computer systems for musical expressive performance.

t-norms, t-conorms, Aggregation Operators and Gaudí’s Columns Amadeo Monreal A Gaudí’s column GC, also known as ’double twist’ or ’double rotation’ column, can be conceived as split into different pieces or spans and each of them is defined as the intersection of two salomonic columns SC1 and SC2 . Thus, the distance of a point in the boundary of a section of GC to the axis of the column GC is the Minimum of the distances of the corresponding points of SC1 and SC2 . The conjunction ’and’ is then modeled by the Minimum

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t-norm. Replacing the Minimum by other t-norms, new Gaudí-like columns can be obtained. In the same way, replacing the t-norm by a t-conorm, and thus considering that our column is the disjunction of both salomonic ones, a new kind of Gaudí-like columns can be generated. The Minimum is also an aggregation operator. Replacing this aggregation by other ones, new families of Gaudí-like columns are obtained. This paper provides a mathematical model of the double twist columns and investigates the effect of the use of different t-norms, t-conorms and aggregation operators in the generation of columns and other longitudinal objects such as capitals, towers, domes or chimneys.

Index

Arcos, Josep LLuís 467 Ausín, Txetxu 157 Barroso, Clara 277 Bradley, Jeremy 257 Carvalho, João Paulo 289 Casasnovas, Jaume 323 Díaz, Bárbara

347

Gander, Lukas 195 García-Honrado, Itziar 125, 237 García-Lapresta, José Luis 301 Heister, Hanns-Werner

423

Kielan, Krzysztof 401 Kosheleva, Olga 267 Kreinovich, Vladik 267 Kwiatkowska, Mila 401 León, Teresa 451 Liern, Vicente 451 Michalik, Krzysztof 401 Monreal, Amadeo 479 Morillas, Antonio 347

Nakama, Takehiko

125

Olivas, José Ángel

175

Peña, Lorenzo 157 Piggins, Ashley 301 Puente, Cristina 175 Reffle, Ulrich 195 Ringlstetter, Christoph

195

Sadegh-Zadeh, Kazem 141, 383 Sanz, Veronica 3, 89 Schlarb, Sven 195 Schulz, Klaus 195 Seising, Rudolf 3, 55 Sobrino, Alejandro 175, 209 Termini, Settimo 39, 113 Trillas, Enric 125, 237 Unterweger, Raphael Vicente Riera, J.

323

195