Fuzziness and Approximate Reasoning Epistemics on Uncertainty, Expectation and Risk in Rational Behavior

Kofi Kissi Dompere Fuzziness and Approximate Reasoning Studies in Fuzziness and Soft Computing, Volume 237 Editor-in-...

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Kofi Kissi Dompere Fuzziness and Approximate Reasoning

Studies in Fuzziness and Soft Computing, Volume 237 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. 221. Gleb Beliakov, Tomasa Calvo, Ana Pradera Aggregation Functions: A Guide for Practitioners, 2007 ISBN 978-3-540-73720-9 Vol. 222. James J. Buckley, Leonard J. Jowers Monte Carlo Methods in Fuzzy Optimization, 2008 ISBN 978-3-540-76289-8 Vol. 223. Oscar Castillo, Patricia Melin Type-2 Fuzzy Logic: Theory and Applications, 2008 ISBN 978-3-540-76283-6 Vol. 224. Rafael Bello, Rafael Falcón, Witold Pedrycz, Janusz Kacprzyk (Eds.) Contributions to Fuzzy and Rough Sets Theories and Their Applications, 2008 ISBN 978-3-540-76972-9 Vol. 225. Terry D. Clark, Jennifer M. Larson, John N. Mordeson, Joshua D. Potter, Mark J. Wierman Applying Fuzzy Mathematics to Formal Models in Comparative Politics, 2008 ISBN 978-3-540-77460-0 Vol. 226. Bhanu Prasad (Ed.) Soft Computing Applications in Industry, 2008 ISBN 978-3-540-77464-8 Vol. 227. Eugene Roventa, Tiberiu Spircu Management of Knowledge Imperfection in Building Intelligent Systems, 2008 ISBN 978-3-540-77462-4 Vol. 228. Adam Kasperski Discrete Optimization with Interval Data, 2008 ISBN 978-3-540-78483-8

Vol. 229. Sadaaki Miyamoto, Hidetomo Ichihashi, Katsuhiro Honda Algorithms for Fuzzy Clustering, 2008 ISBN 978-3-540-78736-5 Vol. 230. Bhanu Prasad (Ed.) Soft Computing Applications in Business, 2008 ISBN 978-3-540-79004-4 Vol. 231. Michal Baczynski, Balasubramaniam Jayaram Soft Fuzzy Implications, 2008 ISBN 978-3-540-69080-1 Vol. 232. Eduardo Massad, Neli Regina Siqueira Ortega, Laécio Carvalho de Barros, Claudio José Struchiner Fuzzy Logic in Action: Applications in Epidemiology and Beyond, 2008 ISBN 978-3-540-69092-4 Vol. 233. Cengiz Kahraman (Ed.) Fuzzy Engineering Economics with Applications, 2008 ISBN 978-3-540-70809-4 Vol. 234. Eyal Kolman, Michael Margaliot Knowledge-Based Neurocomputing: A Fuzzy Logic Approach, 2009 ISBN 978-3-540-88076-9 Vol. 235. Kofi Kissi Dompere Fuzzy Rationality, 2009 ISBN 978-3-540-88082-0 Vol. 236. Kofi Kissi Dompere Epistemic Foundations of Fuzziness, 2009 ISBN 978-3-540-88084-4 Vol. 237. Kofi Kissi Dompere Fuzziness and Approximate Reasoning, 2009 ISBN 978-3-540-88086-8

Kofi Kissi Dompere

Fuzziness and Approximate Reasoning Epistemics on Uncertainty, Expectation and Risk in Rational Behavior

ABC

Author Prof. Kofi Kissi Dompere Department of Economics Howard University Washington, D.C., 20059 USA E-Mail: [email protected]

ISBN 978-3-540-88086-8

e-ISBN 978-3-540-88087-5

DOI 10.1007/978-3-540-88087-5 Studies in Fuzziness and Soft Computing

ISSN 1434-9922

Library of Congress Control Number: 2008935497 c 2009 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 in acid-free paper 987654321 springer.com

To Kenneth Arrow, Herbert Simon, Jacob Marschak, David Blackwell, Maurice Allais, Leonard Savage and Bruno de Finetti; To all dedicated researchers working on stochastic uncertainty and risk; To all dedicated researchers working on fuzzy uncertainty and risk; To all dedicated researchers working to unify both: Stochastic uncertainty and risk on one hand, and fuzzy uncertainty and risk on the other; To the initial members of North American Fuzzy Information Processing Group, may your works yield useful contributions to our knowledge on the phenomena of uncertainty, expectations and risk in the understanding the role of fuzziness in DecisionChoice Rationality. To the Creative Force, the Spirit Force and the Light Force: Three in One, and One in Three.

Preface

We do not perceive the present as it is and in totality, nor do we infer the future from the present with any high degree of dependability, nor yet do we accurately know the consequences of our own actions. In addition, there is a fourth source of error to be taken into account, for we do not execute actions in the precise form in which they are imaged and willed. Frank H. Knight [R4.34, p. 202] The “degree” of certainty of confidence felt in the conclusion after it is reached cannot be ignored, for it is of the greatest practical significance. The action which follows upon an opinion depends as much upon the amount of confidence in that opinion as it does upon favorableness of the opinion itself. The ultimate logic, or psychology, of these deliberations is obscure, a part of the scientifically unfathomable mystery of life and mind. Frank H. Knight [R4.34, p. 226-227] With some inaccuracy, description of uncertain consequences can be classified into two categories, those which use exclusively the language of probability distributions and those which call for some other principle, either to replace or supplement. Kenneth Arrow [R21.5, p.8] The basic need for a special theory to explain behavior under conditions of uncertainty arises from two considerations: (1) subjective feeling of imperfect knowledge when certain types of choices, typically involving commitments over time, are made; (2) the existence of certain observed phenomena, of which insurance is the most conspicuous example, which cannot be explained on the assumption that individuals act with subjective certainty. Kenneth Arrow [R21.5, p. 44]

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The world of human operations relates to three important time stages of past, present and future that are intimately connected for human progress. This is the time trinity. The history of this progress is the work of success-failure decisionchoice actions which are governed by some rationality that constitutes a processing operator on the available knowledge input into the decision-choice modulus to obtain an output as an element of the evolving history. The time trinity presents us with uncertainty that houses contradictions and paradoxes as indispensable elements of human historic journey where informationknowledge structure constitutes the wind that drives the ship of human progress to sail in trouble waters with uncertain outcome. This information-knowledge structure is the connector of past, present and future as they relate to the behavior of cognitive agents. The past presents us with the problem of perceptive knowledge; the present provides us with the challenges of decision-choice actions and the future presents us with expectations of current decision-choice actions that are locked in the jacket of possibilistic belief system. The decisionchoice outcomes are defined by probabilistic belief system and expectations of actualized outcomes on the basis of complex system of beliefs. The possibilistic and probabilistic belief systems through decision-choice actions present potential and actual risks of varying intensity and scope to decision-choice agents. The uncertainty, expectation and risk have continuity with time where decision-choice agents must understand the past-presentfuture dynamics of potential-actual duality. This understanding must be abstracted from the construct of information-knowledge structure in support of possibilistic and probabilistic belief systems for cognitive substitutiontransformation processes on the path of human progress in its complexity. The complex relationships among past, present and future and their impacts on the relationships among information, knowledge and rationality require of us critical reasoning and understanding. In this direction, the statement by Frank Knight is helpful in understanding the time trinity of past, present and future on one hand and the decision-choice process on the other. “We do not perceive the present as it is and in its totality, nor do we infer the future from the present with any high degree of dependability, nor yet do we accurately know the consequences of our own actions” [R4.34, p.202]. Generally, uncertainty, expectations and risk are derived from justified possibilistic and probabilistic belief systems. The justifications of these belief systems are derived from the information-knowledge structure that contains limitations of vagueness, ambiguities, incompleteness in the information and defective knowledge-acquisition capabilities. In understanding decision-choice actions, a number of questions tend to arise. What do we know from the past?

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What is the relationship of what we know to the expected in the future? What effect does the expected have on the current decisions? What is uncertainty and is this uncertainty one or more than one type? What are the possible relationships among uncertainty, expectations and risk? How best can risk be defined and to what extent is the risk definition related to concepts of costs and benefits? Are there as many risks as there are uncertainty types? Are vagueness and ambiguities factors of uncertainty and hence generate risk? What are the generators of paradoxes in decision-choice theories under probabilistic reasoning in the classical paradigm? The search for answers to these questions is one of the motivations to examine the epistemic foundations of uncertainty, expectations and risk under conditions of decision-choice rationality in knowledge production process as we move from the potential to the actual through the possible and the probable. This monograph is the result of this examination from the knowledgeproduction viewpoints of classical and fuzzy paradigms. The focus is on the nature of decision-choice rationality at the presence of uncertainty, expectations and risk; and how these three conceptual elements shape the use of conditions of information-knowledge structure in the general decision-choice process. This monograph is the third in the sequence of a search for meaning and role of fuzzy reasoning and conditions of soft computing in differently defined environments in the different knowledge sectors. It may also be seen as a continuation of a conceptual system developed in my two volumes on costbenefit analysis and theory of fuzzy decisions where the emphasis was on cost-benefit rationality [R7.35], [R7.37]. In those two volumes, the idea is to expand the conceptual system and algorithmic toolbox of cost-benefit analysis to all areas requiring, decision, choice and policy with the view that the essential tool for modern policy research and analysis is cost-benefit analysis. The objective in this volume, however, is to specify the foundations of fuzzy reasoning that will allow us to deal with conditions of vagueness and ambiguities in uncertainty and risk analysis through the methods and techniques of approximate reasoning and fuzzy rationality from the framework of fuzzy paradigm. The work is to examine applicable areas of probabilistic measures of risk and their uses in analytical reasoning in understanding decision-choice behavior in general human conditions. Similarly, it is intended to examine the applicable areas of fuzzy reasoning in decision-choice activities involving risk in substitution-transformation processes that account for time, quantity and quality in the movement from the potential through the possible, and the

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probable to the actual. The essential epistemic point here is that probabilistic reasoning alone is not sufficient to characterize uncertainty, expectations and risk. Furthermore, probabilistic reasoning within the classical paradigm, offers us very little, if any, to accurately examine behavior under uncertainty and risk when vagueness and ambiguities characterize the variables under decision-choice action and logical operations. My hope is that these three treaties on the epistemics on fuzzy rationality and paradigm will be, at least, a defining entry point into theory of knowledge that combines exact and inexact sciences by utilizing the toolbox of fuzzy paradigm. The vision is that some intellectually closed eyes will be opened, especially those of mathematical skeptics and believers of the supremacy of exact science and absolute truth. The added utility of these treaties is to make explicit, the difficult and relatively virgin areas that require extensive research on mathematical theories of fuzzy phenomena. Such areas include fuzzystochastic topological space with fuzzy-random variable and stochastic-fuzzy topological space with random-fuzzy variable. The fuzzy-random variable will allow us to deal with imprecise, or vague or ambiguous probabilities. The random-fuzzy variable will allow us to deal with random fuzzy behaviors. The knowledge of the logic and mathematical tools obtained from these topological spaces will provide new and powerful tools for areas like economics, medical sciences, psychology and others that we characterize as inexact sciences as well as some unresolved problems in exact sciences that are plagued with vagueness. The subject area of fuzzy paradigm is relatively new in logic and mathematical reasoning. It is a paradigm shifting, as such; the monographs involve a number of new and controversial ideas. This paradigm shifting may be understood and appreciated from the duality viewpoint of exact and inexact sciences which is viewed here as mutually supporting in the knowledge-creation process. The introduction of some of the controversial ideas is also intentional. From my entry into Temple University and my course works in symbolic logic, mathematical analysis, philosophy of science and my encounter with economic theories and their application, the position of the Aristotelian logic and the principle of excluded middle were troubling. My background in African thought system instructs me that the elements in the universal system are relational in existence and every statement or proposition is relationally true or false; and that, truth and falsity exist in the same statement or hypothesis as opposing characteristics or attributes which must be reconciled by decision-choice action. An introductory exposition on this approach to theory of knowledge is provided in my monograph entitled polyrhythmicity [R20.21].

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My encounter with multi-valued logic did not relieve me of this troubling logical form of verification and acceptance of truth. I sought some relief from statistical sciences, but my encounter with it and probabilistic reasoning, based on logic of classical paradigm and the principle of exactness, did not help my analytical concerns. It is these concerns that drove me to examine the structure of fuzzy logic, fuzzy sets, systems and fuzzy mathematics in terms of possible analytical relieve. The process of my encounter with the whole fuzzy research program and fuzzy paradigm has been described in my volumes on cost-benefit analysis and the theory of fuzzy decisions [R7.35], [R7.36] and my initial epistemic views on decision-choice rationality is provided in [R17.22]. The question that faces the community of scholars and scientists in the academic and research institutions is whether there is a unified logic and mathematics that will allow a consistent development towards unified knowledge systems that are both exact and inexact in reasoning. For reasons of consistency social acceptance of knowledge element should have the same criteria. The search for an answer to this question encounters series of difficulties with the framework of the classical paradigm. However, our ability to develop logic and mathematical techniques to advance the unification of inexact and exact sciences is the challenge of our time for speeding up the gains in systemicity, cybernetics, artificial intelligence, robotics, informatics, economics, social system’s management, psychology, medical sciences and others. By advancing the epistemic structure of fuzzy rationality a case is made that the needed logic and mathematical techniques can be found in the framework of fuzzy paradigm when the epistemological foundations are understood. Let us keep in mind that information is not knowledge and that uncertaintyrisk reduction takes place on the basis of relevant knowledge. The fact of our current time is that information search and storage are such that we are constantly dealing with complex social system with information deficiency and overflow that make relevant knowledge more difficult to construct on the basis of the classical paradigm. The result of this increasing difficulty is to increase our problem in using information-knowledge structure for risk reduction and to increase decision-choice benefit. Here appears the importance of the statement: We now have a new king of cost-benefit analysis, namely, benefit-risk analysis. The risk of a disutility is itself a cost and a proper subject for measurement along with the direct costs of the usual resource-using type. Similarly, a reduction in risk is to be counted as a benefit [R21.37, p.20]. The information deficiency is not due to quantity limitation but quality deficiency

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generated by vagueness, ambiguities, inexactness and other relevant qualitative elements that may be subjectively defined. The logic and techniques for processing such deficient information to obtain relevant knowledge to reduce social and natural risks are part of the need for understanding the epistemic foundations of fuzzy paradigm and corresponding rationality. Just as we need rationality postulate to measure probabilistic uncertainty and corresponding risk, we need a rationality postulate to deal with fuzzy uncertainty and associated risk. Both of them are not the same, and hence require different logical and mathematical tools for their analyses. Given the task at hand, the monograph is organized in six chapters. Chapter One examines the nature of uncertainty and how it relates to information and knowledge in order to allow us to establish categorial uncertainties, zones of epistemic accessibilities and ignorance. The relationship between pastpresent-future structure on one hand and information-knowledge-decision structure on the other hand is presented and analyzed in relation to expectations formation towards decision-choice actins. The concept and logical technique of knowledge square are introduced in a manner that allows explication of possibility and probability to be advanced and related to the concepts and contents of possibility and probability indexes. The chapter is concluded with discussions on the relational structures among decision-choice rationality, uncertainty and expectations which are then related to the presence of vagueness, ambiguity, inexactness and the analytical structure of fuzzy optimal decision-choice rationality. Chapter Two analyzes the nature and structure of classical sub-optimal decision-choice rationality relative to the classical optimal decision-choice rationality and the conditions that give rise to sub-optimal decision-choice behavior in the classical decision-choice system. The idea of equating classical sub-optimal decision-choice rationality to decision-choice irrationality is dismissed as logically indefensible. The judgment of whether there is decisionchoice rationality is itself a decision-choice governed by some selected criterion conditions also defined by a rationality. The concept and content of classical sub-optimal rationality are given their epistemic structures and related to fuzzy optimal rationality. It is then argued that fuzzy optimal decision-choice rationality provides a covering over both classical optimal decision-choice rationality and sub-optimal rationality. The decision-choice theories that give rise to conditions of classical optimal and sub-optimal decision-choice rationalities are argued to be dealing with rationality as an ideal state of decisionchoice processes conditional on the given optimal rationality as an attribute of decision-choice agents.

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The rise of classical sub-optimal rationality in decision-choice process is explainable by the presence of vagueness, ambiguities, subjectivity and other qualitative factors in the information-knowledge structure and reasoning patterns of decision-choice agents. The process of epistemic explanation and resolution of the effects of vagueness, ambiguities, subjectivities and other qualitative factors that give rise to classical sub-optimal rationality leads to the examination of the roles that fuzzification and defuzzification play in general and specific cases using fuzzy logical reasoning in decision-choice processes. An example is given to show the differential natures in the use of classical and fuzzy mathematical reasoning in the computational space of decision-choice models. The chapter is concluded with discussions and examination of relational structure of fuzzy optimal decision-choice rationality, contradictions and benefit-cost rationality in decision-choice processes. In chapter Three, we present and deal with the structure of critical and important problems of ambiguity, vagueness and other qualitative factors in decision-choice process and their impacts on the understanding of uncertainty and risk and their relationships to fuzzy optimal decision-choice rationality. The role of the concept of probability in understanding ambiguity in decisionchoice process is analyzed in terms of strengths and weaknesses of classical and fuzzy paradigms. The consideration and analysis of simultaneous presence of fuzziness and randomness in the information-knowledge structure lead to the fuzzy-stochastic partition of the decision-choice space into four zones of risk-free, stochastic risk, fuzzy risk and combination of fuzzy and stochastic risk to obtain fuzzy-stochastic risk and stochastic-fuzzy risk. The epistemic analysis of the differences, similarities and relationships among the risk cohorts are individually and collectively discussed where each cohort is relationally structured in terms of total uncertainty, cost-benefit balances and decision-choice motivation conditions. The epistemic structure of the organic paradigm of decision-choice theories is introduced, analyzed and synthesized for clarity of the relational structure of cost-benefit, motivation, knowledge, optimal rationality as an attribute of decision-choice agents and as an ideal state of the decision-choice process The chapter is concluded with reflective discussions on the relative nature of rationality in economics, psychology and decision-choice theories in general. It is then argued that economic theories on decision-choice behavior implicitly assume optimal rationality as an attribute of decision-choice agents and then investigate conditions of rationality as ideal state of decision-choice process while psychological

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theories on decision-choice behavior relate to validity of rationality as an attribute of decision-choice agents. Chapter Four is devoted to the analysis of epistemic problems of concepts and measurement of information, uncertainty and risk in decision-choice processes. The relative differences and similarities of classical theories under uncertainty and theories on risk are then discussed and related to our current structure of theories of information where the central focus of theories of risk and information are argued to be not only similar but find epistemic expressions in stochastic uncertainties, probability space and measures of probability. Epistemic definitions of uncertainty and risk are provided for explication that allows us to introduce concepts of pure uncertainty and risk, principle of information sufficiency, principle of knowledge sufficiency, stochastic uncertainty, fuzzy uncertainty, stochastic risk, and fuzzy risk with complications that they present for both analytical structures of classical and fuzzy decisionchoice rationalities. The relationships among these categories of uncertainty and risk are then related to the spaces of potential, possibility, probability and actual. It is then argued that complete risk is associated with the space of potential, fuzzy risk is associated with the possibility space, stochastic risk is associated with probability space and realized total risk is associated with the space of the actual and related to knowledge-risk square. The meaning of the concepts of necessity and accident, and the roles that they play in the substitution-transformation processes are discussed in relation to risk. Necessity and accident are then argued to be characteristics, first of possibility space and then of the probability space. Connections are established among risk, freedom and decision-choice rationality leading to a discussion on what decision-choice agents’ freedom means under uncertainty and risk as they relate to categories of necessity and accidents. The chapter is concluded with epistemic analysis of the principle of compatibility of necessity and freedom in categorial dynamics of actualpotential duality in substitution-transformation processes, and how they help or hinder our understanding of decision-choice rationality under uncertainty and risk given cognitive agents. In Chapter Five is critical reflections on some decision-choice theories on uncertainty and risk are examined where the theories are grouped into those concern with stochastic uncertainty, probabilistic belief and stochastic risk on one hand and those concerned with fuzzy uncertainty, possibilistic belief and fuzzy risk. The former belongs to the classical class of decision-choice theories with conditions of classical optimal decision-choice rationality while the latter belongs to the non-classical class of decision-choice theories of fuzzy

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type with fuzzy optimal decision-choice rationality. The structure and form of the classical decision-choice theories on stochastic uncertainty and risk and the corresponding decision-choice rationality are analyzed and synthesized in relation to their contributions to our understanding of stochastic risk and behavior under stochastic risk. The concepts of event, outcome, uncertain event, risk, risky event and risky outcome receive fresh explication in terms of deficient information-knowledge structure as it relate to classical decision-choice rationality. A framework is then created to critically examine the strengths and weakness of the conditions of classical optimal decision-choice rationality under uncertainty and risk. The framework involves the relational structure of logical principles of sufficient reason and insufficient reason and the role these principles play in the construct of decision-choice theories on uncertainty and risk as we traverse from the potential to the actual through the possibility and probability. These two principles are further supported by seven others such as principles of sufficient justification, cause, analytical sufficiency, excluded middle and others, as they affect the outcome of the use of methods and techniques of constructionism and reductionism. The conceptual and epistemic foundations of fuzzy risk and decision-choice rationality are presented and discussed by first partitioning the space of decision-choice variables into exact variable, stochastic variable, fuzzy variable and either fuzzy-stochastic or stochastic-fuzzy variable where two mathematical spaces of analytical interests are referenced. The fuzzy risk category is then partitioned into sub-categories of non-stochastic fuzzy risk, fuzzy-stochastic risk and stochastic-fuzzy risk and then related to the conditions of fuzzy optimal decision-choice rationality and approximate reasoning for soft computing. The epistemic analysis presents total uncertainty and total risk as equal to the sum of stochastic uncertainty and fuzzy uncertainty, and the sum of fuzzy risk and stochastic risk respectively that must receive attention in our theories on risk and uncertainty. The phenomenon of risk is then conceptualized in terms of natural risk and social risk with explications that allow the differences and similarities to be established through the corresponding adaptation properties where risk is not characteristic of nature but conceptual element of cognitive agents The discussions and analytical works on the concepts, epistemic framework, and foundations of risk-engineering, social risk-engineering, natural risk-engineering with comparative analyses to risk-taking conclude the chapter. The essential point that must be noted is that total risk is the sum or a weighted sum of fuzzy and stochastic risks almost all important decision-

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choice actions are undertaken under fuzzy-stochastic or stochastic-fuzzy risk. The mathematical knowledge on the behaviors of fuzzy-stochastic topological space with fuzzy-random variable, and stochastic-fuzzy topological space with random-fuzzy variable is not readily available to us to fully pursue the analysis of risk conditions and measures in these spaces. Chapter Six concludes the analytical structure of the epistemic discussions of this monograph regarding problems and relational understanding of uncertainty, expectations and risk in decision-choice process. It is mostly devoted to examining the structure and form of paradoxes that have risen in classical decision-choice theories and models. All the paradoxes in classical decisionchoice theories and the classical paradigm itself come to us as either temporary or permanent, it is argued. Permanent paradoxes are attributed to the presence of problems of subjective phenomena, vagueness, ambiguities and non-acceptance of logical duality in decision-choice behavior and knowledge construction and reduction. Discussions are made as to how the logic and analytical challenges of these paradoxes may be met with the toolbox of fuzzy paradigm and the methods and techniques of fuzzy optimal decision-choice rationality. The Arrow’s paradox is subjected to an epistemic examination within the framework of the fuzzy paradigm for the understanding of the conditions that give rise to it as a permanent paradox in the classical paradigm as well as the decision-choice sub-space that the paradox was created. A suggestion for its resolution is put forward. The relational structures of utility theory, probability and paradoxes in the classical decision-choice theories are analyzed. The role played by deficient information-knowledge structure in the rise of paradoxes is discussed through the analytical structure of fuzzy optimal decision-choice rationality and how the methods and tools of fuzzy optimal decision-choice rationality and the toolbox of fuzzy paradigm can assist us to resolve some, if not all, the paradoxes. The Ellsberg’s paradox and Savage axioms are singled out for a critical examination where the contributions by the presences of the phenomena of subjectivity and vagueness and their implications on optimal decision-choice rationality are examined in relations to Frank Knight’s measurable and unmeasurable uncertainties [R4.34]. The problem of Ellsberg is reframed to show its analytical defect relative to the decision-choice axioms of Savage by examining the problem in both possibility and probability spaces. It is then argued that the paradox arises by using two incompatible analytical frames. The Savage frame is in the probability space conditional on the assumption of existence of required possibility space. The choice variable here is the classical exact random variable. The Ellsberg frame is derived from both the possibility and probability spaces. The choice

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variable is fuzzy-random variable requiring a non-classical framework for its analytics. The understanding and appreciation of the differences are very important in order to understand the Ellsberg’s paradox. Savage deals with existence of probabilistic belief while Ellsberg deals with simultaneous existence of possibilistic and probabilistic beliefs. Reflections and criticisms of some suggested resolutions of the Ellsberg’s paradox are provided with a conclusion that the use of correction and adjustment factors within the classical paradigm is logically unjustifiable. On the basis of the critique, a resolution of the Ellsberg’s paradox is provided through the use of the toolbox of fuzzy paradigm and methods and techniques of fuzzy optimal decision-choice rationality. A numerical example is given to illustrate the nature of the resolution of the Ellsberg’s paradox. The chapter is concluded with an epistemic examination of relational structure of fuzzy optimal decision-choice rationality and paradoxes in general and how the methods and techniques of the knowledge square and quadrangle logical pyramids will assist in our critical understanding of paradoxes as we seek knowledge of the process of moving from the potential to the actual through the possible and the probable. In general and specific frameworks, the phenomena of uncertainty, expectations and risk must retain their inter-relational foci with themselves, information-knowledge structure and decision. In this respect, the analytical process on rationality should not lead to a situation where the explicate words are devoid of the concepts of the phenomena in such a way as to create phantom problems or irresolvable paradoxes. The reflection by Claude Bernard is useful at this point. In creating a word to define a phenomenon, the idea it expresses is generally specified at that time together with its exact meaning. However, with the passage of time and the progress of science, the meaning of the word changes from but keeps its initial significance for others. As a result there is often such a discordance that persons employing the same word mean very different ideas. Our vocabulary is only approximate and so imprecise, even in science, that if we focus on words rather than phenomena, we stray quickly from reality. Science can only suffer when we discuss to keep a word which can only induce error because it does not convey the same meaning to all. Let us conclude that one must always focus on phenomena and view the word only as an expression void of meaning if the phenomena or if it happens that they do not exist.

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The mind of course moves systematically, which explains why we tend to reach an agreement on words rather than on phenomena. This leads experimental criticism in the wrong direction, confuses issues, and suggests the existence of dissidences; but these relate most often to the interpretation of phenomena, instead of the existence of facts and their true importance. Claude Bernard [R21.16, p.5]) The statement by Bernard may be read in conjunction with that of Faraday in relation to cognitive anchorage on the road to discovery of scientific truth. We must always guide ourselves against the tyranny of established schools of thought and the acceptable boundaries of reasoning. The scientist should be a man (person) willing to listen to every suggestion, but determined to judge for himself. He should not be biased by appearances; have no favorite hypothesis; be of no school; in doctrine have no master. He should not be a respecter of persons but of things. Truth should be his primary object. Michael Faraday, [R21.16, p. 6]

Acknowledgements The epistemic foundations of analysis of uncertainty, expectations, risk and the theory of knowledge are intimately connected to the theory of rational process of human action and rules of reason. Any serious analytical work on human understanding of nature and society and epistemic justification of belief system that correspond to rationality cannot neglect these phenomena. There are many works on these phenomena which we must be grateful. The phenomena of fuzziness and approximate reasoning as they relate to expectations, uncertainty and risk have received almost no attention. The works, however, by a number of scholars and researchers on fuzzy logic, mathematics, decision theory and category theory under fuzzy information-knowledge constraint provide us with a new toolbox for better understanding of rational behavior. We are thankful for all these dedicated people. However, the greatest danger to the progress of fuzzy research, we may note, is ideological and scientific credulity that finds expression in the classical paradigm with the principle of exactness that ties uncertainty, expectation and risk to only probabilistic belief, probability and stochastic processes. The danger may be diminished by acknowledging the fundamental idea that the vocabularies of our

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languages and linguistic reasoning are approximations that require explications in communications. Extreme explication, however, may increase the irrelevance and misunderstanding of the communication process leading to increased penumbral region of decision-choice actions. In this respect, in the development of fuzzy paradigm, enhancement of its logic, expansion of its mathematical domain, and the integration of fuzziness and randomness, we are thankful for all researchers and scholars who have freed themselves from the ideological grip of the classical paradigm. This monograph has benefited from their works and contributions. Special thanks go to Dr. Thomas Ditzinger, Dr. Janusz Kacprzyk and their supporting staff for the editorial work and the publications of the series on fuzziness and soft computing. I also thank all my friends whose have provided me with positive encouragements in my research. I express my thanks to Professor James A. Momoh and his team in the Center for Energy Systems and Control at the Electrical Engineering Department of Howard University for inviting me to present to them some of my work on fuzzy phenomena. I also appreciate my association with Professor Tepper L. Gill of both Departments of Mathematics and Electrical Engineering for his encouragement and interest for my initial reflections on the problems of exactness and fuzziness in mathematics and science. My thanks go to Professor Mohammad Mahmood for the invitation to present some of my works on fuzzy mathematics in the Colloquium organized by Mathematics Department of Howard University. I also thank the Economics Department at Fox Business School, Temple University, Philadelphia, PA, USA, for their flexible program that allowed me to study philosophy of science, mathematics, operations research in addition to the core courses in economic theory and econometrics. I also like to acknowledge the benefits that I obtained from my interactions with the faculty members in charge of probability and mathematical statistics in the Mathematics Department at Temple University. I express grate appreciation to Dr. Grace Virtue for providing editing advice that has allowed this work to be clearer and easier to read. Some ideas that have been introduced in this monograph and previous ones are controversial and intentional for further reflections and research. All errors and mistakes are mine.I fully take the responsibility. November 2008

Kofi Kissi Dompere

Table of Contents

1

Fuzzy Rationality, Uncertainty and Expectations................................1 1.1 Expectations and Rationality.............................................................4 1.1.1 Information, Knowledge and Uncertainty...............................5 1.1.2 Explications of Possibility and Probability in Decision-Choice Rationality .................................................11 1.2 Rationality, Expectations and Uncertainty ......................................22

2

Fuzzy Rationality and Classical Sub-optimal Rationality.................27 2.1 The Relationship Between Fuzzy Optimal Rationality and Classical Sub-rationality..................................................................27 2.2 Fuzzification, Defuzzification and Fuzzy Rationality .....................34 2.2.1 Fuzzification in General Fuzzy Rationality ..........................34 2.2.2 A Note on Real Number System, Fuzzy Number System and Fuzzy Rationality...............................................38 2.2.3 Defuzzification, Fuzzy Rationality and Classical Rationality .............................................................................40 2.3 Fuzzy Optimal Rationality, Contradiction and Cost-Benefit Rationality .......................................................................................45 2.3.1 Fuzzy Optimal Rationality and the Theory of Contradiction .........................................................................45 2.3.2 Cost-Benefit Rationality and Fuzzy Optimal Rationality .....46

3

Fuzzy Rationality, Ambiguity and Risk in Decision-Choice Process....................................................................................................51 3.1 Ambiguity, Probability and Decision ..............................................53 3.2 The Epistemic Analyses of the Decision-Choice Cohorts...............55 3.2.1 COHORT I: Non-fuzzy and Non-stochastic DecisionChoice Systems .....................................................................56

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3.2.2 COHORT II: Non-fuzzy and Stochastic DecisionChoice Systems .....................................................................58 3.2.3 COHORT III: Fuzzy and Non-stochastic DecisionChoice System.......................................................................62 3.2.4 COHORT IV: Fuzzy and Stochastic Decision-Choice System...................................................................................67 3.3 The Organic Paradigm of Decision-Choice Theories......................75 3.4 A Brief Reflection on Optimal Rationality in Economics and Psychology ......................................................................................79 4

Epistemics of Risk and Optimal Decision-Choice Rationality ..........83 4.1 Introduction .....................................................................................83 4.2 The Conceptual-Measurement Problem of Information, Uncertainty and Risk .......................................................................84 4.3 Epistemic Definitions of Uncertainty and Risk...............................87 4.4 Accident, Necessity and the Knowledge-Risk Squares...................92 4.5 Risk, Freedom and Decision-Choice Rationality ............................94 4.5.1 Freedom under Uncertainty and Risk....................................95 4.5.2 The Principle of Compatibility of Necessity and Freedom ................................................................................98

5

Reflections on Some Decision-Choice Theories on Uncertainty and Risk ..........................................................................105 5.1 The Classical Decision-Choice Theories under Stochastic Uncertainty and Risk .....................................................................106 5.2 Principle of Insufficient Reason, Principle of Sufficient Reason and Uncertainty-Risk Decision-Choice Theories .............113 5.3 Conceptual Foundations of Fuzzy Risk and the DecisionChoice Rationality .........................................................................116 5.4 Risk Engineering, Risk Bearing and Decision-Choice Rationality .....................................................................................119 5.4.1 Social Risk Engineering......................................................120 5.4.2 Natural Risk Engineering....................................................125 5.5 Costs-Benefit Rationality, Risk-Taking and Risk-Engineering.....127

6

Fuzzy Decision-Choice Rationality and Paradoxes in Decision-Choice Theories ...................................................................133 6.1 The Four Structures of Decision-Choice Rationality ....................133

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6.2 Fuzzy Decision-Choice Rationality and the Arrow's Paradox (Impossibility Theorem)................................................................137 6.3 Fuzzy Optimal Decision-Choice Rationality, Utility Theory, Probability and Paradoxes .............................................................140 6.3.1 The Ellsberg Paradox and Savage Axioms .........................144 6.3.2 A Fuzzy Reflection on Some Suggested Resolutions of Ellsberg Paradox .................................................................151 6.3.3 Fuzzy Optimal Rationality and Ellsberg DecisionChoice Experiment..............................................................158 6.4 A Numerical Example for the Ellsberg’s Experiment...................161 6.5 Fuzzy Optimal Decision-Choice Rationality, Other Decision Paradoxes and Paradoxes in General.............................................163 References....................................................................................................171 R1 On Aggregation and Rationality....................................................171 R2 Cost-Benefit Rationality and Decision-Choice Processes.............174 R2.1 On Cost-Benefit Rationality and Accounting Theory.........176 R2.2 On Cost-Benefit Rationality and Real Economic Costing ................................................................................176 R2.3 On Cost-Benefit Rationality and Decision-Choice Criteria ................................................................................180 R2.4 On Cost-Benefit Rationality and Pricing ............................182 R2.5 On Cost-Benefit Rationality and Discounting ....................184 R2.6 Cost-Benefit Rationality and Contingent Valuation Method (CVM) ...................................................................186 R2.7 Cost-Benefit Rationality and the Revealed Preference Approach (RPA)..................................................................189 R3 On Rationality and Social Decision-Choice Process.....................191 R4 On Expectations, Uncertainty and Rationality ..............................199 R5 On Decision-Choice Process, Fuzziness and Rationality ..............203 R6 On Fuzzy Decisions, Applications and Rationality.......................213 R7 On Game Theory, Risk Analysis. Fuzziness and Rationality........219 R8 On Fuzzy Logic and Rationality....................................................225 R9 On Fuzzy Optimization and Decision-Choice Rationality ............229 R10 On Fuzzy Mathematics and Optimal Rationality ..........................233 R11 On Fuzzy Probability, Fuzzy Random Variable and Random Fuzzy Variable ..............................................................................242 R12 On Rationality and General Decision-Choice Processes...............245

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R13 On Rationality, Ideology and Decision-Choice Process ...............247 R14 On Rationality, Information and Knowledge ................................248 R15 Rationality and Category Theory in Mathematics, Logic and Sciences .........................................................................................253 R16 On Rationality, Probabilistic Concepts and Reasoning.................254 R17 On Classical Rationality, Optimality and Equilibrium..................257 R18 On Bounded Rationality ................................................................262 R19 On Rationality, Information, Games and Decision-Choice Conflicts ........................................................................................264 R20 On Rationality and Philosophy of Science ....................................266 R21 On Rationality, Riskiness, Decision-Choice Process and Paradoxes.......................................................................................272 R22 On Theories of Utility and Expected Utility .................................274 R23 On Vagueness and Approximation................................................275 R24 On Rationality, Prescriptive Science and Theory of Planning ......280 Index.............................................................................................................281

1

Fuzzy Rationality, Uncertainty and Expectations

Some important questions were raised in Chapter two of [R17.24] whose answers will allow us to epistemically connect the relational structure of decision, choice, information, knowledge and rationality. It will further allow us to explicate the relationship between the Euler’s mini-max principle of universal event decision-choice rationality under various conditions of linguistic communications It was pointed out that the structure of decision-choice process depends on information and knowledge where knowledge is a derived category from characteristic-based information as the primary category of reality. The decision-choice activity is an input-output process. Information and knowledge are the driving inputs of the decision-choice actions. The same information and knowledge are the outputs of the decision-choice actions. Thus the decision-choice process presents itself as an input-output duality whose unity is information and whose conflict is knowledge. Epistemically, information is an input into reasoning while knowledge is its output; knowledge is input to decision whose output is decision rationality; decision rationality is an input to choice action and substitution-transformation is its output which then becomes input into information. The process provides us with never-ending system of information, knowledge, decision, choice, and information that spin the circle of substitution-transformation processes over the potential-actual dualities. The circular process is illustrated in Figure 1.0.1. To examine the relative roles played by information and knowledge in the decision-choice process and how such roles may be related to rationality we presented epistemic definitions and conceptual contents of the two and pointed out where conflicts tend to arise in the substitution-transformation process of cognitive agents. From the definitions and conceptual contents it was argued that information is epistemically different from knowledge and that knowledge is derived from K.K. Dompere: Fuzzy Rationality, STUDFUZZ 237, pp. 1–26. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

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1 Fuzzy Rationality, Uncertainty and Expectations

CHARACTERISTIC – BASED INFORMATION

RELATIONSBASED INFORMATION

KNOWLEDGE

DECISIONCHOICE RATIONALITY

SUBSTITUTIONTRANSFORMATION PROCESS

CHOICE

DECISION

Fig. 1.0.1. The Input-Output Process for Decision-Choice Rationality

information by the method of constructionism. The social knowledge structure is a unity of objective and subjective phenomena of elements in the universal system. The objective information is defined by characteristics-based information. The subjective information is defined on the basis of relations-based information. Subjective knowledge is a justified belief. The justification is produced by the interplay of attribute signals and cognitive conversion processes. Subjective knowledge is not necessarily true knowledge of reality. It is a model of reality produced through the prism of the perception characteristics set. It is pointed out that for a subjective knowledge to have a social claim to objective knowledge about reality; it must satisfy a verification principle. Additionally, the conditions of justification must satisfy the corroboration principle with a stated degree of collaboration. Importantly, we distinguished concept of corroboration from concept of verification and their role in knowledge production. Together, the principles of corroboration and verification work collaboratively to allow a test of objectivity of subjective knowledge. Here lies the usefulness of probabilistic and statistical reasoning or any reasoning mode that can be brought to bear on decision-choice process in accepting subjective knowledge as true social knowledge of reality. The acceptance of sub-

1 Fuzzy Rationality, Uncertainty and Expectations

3

jective knowledge as true knowledge of reality is itself choice-decision problem that must either be driven by the intelligence of cognitive agents or some rationality defined in terms of best or maximum content of reality. The approved objective knowledge set is, by the construct, a collective acceptance that depends on the social knowledge-production culture on the basis of endowed intelligence of decision agents. Decision-choice intelligence that characterizes the rationality of behavior in the decision-choice space involves all aspects of human endeavor in productionconsumption space. It is defined in relation to information in the final analysis, in that the decision-choice process is a work of an interactive informationknowledge process. Decisions are made and choices are undertaken with either individually or collectively accepted subjective knowledge that may fail to satisfy the condition of objective knowledge in the sense of corroboration with the relevant element in the universal objective knowledge set. Both individually and collectively approved subjective knowledge is defined in perception space on the basis of which decisions and choices are made in societies by cognitive agents. Thus decision-choice intelligence is driven by subjective knowledge that is abstracted by a cognitive process from information. All concepts of rationality including optimal rationality and sub-optimal rationality of decision-choice behavior are information defined and operate through knowledge-processing module. There are different types of decision-choice problems for decision-choice agents depending on the element of interest in endeavors of the cognitive agents. Corresponding to the decision-choice problems, different structures of information are projected by perception onto the decision-choice space in addition to different knowledge structures that are also projected onto the decision-choice space through cognitive information-processing capacity. The types of decision-choice problem are defined in relation to the types of received information structure. While the decision-choice problems are defined by information structures, the rationality embodied in decision-choice outcomes is defined in relation to subjective knowledge as a derivative of information through a cognitive process. By partitioning the decision-choice problems with the types of information structure; and defining rationality by means of subjective knowledge, we place rationality on the same epistemological footing as what there is and the knowability of what there is through information processes. The search for and knowability of what there is in natural and social processes involves expectations on the part of cognitive agents to which we turn our attentions.

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1 Fuzzy Rationality, Uncertainty and Expectations

1.1 Expectations and Rationality By relating the existence and knowability of what there is to information, knowledge and rationality, the decision-choice intelligence of cognitive agents and the motivation of decision-choice activities are made to rest on expectations of decision-choice agents as cognitive behavior in the decisionchoice space. Such expectations are span by decision-choice agents’ understanding of the past-present-future dynamics of potential-actual duality. Expectations, decisions and choices constitute a trinity and appear as inseparable unit and yet may be analytically viewed as separates but interdependent activities by cognitive agents in the substitution-transformation process. Let us now turn our attention to expectations in decision-choice rationality. This requires us to examine the concept of expectation at the levels of general human thought and enterprise of knowledge production. At the level of human behavior, the concept of expectation demands meaning. At the level of thought, the concept of expectation demands a definition; and at the level of knowledge enterprise the concept of expectation demands of us knowledge representation. At the level of scientific enterprise, the meaning, definition and representation must lead to the establishment of explication of expectation and its use in the knowledge enterprise in a manner that allows us to establish the role of expectation in decision-choice rationality and its link to the past-presentfuture time chain. At the level of application in decision-choice activities, the concept of expectation demands measurement. Expectation may be viewed as a cognitive disposition toward decisionchoice behavior. Thus expectation is a surrogate representation of decisionchoice activity. Behind every decision-choice action is an expectation and that every expectation has decision-choice action behind it. We may note that not deciding is itself a decision. This is consistent with the notion that the null set is contained in every set of alternative decision-choice elements. Expectation in decision-choice behavior involves a consideration of four elements of 1) decision-choice agent, 2) conditions of justification, 3) prospect and 4) time. Time is a collection of time elements that together constitutes a time set, T , whose construction may show it to be a monoid (semigroup) [R7.29], [R7.36], [R10.123] with left and right continuities and interconnectedness. The time set, T , may be partitioned into three broad categories (sets) of past Q , present, P and future, F such that if t1 ∈ Q, t2 ∈ P and t3 ∈ F then t1 < t2 < t3 for crisp partition and t1 ≤ t2 ≤ t3 for fuzzy partition. Decision-choice agents are cognitive agents that produce knowledge at P (present) from Q (past) and F (future) to decide and choose a choice-decision element at time

1.1 Expectations and Rationality

5

t ∈ P (present) whose outcome effect is to be observed at t ∈ F (future). Justification is a set of conditions that generate a belief in the outcome effect of decision-choice element from present decision-choice action whose outcome will be observed at t ∈ F . The justification is subjectively evidential support for introspection at t ∈ F . A prospect is simply a possible outcome subjectively drawn from the potential space. It is subjective view of what ought to be under the dynamics of decision-choice process. It is first an element of the potential space, U and then brought to the possibility space, P by cognitive transformation, and then to be transformed into an element of space of the actual, A by decision-choice action relative to held expectations. The decision-choice agent is one who constructs a justification at time t ∈ Q from the available total knowledge of the present Q and past P in support of held expectations at time t ∈ Q for a particular potential whose outcome is expected at time t ∈ F . The nature of decision-choice behavior of cognitive agents thus depends on expectations, E whose inputs are justification set, G prospect set, Π and time set T and may symbolically be defined as: E = G⊗Π⊗T =

{( g, π,t ) | g ∈ G , π ∈ Π , t ∈ T}

The characterization of expectations as composed of justification G , prospect (potential outcome) Π and time T = F ⊗ P ⊗ Q raises a question as to how expectations arise in cognition. Further questions arise as to the similarities and differences among concepts of expectation, forecasting and prediction on one hand and among goal, objective and prospect on the other and the relational structure that they mutually induce. Does expectation relate to prediction and how does prediction differ from forecasting? Similarly, how is a prospect (potential) related to or different from goal and objective in decisionchoice processes? Answers to these questions require us to turn our attention to information, knowledge and uncertainty and how they may help to ascertain the meaning and knowledge of rationality in general. 1.1.1

Information, Knowledge and Uncertainty

Different concepts of information have been introduced in a companion volume [R17.23].Objective type and subjective type of information was distinguished and analyzed and related to knowledge creation. Objective information is related to reality that exists independently of awareness of cognitive agents. Subjective information is related to perceived reality that is dependent on awareness of cognitive agents. Total information is thus composed of the

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1 Fuzzy Rationality, Uncertainty and Expectations

Boolean sum of these components, whose basic structure is presented in Figure 2.1.1 of [R17.23]. Given the structure of information, the knowledge set is constructed from the subjective component of information and verified in the space of objective information in terms of corroboration with reality. The knowledge structure is composed of socially accepted knowledge set that is derived from the perception-to-reality set. The social knowledge structure has a time dimension and we accept the postulate that the spaces of both subjective information and knowledge increase with time and experience of endeavors of cognitive agents. Alternatively stated the knowledge set at any time is a family of nested sets. The knowledge structure at each time point becomes an input into, as well as initializing condition for further knowledge production. The same knowledge structure becomes input into decision-choice process; thus every decisionchoice action has its knowledge support that forms its justification. In an epistemic setting every decision-choice activity has as its objective the achievement of a goal or a set of goals. Similarly, every decision-choice process is intended to preserve the actual or to change the actual. Alternatively stated, every decision-choice process is intended either to prevent the potential from coming into being or to actualize it. The two alternative statements are embedded in the dynamics of actual-potential duality that spins the substitutiontransformation process for the emergence of the new and disappearance of the old. Here, a question arises as to the degree of surety that we can attach to the goal been achieved leading to the realization of the defined objective. The degree of surety depends on the input of our subjective knowledge that gives rise to categorial uncertainties. These categorial uncertainties are produced from the zone of epistemic accessibility ([ZEA]: what we claim we know) and the zone of epistemic ignorance ([ZEI]: what we do not know). Categorial uncertainty arises from two major sources of 1) incomplete or lack of knowledge and 2) vagueness in knowledge due to a number of problems associated with cognition. This number of problems in cognition will include inconclusive evidence, imprecise evidence, ambiguous evidence, approximate statements, ill-defined explication, qualitative evidence and others that rob us of exactness. The sources of these problems and characteristics may arise from the inaccuracies of our linguistic structures, poor reception that affects signals and meaning, cognitive limitations in interpretation and retention and others. The presence of incomplete knowledge or lack of knowledge presents a particular type of uncertainty which may be called stochastic or probabilistic uncertainty [R11], [R11.27], [R16], [R16.16], [R16.27], [R16.31]. The presence of vagueness in the knowledge structure and explications of

1.1 Expectations and Rationality

7

terms and concepts presents another type of uncertainty which may be referred to as fuzzy uncertainty [R11], [R11.27], [R11.34], [R11.46]. Uncertainties have continuity with time in terms of linking the past to the future through the present at each point of time relative to knowledge production and use. The past and present have knowledge structures that may be incomplete, vague or both. The future presents cognitive ignorance for the present decision-choice action except by cognitive relational projections of the future unto the present on the basis of the past. Similarly decision-choice processes have time point continuity where the present decision and choices are linked to the past through the present deliberations of the relative desirability of actual and potential which are connected to future outcomes. Both uncertainties and decision-choice processes traverse through time in states with knowledge as the controlling force (controller) where the future fades into the present and the present fades into the past through knowledge production and decision-choice process, all under the conditions of uncertainty that operates through decision-choice-information-interactive processes. Experiential knowledge is characteristic of the past and present and the lack of knowledge is a characteristic of the future. Decisions are of the present that are linked to the established or non-established goals whose realizations depend on the quality of knowledge input and the best path of decision choice process. The presence of uncertainties in the knowledge input affects the behavior of decision-choice agents in a fundamental way involving their perception of the dynamic behavior of actual-potential duality in substitutiontransformation processes. Decision-choice processes generate prospects (potential elements) seeking to actualize a potential and potentialize an actual through substitution-transformation process by fading the actual into the potential by the method of reductionism. The potential is embedded in cognitive uncertainties that flow from the subjective knowledge structure. The uncertainties in turn generate cognitive expectations on the part of decision-choice agents about the potentials to be actualized. The cognitive expectations about potential outcomes (prospects) become the motivation, as the driving force in the direction of the decisionchoice process which allows the actualized potential (outcome) to be compared with the perceived goal or the existing actual. The derived perception and interpretative relationship of either the past and present knowledge to the future outcomes, or the dynamic behavior of the actual-potential duality in the substitution-transformation process constitutes the justification conditions for the held expectations. The complex relationships among past, present and future on one hand and their impacts on relationships among information, knowl-

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1 Fuzzy Rationality, Uncertainty and Expectations

PAST

INFORMATION

ZEI

KNOWLEDGE

EXPECTATIONS

ZEA

PRESENT

FUTURE

RATIONALITY

Fig. 1.1.1.1. The Complex Relationship among Past, Present and Future on One Hand and Information, Rationality and Knowledge on the Other Hand.

edge and rationality may be presented as interactive epistemic geometry in pyramidal logic as in Figure 1.1.1.1. The past, present and future constitutes a pyramid. The information, rationality and knowledge constitute another pyramid. The two pyramids interact to define the conditions of expectations formation given rationality as attribute to decision-choice agents. We have argued that rationality is defined in terms of the best-path action in behavior of cognitive agents operating in decision-choice space. The bestpath action in the classical logic involves either exact maximum or exact minimum depending whether benefits or costs are under decision-choice consideration. Our discussions, so far, on rationality seem to imply certainty in the knowledge input for the decision-choice behavior. Our epistemic calculus to abstract the best path of decision-choice action is less complete in the sense that the rationality that emerges is what is referred to as the classical rationality involving only certainty outcomes.

1.1 Expectations and Rationality

9

Generally, however, the knowledge controller of the best-path action is clouded in uncertainties of stochastic and fuzzy types that give rise to cognitive expectations. These uncertainties must be taken into account in decision making and choice implementation. We are therefore confronted with the problem of how best to integrate uncertainties that give rise to expectations as well as how to include the family of set values in the worst-to-best possibilities into the decision-choice processes involving the behavior of actualpotential duality in the substitution-transformation process. It is precisely the presence of both stochastic and fuzzy uncertainties in our knowledge structure and cognitive capacity that provides us with explanation of the principle of unintended consequences in behavior of actual-potential duality and substitution-transformation processes. Thus, any serious study of decision-choice rationality or theories of decision-choice behavior must deal with the structure of uncertainty and expectation formation as mutually inseparable concepts in the analysis. These uncertainties and expectation formation are dealt with in a number of theories about decision-choice behavior by assumptions that simplify the complexities in the environment of behavior. Attempts to deal with expectations in decision-choice processes, as we examine the epistemic conditions of setting the potential against the actual, give rise to the study of behavior and measurement of uncertainties. The need to integrate expectation formation into decision-choice processes forces us to examine the best path of cognitive behavior under conditions of uncertainties span by the social knowledge diameter and potentiality induced by natural order. Expectations, therefore, involve two sets of conditions of possibility in the space of potentials, U and probability in the space of the actual, A . The space of potential elements is infinite. However for every statement for explicit substitution-transformation act in the dynamics of actual-potential duality there is a finite set of possibilities, P , contained in the potential space, U , that is ( P ⊂ U ) . We shall refer to P as the possibility space whose construct depends on the explicit statement or hypothesis in the substitutiontransformation process and the accumulated knowledge. The construct of the possibility space, P is based on ranking by the degree of possibility in the space of potentials, U , given the explicit statement of substitutiontransformation process. This degree of possibility may be constructed by the use of fuzzy logic where grades of truth, shades of meaning, and degrees of belonging and values of fading are logically and mathematically admissible [R4], [R7], [R7.31], [R8], [R8.7], [R8.15], [R10], [R10.54], [R10.55], [R10.62], [R11.47].

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1 Fuzzy Rationality, Uncertainty and Expectations

U ϕ4 (U )

A

ϕ1 ( U )

SubstitutionTransformation process in dynamics of Actual-potential Duality

ϕ3 ( B)

P ϕ2 ( P )

B

ϕ

Fig. 1.1.1.2. Relational Geometry of Potential to Possibility to Probability to Actual: The Knowledge Square

Given the possibility set or space, we are confronted with a fundamental question as to which possible event from the possibility space must command our decision-choice action. The answer to this question is dealt with in a different space of how probable can each possible element be brought into occurrence or can be actualize by decision-choice action by moving the element from the possibility space, P to the space of the actual A . The cognitive space to answer the question of how probable can an element in possibility space be actualized is called the probability space, B where ( B ⊆ P ) . The decisionchoice process gives us a rectangular logical relational structure in square of the form shown in Figure 1.1.1.2. We refer to this as the knowledge square. Definition 1.1.1.1: The Knowledge Square The knowledge square is a geometric representation of cognitive transformation processes that present decision-choice movements from the potential to the possible, from the possible to the probable, and from the probable to the actual in the knowledge construction process in support of our beliefs, ideas and facts about nature and society through category formations and categorial conversions in potential-actual duality.

1.1 Expectations and Rationality

11

The transformation process may be rational or non-rational depending on the conceptual definition of rationality in behavior and the knowledge enterprise. The functions ϕ1 ( i ) , ϕ2 ( i ) , ϕ3 ( i ) , and ϕ4 ( i ) constitute an ordered sequence of cognitive substitution-transformation functions that take an element in the potential space and convert it into a potential element where ϕ4 ( B ) = ϕ3 ϕ2 ( ϕ1 ( B ) ) . Our cognitive process and decision-choice technology of categorial conversion do not allow us to go directly from ϕ4 ( i ) to ϕ3 ( i ) and bypass the intermediate sequence ϕ1 ( i ) and ϕ2 ( i ) . The categorial conversion sequence is such that ϕ1 ( i ) creates the possibility set from the potential. This is taken over by ϕ2 ( i ) and creates a probability pace from the possibility space. The conversion continuity is maintained where ϕ3 ( i ) takes over and creates the perceptive accrual category of reality. It is this substitution-transformation process that allows the potential space to be linked to the space of the actual through decision-choice behavior involving expectations and motivations in the possibility space. For each potential sub-space, relative to an explicit statement about the substitution-transformation process in the dynamics of actual-potential duality, the elements may be ordered first in terms of degree of possibility and then rearranged in terms of degrees of probability of actualization. Thus the concept of probability occupies a very important position in selecting the best path of decision-choice process that will allow the potential to be set against the actual. It may be noticed that the element with the highest degree of possibility may have a low degree of probability in the substitution-transformation process within a defined constraint set. Furthermore, we may note that the potential space constitutes the primary logical category of the epistemic analysis. The possibility space is a derived logical category while the probability space is a logical derivative from the possibility space and the space of actual is a logical derivative of the probability space. Thus in the final analysis the space of the actual is derived from the potential space by the method of constructionism while the elements in the space of the actual are traceable to the potential space by the method of reductionism.

(

1.1.2

)

Explications of Possibility and Probability in Decision-Choice Rationality

We have introduced the concepts of possibility and probability into the epistemic analysis of Rationality as if the meanings and their importance are clear as they relate to cognitive expectations in the dynamics of actual-potential duality. Let us turn our attention to the importance of expectation formation

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1 Fuzzy Rationality, Uncertainty and Expectations

after explication of the concept of expectation. First, we may note that the concepts of possibility and probability have epistemic difference as we examine decision-choice rationality. They play different but inter-supporting roles in decision-choice process in all aspects of endeavors of cognitive agents. The distribution of the degrees of possibility associated with the elements in the potential space presents a different decision-choice meaning from the degrees of probability associated with the element in the possibility space. Both distributions may substantially differ from one another. To understand the epistemic difference between possibility and probability let us conceptualize a six-faced dice whose numbers are in a constant state of substitution-transformation process between one and infinity. The question is: what are the different degrees of possibility associated with these numbers such that six of these numbers will appear on the six-faced dice through the substitution-transformation process? The numbers from one to infinity constitute the potential space. The decision-choice action on these numbers is imposed by an order of degrees of possibility where such degrees of possibility are the available knowledge structure. Let such degrees of possibility as constructed from the available knowledge structure range between zero and one inclusive ([ 0 , 1]) with the following distribution {(1, 0.9 ) ,( 2, 0.6 )( 3, 0.8)( 4, 0.7 )( 5, 0.9 )( 6,0.8)} where the first numbers in the parenthesis are the possible numbers and the corresponding numbers are the degrees of possibility. The degrees of possibility of numbers ranging from seven to infinity are less than equal to (0.5). Thus by fuzzy optimal principle of α − level set construct [R7.35], [R9], [R9.9], [R9.20], [R9.31], [R10], [R10.6], [R10.13], [R10.56], [R10.57], [R10.75] for abstracting candidates for the possibility space we obtain the possibility set of {1, 2, 3, 4, 5, 6} . The elements in the possibility space need not be quantitative. They may be either propositions or hypothesis of qualitative in nature. For example, consider a simple statement that “it is more probable that a cure for cancer will be found through research in understanding the mechanism of growth of normal cells than through research on experimental treatment of cancer patients”. There is an important implicit assumption of conceptual meaningfulness of this statement. The assumption is that the cure of cancer is possible without which the concept of probable has no material meaning and the comparative analysis of methods of research for cure is unintelligible. If cure for cancer is a potential that is impossible then the decision choice action between the two methods is cognitively void. The material meaning of “probable” and the decision-choice action on the directions of research find expression in the possibility space in providing a justification for

1.1 Expectations and Rationality

13

committing “resources” to research while the degree of actualizing cure finds expression in the concept of “probable” whose relative degree provides decision-choice action on a research path. Cure as a characteristic of the universal object set is first defined in the potential space and then cognitively transformed into the possibility space and then moved by categorial conversion into the probability space. The possibility space performs a linkage between the potential space and the probability space in the decision-choice process and the rationality that may be induced. Now given the possibility set that represents a relationship between present decision-choice action and future actual or outcomes, a selection of an element from the possibility set must be based on the degree to which a potential element defined in the possibility space can be actualized. The degree to which an element in the possibility space can be actualized has traditionally been handled as a question of probability. This is one way but not the only way to enter into epistemological questions and the calculus of probability [R4], [R16], [R16.12], [R16.34], [R16.37], [R16.46]. The degree to which the potential can be actualized has been viewed in terms of degrees of confirmation, conviction, intuition or a degree of belief that one way or the other may be linked to induction, deduction and others as we move from possibilistic set of events to probabilistic set of propositions in the decision-choice activities in the knowledge accumulation process. The probabilistic induction or deduction in substitution-transformation process involves relation of the present decision-choice activity to the knowledge about the future in terms of ranking the elements in the possibility set conditional on a given hypothesis. The acquisition of knowledge about the future is to reduce uncertainties that place partial shadow on present decisionchoice activities whose outcomes are hoped to be actualized in the future date. The manner in which the knowledge about the future is obtained, relative to the decision-choice action will affect the structural understanding of uncertainties and the degree of belief in the coming into being of that which is expected from the possibility space. The knowledge about the future, as an input into present decision and choice, constitutes the justification for the expected. The use of knowledge that is derived from the future about the possible events depends on the degrees of belief attached to the validity or confirmation or truth of this knowledge which may be presented as given, conditional on a hypothesis. The expected (prospect) becomes the motivation for the present decision-choice action on the elements in the possibility set. These elements in the possibility set are reordered by the degree of belief that the elements can be actualized by decision-choice action in the substitution-transformation process.

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1 Fuzzy Rationality, Uncertainty and Expectations

It must be clear from the epistemic discussion that every probability space, either mathematically or non-mathematically conceived, assumes the existence of possibility space. The construct of such possibility spaces are not logically presented as part of the theory of probability. We have indicated that the possibility space links the probability space to the potential space by the logic of reductionism and that the possibility space may be constructed by the method of fuzzy logical reasoning [R8], [R8.20], [R8.64]. A conceptual system is thus required to establish a logical connection between concepts of possible and probable. Such a conceptual system requires the development of probability theory conditional on possibility theory through some best practice of science. This best practice, we shall show to be contained in what we are referring to as fuzzy rationality. In the possibility space, we speak of degree of belief in terms of possibility in the potential space. In other words, we deal with degrees of belief in the cognitive transformation from the potential space to the possibility space on the basis of some justification principle that may be constructed from the established knowledge base. For matters of distinction, the degree of belief associated with cognitive transformation of elements in the potential space is here referred to as the degree of convertibility (or transformability) of the potentials to the actual. The degree of convertibility merely indicates the justified degree of belief that a potential has a possibility of being transformed by decision-choice action into an actual. We may now define a possibility index or the degree of convertibility. Definition 1.1.2.1: Possibility Index/Degree of Convertibility A possibility index or the degree of convertibility as a measure of the extent to which a potential, on the basis of accumulated present knowledge, and by substitution-transformation process, and a necessity justifies a held idea that something is a possible element that can be actualized. The possibility space is, therefore, a schedule made up of potential elements with the corresponding degrees of convertibility to the actual. The conditions of conversion and justification of the degree of convertibility depend on the expected and presently accumulated knowledge about how old attributes, conditional on a given hypothesis, are lost and new attributes are acquired through substitution-transformation processes of elements of similar and dissimilar categories. The concept of substitution-transformation process acknowledges the basic principle of duality that two elements occupy the same space and time only in mutual conflict. The categorial dynamics of actualpotential duality point to an epistemic position that the only thing permanent in nature is change and the only thing that is not part of the substitution-

1.1 Expectations and Rationality

15

transformation process in the categorial dynamics of actual-potential duality is impossibility. This statement denies transformational impossibility in the universal object set [R17.23], [R17.24]. It does not deny the existence of impossibility which will cause a logical violation of duality. The statement does not mean that the characteristic set of impossibility is null which will logically imply the non-existence of impossibility which in turn will deny the existence of possibility as required principle of duality. Its epistemic meaning is that the characteristic set of impossibility is continually being transformed to increase the characteristic set of possibility to ensure the required conditions of categorial convertibility as we move from the potential to the actual. This leads to a logical position that every thing in the universal object set is under categorial conversion through substitution-transformation, and governed by principles of constructionism and reductionism to ensure the epistemic claim that the universal object set is infinitely closed under transformations. In the probability space, we speak of degree of probability in the possibility space. In other words, we deal with degrees of belief in decision-choice action that induce substitution-transformation activities from the space of possible elements to the actual on the basis of some justification principle that may be constructed from the knowledge about the future. The degree of belief associated with the success of decision-choice action in substitution-transformation or potential to the actual has many names such as the degree of likelihood, degree of confirmation or degree of conviction or degree of rational belief or simply probability. Customary thinking suggests that events in the possibility space are such that the degree of belief associated with an element of decision-choice action may be either objective or subjective and hence we have objective and subjective probability. The knowledge formation about the future in support of the present degree of belief in actualizing any possible element may be logically abstracted on the basis of present accumulated knowledge on how, and the rate of success in destroying existing attributes and creating new attributes. A conceptual definition of probability is required in other to distinguish it from possibility. Definition 1.1.2.2: Probability Index Probability may be defined as a measure of the extent to which an element in the possibility space, either by human or natural action, and on the basis of knowledge of the future, is believed to be actualized by decision-choice action through the substitution-transformation process in the dynamics of actualpotential duality in order to become an element in the space of the actual.

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1 Fuzzy Rationality, Uncertainty and Expectations

Some modifications of the definition of probability may be required in order to accommodate natural substitution transformation processes that are not dependent on human decision-choice action. The distinction between possibility and probability is made through the understanding of stages of decisionchoice actions in the substitution-transformation process involving the dynamics of actual-potential duality in addition to the nature of time-point knowledge that is used in assessing and justifying the degrees of present beliefs underlying possibility and probability respectively. The distinction allows us to speak of possibilistic belief and probabilistic belief. Similarly, the distinction that may be established between subjective belief (probability) and objective belief (probability) depends on the degree to which personal judgment and justification are allowed to enter into the process of constructing the probability indexes over a given possibility space as we consider decision-choice action in the substitution-transformation process involving the dynamics of actual-potential duality. There is an intuitive notion which is taken as immutable assumption or postulate of natural order in the construct of the spaces of possibility and probability. The intuitive notion is that cognitive agents have innate ability as well as capacity to rank objects by compare and contrast processes in the decisionchoice fields. This innate ability and capacity for ranking flows from rationality as cognitive attribute of decision-choice agents. The postulate is then expressed in terms of capacity of ranking of elements by degrees of belief over the potential elements that allow the cognitive construct of the possibility space. The postulate also allows an expression of second ranking by degrees of belief of the possible elements that allow the probability space to be constructed for the decision-choice activities within the same substitutiontransformation process. The point to be emphasized is that the constructions of both possibility and probability spaces are span by decision-choice actions that cannot avoid subjectivity. Such decision-choice activities may be rational or non-rational relative to the best human action relative to accumulated knowledge; in other words based on some rationality. The linguistic reasoning of establishing ranking implies the existence of some form of ranking capacity in our perception of the elements in the universal object set where such a conceived ranking may or may not lead to best decision-choice action (optimal rationality) conditional on the accumulated knowledge. As it has been argued, the accepted knowledge structure and the claim to objectivity are the result of collective agreement on the basis of socially accepted rules for discovering knowledge and verifying the validity of knowledge element. Individual subjectivity is assessed around this collectively or socially accepted principle in terms of the content of the subjective view.

1.1 Expectations and Rationality

17

Again, the emphasis here must be referenced to the proposition that the whole social enterprise of human actions and the path of history that emerges in life and knowledge sectors are decision-choice activities whose foundations rest on substitution-transformation process operating within the dynamics of actual-potential duality. The individual and collective belief is that this enterprise will take place on the path of the most efficient decision-choice process whose effective constraint is the capacity of cognitive agents to create and use knowledge. The condition of the most efficient path of cognitive action is optimal rationality which may be individual or collective. This is also consistent with Euler’s idea that nothing happens in this universal system without a sense of maximum or minimum. The epistemic structure that emerges to define fuzzy rationality relative to linguistic reasoning in decision-choice actions is such that there is a cognitive system that generates two interdependent functions and action function in the substitution-transformation process between the potential and the actual. First, there is cognitive function, ϕ1 ( i ) , that takes elements in the potential space, U , into the possibility space, P , on the basis of the accumulated present knowledge, K P . Secondly, there is another cognitive function ϕ2 ( i ) that takes the elements in the possibility space P , into the probability space, B , on the basis of knowledge about the future K F . Thirdly, there is an action function, A ( i ) or another cognitive function ϕ3 ( i ) that takes the best elements in the probability space, B into the space of the actual, A , on the basis of knowledge on probabilistic ranking over the elements in the possibility space. Both possibilistic and probabilistic beliefs have corresponding justification principle on the basis of knowledge that may be constructed as conditional beliefs. Let us take the possibility space as given and on this basis let H = {H1 , H 2 , , H i , H n −1 ,H n } be an n-dimensional set of probabilistic hypotheses defined over the possibility space and; let K F be the knowledge about the future that one has to justify the held probabilistic belief. Generally, the system is such that P ( H | K F ) is one’s degree of belief in hypothesis H , given the knowledge justification, K F where and if A is an event to be actualized relative to the hypotheses H then P ( A | H ∩ K F ) is the degree of probabilistic belief associated with actualization of A given that the hypothesis H is justified on the basis of knowledge structure K F and that A ∩ K F = ∅ . Note that the event A is not part of the knowledge set about the future. Now if we consider individual hypothesis Hi then P ( Hi | A ∩ K F ) is the degree of probabilistic belief in hypothesis Hi in supporting the outcome of event, A, given the justification K F . It may be noted that if {H1 , H 2 , , H i , H n −1 , H n } are different individual hypotheses with a common justification, K F and A is an event that has

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1 Fuzzy Rationality, Uncertainty and Expectations

already occurred such that {H1 , H 2 , , H i , H n −1 , H n } are different possible explanations that may be associated with the outcome A, then {P ( H1 | A ∩ K F ) , P ( H n | A ∩ K F )} is the distribution of degrees of probabilistic belief associated with hypothesis H i in explaining the event, A, on the basis of knowledge, K F . The Hi ’s may be interpreted as either conditions of explanatory hypothesis or prescriptive hypothesis. When the system is reversed for a given knowledge base K F then the degrees of our probabilistic belief shifts cognitively from hypothesis to events such that {P ( A | H1 ) , P ( A | H n )} constitutes probabilistic degree of belief associated with event A, occurring under various postulated hypothesis with a common justification support, K F . The relational structure among the universal object set, potential, possible, probable, optimal decision and choice may be presented in terms of pyramidal logic as in Figure 1.1.2.1.

Universal Object Set

Perceptive Actual

Cognitive Agent

SubstitutionTransformatio

Potential Space Expectations

Actual-Potential Duality

Decision

Probability

Choice

Possibility

Given Hypothesis

Impossibility

Fig. 1.1.2.1. Relational Structure among the Universal Object Set, Potential Space, Possibility Space, Probability Space, Expectations, Decisions and Choice

1.1 Expectations and Rationality

19

It may be noted that given the common justification conditions K F , the possibility space must meet order conditions for comparability, consistency and asymmetry. In other words, for all ( ∀i and j; i ≠ j ) it must be the case that either P ( Hi | A ) ≥ P ( H j | A ) , or P ( Hi | A ) ≤ P ( H j | A ) , or P ( Hi | A ) =P ( H j | A ) in the case where the degree of probabilistic belief in explanation, Hi and H j ex post event A or in prescription Hi and H j ex ante event, A. Here different explanatory hypothesis about event, A, that has occurred or prescriptive hypothesis about event, A, that may occur are ordered by the probabilistic degree of belief where {P ( H 2 | A ) ≥ P ( H1 | A )} implies that hypothesis H 2 is believed to explain A better than hypothesis H1 explains A in the case of explanatory construct or the event A, is better believed to occur in conditions, H 2 than in H1 Similarly, P ( A | H i ) P ( A | H j ) holds in the case where the degree of belief is about actualizing an event, A under different prescriptive hypothesis or a set of conditions. Here the degrees of probabilistic belief in A occurring under different conditions become the focus under the same knowledge set, K F . The former justified belief structure involves the conditions of explanation after actualization. The latter justified belief involves the possible actualization of event, A in a given explanation or conditions. In both cases, the decision-choice problem is defined over the hypothesis or condition, H i given the conditions of justification embodied in K F . It must be pointed out at this juncture that the construct and the use of the justification support of the belief system define conditions of decision-choice rationality. An examination has been undertaken on the epistemic meaning of probabilistic belief and degrees of probabilistic belief associated with a single event, A, and a set of multiple differential conditions or hypotheses. This is not the only case in decision-choice analysis as seen in substitution-transformation process in the dynamics of actual-potential duality. There are some cases in the decision-choice process where we have a defined set of conditions or hypothesis, H, and multiple events {A1 , A 2 , , A i , A m −1 , A m } to be actualized by decision-choice action. Just like the previous structure, two analytical situations are opened to us. One of them is to examine the degrees of probabilistic belief associated with different events in the possibility space under the given and common hypothesis, H, and the justification condition K F . Symbolically, this may be presented as P ( A i | H, i ∈ I ) where I is an index set of events. The problem of interpretation is simply which of the events has the highest degree of probabilistic belief to be actualized in a given condition or hypothesis, H at a justification, K F . The conceptual problem may be switched to give an alternative interpretation where we examine the degrees

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1 Fuzzy Rationality, Uncertainty and Expectations

of probabilistic belief associated with hypothesis H in supporting the given occurrences of alternative events {A1 , A 2 , , A i , A m −1 , A m } in the possibility space with justification, K F . Symbolically this may be presented as P ( H | Ai , K F , i ∈ I ) . The conceptual problem associated with P ( H | A i , K F , i ∈ I ) may be interpreted as what is the degree of probabilistic belief associated with hypothesis H, in explaining the outcome of each event in {A i , i ∈ I} . This is an explanatory notion of degree of probabilistic belief under justification support K F . The rationality involves a decision-choice action on the outcome that is best explained by hypothesis, H. Alternatively, P ( H | A i , K F , i ∈ I ) may be interpreted as: given events {A i , i ∈ I} in the possibility space what are the degrees of probabilistic belief associated with hypothesis, H that will motivate decisionchoice action on one of the stated events in {A i , i ∈ I} . This is the prescriptive notion of degree of probabilistic belief under justification support K F . In both cases of explanatory and prescriptive notions of degrees of probabilistic belief, the decision-choice action is on {A i , i ∈ I} and the rationality is on the event {A i , i ∈ I} that has the maximum degree of probabilistic belief in the case of benefit and minimum degree of probabilistic belief in the case of cost. The whole discussion of probability in terms of degrees of belief in the substitution-transformation process in the dynamics of actual-potential duality involves rationality in two areas of decision-choice actions. One area is decision-choice action regarding the best knowledge construct (maximum knowledge) for decision-choice action. Given the maximum knowledge structure, the decision-choice action is exercised to obtain either the event that has maximum degree of probabilistic belief for actualization in a defined environment in the case of prescriptive science, or maximum degree of belief in the hypothesis in explaining events that have already occurred in the case of one hypothesis in explanatory science. In the case of multiple hypotheses and one event the process of rationality is a decision-choice action on the hypothesis that has the highest degree of probabilistic belief as an explanatory hypothesis when explanatory science is the focus. When prescriptive science is under consideration, rationality requires a decision-choice action on an environment or hypothesis that will support the actualization of the event under consideration. In all these cases, the epistemic calculus of probability derives its importance from what is not known, that is the future history of outcomes whether the focus is on explanatory or prescriptive science. The epistemic analysis of probability concerns future knowledge while the calculus of probability concerns the measurement of what is believed to be known (perceptive knowl-

1.1 Expectations and Rationality

21

edge) and what is believed not to be known (ignorance). The epistemics and calculus of degree of probabilistic belief concern the perceptive knowledge of expectations of future realizations that are linked to decision-choice action of the present irrespective of whether explanation or prescription is demanded by conditions of action on the basis of held knowledge at any decision-choice time. In this interpretive framework, zero degree of belief, that is P ( A i | H, i ∈ I ) = 0 as impossible event does not imply no knowledge. Rather, it implies that the element whose degree of probabilistic belief is zero does not belong to the possibility set under hypothesis, H. It, however, belongs to the set of elements in the potential space. It may however belong to another possibility space under a different hypothesis. Furthermore, the classical relative frequency approach to the analysis of probabilistic belief cannot necessarily be sustained in this interpretive framework. The degree of probabilistic belief is supported by justification principle constructed from a perceptive knowledge structure. Hence, the use of Bernoullian principle of insufficient reason, or Keynesian Principle of indifference, leading to the assumption of equally likely, is not admissible in the possibility space that is constructed on the basis of degrees of possibilistic belief. To admit it is to ask for a logical exception against the Principle of Sufficient Reason and hence Principle of Decision-Choice Rationality. To say that degrees of probabilistic belief arises from knowledge and hence a surrogate representation of perceptive knowledge is not to say that probability is perceptive knowledge. The epistemic analysis projects the notion that the structure of the universal reality is divided into actual and potential spaces by our perceptive knowledge. The potential space is induced into possibility and impossibility spaces (see Figure 1.1.2.1) by some logical construct around a given hypothesis. In this respect, the possibility space is logically closed in the sense that no event relative to the given hypothesis can become actualized if such event does not entirely belong to the possibility set. In other words, the probability distribution over the possibility space is derived on the Principle of Sufficient Reason on the basis of our perceptive knowledge but nothing else. The possibility space may be constructed on the principles of fuzzy optimal rationality that allows subjectivity, vagueness and quality to be introduced in the knowledge-decision-choice process. Without a detailed discussion, it may be pointed out that some level of initial perceptive knowledge, conditional on subjectivity, is required in order for interpretations of probability to be conceptually meaningful and operationally practical when one ascribes to the viewpoint of classical relative frequency approach. Here relative frequency as representing probability is defined as

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relative value of the number of favorable cases to the total number of equally likely cases in the possibility space. The definition requires knowledge of how the possibility space is constructed as well as the knowledge of number of favorable cases. Such knowledge cannot be assumed to be objective in terms of experimentation. As it has been argued and analytically presented in previous chapters, perceptive knowledge is subjective relative to objective knowledge as presented by the universal object set. Subjective knowledge becomes socially objective knowledge by collective agreement where propositions of perceptive knowledge are made parallel to propositions true of some aspects of objective knowledge induced by the corresponding characteristics of the universal object set. In this respect, the study of “data” (elements) in the space of the actual is nothing more than the study of how new subjective knowledge reduces uncertainty and hence alters the degree of probabilistic belief associated with elements in the possibility space that give rise to expectations under a defined rationality. Subjective knowledge is both experiential and abstract. It is in this respect that the concepts of subjective and objective probabilities acquire logical unity.

1.2 Rationality, Expectations and Uncertainty Given the explication of possibility and probability, the degree of probabilistic belief is a cognitive calculus concerning expectations of actualizing elements in the possibility space where such elements may be hypotheses or objects or others. The importance of the explication is its direct link of expectations to uncertainty and then to decision-choice process. Such a direct link shows itself as motivation to act on the present possibility that will bring about future outcome in the actual space. Motivation is a derivative of expectations which is a derivative of degree of probabilistic belief that is associated with elements in the possibility space with a justification based on knowledge about the future. This knowledge about the future for present decision-choice action may be acquired scientifically on the basis of optimal rationality or non-optimal rationality supported by accumulated knowledge. It may also be acquired nonscientifically. We shall assume that knowledge is scientifically acquired in accordance with methods and rules that are collectively and socially agreed upon. Within this epistemic framework, emphasis may be placed on methods of expectation formation required for current decision-choice action. For consistency of decision-choice behavior the method of expectation formation cannot deviate from the method of probabilistic belief formation. Expectation forma-

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23

tion is rational in the sense of best practice to the extent to which formation of degree of probabilistic belief is rational. The strength of probabilistic belief depends on the quality of perceptive knowledge about the future which is constrained by cognitive limitations on capacity to receive, process and interpret information as knowledge input in the present decision-choice activity. Now if expectations are cast in an axiomatic formal system (as it is done in many cases) that allows deduction or induction to be made by a set of admitted rules of logical inference in order to prove all the propositions claimed about expected behavior of values in the formal system, then such a system of expectations would be said to be logically complete. If the admitted set of propositions about expectations in the system is made parallel (isomorphic) to the set of admitted rules of degrees of probabilistic belief and if the set of the rules of inferring degrees of probabilistic belief is also made parallel (isomorphic) to the set of logical rules of accumulation of perceptive knowledge, and if the initial set of axioms of expectations is made to conform to initial perceptive knowledge while the set of logical rules of perceptive knowledge is made to conform with objective knowledge then the axiomatic system of expectations becomes derivable from perceptive knowledge which then constitutes its justification. Simply stated; expectations are derived from perceptive knowledge or lack there off which constitutes its support. The degree of fulfillment of justified expectations depends on the degree to which the elements in the subjective knowledge set correspond to their counterparts in the objective knowledge set. If the subjective knowledge set is constructed on the basis of rules of decision-choice rationality (that is, a selection of the best path for knowledge accumulation as has been discussed in [R17.23] and that such best path will have the highest ranking index that is consistent with optimal rationality) then by logical extension, expectation formation must be rational in such a way that expectation rationality will be isomorphic to decision-choice rationality. Viewed in terms of categories of reality, and substitution-transformation processes that are logically linked to decision-choice activities in the dynamic behavior of actual-potential duality, objective knowledge is held as the primary category of reality and subjective knowledge as cognitively categorial derivative of primary reality. Relative to decision-choice process, subjective knowledge is held as primary category of logical reality and degrees of probabilistic belief as a derived category of logical reality which is then held as primary category of logical reality from which expectation formations are categorial derivatives. If the knowledge construct is taken to be optimally rational (in the specific sense of best practice) then it must be the case that expectations constructed

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on the basis of accumulated knowledge as their justification support must be optimally rational by the principle of logical extension. This is one way of conceiving the concept of rational expectations that may come to influence the decision-choice process. The set of conditions of rationality is more or less made parallel or isomorphic to the set of conditions of best (maximum) use of knowledge in order to arrive at the best element in the possibility space (best goal). The degree to which such a best goal in the possibility space is actualized depends on cognitive capacity for explication and computability in general. This may be taking as an entry point in analysis of the structures of Muthian rational expectations hypothesis [R4], [R4.33], [R4.35], [R4.43], [R4.47] and Simonian bounded rationality hypothesis [R18], [R18.9], [R18.25], [R18.28] where expected values under conditions of uncertainties are obtained in terms of certainty equivalences by an acceptable logical inference. Both rational expectations and bounded rationality have no scientific meaning unless the concept of rationality is explicated in a manner that allows a logical move from its pre-scientific use to scientific term. Both concepts arise from decision-choice actions in substitution-transformation processes within the dynamic behavior of actual-potential duality. As such, both concepts find meaning around optimal rationality in the dynamics of actual-potential duality in a manner that allows decision-choice interactions between expectations and reality (actual) on one hand and the degree of probabilistic belief and knowledge on the other hand. Criticisms against optimal rationality on the basis of cognitive capacity constraint in all stages of decision-choice process do not invalidate the epistemic concept of optimal rationality in decision-choice actions. This is the futility of criticizing optimization principle in decision-choice rationality to which all decision-choice agents ascribe to. In fact, a properly formulated complete decision-choice theory must incorporate this cognitive capacity constraint in order to abstract an optimal solution. In other words, decision-choice theories developed without the inclusion of this cognitive capacity constraint are incomplete and hence fail to be optimally rational. The remedy to this theoretical difficulty is to initialize the construct of the decision-choice theories by assuming away the existence of cognitive-capacity constraint in addition to the assumption of a given knowledge structure devoid of vagueness. The relaxation of the two assumptions leads to decision-choice theories whose solutions establish cognitively constrained optimal rationality which has come to be known in the decision-choice theories as bounded rationality as advanced by Professor Simon [R18], [R18.27], [R18.28]. Thus any theory on bounded rationality must, by logical necessity, incorporate knowledge forma-

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25

tion and its use in order to be complete [R18.3], [R18.7], [R18.25]. Models of bounded rationality may thus be interpreted as models of constrained classical optimal rationality. The solutions to decision-choice problems that satisfy the conditions of bounded rationality fall under the classical sub-optimal rationality that is completely accommodated by fuzzy optimal rationality. To what extent then are the models of bounded rationality subsumed under models of fuzzy optimal rationality? In psychological approaches to the study of actions by decision-choice agents attempts are made to abandon the optimization postulate in favor of concepts like aspiration level or satisficing level or reasonable level, all of which seem to be covered by bounded rationality. Should we consider the content of these concepts as attributes of decision- choice agents thus defining the internal conditions of behavior or should we consider them as ideal state of decision-choice behavior thus defining the external conditions of rationality? What level should a decision-choice agent consider as satisficing level or aspiration level or reasonable level? Will these levels be greater than or lower than the optimal level? What are the conditions if any for these levels to be equal to the optimal level? Alternatively, should we consider aspiration level or satisficing level as rational that provides us with aspiration rationality or satisficing rationality? Are the aspiration rationality and satisficing rationality attributes of decision-choice agents or are they expressions of ideal decisionchoice state in expressing optimal rationality as an attribute of decision-choice agents in decision-choice problem-solving process? The concepts of satisficing level, aspiration level and reasonable level are vague that need some explication. On the basis of degrees of probabilistic belief in the likelihood of outcomes from the possibility space and in addition to their justification conditions, expectations are rationally formed in the sense of best path to knowledge about the future. The justification condition is constructed from the knowledge about the future which may be logically derived from the accumulated knowledge. Rational expectations in decision-choice process, therefore, may be viewed in terms of rational summary of the complexities of future knowledge about the behavior of the elements of future possibilities where such knowledge becomes an input into the present decision-choice action whose actualization is anticipated in the future. Expectations as summary knowledge about the future possibilities that become inputs into rational decision-choice process take claim to rationality by logic of parallel or isomorphic constuctionism or reductionism of the knowledge structure. The logical process of rationality establishes connections among critical points of decision-

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choice actions in the substitution-transformation process under the dynamics of actual-potential duality. The quality of the expectation, that is, the degree to which the expected event may be actualized or the expected deviations from the actual may be registered, will depend on the abstracted future knowledge. Such future knowledge may be forecasted on the basis of past and present knowledge. Thus forecasting is a rational knowledge transformation process that links the past to the present and to the future in order to define the knowledge about the future as the justification of expected outcome from the possibility space that is not contained in the available knowledge set. Here we must be careful in distinguishing between the forecasting of the future knowledge, K F that justifies expectation formation about an event E ∉ K F and the forecasting of event E. For example, in economics the future behavior of prices may be forecasted to obtain knowledge about the behavior of future prices. This knowledge becomes the justification condition for expected output that may be actualized in either production or consumption sector. Forecasting may be viewed as justified rational prediction of knowledge about the future on the basis of past and present knowledge while rational expectation is a justified prediction about future event on the basis of forecasted knowledge about the future. In this epistemic construct, expectations are rational in the sense that they are justified on the basis of maximum (best) available knowledge. If this maximum subjective knowledge is in strict relation with objective knowledge (that is, perfect foresight) and if the logic of inference is on the best scientific path then uncertainties will be reduced to zero and that an event which is to be expected will be actualized. To put another way, future outcomes are ordained and are independent of human cognition. Such future outcomes are unknown to cognitive agents because there is a lack of complete and exact information as well as computational and processing capacity of cognitive agents to arrive at the required knowledge about the future for forecasting outcome of future events. The knowledge about the future must be obtained in terms of best scientific practice. As discussed, optimal rationality in expectation formation is consistent with subjective view of probability since this subjective view is constructed from degrees of belief whose justification is a categorial derivative of subjective knowledge. The subjective knowledge is not only incomplete and vague but it is not isomorphic to the true knowledge as established by the information characteristics set of the elements of interest in the universal object set (objective reality that exists outside human cognition). This leads us to discuss the problems of classical sub-optimality seen from conditions of fuzziness.

2

Fuzzy Rationality and Classical Sub-optimal Rationality

In this Chapter attention will be shifted to some issues that have emerged in our discussions on rationality. This shift of attention will allow us an entry point into discussions on the epistemic nature of fuzzy rationality. In the previous discussions, the concepts of optimal rationality and sub-optimal rationality were introduced. Decision-choice rationality in the framework of classical paradigm was defined in terms of optimal behavior in all aspects of activities of cognitive agents. The deviations of this optimality were defined as suboptimal rationality. The concept of irrationality was bypassed since its explication, meaning and usefulness depend on those of the concept of rationality. It will be argued that irrationality and rationality constitute polarity (or extreme) of cases of decision-choice behavior between which deferential degrees of sub-optimal rationality fall. Such degrees of sub-optimality lend themselves to fuzzy topology and hence best handled in the framework of fuzzy paradigm composed of logic and corresponding mathematics.

2.1 The Relationship Between Fuzzy Optimal Rationality and Classical Sub-rationality The introduction of sub-optimal rationality allows us to deal with a wide range of decision-choice behavior including some in the literature such as bounded rationality, procedural rationality, aspiration level, satisficing level, substantive rationality and others. These qualifying rationalities are here defined in terms of degrees away from optimal rationality. It is useful to keep in mind that the concept of polarity is viewed in terms of dualities where the opposites mutually reside in each other through the resident dualities. This position points to the notion that the concept of irrationality is subjective by logical K.K. Dompere: Fuzzy Rationality, STUDFUZZ 237, pp. 27–49. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

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2 Fuzzy Rationality and Classical Sub-optimal Rationality

CLASSICAL OPTIMAL RATIONALITY: Distinguishing Assumption:Exactness, Absolutism in Knowledge Construct Logic: Classical Two-value Mathematics: - Classical Conclusions: Principle of Non-acceptance of Contradictions

BOUNDED RATIONALITY Distinguishing Assumption: Exactness, Human Capacity Limitations and Absolutism in Knowledge Construct Logic: Classical Two-value Mathematics: - Classical Conclusions: Principle of Nonacceptance of Contradictions

RATIONALITY AS AN IDEAL DECISIONCHOICE PROCESS

FUZZY OPTIMAL RATIONALITY Distinguishing Assumption:- Vagueness, QualityQuantity Duality ,Subjectivity-objective Duality, True-false Duality, Grades of truth and false in Knowledge Construct. Logic: Fuzzy Logic of infinite value Mathematics: Fuzzy Conclusions Principle of Acceptance of Contradiction

FUZZY OPTIMAL RATIONALITY

COGNITIVE ATTRIBUTE

IDEAL DECISIONCHOICE STATE

DecisionChoice System BOUNDED RATIONALITY

CLASSICAL OPTIMAL RATIONALITY

DECISION-CHOICE AGENTS

Fig. 2.1.1. Relational Structure of Rationalities, Optimality and Decision-Choice System and Properties

2.1 Fuzzy Optimal Rationality and Classical Sub-rationality

29

and linguistic construct in decision-choice process. In other words, irrationality and rationality are products of decision-choice action. The classical optimal rationality, the bounded rationality and fuzzy optimal rationality stands in pyramidal relation. It is epistemically useful to view bounded rationality to include aspiration level, satisficing level, reasonability and others. The relational structure is presented in Figure 2.1.1. It may be observed that much of the criticisms offered against optimal rationality involves criticisms of a) assumptions about values of decision-choice agents, b) assumptions about the goals and objectives held, c) assumptions about incentives and motivation that drive the decision-choice engine, d) assumptions about the computational and information processing capacity of decision agent and, e) assumptions about the knowledge structure of the decision-choice system that allows the decision-choice engine to function smoothly leading to the implementation of decision-choice actions. Some of these criticisms directly and indirectly extend to the framework of classical paradigm with its logic and mathematics. It is also the case that all other alternatives of theoretical decision-choice analysis offered in place of the optimal rationality share in common with optimal rationality the problems and limitations of classical paradigm of logic and its mathematics. This classical framework, we have pointed out, makes it difficult to deal with and integrate simultaneously qualitative, quantitative, objective and subjective phenomena. Within the classical paradigm of two-valued logical system, corresponding mathematics that project objectivity, what meanings must be given to behavioral concepts as aspiration level, satisficing level, reasonable decision and others? How do we represent these in the mathematical analysis of decision and how do they relate to optimal rationality? All these alternative approaches including bounded rationality require an explicit representation of the knowledge structure of the decision-choice module. Some of the discussions and representation of knowledge structure in modeling decision-choice process with bounded rationality may be found in a number of works including [R18], [R18.3], [R18.7], [R18.24]. Furthermore, these approaches may not be considered as a methodological departure from the classical paradigm on the basis of which optimal rationality derives its reasoning tools. The problem confronting the classical paradigm of logic and its mathematics in all decision-choice processes is simply how to account for the basic idea that vagueness, imprecision and linguistic approximations are the basic characteristics of subjective phenomena, explication, knowledge construction and nature of language as it is related to categories and linguistic reasoning in assessing truth and interpreting meaning. Language admits vagueness of verbal

30

2 Fuzzy Rationality and Classical Sub-optimal Rationality

meanings and the nature of linguistic reasoning by decision-choice agents admits of shades of meaning, truth, and approximations that operate through subjectivity. Shades of meaning and degrees of truth do not lend themselves easily, if at all, to analysis in the paradigm of classical logic and its mathematics. The classical epistemic paradigm fails to acknowledge the fundamental element that decision-choice process is essentially linguistic and subjective, and that social knowledge takes claim to objectivity by social consensus with reasoning toolbox that is collectively agreed upon. It is this classical epistemic paradigm that forms the legs on which optimal rationality stands. While the criticisms against optimal rationality are legitimate in their own right the alternatives offered are developed in the same logical framework and hence suffer from the problems and shortcomings of the classical paradigm. The shortcomings of the classical epistemic paradigm create logical barriers that make it difficult or perhaps impossible to efficiently deal with new theoretical and applied problems that arise within it in relation to frontiers of knowledge, decision-choice processes and sciences in general. It must be kept in mind that when problems are solved within a paradigm they give rise to new problems. Some of these new problems are so stubborn that their solutions must be sought at higher logical plane if we are to avoid chasing phantom problems. Some of these new stubborn problems in science of decisionchoice and techno-scientific processes are vagueness that emerges from explication, imprecision in explanatory, predictive and prescriptive conclusions, approximations in linguistic reasoning and others that constantly appear and are considered as anomalies which are dealt with by assumptions in order to retain the protective cognitive belt of the classical paradigm of logic and its mathematics which may be viewed generally in terms of set theory. To deal with these new problems a new epistemic paradigm that establishes a new set of conditions of rationality is required. The new paradigm must be able to provide solutions to already solved problems as well as unsolved problems in the classical paradigm. It should foster the development of theories and applications that integrate simultaneously behaviors of objective, subjective, quantitative and qualitative phenomena in a manner that incorporates the classical two-valued truth logical system as well as allow a rigorous mathematics to be constructed for general problem formulation, solution and analysis. It is at this juncture that the epistemic paradigm of fuzzy rationality enters into the whole theory of growth of knowledge and methodology of science. Just as classical rationality may be viewed in terms of crisp set theory, fuzzy rationality may be viewed in terms of fuzzy set theory. Its development must contain the classical crisp set theory. The category of its logic must also con-

2.1 Fuzzy Optimal Rationality and Classical Sub-rationality

31

tain the classical logic as a sub-category. In other words, for every definable sub-category of classical crisp set there must be a corresponding category of fuzzy set. However, there exist fuzzy sub-categories that do not have a corresponding classical sub-category. Alternatively stated, fuzzy sets contains the classical crisp set and hence fuzzy rationality encompasses classical optimal rationality and other subjective and vague decision-choice situations such as satisficing level, aspiration level and others. We may note that {0,1} ⊂ [ 0,1] , similarly ( 0 ,1) ⊂ [ 0,1] but {0,1} ⊄ ( 0,1) . The paradigm shift from classical crisp set to fuzzy set increases the number of degrees of freedom in knowledge creation from perceptive knowledge involving truth verification and acceptance on the basis of linguistic reasoning and scientific research into social and natural phenomena. The paradigm shifting involves two cognitive transformation functions ξ 1 of classical twovalued logic of knowledge creation and ξ 2 that is associated with fuzzy logic and its knowledge creation. The cognitive transformation functions, ξ1:A12 → A1 where A1 = {0 ,1} and ξ2 :A 22 → A 2 where A 2 = [ 0,1] . The classical paradigm offers us the analytical channels to study the behavior and properties of transformation functions ξ 1 . The fuzzy paradigm, on the other hand offers the channel to study the behavior and properties of transformation functions, ξ 2 that encompasses the behavior and properties of ξ 1 . The set A1 contains two elements which may be translated as true and false values or defective and non-defective and similar structures. The set A 2 contains infinite elements that may be interpreted as grades of truth and false where truth and false reside in the same element admitting the acceptance of contradiction. In the classical logical system truth and false cannot reside in the same element and hence contradiction is not allowed. The differences and similarities of the two paradigm may be illustrated by the introduction of membership functions µ A1 ( x ) and µ A2 ( x ) where A1 ∈ E and A 2 ∈ E with x ∈ E as a reference set. We can write the classical two-valued system as ⎧1 if x ∈ A1 or statement, x is true µ A1 ( x ) = ⎨ ⎩0 if x ∉ A1 or statement x is false

By fuzzy set representation we will write A1 = {( x,1) ,( x,0 )} as the classical truth set where µ A1 ( x ) = 1 and µ A1 ( x ) = 0 are degrees of truth embodied in the statement x and A1 is referred to as crisp set. If x is one statement or hypothesis then A1 is a singleton set whose degree of belonging to A1 is one. Given the reference set of independent and separate statements or hypothesis E = { x1 , x2 x n } then A1 is a set covering of true statements. Thus xi be-

32

2 Fuzzy Rationality and Classical Sub-optimal Rationality

longs to the set of true statements if µ A1 ( xi ) = 1 for all i' s while x j ' s, j ≠ i are false statements if µ A1 ( x j ) = 0 for all j' s . In terms of classical logic of set construction A1 = xi | µ A1 ( xi ) = 1, xi ∈ E and A1′ = xi | µ A / ( xi ) = 0, xi ∈ E . 1 On the basis of these two mutually exclusive sets, and under appropriate qualifications and data, all hypothesis can be verified or falsified or check for corroboration in sciences, logic, knowledge construction and mathematics where A1 ∩ A1′ = ∅ and A1 ∪ A1′ constitutes the universal system of knowledge by this classical logic. They also define the epistemic foundations of the classical rationality. Thus classical method of reasoning deprives us of linguistic approximations. It also closes the channels of logically integrating quality and quantity changes that are characteristics of the unity of universal system of substitution-transformation process in the dynamic behavior of quality-quantity duality. The distinguishing factor that allows continuity of categories, groups and sets to be established over the universal object set as well as allow transformation-substitution process to be traced in the evolutions of the emergence of the new from the potential and the disappearance of the old from the actual is taken away from the natural cognitive process. The effective study and linguistically scientific description of chemical interaction and changes, and the fading spectrum of color changes, for example, becomes problematic in our mathematical representation where problem resolutions are arbitrarily imposed. We may note that our currently established number system is continuous at all points by allowing the interplay of zero and infinity such that one whole number fades into the other in both directions as we travel between two whole numbers. If our number system is the foundation of our mathematical construct then the extension of this mathematical reasoning to other areas of thought must preserve its logic by consistency principle. Our mathematics must be isomorphic to the universal object set and our mathematical logic must asymptotically approximate the isomorphism of our linguistic reasoning. In other words “somewhat red” or “approximately red” in the color spectrum in mathematical representation must not be linguistically different from “somewhat two” or “approximately two”. In the classical logical system a color is either red or not red but not both or an element is either two or not two but not both. A problem arises within this classical logic when we seek knowledge about changes and their impacts on the observable (seeing, feeling, hearing and others) elements in the universal object set since every attempt is made to objectify subjective phenomena as well as quantify qualitative phenomena. The class of problems that may be dealt with in this classical paradigm is reduced to a small subset far below that which is within our cognitive

{

}

{

}

2.1 Fuzzy Optimal Rationality and Classical Sub-rationality

33

capacity in understanding elements in the universal object set, categories, categorial conversions and substitution-transformation processes. The paradigm of fuzzy logic and its corresponding mathematics seek to provide a larger logical system that contains the classical logical system in addition to many other linguistic reasoning about problems that fall outside the solution set of the classical paradigm. The fuzzy paradigm advances a cognitive framework that is intended to provide a logically consistent representation of linguistically constructed knowledge structure. The framework is advanced to allow the utilization of precise logical operators for problem formulations in imprecise, uncertain, vague, qualitative and complex environments that include subjective assessments as well as the development of mathematical and non-mathematical algorithms for abstracting precise solutions and exact conclusions with subjective qualifications as to the degrees of their acceptances or belonging. The analytical power of the fuzzy paradigm, thus, finds expression in the statement that given any theory in any knowledge category (sector) constructed from the classical logic as exact sets, one can fuzzify the knowledge category and all theories within it through the substitution-transformation process of knowledge acquisition by replacing all exact sets induced by classical logic by fuzzy sets induced by fuzzy logic. We may note that by this cognitive transformation, fuzzy categories are not only created but they become the primary categories of logical reality while the classical logical categories become derived categories by the method of reductionism. Alternatively, one may view the fuzzy logical category as an extension or amplification of classical category by the method of constructionism where the classical logical categories are ideal types. Linguistic reasoning is the foundation no matter how imprecise it is. The fuzzy logic is to humanize our abstract logical and mathematical reasoning toward an isomorphism with our linguistic reasoning. When one looks at our contemporary knowledge production under fuzzy scientific research program one finds at both basic and applied sectors of knowledge production, fuzzy arithmetic [R10.22], [R10.39], [R10.54], [R10.87] fuzzy set theory, [R10.55], [R10.63], [R10.72, fuzzy probability theory, [R7.27], [R10.42], [R11.15], [R11.28], fuzzy topology, [R10.10], [R10.21], [R10.38], [R10.98], [R10.124], [R10.125], [R10.126], fuzzy economic theory, [R7.5], [R7’11], [R7.25], [R7.31], [R7.60 ], fuzzy statistic and data analysis, [R11.2], [R11.3], [R11.6], [R11.30], [R11.46], fuzzy neural network, [R10.103], [R10.115], [R10.133], fuzzy optimal control theory [R10.12], R10.132], fuzzy optimization and mathematical programming [R9.25],

34

2 Fuzzy Rationality and Classical Sub-optimal Rationality

[R`9.31], [R9.33], [R9.41], [R9.38] and others. Additionally, possibility theory is advanced [R10.130], [R11.28], [R11.43] where such a theory provides a linguistic reasoning for understanding possibility set in probability theory and dynamics of evolution where one element in one category is naturally or socially transformed to another category through categorial conversion. The gains in this paradigm shifting leading to fuzzifications of concepts, measurements and logic is simply ability to handle solved problems in the classical paradigm in addition to exploiting the best inclusion of imprecision, vagueness, linguistic approximations, subjectivity, quality, linguistic magnitudes and others in scientific works and general knowledge production. The additional gain is the drive to achieve logical isomorphism to human linguistic reasoning that prevents extreme explication. Thus the paradigm shifting draws its strength and justification from the basic difficulty characterizing the twovalued logic and corresponding mathematics of the classical paradigm on the basis of which much of our sciences and knowledge have been constructed at the expense of inadequacy to account for vagueness, subjectivity, quality, complexities and relational equations contained in our linguistic structure and perceptive knowledge about the universal object set. The whole paradigm shifting leads to the establishment of fuzzy rationality that contains or become more or less a fuzzy-set covering of the classical rationality that whose values occupy extremes in all areas of decision-choice processes.

2.2 Fuzzification, Defuzzification and Fuzzy Rationality To deal with the role of fuzzy rationality in decision-choice processes that present an important foundation of paradigm shifting in all areas of knowledge, growth of knowledge and scientific discovery, we need to understand the concepts of fuzzification, defuzzification and their cognitive principles relative to classical concept of exactness and its cognitive principles that impose extreme explication in moving from ordinary language, concepts and linguistic reasoning to those of sciences. 2.2.1

Fuzzification in General Fuzzy Rationality

A fuzzification is a logical process of integrating vagueness as broadly defined (including subjectivity, quality, imprecision and others) into the classical exact propositions and reasoning reflecting claims of objectivity and quantity in such a way as to closely account for true properties of our natural languages and approximate reasoning that characterize human cognitive behavior

2.2 Fuzzification, Defuzzification and Fuzzy Rationality

35

that also incorporates subjective and qualitative phenomena. The fuzzification leads to replacing the classical crisp set, category, or group with fuzzy set, fuzzy category and fuzzy group. The core idea of fuzzification is, thus, to develop a logical approach and corresponding mathematics that will assist the process of knowledge production in understanding human cognitive limitations and the uncertainties that are imposed on the outcomes of decisionchoice activities of substitution-transformation processes in the dynamics of actual-potential duality as our knowledge sectors increase in numbers through differentiation and mutation. The paradigm shift is to change the two-valued truth system of statements and propositions into multi-valued truth with fuzzy logical extension to infinitevalued truth system. The classical two-valued truth set {0 ,1} is now replaced with fuzzy infinite-valued truth set of the form [ 0,1] that allows linguistic expression such as almost true or almost false or it is somewhat red, to be analyzed with mathematical operators; as well as to give mathematical contents to linguistic numbers such as small, medium, large, and many other qualitative and quantitative statements that has hedges in terms of linguistic qualifiers and subjective statements (for extensive examples of hedges see [R 23.32]). Definition 2.2.1.1: Fuzzification The fuzzification is a cognitive process, approximate and linguistic reasoning that work through the introduction of fuzzy operator call the membership function, µ ( i ) , that operates on our concepts to extend the working mechanism of the classical crisp set generated by the classical two-valued truth logic to a new mechanism of fuzzy set that is generated by fuzzy infinite-valued truth logic. Instead of µ:{0,1} → {0,1} in the classical system we have µ:{0,1} → [ 0 ,1] or µ:[ 0,1] → [ 0 ,1] in the fuzzy system where zero and one are values, for example, for false and true respectively. In this case, the classical crisp set is expanded to fuzzy set for any classical statement or hypothesis or concept whose truth value is of decision-choice interest. On the basis of the fuzzy paradigm, the classical crisp set is always constructible by the method of approximate reasoning through fix-level set ( α − level set ) that allows inclusion of subjective judgment and differential expert decisions. In the fuzzy paradigm, a set and its complement assume different topological structures where the complement is constructed from the membership function such that if µ A ( x ) is the membership function of set A with x ∈ A then the set A′ is the complement of set A if its membership function is µ A′ ( x ) = 1 − µ A ( x ) . In other words, if A is a set of true statements or hy-

36

2 Fuzzy Rationality and Classical Sub-optimal Rationality

pothesis then its complement, A′ is the set of all false statements or hypothesis with a membership function defined as µ A′ ( x ) = 1 − µ A ( x ) for a given reference set E where x ∈ E . A certain epistemic understanding must be brought to link the construct of classical crisp set to that of fuzzy set and to show the differences and similarities of their topologies. We note that the reference set E may be the set of all statements or hypothesis or a set of validity or truth of a statement. If the reference set E is the set of truth-values of a given statement or hypothesis and A is a set of true statement then A is either a singleton set or null set and its complement A′ is either a null set or singleton set correspondingly in the classical paradigm. It may be noted that while the singleton set contains abstractly the null set, the null set does not contain the singleton set. In this respect, if A = 1 then µ A ( x ) = 1 and hence A′ = 0 ⇒ µ A ( x ) =0 and conversely; that is, if A′ = 1 then µ A′ ( x ) = 1 with A = 0 and µ A ( x ) = 0 . On the other hand if the reference set E is the set of statements or hypothesis and A is the set of true statements with the complement A′ as the set of false statements then in general A ≥ 0 and A′ ≥ 0 with extreme cases where if E ≠ ∅ and A = 0 then A′ ≥ 1 in the sense that all statements in E are false. Similarly, if A′ = 0 then A ≥ 1 relative to the reference set, E where in this case all statements in E are true. The membership function, however, retains the same structure where µ A ( x ) = 1 or 0 and µ A′ ( x ) = 0 or 1 or generally speaking µ A ( x ) ∈ {1,0} and µ A′ ( x ) ∈ {1,0} where µ A ∩ A′ ( x ) = µ A ( x ) ∧ µ A′ ( x ) = 0 with x ∈ E . This is the principle of non-acceptance of contradiction in classical logic and corresponding mathematics. The reasoning structures of these sets are, however, different in the case of the paradigm of fuzzy logic and its mathematics. For any single statement or hypothesis, x ∈ E whose truth-valued sets are A and A′ the membership functions are defined as µ A ( x ) ∈ [1,0] and µ A′ ( x ) = (1 − µ A ( x ) ) ∈ [1,0] . To preserve the properties of the crisp set the scalar cardinality, i of the fuzzy set is defined in terms of the sum of the membership values as A = ∑ µ A ( x ) and A′ = ∑ ⎡⎣1 − µ A′ ( x ) ⎤⎦ for a finite reference set E . The x∈A x∈A ′ ⎛A ⎞ relative cardinality i is then computed as A = ⎜ and likewise E ⎟⎠ ⎝ with the complement. For example, consider a finite reference set, E of ten hypotheses or statements and let A be the set of fuzzy truth statements and A′ a set of fuzzy false statements. Let E = {1, 2, 3, 4 , 5, 6 , 7 , 8, 9 , 10} represents ten hypotheses such that the fuzzy sets A and A′ appear in a schedule of the form

2.2 Fuzzification, Defuzzification and Fuzzy Rationality

37

Table 2.2.1. Schedule of Fuzzy Set, Its Complement, Union and Intersection A µA ( x ) 1 0.2 2 0.5

A′ µ A′ ( x ) 1 0.8 2 0.5

A ∪ A′ µ A ∪ A ′ ( x ) A ∩ A′ µ A ∩ A ′ ( x ) 1 0.8 1 0.2 2 0.5 2 0.5

3 4 5 6 7

0.8 1.0 0.7 0.3 0.0

3 4 5 6 7

0.2 0.0 0.3 0.7 1.0

3 4 5 6 7

0.8 1.0 0.7 0.7 1.0

3 4 5 6 7

0.2 0.0 0.3 0.3 0.0

8 9 10

0.0 0.0 0.0

8 9 10

1.0 1.0 1.0

8 9 10

1.0 1.0 1.0

8 9 10

0.0 0.0 0.0

The scalar cardinalities are A = 3.5 , A′ = 6.5 , A ∪ A′ = 8.5 and A ∩ A ′ = 1 .5 the relative cardinality is A = 0.35 , A′ = 0.65 , ′ ′ A ∪ A = 0.85 and A ∩ A = 0.15 . There are some important conceptual elements that must be observed within the fuzzy paradigm. We can no longer speak of non-acceptance of contradictions in our reasoning in both logic and mathematics since the intersections of the set, A and its complement, A′ is non-empty. This property of fuzzy paradigm allows us to study conflicts and contradictions as a normal process of our mathematical systems and the system of our linguistic reasoning. In the case of the fuzzy set construct the set ( A ∩ A′ ) has a membership function of the form µ( A ∩ A′) ( x ) = µ A ( x ) ∧ µ A′ ( x ) with A ∩ A′ = 1.5 and hence the complement of ( A ∩ A′ ) , ( A ∩ A′ )′ has a membership function of the form µ ( x ) where µ( A∩ A′)′ ( x ) = 1 − ⎡⎣µ A ( x ) ∧ µ A′ ( x )⎤⎦ and ( A ∩ A′)′ = 8.5 = ( A ∩ A ′)′ A ∪ A′ = 8.5 . The simple implications in the fuzzy logical system are that truth and false are admitted to exist in the same space and time as a natural process without the classical non-acceptance of contradiction. This cognitive idea is inconsistent with the traditionally accepted notion that two things cannot exist in the same space and time, particularly when the qualitative characteristics are the focus. The principle of non-acceptance of contradiction in classical logic and corresponding mathematics is disposed of as inconsistent with linguistic reasoning where, "( A and not A )" is a false statement. The

38

2 Fuzzy Rationality and Classical Sub-optimal Rationality

cognition of acceptance of contradiction in fuzzy logic and the corresponding mathematics is that "( A and not A )" contains some true elements as well as some false elements and hence ( A ∩ A′ ) ⊂ A and ( A ∩ A′ ) ⊂ A′ . This may be called the principle of acceptance of contradiction in human reasoning. The statement "( A and not A )" or "( A and A′ )" simultaneously contains some degree of false and some degree of truth. The principle of acceptance of contradiction in fuzzy logic and corresponding mathematics provides an important channel to rigorously study and understand decision-choice problems that result in sub-optimal rationality and maintained decision-choice behavior in disequilibrium states in economics and other sciences that fall into the domain of classical optimal rationality. The principles of non-acceptance of contradiction as a truth-valued statement among a set of true or false statements denies us the channels of mathematical understanding of internal self-motion induced by internal force generated by the dynamics of duality and polarity induced by conflicts of simultaneous existence of truth and false in the same statement or hypothesis. A logical extension of the principle of non-acceptance of contradiction in the classical paradigm may be made to the rejection of the concept of creative destruction where construction and destruction exist in the same entity. Here a problem arises as to the relational interactions of qualitative and quantitative categories, and similarly between categories of subjectivity and objectivity in substitution-transformation process, and hence characterization of gradual change, that may be allowed naturally and linguistically without artificial grafting and patchwork. Much of these epistemic problems and difficulties in the classical paradigm are resolved in the fuzzy paradigm where fuzzy logic and the corresponding mathematics offer us a rigorous analytical structure to the study and understanding of contradiction and qualitative motion governed by mathematical structure of change as an internal rational process in substitution-transformation activities that take place in the dynamics of actualpotential duality. 2.2.2

A Note on Real Number System, Fuzzy Number System and Fuzzy Rationality

Let us note that rationality requires comparability of decision-choice elements in a completely ordered decision-choice space which when defined in the realnumber space forms a lattice but not a complete lattice under minimum and maximum operators. To achieve a complete lattice, negative infinity ( −∞ ) and positive infinity ( +∞ ) are introduced to extend the real number system to ob-

2.2 Fuzzification, Defuzzification and Fuzzy Rationality

39

tain a complete lattice where the maximum or upper bound is ( +∞ ) and the minimum or lower bound is ( −∞ ) . Let the complete lattice of the real number system be R = { x | −∞ ≤ x ≤ +∞} . The ordering of the decision space, Ω must be made isomorphic to the real number system where such ordering is induced by some functional process that admits of continuity defined in terms of comparison of classical epsilon neighborhood set coverings, N ε ( x ) , of all decision-choice elements, or rational numbers, x in either Ω or R . In other words, if ∀x0 , y0 ∈ R then ∀ x ∈ N ε ( x0 ) and ∀y ∈ N ε ( y0 ) , x y and hence N ε ( x0 ) N ε ( y0 ) . This property of continuity of R = { x | −∞ ≤ x ≤ +∞} is a useful entry point into fuzzification of the real number system that will allow us to construct the fuzzy number system required for decision-choice analysis in qualitative and quantitative environment in all areas of knowledge production. Notice that both N ε ( x0 ) and N ε ( y0 ) are classical sets that in the real number system are basically intervals. Our number system is such that for every number, x0 ∈ R there is a classical epsilon-neighborhood set, N ε ( x0 ) which covers the number. This epsilonneighborhood N ε ( x0 ) is a classical set covering that must have its own structure, properties and rules of behavior that follow the dictates of the classical paradigm. It offers us another direction into the understanding of fuzzification of R = { x | −∞ ≤ x ≤ +∞} to obtain the fuzzy number system. The epsilonneighborhood set covering is induced by an arbitrary selection of epsilon that depends on subjectivity, other information and decision that are outside the real number system. This epsilon-neighborhood set covering is externalized from R = { x | −∞ ≤ x ≤ +∞} and not easily amendable to deal with a number of linguistic quantities such as big, tall, fat, medium small or qualitative characteristics that are produced by hedges in linguistic reasoning and the study of meaning. The fuzzification allows us to internalize the classical epsilonneighborhood set covering with fuzzy set covering to obtain a fuzzy number system, e , consistent with fuzzy logic by explication of the classical epsilonneighborhood set. Here an epistemic definition of a fuzzy number is requires. Definition 2.2.2.1: Fuzzy Number and Fuzzy Number System A fuzzy covering of a classical number x0 ∈ R is a fuzzy set which is a fuzzy number X ∈ e formed by a membership characteristic function (or operator), µX ( x ) ∈ [ 0,1] , that specifies the degree to which any classical number x ≠ x0 may belong to X ∈ e relative to x0 ∈ R . We may note that every fuzzy number X ∈ e appears as a schedule of values of classical numbers x relative to x0 ∈ R and corresponding degrees of belonging µX ( x ) ∈ [ 0 , 1] . Furthermore if {X i | i ∈ I ∞ } ∈ e then e is a

40

2 Fuzzy Rationality and Classical Sub-optimal Rationality

power set which is a family of fuzzy sets where

∩ {X

i ∈I

∪ {X

i

| i ∈ I ∞} = e ,

i ∈I∞

i

| i ∈ I ∞ } ≠ ∅ and X i ∩ X i +1 ≠ ∅ . The fuzzy number may be

∞

viewed as epsilon neighborhood set such that the fuzzy number system is a fuzzy system’s covering of the classical number system where R ⊂ ∪ {X i | i ∈ I ∞ } = e (for alternative definition of fuzzy numbers i ∈I

∞

see [R10.55], [R11.27]. To deal with rationality in the fuzzy number system it becomes necessary to establish the structure and rules of operations (addition, subtraction, multiplication and division) on classical numbers. This need has given rise to fuzzy arithmetic [R10.52], [R10.8] fuzzy topology, fuzzy probability and other theories of fuzzy phenomena [R11], [R11.16], [R11.43], that we have already alluded to. The fuzzy rationality constructed from fuzzy logic and its corresponding mathematics is a fuzzy-set covering for classical optimal rationality constructed from classical logic and its corresponding mathematics. 2.2.3

Defuzzification, Fuzzy Rationality and Classical Rationality

From the fuzzy real number system to the classical real number system we observe that every value of degree of belonging as expressed by the membership characteristic function has a corresponding classical real number. In other words every fuzzy number has a set of classical numbers and a corresponding set of degrees of belonging that together define the uniqueness of the fuzzy number. Thus knowing the value of the membership characteristic function relative to its support allows us to find the corresponding real number. The analytical structure points to the idea that since the classical real number system is contained in the fuzzy real number system by the method of constructionism we can defuzzify the fuzzy numbers to obtain the classical real numbers by the method of reductionism. The process is called defuzzification. Definition 2.2.3.1: Defuzzification Defuzzification is a cognitive process, supported by approximate and linguistic reasoning methods in addition to fuzzy algorithms, that works through the introduction of inverse operator of the membership characteristic function µ ( x ) ∈ [ 0,1] and fuzzy infinite-valued truth logic that operate on our fuzzy and linguistic concepts to reduce them into the classical two-valued truth logic of reasoning to obtain solutions of crisp-value equivalences, x* with a clearly defined membership values of the form α = µ ( x* ) ∈ ( 0 ,1) and

2.2 Fuzzification, Defuzzification and Fuzzy Rationality

41

where by the method of fixed-level set we obtain µ ( x* ) ∈ {0,1} ⊂ [ 0,1] as a solution with fuzzy optimal rationality. The epistemic implication for decision-choice rationality is that the problems of classical optimal rationality and sub-optimal rationality can be logically extended to fuzzy optimal rationality that contains bounded rationality, aspiration level, satisficing level and others and where crisp solutions to these fuzzy problems can be abstracted by methods of fuzzy algorithms and soft computing [R10.36], [R10.40], [R10.52], [R10.65]. We immediately observe that fuzzy optimal rationality is a fuzzy set covering of both classical optimal and sub-optimal rationalities. Thus if P is a set of problems of classical optimal rationality, Q the set of problems of classical sub-optimality and c the set of problems of optimal fuzzy rationality then ( P ∪ Q ) ⊆ c such that ( P ∩ Q ) = ∅ . Thus the fuzzy optimal decision space is partitioned by the classical rationality into optimal and sub-optimal behavior at a given knowledge set, K . As presented every decision-choice problem has its fuzzy covering that allows us to study a wide range of decision-choice behavior of cognitive agents in both explanatory and prescriptive sciences. All decisionchoice behavior can be related to goals and constraints considered in either the classical or fuzzy logical systems, all of which may be presented in terms of membership characteristic functions by methods of fuzzification and defuzzification. Decision-choice activities involve goals and constraints. Let us suppose that A is the set of goals and B the set of constraints on A relative to a given knowledge set K and a set of personality characteristics P of the decisionchoice agents then the optimal rationality in general is obtained by solving a set equation of max x ∈ A , st. x ∈ B . In terms of fuzzification through the membership functions with a given reference set E , we write the classical optimal decision system as:

⎧1, if x ∈ A ⊂ E µA ( x ) = ⎨ ⎩0, if x ∉ A ⊂ E ⎧1, if x ∈ B ⊂ E µB ( x ) = ⎨ ⎩0, if x ∉ B ⊂ E and hence in the classical system the optimal element is obtained as

⎧⎪1, if x ∈ ( A ∩ B ) ⊂ E µA∩B ( x ) = ⎨ , ⎪⎩0, if x ∉ ( A ∩ B ) ⊂ E

42

2 Fuzzy Rationality and Classical Sub-optimal Rationality

which may simply be written in a fuzzified structure constituting the fuzzy decision problem as: µ( A ∩ B) ( x ) = µ A ( x ) ∧ µ B ( x ) , x ∈ ( A ∩ B ) ⊂ E .

The defuzzification process requires us to construct algorithms to find an optimal element of decision-choice problem that satisfies the goal-constraint configuration of ( A ∩ B ) ⊂ E . The search for the optimal element in ∆ = ( A ∩ B ) proceeds by maximizing the membership function µ A ( x ) in a negative functional set configuration or minimizing µ A ( x ) in a positive functional set configuration as a dual in the form max µ( A ∩ B ) ( x ) x∈E

s.t. ⎡⎣µ A ( x ) − µ B ( x ) ⎤⎦ ≤ 0 The dual decision problem may be written as: min µ A ( x ) x∈E

s.t. ⎡⎣µ A ( x ) − µ B ( x ) ⎤⎦ ≥ 0 The need for algorithms to abstract the solution constitutes the motivation for the development of fuzzy optimization or fuzzy mathematical programming that allows defuzzification of the fuzzified decision-choice problem in order to obtain crisp-value equivalences of fuzzy optimal rationality that may be different from crisp value of classical optimal rationality.The value of fuzzy optimal rationality appears as a pair in the form ⎡⎣ x* ∈ A ∩ B, µ A ∩ B ( x* ) ∈ [ 0 , 1]⎤⎦ In the case of the classical paradigm the solution to the problem of the optimal rationality in terms of fuzzy representation is simply ⎡⎣ x* , µ A ∩ B ( x* ) = 1⎤⎦ . Here discrete mathematics and optimization become useful tools depending on further problem complications that may require the use of mathematics of continuous process. The value µ A ∩ B ( x* ) = 1 may be interpreted in many epistemic settings that fit the classical paradigm depending on the problem and the knowledge sector of relevance. It may represent any of the following such as either complete exactness or objectively sure or optimally complete or lack of approximation or absence of qualitative element and lack of many other computational and conceptual vagueness that may caste doubt on optimal rationality. The implication from the classical paradigm is that the degree of acceptance of x* ∈ ∆ = ( A ∩ B ) as defining the optimal decision-choice rationality

2.2 Fuzzification, Defuzzification and Fuzzy Rationality

43

is perfect without question where, µ A ∩ B ( x* ) = 1 is the degree of acceptance or exactness. Decision-choice elements of suboptimal rationality are computationally not admissible in the classical setting. This excludes channels that may allow us to examine and analyze decision-choice problems whose solutions may meet conditions of satisficing and aspiration levels. The shifting of the classical paradigm to fuzzy paradigm leads importantly to solutions of decision-choice problems of the form x*, µ( A ∩ B ) ( x* ) ∈ [ 0 ,1] that may represent all possible solutions to problems of optimal and suboptimal rationality where x* ∈ A ∩ B , is a decision-choice solution that satisfies the fuzzy optimal rationality in the sense that it satisfies the optimal element in the goal-constraint configuration with a defined degree of belonging to the set of optimal solution µ( A ∩ B) ( x* ) = α ∈ [ 0,1] . When µ A ∩ B ( x* ) = α = 1 we obtain solutions to decision-choice problems that meet the requirement of classical optimal rationality with full confidence that x* is an exact optimal value. The solution to the decision-choice problem falls outside the fuzzy optimal rationality if µ A ∩ B ( x* ) = α = 0 and hence x* ∉ A ∩ B . In general, therefore, 0 ≤ µ A ∩ B ( x* ) ≤ 1 for all decision-choice problems defined within the fuzzy paradigm. Thus the fuzzy optimal rationality incorporates the classical optimal rationality with general solution set, S , defined as:

{

{

S=

{( x*,µ

∆

}

}

( x* ) ) | x* ∈ A ∩ B, µ A ( x* ) = µ B ( x* ) = µ ∆ ( x* ) = α* ∈ [ 0,1] }

KNOWLEDGE SPACE

CRISP VALUE EQUIVALENCE

DEFUZZIFI CATION PROCESS

INFORMATION SPACE

ANALYSIS OF CONDITIONS OF FUZZINESS

FUZZY ALGORHYTHMS

FUZZY MODLING MODULES

FUZZIFICATION PROCESS

FUZZY SET EQUIVALENCE

MODULE OF EXAMINATION OF FUZZY LOGICAL CONSISTENCY

VALIDITY PROCESS

Fig. 2.2.3.1. Geometry of General Fuzzification and Defuzzification Process for the Study of Fuzzy Optimal Rationality in Classical and Fuzzy Logical Spaces

44

2 Fuzzy Rationality and Classical Sub-optimal Rationality

The value { α* ∈ [ 0 ,1]} may be interpreted as optimal degree of aspiration, or satisficing or closeness to the classical ideal where x* = µ −1( A ∩ B ) ( α * ) − µ −A1 ( α * ) = µ −B1 ( α * ) . If x** is the value of the classical optimal rationality with µ A ∩ B ( x** ) = 1 and x* is the value of fuzzy optimal value with µ A ∩ B ( x* ) = α* ∈ ( 0,1] then the difference between solutions of the classical optimal rationality and fuzzy optimal rationality is µ −A1∩ E (1) − µ A−1∩ B ( α * )

{

}

{

}

The difference relative to the value of classical optimal rationality is called relative classical optimal rationality ratio which is computed as V ( x**,x* ) = min ⎡⎣ µ A−1∩ E (1) − µ A−1∩ B ( α * ) µ A−1∩ E (1) ⎤⎦ . The relative fuzzy x∈E optimality ratio is the difference relative to the value of fuzzy optimal rationality which may in turn be specified as Y ( x**,x* ) = −1 −1 −1 min µ A ∩ E (1) − µ A ∩ B ( α * ) µ A ∩ B ( α * ) . These two relative values may offer x∈E us channels of studying properties of contradictions in general (for further discussions see [R17.24]).

{

}{

{

}{

KNOWLEDGE SPACE OF MATHEMATICA L SYSTEM

}

CRISP VALUE EQUIVALENCE OF

}

FUZZY LOGICAL OPERATORS

DEFUZZIFICATION PROCESS OF FUZZY NUMBERS,

e

e

GENERAL MATHEMATICAL SPACE

FUZZY ARITHMATIC AND FUZZY ALGEBRA ON,

e

ANALYSIS OF CONDITIONS OF FUZZINESS IN THE REAL NUMBERS

R

FUZZIFICATION PROCESS OF THE

FUZZY EQUIVALENCE: THE FUZZY NUMBER

REAL,

SYSTEM,

R

e

EXAMINATION MODULE OF FUZZY LOGICAL CONSISTENCY

LOGICAL VALIDITY PROCESS

Fig. 2.2.3.2. Fuzzification and Defuzzification of Classical and Fuzzy Number Systems for Cognitive Rationality in Classical and Fuzzy Logical Spaces

2.3 Fuzzy Optimal Rationality, Contradiction and Cost-Benefit Rationality

45

2.3 Fuzzy Optimal Rationality, Contradiction and Cost-Benefit Rationality It was pointed out in the previous section that the concept and the measure of relative fuzzy optimality that emerged from comparative analysis of fuzzy optimal rationality and classical optimal rationality offer us decision-choice logic to an epistemic study of the theory of contradictions. This theory of contradictions will reveal the structure and analytical importance of cost-benefit rationality as a general intelligence of decision-choice behavior and the theory of cost-benefit analysis as o general theory of decision-choice behavior. Here emerges the classical cost-benefit rationality [R2], [R3] as a generalized approach to classical optimal rationality and fuzzy cost-benefit rationality as a generalized approach to fuzzy optimal rationality that contains the classical optimal rationality in the analysis of decision-choice activities [R728], [R7.29], [R7.32], [R7.33], [R7.35], [R7.36]. 2.3.1

Fuzzy Optimal Rationality and the Theory of Contradiction

In terms of theory of contradiction, since from any fuzzy true statements or hypotheses about a given explanatory or prescriptive event {A , µ A ( i ) ∈ [ 0 ,1]} in a given reference set E we can derive its complement, fuzzy false statement as A′, 1 − µ A′ ( i ) ∈ [ 0,1] where {( A ∩ A′ ) , µ A ∩ A′ ( i ) ∈ [ 0 ,1]} is the fuzzy contradiction in both fuzzy true and false statements which is non-empty. The membership characteristic function is computed as µ A ∩ A′ ( i ) = µ ∆ ( i ) = µ A ( i ) ∧ µ A′ ( i ) where ∆ = A ∩ A′ . The fuzzy set of contradictions constitutes the fuzzy decision-choice set. The decision then is to select a hypothesis that minimizes the contradiction in both false and true statements. The decisionchoice problem may be formulated as find the maximum fuzzy true hypothesis subject to a set of fuzzy false hypothesis. This can be written as:

{

}

max µ( A ∩ A′) ( x ) x∈E

s.t µ A ( x ) − µ A′ ( x ) ≤ 0

The solution to this problem appears as a pair of the form

{( x*,µ

∆

( x* ) ) | x* ∈ ∆ = A ∩ A′ ⊂ E , µA ( x* ) = µA′ ( x* ) = µ ∆ ( x* ) = α* ∈ [0,1] }

The existence of the contradiction arises from the basic character of linguistic reasoning and approximation, or from the presence of vagueness or from the

46

2 Fuzzy Rationality and Classical Sub-optimal Rationality

principle of simultaneous existence of characteristics of truth and false in the same space and that statements and hypothesis are not completely false or true as seen from terms in our languages and linguistic reasoning. This notion of contradiction is the result of the nature of the fuzzy paradigm involving fuzzy logic and fuzzy mathematics in the construct of knowledge set. When our social knowledge structure is properly constructed, it will contain knowledge elements that are partially true and partially false and where these elements have the lowest degree of contradiction as a criterion of acceptance. This criterion of acceptance is constructed from the method of fuzzy optimal rationality as a general criterion for decision-choice activities in substitutiontransformation process that arises from the dynamics of actual-potential duality. The substitution-transformation process in nature and society derives its energy from the presence of contradiction in truth-false duality where truthfalse concept is an epistemic representation of opposites (for extensive discussion on duality and opposite see [R20.20]). Contradiction is an integral part of universal dynamics without which change and stability are impossible. Our construct of logic and mathematics must therefore reflect this important property of our universe and knowledge construction. This important property is revealed in fuzzy paradigm where fuzzy logic and mathematics offer us fuzzy toolbox for its study, analysis and synthesis. The theory of contradiction in the knowledge production suggests to us that the enterprise of knowledge production is about truth-false decision-choice balances. This suggests that on the road to knowledge discovery we must study the dynamics of contradiction in truth-false duality that governs the knowledge substitution-transformation processes. The knowledge enterprise and scientific activities must not be guided by the ideology of absolutism of truth-false elements characteristic of classical optimal rationality. They must be guided by fuzzy optimal rationality that is consistent with the behavior of nature and society where qualitative and quantitative motions and time are permanent elements to ensure natural and social continuities. 2.3.2

Cost-Benefit Rationality and Fuzzy Optimal Rationality

How does fuzzy optimal rationality and contradiction relate to cost-benefit rationality? To answer this question we observe that the principle of cost-benefit analysis is based on the acceptance of the ideas that in every decision-choice element reside simultaneously cost and benefit components. The presence of cost and benefit characteristics in the same element shows itself as contradiction. This puts cost-benefit analysis into the epistemic notions of polarity and

2.3 Fuzzy Optimal Rationality, Contradiction and Cost-Benefit Rationality

47

duality with contradiction produced by conflicts between costs and benefits that must be carefully balanced to assist in the decision-choice activity. Thus costbenefit analysis takes its decision-choice logic, therefore, from dynamics of cost-benefit duality. The theory of cost-benefit analysis is thus the development of cost and benefit information and algorithms that will allow best decisionchoice activity to be undertaken on the basis of the cost and benefit balances. Another way of looking at cost-benefit rationality is the natural proposition that every substitution-transformation process involves costs and benefits. Thus in decision-choice processes every cost has its benefit support and every benefit has its cost constraint. The cost in real terms is the resource commitment (broadly defined) and the benefit is the realized output. In economic measurements, the cost to a decision-choice action is seen in terms of benefit forgone by not selecting the next best decision-choice action. Given the benefit and cost information the cost-benefit rationality involves an optimization procedure to abstract the best the best element in the decision-choice space. If we suppose that Ai is a set of benefit characteristics then A i′ is the set of cost characteristics associated with ith-alternative. The decision-choice problem can be examined from the set of cost-benefit configuration of each alternative defined as ∆ i = A i ∩ A′i which defines the cost-benefit contradiction in the cost-benefit duality for each alternative. By this interpretation all decision-choice alternatives will be ranked in terms of an index of costbenefit contradiction. The cost-benefit rationality is to select the alternative with minimum index of contradiction. The index and the computation of the level of contradiction may proceed in terms of membership of characteristic function of belonging. In other words, we must combine µ Ai ( x ) , µ Ai′ ( x ) and µ Ai ∩ Ai′ ( x ) to obtain the cost-benefit index of balance, xi * ∈ E with fuzzy index of contradiction αi * = µ ∆ ( xi * ) where the general reference set of cost and benefit characteristics is E ⊇ ( A ∪ A′ ) . The values ( xi *,α i * ) are obtained by solving a fuzzy mathematical programming problem of the form max µ( A ∩ A′) ( x ) s.t. µ A ( x ) − µ A′ ( x ) ≤ 0 . If the index set of alternative is I and x∈E

A* = {αi | i ∈ I} then the fuzzy optimal cost-benefit rationality is defined as

( )

∨ α i . If the maximum value, α i = 1 , then we obtain the classical optimal i∈I cost-benefit rationality [R7.35], [R7.36]. The cost-benefit rationality holds for both static and dynamic decision-choice problems in either classical or fuzzy paradigm. The important epistemic point that we would like to emphasize is that all decision-choice problems in human endeavors are driven by costbenefit balances and that all decision-choice rationalities are reducible to costbenefit rationality whether the cost and benefits are measurable or not.

48

2 Fuzzy Rationality and Classical Sub-optimal Rationality

It is precisely the limitativenes or limitationality of human capacity to correctly identify measures and calculate the costs and benefits associated with different decision-choice alternatives that leads to objections often raised against optimal rationality. As we have discussed in a different setting, the objections are not sustainable within the fuzzy paradigm. The fuzzy α − level optimality and its closeness relationship to classical optimality allow any decision-choice criterion that represents ranking behavior to be related to fuzzy optimal rationality and classical optimal rationality. The point is simple in that given { $20, $60, $100}as three alternative, each of which is a benefit over cost, how many decision choice agents will select any benefit that is less than $100? Similarly if { $20, $60, $100}is cost over benefit, how many decisionchoice agents will select any cost greater than $20? The fact that there are constraints and limitations on cognitive capacity of decision-choice agents to correctly reach the value $100 in case of benefit and $20 in the case of cost can not constitute a legitimate criticism and hence rejection of optimal rationality in behavior of decision-choice agents. The idea of optimal rationality in the epistemic setting is that increasing benefits over costs and decreasing costs over benefits are always preferred and will always be selected by decision-choice agents. The maximum and minimum (optimum) in this continuum of decision-choice outcomes may be taken as a postulate of existence in the meta-theoretic analysis even if we do not know them. This is another way of connecting to the Euler’s statement that “nothing happens in the universe that does not have a sense of either certain maximum or minimum” [R14.75, p.1]. Decision-choice agents behave to reach the optimum and if they cannot reach the optimum as a result of knowledge constraints and capacity limitations on cognitive computations, they behave to reach the best which may be referred to as aspiration level or satisficing level whose degree of belonging to the set of optimum or the best one with an optimal α − value of the membership characteristic function that satisfy the real cost-benefit configuration in decision-choice processes. The knowledge constraint reveals itself in explications, vagueness, expectations and expectation formations about cost-benefit information sets. The capacity limitations on cognitive computation reveal themselves in cost-benefit information processing involving the establishments of correct benefit criterion index, cost constraints and cost-benefit imputations. It is these knowledge constraints, and limitations on cognitive computations that give rise to fuzzy optimal rationality. The classical optimal rationality is obtained in theory by assuming away knowledge constraint through the assumption of perfect and exact information

2.3 Fuzzy Optimal Rationality, Contradiction and Cost-Benefit Rationality

49

in addition to the assumption of perfect and exact computable cognitive skills. When this important assumption is relaxed and the theories about the classical rationality are presented with knowledge constraint and limitations of cognitive capacity then the bounded rationality is obtained in the same classical paradigm. The classical two-valued truth system is still in operation where the decision agent is either satisfied (truth-value = 1), or not satisfied (truth-value = 0) but not both. By shifting the classical paradigm to the fuzzy paradigm we extend the theory of the classical optimal rationality to fuzzy optimal rationality we obtain an optimal α * −level ∈ [ 0,1] of satisficing or aspiration for decision-choice agents. In this fuzzy optimal rationality the decision agent has both satisfaction and un-satisfaction in the same decision-choice situation with α − level of satisfaction and (1 − α ) − level of un-satisfaction which may be considered as the optimal fuzzy risk covering of the decision-choice activity whose main risk ρ ( x* ) may be computed as:

{

} ⎤⎥

⎡ (1 − α * ) µ −A1∩ E (1) − µ A−1∩ B ( α * ) ρ ( x* ) = min ⎢ x∈E α * µ A−1∩ E (1) ⎢⎣

{

}

⎥⎦

The point here is that every decision has a satisfactory characteristics (benefits) that make it attractive for choice, and unsatisfactory characteristics (costs) that make it unattractive for choice. Every decision-choice element is a duality. It is through this cost-benefit duality that decision and choice under risk is reflected in fuzzy optimal rationality (see an extensive discussions in a companion volume [R17.24]). This principle of simultaneity of cost-benefit characteristics must be reflected in optimal decision-choice rationality. The shifting to fuzzy paradigm allows us the channels to analyze duality and conflict as well as institute a stopping rule in computable models of bounded rationality or computable models of aspiration level in decisionchoice processes such that the stopping rules are not arbitrarily imposed but emerges from within the theory itself. Notice that the fuzzy risk vanishes when ( α = 1) signifying complete exactness in the information-knowledge structure and methods of reasoning. The measure of fuzzy risk is one component of the possible measure of total risk which is composed of fuzzy risk and stochastic risk whose implicit or explicit sum will be greater than each one of them. A theory of risk measurement must show us how the two are measure and combined.

3

Fuzzy Rationality, Ambiguity and Risk in Decision-Choice Process

In chapter One of this monograph we dealt with the nature of fuzzy optimal rationality as it relates to uncertainty and expectation formation of decisionchoice agents. This fuzzy rationality is also intended to resolve the problems of vagueness and ambiguity that produce contradiction. Total uncertainty in the decision-choice processes including knowledge construct was separated into probabilistic and possibilistic uncertainties. Both uncertainties relate to cognition. The probabilistic uncertainty is due to knowledge limitativeness and limitationality in the decision-choice process. The probability space is said to be limitational (limitative) if a knowledge production is necessary (sufficient) condition to reduce uncertainty and hence increase probabilistic belief of decision-choice outcome. The possibilistic uncertainty is due to the presence of vagueness (broadly defined) that includes blurredness and penumbral regions in knowledge, language and linguistic reasoning in the decisionchoice process. This knowledge constraint is due to the idea that in all decision-choice circumstances knowledge is incomplete and provisional, whose acceptance is based on socially acceptable methods of knowledge production. Expectation formation has traditionally been linked with stochastic uncertainty and our discussion on the subject maintained the traditional connection between expectation formation and probabilistic uncertainty. This allows us to discuss the cognitive relationship between probability and the measure of ignorance. In general the circumference of expectation formation must be span by the diameter of total uncertainty of both probabilistic and possibilistic nature. The mathematical and logical tradition is such that the possibilistic characteristics are neglected or done away by assumption in the sense that what we know is exact. The probabilistic uncertainty is analytically conceived in terms of behavior of elements in the possibility space that allows cognitive transformation to K.K. Dompere: Fuzzy Rationality, STUDFUZZ 237, pp. 51–82. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

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3 Fuzzy Rationality, Ambiguity and Risk in Decision-Choice Process

probabilistic space and hence logical analysis of expected outcome from the possible outcomes. The general analytical structure is such that the decisionchoice space may be partitioned into four categories by probabilistic and possibilistic uncertainties that define the conceptual system of decision-choice problems. Possibilistic uncertainty is related to fuzzy uncertainty and decision-choice problems while the probabilistic uncertainty is related to stochastic uncertainty and decision-choice problems. First classified category is nonfuzzy and non-stochastic decision-choice problems. There are no uncertainties in this class of decision-choice problems and there are no risks to contend with. This class of decision-choice problems may be viewed as an ideal category. The second category is the class of non- fuzzy and stochastic decisionchoice problems. The major characteristics are stochastic uncertainties that are involved with both the control and state variables. This class has stochastic risk but devoid of fuzzy risk. The third category is composed of a set of fuzzy and non-stochastic decision problems. The problems in this category are characterized by fuzzy knowledge in control and state variables as well as the methods of reasoning. The problems in this class have fuzzy risk but no measurable stochastic risk. The fourth category is composed of a set of problems characterized by fuzzy and stochastic elements in the sense of vagueness and incomplete knowledge about both the state and control variables that together constitute possibilistic-probabilistic uncertainty in the decision-choice process. The presence of possibilistic and probabilistic uncertainties amplifies the risk of decision-choice process. There are two types of interactive risks of fuzziness and stochasticity in terms of separability and non-separability. The simultaneous presence of fuzziness and stochasticity presents special difficulties and challenges in the use of logic and mathematical methods in decisionchoice theories. The classical optimal rationality deals with exact decision-choice problems in the non-fuzzy and non-stochastic in addition to non-fuzzy and stochastic environments. The principle of exactness in concept and cognition allows the utilization of the analytical toolbox of classical paradigm composed of its logic and corresponding mathematics. This analytical toolbox used in the classical optimal rationality offers us very little help, if any, when we are confronted with the fuzzy category of decision-choice problems composed of non-stochastic and fuzzy, and stochastic and fuzzy. We have discussed the epistemics and knowledge-creation role of the toolbox of classical optimal rationality in dealing with completely certainty conditions of the first kind and non-fuzzy and stochastic uncertainty of the second kind. The epistemic discussions on the usefulness of the toolbox of fuzzy paradigm as embodied in

3.1 Ambiguity, Probability and Decision

53

the fuzzy optimal rationality was conceptually applied to fuzzy and nonstochastic category of the decision-choice problems. In all, there is one category that we have not discussed. This category involves the simultaneous occurrence of fuzziness and stochasticity. We shall now turn our meta-theoretic attention to deal with the epistemics of this class of decision-choice problems where simultaneity of fuzziness and stochasticity are the basic characteristics of the decision-choice environment. In dealing with the non-fuzzy and stochastic class of the decision-choice problems, it is a tradition to use the classical toolbox since this is what is widely known. This means that we take the measures of probability that connect expectations of current decision-choice activity to knowledge about the future to be exact. The implication here is that there are no ambiguities or fuzziness in the probability values no matter how they are derived. This includes probabilities constructed from axiomatic systems as well as those derived from utilities involving decision-choice actions among gambles in such a way as to preserve the axioms of probability that allow independence and summability. The probability space is composed of crisp sets. The presence of fuzziness is incompatible with the postulate of independence and summability of probabilities. We will find out that the postulate of independence and additivity with the principle of exactness leads to a number of paradoxes that are not easy to reconcile within the classical paradigm.

3.1 Ambiguity, Probability and Decision In conceptual discussions on non-fuzzy and stochastic class of decision-choice problems we argued that the tradition is such that the measures of probability are not only inferred as exact values but they enter as exact weights that connect cognitive expectations of current decision-choice activities to knowledge about the future to be exact. One may view the probability values in decision-choice processes as future-present conversion factors that are exact in values and logic of derivation. The exact probabilities are derived from stochastic crisp sets of available information as consistency demands of the classical paradigm. The exactness of the probability values is a reflection that we have hundred percent confidences attached to the values. The implication of the assumption of exactness and crisp sets in both possibility and probability spaces is that there are no ambiguities or fuzziness in the concept and measurement of probabilities. In section 1.1.2 of this monograph we provided epistemic analysis of explications of both probability and possibility concepts as they affect the optimal decision-choice rationality. Implied in the process of explication are the

54

3 Fuzzy Rationality, Ambiguity and Risk in Decision-Choice Process

requirements of similarity between the explicatum (exact or precise scientific concept) and its explicandum (inexactness and pre-scientific concept), exactness of the concept, the usefulness of the concept and finally the simplicity of the concept toward knowledge construction and decision-choice actions. These requirements of explication themselves lead to ambiguities in the knowledge construct and linguistic reasoning in the construct and measurement or inference and measurement of the needed probabilities. Ambiguities arise not from limited knowledge of the process that generates outcomes as stated in [R21.11] but rather from linguistic structure and explication process of concept formations that give rise to hedges or multiple meanings. These are further complicated by un-sureness of linguistic reasoning and interpretations of derived conclusions within penumbral regions of cognition as we seek greater degree of exactness to fit into the classical paradigm. We have explained the idea that total uncertainty is made up of probabilistic component and possibilistic component. The probabilistic or stochastic uncertainty result from limited knowledge (ignorance) involving all aspects of decision-choice activities. The possibilistic or fuzzy uncertainty is composed of vagueness, un-sureness, reservations and others that result from language formation, explication, imprecision in thought, ill-defined problems, and approximations in reasoning and interpretations in conclusions. The fuzzy uncertainty gives rise to two situations in the knowledge construct, knowledge verification and decision-choice process. One situation is where

Non-Fuzzy

Non-Stochastic

Stochastic

Non-Fuzzy and Non-stochastic

Non-Fuzzy and Stochastic (Stochastic Risk Zone) II

(Risk-less Zone) I Fuzzy

Fuzzy and Non-Stochastic (Fuzzy Risk Zone) III

Fuzzy and Stochastic (Fuzzy Risk and Stochastic Risk Zone) IV

Fig. 3.1.1. Fuzzy-Stochastic Partition of Uncertainty Space

3.2 The Epistemic Analyses of the Decision-Choice Cohorts

55

accepted knowledge contains fuzzy characteristics and hence fuzzy uncertainty that produces non-stochastic and fuzzy (imprecise and vague) environment. The other situation is where we find ourselves in a fuzzy zone of incomplete knowledge giving rise to simultaneous existence of stochastic and fuzzy uncertainties. This zone (IV) is what reality seems to reside. It is in this category that complexities in ignorance and risk tend to arise and accident and necessity acquire their complete definitions. Their partitioning structure is presented in Figure 3.1.1. The category IV is actually the primary category of reality. All other categories are logical derivatives that are obtained through explication and simplification with assumptions for cognitive simplicity and manageability. These logical categorial derivatives mainly reduce complexities but do not get rid of the uncertainties and risk. Here complexity, uncertainty, simplicity and risk interact in human thought, linguistic reasoning, knowledge use and decision-choice actions that are taken in penumbral regions of cognition.

3.2 The Epistemic Analyses of the Decision-Choice Cohorts The partition and the cohorts of Figure 3.1.1 that it presents create epistemic categories for zonal analysis of theories of decision and choice under risk and no risk. The general zonal analyses allow us to abstract the cognitive relevance of applicable areas of theory constructions about certainty decisions, fuzzy decisions, statistical decisions and fuzzy-stochastic decisions, and how they are connected to the maintained assumptions and the utility characterization of decision-choice activities. The epistemic zonal analyses allow us to place various theories, methods of reasoning and involved mathematics in appropriate cohorts that will allow us to appreciate their respective contributions to our knowledge structure since uncertainty and risk of decision-choice activities tend to interact to define their effects on optimal decision-choice rationality. We shall consider individually these categories of the decisionchoice environment one after the other. Our discussions of rationality under cohorts I, II, III and IV are about rationality as an ideal state of decisionchoice processes but not rationality as attribute of decision agents. Optimal rationality as an attribute of decision-choice agents is accepted as immutable element in the universal object set as defined in Chapters One and Two. The theories of optimal rationality are either to explain or to find the paths of ideal states of decision-choice processes consistent with rationality as an attribute of decision-agents.

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3 Fuzzy Rationality, Ambiguity and Risk in Decision-Choice Process

3.2.1

COHORT I: Non-fuzzy and Non-stochastic Decision-Choice Systems

The category I contains decision-choice problems whose solutions are obtained under complete certainty and with no risk. This is the risk-less environment of decision choice actions. The environment of decision-choice actions is characterized by non-fuzzy and non-stochastic conditions by assumption with no contradictions in preferences and reasoning by the required logic. This is the fundamental assumption of characterizations of decision-choice problem of this cohort given optimal rationality as attribute of decision-choice agents. All decision-choice theories that assume perfect and exact knowledge fall under Cohort I. The theories in this class are initialized with a fundamental assumption that decision-choice activities are carried on in an environment characterized by non-fuzzy and non-stochastic processes. The goals, objectives and resource limitations are exactly and completely known with every measurement crisply defined. The implied exactness offers us the convenient use of classical logic and its corresponding mathematics to study decision-choice behavior under no risk. In other words, to study optimal rationality as an ideal state of decision-choice processes given rationality as an attribute of decision-choice agents. The process involves the establishment of the framework that fixes the boundaries of acceptable theories that either explains or prescribes the path to the ideal decision process. The basic characterizations of the decision-choice environment are established by the following assumptions that are meant to capture consistency between optimal rationality as ideal behavior and optimal rationality as attribute of decision agents: I.

Knowledge Structure or the Nature of Uncertainty

1. Characterization of decision-choice space without knowledge constraint with a complete and exact knowledge (this may be viewed as the postulate of perfect information) II. Cost-Benefit Structure or Goal-Constraint Structure

2. Characterization of the reward and cost (real benefit-cost) spaces with exactness (this may be viewed as the postulate of cost and benefit information sets); 3. Characterization of the function space with exact measures (postulate of exact Measures for relating real costs and benefits);

3.2 The Epistemic Analyses of the Decision-Choice Cohorts

57

III. Motivation and Action Structure or Preference Structure

4. Characterization of the action space with exact decision-choice variables (this may be viewed as the postulate of goal-constraint relation with reference to real cost-benefit balances); 5. Characterization of exact preference ordering consistent with classical ordered crisp set that contains minimal and maximal elements for decision-choice action (this may be viewed in terms of postulate of optimal process). The knowledge obtained from the analyses of the problems in this zone of cognition involves explanatory or prescriptive propositions and hypotheses about how decision-choice agents working in ideal state must behave under risk-free conditions, perfect and exact knowledge with no cognitively computable capacity constraint. The theories obtained in this cohort may be viewed as ideal types that provide us with the conditions of classical optimal non-fuzzy and non-stochastic rationality and that decision-choice agents have attribute of rationality in accordance with Eulerian mini-max postulate of universal order. When one accepts the characterization of this decision-choice environment, the criticisms must be restricted to the usefulness of supporting assumptions, logical constancies and correctness in the computable system. It is not useful to discredit the theory on the basis of the fundamental assumptions regarding attribute of decision-choice agents if such fundamental assumption has no alternative. We can challenge the role of the supporting assumptions and replace them or some of them in order to improve a theory or add to the cluster of theories within this cohort. The epistemic nature and the treatment of rival or competing theories in the same cohort or different cohorts are discussed in a companion monograph [R17.23]. The fundamental assumption of risk-free condition constitutes the epistemic core of the research program with cluster of theories in this cohort. One can reject the fundamental assumptions that structure the permissible range of cluster of theories constructible within the cohort under the Eulerian mini-max principle relating cost to benefit. When the cohort is defined by the fundamental assumptions our research on and the development of logic and mathematics are then restricted and directed by conditions permissible within the cohort.

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3 Fuzzy Rationality, Ambiguity and Risk in Decision-Choice Process

3.2.2

COHORT II: Non-fuzzy and Stochastic Decision-Choice Systems

When the fundamental assumption of perfect and exact knowledge about the decision-choice environment is relaxed and replaced with incomplete but exact knowledge then all decision-choice theories constructed in this environment fall under Cohort II. The theories constructed in this environment are intended to deal with non-fuzzy and stochastic processes. Stochasticity implies incomplete knowledge not in terms of vagueness but in terms of partial ignorance about the object of decision-choice action. The implied incomplete knowledge leads to stochastic uncertainty even though the available knowledge to decision-choice agents is taken to be exact. The stochastic uncertainty presents the environment with risky decision-choice activities that are constrained by partial ignorance. The type of risk generated in this environment may be distinguishably called stochastic risk. The stochastic risk is due to the random behavior of the parameters that characterize the environment of decision-choice activities in our cognitive perception. The stochastic risk is simply due to ignorance or incomplete subjective knowledge regarding the object of decision-choice action. The risk can be minimized by reducing stochastic uncertainty through acquisition of more exact knowledge to expand the knowledge sector. The non-fuzziness implies exactness defined in terms of absence of vagueness, ambiguity, linguistic approximation, hedges, penumbral regions and others in the cohort and methods of reasoning. The principle of exactness with crisp sets in Cohort II just as in Cohort I allows us to employ the classical logic and its corresponding mathematics to study decision-choice behavior under stochastic risk. The studies in this category of knowledge growth will include probability [R16], [R16.4], [R16.17], [R16.27] statistical decision theory [R16.15], [R16.17], [R16.45], [R19.21] expected utility theories [R22], [R22.2], [R22.12], [R22.13] static and dynamic stochastic games [R19], [R19.2], [R19.8], [R19.24] portfolio theory [R21.5], [R22.5], theories of stochastic optimal control [R24.2], [R24.5], [R24.1] and theories of stochastic risk-bearing [R21], [R21.1], [R21.3], [R21.16] Theories developed within this cohort yield a set of propositions and hypothesis that define decision-choice intelligence that is claimed to be consistent with optimal rationality as an attribute of decision-choice agents. We shall refer to this decision-choice intelligence as optimal classical non-fuzzy stochastic rationality. It is classical in terms of the methodological paradigm used in formulating and abstracting the required propositions and hypothesis about decisionchoice behavior on, or toward the path of the ideal state.

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The new theoretical knowledge obtained in this zone involves explanatory or prescriptive propositions and hypotheses about how optimally rational decision-choice agents behave or must behave with limited but exact knowledge in the decision-choice space. The phrase behave or must behave connotes explanatory or prescriptive propositions about optimal behavior. From the viewpoint of explanatory science, the theories developed in this environment are intended to explain decision-choice behavior under general conditions of limited but exact knowledge without computational limitations. At the level of prescriptive science, the theories are intended to develop optimal prescriptive decision-choice rules consistent with non-fuzzy stochastic rationality to guide decision-choice agents under conditions of stochastic risk and non-fuzzy stochastic uncertainty. The presence of stochastic uncertainty allows the examination of conditions of optimal risk-taking under the assumption that riskbenefit preferences of decision-choice agents are measured by utility [R22.2], [R22.9], [R22.12], [R22.15]. A theoretical problem arises in terms of whether probability values can be inferred from the utilities defined over risky decision-choice alternatives. This theoretical problem is important to the extent to which we relate it to classical non-fuzzy stochastic rationality where the measure of stochastic risk is specified indirectly in terms of probability or expected utility. The environment of decision-choice actions, for which explanatory or prescriptive theories are being constructed about agents’ best decision-choice behavior, is characterized by non-fuzzy and non-stochastic conditions where contradiction in reasoning and choice are not allowed. This is the fundamental assumption of the problem characterizations of the cohort II. All decision-choice theories that assume exact and limited knowledge fall under Cohort II. The theories in this class are initialized with a fundamental assumption that decision-choice activities are carried on in an environment characterized by non-fuzzy and stochastic processes and hence with stochastic risk. Again when one accepts the fundamental characterization of the decisionchoice environment as non-fuzzy and stochastic then the criticisms must be restricted to logical inconsistencies (such as paradoxes), correctness of theoretical specifications (such as axioms), correctness of computable algorithms, knowledge representations, appropriateness of designed measures of degree of ignorance or degree of knowledge held by agents for decision-choice activities. Any claimed paradox must arise from within the confines of the parametric boundaries of the categories of decision-choice problems. For example, the St Petersburg paradox, [R4.45], [R21], [R21’26], arises within the non-fuzzy and stochastic environment for theoretical construct and its resolution must be

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within the same parametric boundaries of this cohort. It was precisely the satisfaction of this epistemic requirement that led Bernoulli to resolve the St. Petersburg paradox by advancing the mean-value theory not in terms of monetary pay-offs but in terms of utility of money constrained by diminishing marginal utility of money [R4.45], [R21.26]. The Bernoulli’s resolution to St. Petersburg paradox gives rise to optimization of expected utility and inference of probability values derived from choice behavior among gambles or decision-choice activities in non-fuzzy and stochastic environment given that decision-choice agents are endowed with rationality consistent with Eulerian mini-max principle in the universal object set. The theoretical environment stayed the same while `the specification of the problem was restructured to retain the use of the classical paradigm and its mathematics in order to resolve the paradox that retained the optimal non-fuzzy and stochastic rationality. The non-fuzzy and stochastic rationality received an axiomatic description that allowed behavior under stochastic uncertainty and risk to be portrayed by expected utility ranking of decision-choice alternatives. Again the process involves the establishment of the logical framework that fixes the boundaries of acceptable theories that either explain or prescribe the path to the ideal decision process under the fundamental assumptions of exact but information incompleteness. The basic characterizations of the decision-choice environment are established by the following assumptions that are meant to capture consistency between optimal rationality as ideal behavior and optimal rationality as attribute of decision agents in Cohort II: I.

Knowledge Structure or Nature of Uncertainty

1. Characterization of the probability space under knowledge constraint (this may be viewed as the postulate of incomplete and exact information); 2. Characterization of the induced exact probability measures (postulate of exact measure of ignorance); and II. Cost–Benefit Structure or Goal-Constraint Structure

3. Characterization of the of reward and cost (benefit-cost) spaces with exactness (postulate of cost and benefit information sets); 4. Characterization of the function space with exact measures (postulate of exact measures relating real costs to benefits);

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III. Motivation and Action Structure or Preference Structure

5. Characterization of the of action space with exact but random decision-choice variables (postulate of goal-constraint relation with reference to real cost-benefit Balances); 6. Characterization of exact stochastic preference ordering consistent with classical ordered set has minimal and maximal elements (postulate of existence of optimal decision-choice element). This set of characterizations is nothing but a protective shield of the classical decision-choice paradigm that allows the sustainability of the fundamental assumptions that create an environment of stochastic risk associated with decision-choice behavior in cohort II that will satisfy the Euler mini-max principle. The axiomatic description of the optimal non-fuzzy and stochastic rationality has given rise to other paradoxes such as Allais’ paradox [R21], [R21.1], [R21.2], [R21.26]Newcomb’s paradox [R21.6], [R21.15], Ellsberg’s paradox [R21.12], [R21.28], [R21.29]. The Ellsberg’s paradox, for example, shows that probabilities inferred and derived from decision-choice behavior among gambles are incoherent and ambiguous [R21.11], [R21.12]. The explanation of the existence of Ellsberg’s paradox in the non-fuzzy and stochastic rationality is then attributed to the exactness of assumption in the probability space that leads to the use of exact of probability values in the expected value optimization and computations. The line of resolution is first to appeal to the presence of ambiguities in the information processing and cognitive computations of decision-choice agents and second to remove these ambiguities. Other paradoxes such as sorites emerging out of conflict between vagueness and the classical paradigm may be found in [R21.27], [R21.31]. The refinement is in response to the Bayesian claim that subjective probability that measures the degree of belief of decision-choice agents can be exact and defined operationally by observing decision-choice activities among gambles [R21.19], [R21.27], [R22.7]. The attempted resolution of this paradox has given rise to theories of decision making under ambiguity and the ensuing debate on the Ellsberg’s paradox. Here, an epistemic problem arises in that the integration of ambiguity into non-fuzzy and stochastic rationality is inconsistent with the use of classical logic and its mathematics except by some artificial grafting process [R21.19], [R21.29]. This paradox is an anomaly that challenges the exactness assumption. The fundamental assumptions in Cohorts I and II regarding exactness, in measurements and reasoning, crisp sets and non-contradiction must be changed and replaced with broadly defined

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inexactness in knowledge and contradiction (incoherence) in reasoning where shades of meaning, doubt, subjectivity, approximations, ill-definedness, computational limitations and others are allowed to interplay in the decisionchoice processes. The introduction of ambiguities in probabilities in attempt to resolve the Ellsberg’s paradox brings us to the zonal analysis of Cohort III and Cohort IV. The analyses of effects of fuzziness in the knowledge structure and the reasoning process with full knowledge place us in Cohort III. The analysis of the effects of simultaneous existence of ambiguity and uncertainty in probabilistic inference from decision-choice behavior over gambles logically belongs to Cohort IV where we have both fuzzy and stochastic characteristics in the knowledge structure, specification of decision-choice problems and reasoning methods to arrive at decision-choice act. It may be pointed out that a distinguishing characteristic of non-fuzzy-stochastic optimal rationality is that it is characterized by a pair of optimal decision-choice value and corresponding optimal probability value. The non-fuzzy-stochastic optimal rationality implies that given probability values we seek an alternative that will provide the decision-choice agent with optimal decision-choice alternative. Alternatively, given the decision-choice alternative we seek the optimal probability distribution that will allow the alternative to be actualized. 3.2.3

COHORT III: Fuzzy and Non-stochastic Decision-Choice System

Let us now turn our attention to Cohort III. The Cohort III is characterized by decision-choice space composed of fuzzy and non-stochastic processes. In this cohort the assumption of complete knowledge as in the Cohort I is retained but the assumption of exact knowledge is replaced by a knowledge structure that contains vagueness, imprecision, hedges, linguistic approximations and others under the general concept of fuzziness. The decision-choice theories constructed in this environment are currently grouped under the category of fuzzy decision theories [R.5], [R5.17], [R5.26], [R5.71], [R5.78].The theories may be viewed in two ways in terms of explanatory and prescriptive structures. As theories in explanatory science, they are intended to explain decision-choice behavior under conditions of fuzziness and complete knowledge given that decision-choice agents are endowed with capacity for rational behavior in terms of the mini-max principle. As theories in prescriptive science they are intended to prescribe a set of optimal rules for the best path of decision-choice behavior under conditions of full knowledge and fuzziness that

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will place the decision agent on the ideal path of the decision-choice activities under substitution-transformation process in the dynamics of actual-potential duality. In both explanatory and prescriptive structures the goal to locate the ideal and then examine the departures from the ideal as well as constructing the reasons for the departures from the ideal if decision-choice agents are endowed with optimal rationality as an attribute. The characterization and measure of level and degree of fuzziness (ambiguity) are done by means of membership characteristic functions or possibility distribution functions. These measures allow us to objectively and subjectively quantify a number of elements in conceptual doubt such as characterization of grades of quality, subjectivity, shades of truth, differential meanings implied in linguistic hedges, interval values, qualitative values, approximations to quantitative values and many others that give rise to penumbral regions in the decision-choice and knowledge spaces. The presence of fuzziness, the use of linguistic reasoning, the application of linguistic numbers such as large, medium or small, allowance of subjectivity in conceptual approximations, and limitations of capacity of cognitive computations lead to non-stochastic fuzzy uncertainty. We must keep in mind that there are stochastic uncertainties that generate stochastic risk, and there are fuzzy uncertainties that generate fuzzy risk. The manner in which they are combined in an analytic construct to understand or prescribe the ideal path of decision-choice behavior, consistent with optimal rationality as attribute to decision-choice agents, will depends on the nature of fundamental assumptions imposed on the environment for developing decision-choice theories. Such environment is designed by the analyst or theorist. The fundamental assumptions when they are made and accepted have preponderating effect on the paradigm in which the theories are constructed as well as the kind of theories that are constructible within the paradigm that shapes reasoning. It may be pointed out that one important characteristics in Cohorts III and IV is the built in capacity to deal with the presence of qualitative-quantitative duality and contradiction as natural process of decision-choice activities. In this way, reasons for deviations from the ideal may be constructed or explained within the paradigm. The non-stochastic fuzzy uncertainty in this environment presents to the decision-choice agents non-stochastic fuzzy risk due to the presence of vagueness and other elements that give rise to penumbral regions in knowledge and decision-choice spaces. It is not due to ignorance, and it is different from nonfuzzy stochastic risk as generated under non-fuzzy and stochastic conditions. Just as the stochastic risk can be minimized by reducing stochastic uncertainty through widening the knowledge structure, the fuzzy risk can be minimized

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by reducing fuzzy uncertainty through the improvement of knowledge exactness or reducing vagueness and ambiguity in the complete knowledge structure as well as improving cognitive computational capacity of decision-choice agents. The reduction of fuzziness is a minimization of the size of the penumbral region of knowledge and decision-choice spaces. The required measure of fuzzy risk cannot be computed and analyzed in terms of probabilities and probability measures since randomness is completely out of the assumed space of decision-choice activities. The uncertainties and risk in this category of decision-choice problems must be measured and computed in terms of fuzzy measures [R7.35]. The principle of fuzziness (broadly defined) presents a particular logical difficulty that prevents us to employ the classical logic and its mathematics to study decision-choice behavior under fuzzy risk including ambiguities as it is presented in [R21.1], [R2112], [R21.28]. In other words, the classical paradigm does not offer us a way out of ambiguity dilemma since the logic and mathematics deal with exactness, precision and non-contradiction or coherence. By introducing the concept of fuzziness in the full knowledge structure we have altered the fundamental assumption as well as the epistemic core of the paradigm of Cohort I. An ambiguity dilemma arises in this cohort leading to paradoxes and cognitive difficulties. A new logical framework with corresponding mathematics is required if we are to overcome this cognitive difficulty and resolve the ambiguity and vagueness dilemma. This new framework is fuzzy logic and its corresponding mathematics that presents a new paradigm of scientific reasoning, analysis and knowledge construct. The new paradigm is the fuzzy paradigm. The framework allows us to undertake a new look at utility theory through the construct of fuzzy preferences and fuzzy utility theory [R5], [R7], [R7.5], [R7.42], [R5.18], [R5.81], [R5.147]. Furthermore, it allows us to examine the nature of fuzziness as well as optimal risk-taking behavior under fuzzy preferences and decisions in the penumbral regions. The quantitative and qualitative values that enter into the reasoning are fuzzy numbers in the fuzzy real numbers that allow linguistic reasoning that is a characteristic of decision-choice agents irrespective of the level of formal education. Theories constructed in Cohort III present different cognitively zonal analysis of our knowledge bag. At the level of decision-choice activities they lead to optimal fuzzy nonstochastic rationality as the intelligence of decision-choice behavior either for the cognitive task of explanation or cognitive task of prescription irrespective of a particular knowledge sector of special interest.

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The major characteristics of this cognitive zone of theoretical construct are vagueness, errors, ill-definedness, imprecision, subjectivity, inexactness, and many others under the umbrella of fuzziness in our knowledge structure, concept explication, measurement and linguistic reasoning. These characteristics are taken to be subjective realities. These characteristics in our knowledge construct and mode of reasoning cannot easily, if at all, be handled by the classical logic and the corresponding mathematics without imposing conditions outside the classical paradigm. These characteristics in our knowledge structure and linguistic reasoning have been pointed out by some important scientists and philosophers of science [R20.75], [R23.10], [R23.11], [R23.29], [R22.15], [R23.55], [R23.56], [R23.66], [R23.67]. For example, Russell states “vagueness or precision are characteristics which can only belong to representation…They have to do with the relation between a representation and what it represents” [R23.51, p.85]. A similar reflective statement by Max Black will be useful in illustrating the troubling problems of vagueness, ambiguity and classical logic. The contradictions inhere in the very principles of science, produced by the inevitable vagueness of the concepts it employs. However much reflection and experiment by inventors of theories may mitigate the opposition of mutually contradictory opinions by modification and elimination of obscurity, contradictions remain even in scientific theories which find widespread acceptance [R20.4, p.3]. The epistemic relevance of Russell-Black statements must be viewed in terms of relation between knowledge representations (accepted knowledge) and elements of reality on one hand and the relation between decision-choice representations and actual decision-choice behaviors. It may also be viewed in terms of relation between formal logic of reasoning and linguistic reasoning in general. If we accept the fundamental characterization of the decision-choice environment as defined by fuzzy and non-stochastic (complete knowledge) processes then the problem of fuzziness is not simply a dilemma or paradox or anomaly in the classical paradigm but it is an important limitation and constraint which forces us to exist from, or search our way out of the established classical paradigm. The presence of fuzziness in this category of decisionchoice problems forces us to accept the fuzzy paradigm composed of fuzzy logic and corresponding mathematics to reason and solve the problem of

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fuzziness and the construction of theories of decision-choice activities under fuzzy risk that is guided by optimal fuzzy non-stochastic rationality. To be convincing and analytically potent, the framework must be able to solve problems in non-fuzzy and non-stochastic nature of Cohort I. This is exactly what fuzzy decision theories offer [R5], [R6], [R7], [R8], [R9], [R10]. Again the process involves the establishment of the logical framework that that fixes the boundaries of acceptable theories that either explain or prescribe the path to the ideal decision process under the fundamental assumptions of fuzziness and information completeness. The basic characterizations of the decision-choice environment are established by the following assumptions. The axiomatic and non-axiomatic description of fuzzy non-stochastic rationality as presented in fuzzy decision-choice theories on decision-choice alternatives that are meant to capture consistency between optimal rationality as ideal behavior and optimal rationality as attribute of decision agents in Cohort III: may be stated as: I.

Knowledge Structure or Nature of Uncertainty

1. Characterization of membership set that defines the fuzzy decision space (postulate of fuzzy knowledge and penumbral region); 2. Characterization of fuzzy measures or possibility measures (postulate of inexact measures on quantity and quality characteristics); II. Cost-Benefit Structure or Goal-Constraint Structure

3. Characterization of fuzzy reward (fuzzy objective) space (postulate of fuzzy goal space); 4. Characterization of fuzzy cost (fuzzy constraint) (postulate of fuzzy constraint space); III. Motivation and Action Structure or Preferences Structure

5. Characterization of space of decision-choice action (postulate of benefit-choice conflict and contradiction); 6. Characterization of fuzzy preference ordering consistent with ordered fuzzy set that contains minimal fuzzy element and maximal fuzzy element with membership values (postulate of optimal process). These characterizations together and operating under fuzzy logic and mathematics lead to the optimal fuzzy non-stochastic rationality. The optimal fuzzy

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non-stochastic rationality is described by two elements of optimal decisionchoice value and optimal degree of acceptance of optimal rationality which is obtained from the membership characteristic function or possibilistic distribution function. The optimal degree of acceptance must be interpreted in relation to subjective classification of vagueness, ambiguity and other related concepts as they give rise to penumbral region of knowledge and decision-choice activities. An important epistemic lesson from the decision-choice theories in the fuzzy nonstochastic environment is that we may observe certain decision-choice behaviors to go counter to the established axioms and theories in cohort I and II simply because either the full knowledge sector or incomplete knowledge and linguistic reasoning that support decision-choice processes contain fuzzy characteristics that lead to contradictions and incoherence as are observed in the experimental results on decision-choice studies in psychology and economics under classical logic and mathematics. These contradictions and incoherence generates fuzzy risk that is assumed away in the classical paradigm. In general, we should expect that decision-choice actions take place under combined conditions of vagueness and incompleteness of knowledge that takes us away from decision-choice activities and corresponding theories in Cohorts I, II, and III into those of Cohort IV. Let us turn our attention to Cohort IV and examine the decision-choice environment and possible theories that may be constructed to either explain or prescribe the ideal paths of decision-choice processes with corresponding types of intelligence. 3.2.4

COHORT IV: Fuzzy and Stochastic Decision-Choice System

The epistemic zonal analyses of cohort I, II, and III bring us to Cohort IV where the decision-choice environment contains both fuzzy and stochastic characteristics. The Cohort IV presents a decision-choice environment that represents total uncertainties in the substitution-transformation processes with the dynamic behavior of actual-potential duality as induced by decisionchoice actions. It contains fuzzy characteristics that give rise to fuzzy uncertainties and fuzzy risk. It simultaneously contains stochastic characteristics that give rise to stochastic uncertainties and stochastic risk. In other words, the penumbral region of knowledge and decision choice activities is generated by both fuzziness and stochasticity. Epistemologically, therefore, the Cohort IV presents a knowledge space that has isomorphic representation of human interpretive experiences of decision-choice activities where vagueness, errors,

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measurement approximations, knowledge incompleteness, linguistic reasoning, grades of truth and others are the fundamental characteristics of totality of human experiences, and where every aspect of human activity and decisionchoice processes involve fuzzy and stochastic uncertainties. Fuzziness is defined in a broad general term of characteristics opposed to exactness or crispness and stochastic defined in terms of knowledge completeness (fullness). The presence of simultaneity of stochastic and fuzzy uncertainties is theoretically troublesome, analytically challenging and very difficult to deal with in the classical paradigm. This of course is the case of the actual world of human operations even though we evidentially perceive and sympathetically accept that fuzzy expressions are defective rather than natural characteristics of language structures and language formation. And yet such fuzziness without explication gives meaning to account of human understanding in verbal communications, written records of ideas and interpretations of symbolic representations of events. Similarly, we evidentially perceive with some degree of sympathy and comfort that our knowledge of operations in our perceived universe is always incomplete and even the part that we claim to be knowledge is riddled with fuzzy characteristics that are subject to relational and subjective interpretations. The rise of theoretical difficulties in this zone of knowledge acquisition may be traced to the classical paradigm of knowledge construction, its acceptable method of truth verification or falsification where shades of truth and presence of contradiction in evidence and reasoning are considered false and unacceptable. The perceptive fact remains that our knowledge construction of the universe is composed of objects and processes that are made up of dualities and particularities with contradictions and continual changes that tend to characterize our perceptive knowledge as we have explained and represented in Chapter Two of this monograph. A claimed scientific truth of today may be a fiction of tomorrow or may be subject to substantial refinement by tomorrow’s perception due to either errors or vagueness or general fuzziness. A claim to absolute truth in knowledge is cognitive delusion that becomes an important ideological barrier to knowledge expansion through further scientific discoveries. Shades or grades of truth are characteristics that are reflected in relations between representation of perceptive knowledge and categories of reality. Such representation is constantly being refined by our knowledge enterprise. The above statement about perceptive knowledge is also reflected in Parrat’s statement that Every fact in science, every law of nature as devised from observations is intrinsically ‘open-ended’, i.e. contains some uncertainty and is subject to future improvement [R23.48, p.1]. This uncertainty is due to

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fuzziness and stochasticity that affect the optimal decision-choice rationality as the ideal state in the knowledge production. An epistemic clarification regarding two questions is necessary here. 1) What is fuzziness and what is exactness? And 2) are fuzziness and exactness characteristics of objective reality or subjective reality? Fuzziness, as has been explicated in this monograph, may be viewed in terms of cognitive indeterminacy whose meaning is clarified by subjective interpretations. Fuzziness, therefore, is a characteristic of subjective reality. It is not a characteristic of objective reality as seen from the universal object set. This leads to a philosophical question as to whether there are fuzzy or vague elements in the universal object set. This question is not of immediate interest. Decision-choice activities are supported by what is perceived to be true, but not what truth actually entails. We have also pointed out that stochasticity or randomness is the result of knowledge incompleteness but not necessary a characteristic of the universal object set. As presented, risk and accident are viewed in terms of perceived knowledge and constructed logic of reasoning but not necessarily characteristics of natural processes [R20.20]. The presences of simultaneity of stochastic and fuzzy uncertainties must lead to different decision-choice responses by decision-choice agents. The studies of these decision-choice responses and human intelligence that may be brought to bear on them require analytical approach different from the ones used in non-fuzzy and stochastic problems and fuzzy and non-stochastic problems in order to examine rationality of the decision-choice process in this cohort. The analytical procedure is such that, first, it involves the assessment and analysis of stochastic risk relative to the decision-choice variables. Second, it will involve assessment and analysis of knowledge about fuzzy risk. The availability of the method for dealing with non-fuzzy stochastic uncertainty (classical paradigm) and fuzzy non-stochastic uncertainty (non-classical paradigm) offers a possible way to deal with the simultaneity of fuzzy and stochastic uncertainties by methodological and logical integration of compatible elements. The decision-choice intelligence that emerges is complex. It presents itself either as optimal fuzzy-stochastic rationality or optimal stochastic-fuzzy rationality. The epistemic complexities, and the possible cognitive computational difficulties associated with either optimal fuzzy-stochastic rationality, or optimal stochastic fuzzy rationality, cannot be underestimated. This category of decision-choice problems presents theoretical challenges for all areas of scientific investigations as they involve fuzzy-stochastic process or stochastic-fuzzy process in humanistic and non-humanistic systems.

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PRIOR

POSTERIOR

FUZZY

PRIOR

POSTERIOR FUZZY

STOCHASTIC

PRIOR STOCHASTIC

POSTERIOR STOCHASTIC

Fig. 3.2.4.1. Prior-Posterior Partitioning of Fuzzy-Stochastic Uncertainties

At the level of theoretical construct, there are two situations from which we can analytically view the decision-choice activities regarding the simultaneous presence of fuzzy and stochastic uncertainties in the environment of decisionchoice activities. The two situations are conditions of separability and nonseparability of uncertainties. If the uncertainties are separable, then we can speak of the sum of the effects of stochastic risk and fuzzy risk on the decision-choice rationality and these must be reflected in the structure of the theory of decision-choice behavior under fuzzy and stochastic risks. In this respect, we must deal with aggregation problem of fuzzy and random variables On the other hand if the uncertainties are not separable into respective components, how then do we represent and analyze their joint interactive effects in our theoretical construct of optimal decision-choice rationality. How do we relate them in terms of theoretical order of occurrence within prior and posterior conditions if they are separable? The separability of total uncertainties into fuzzy and stochastic components requires that the uncertainty space be partitioned into prior and posterior relative to fuzzy and stochastic elements as shown in Figure 3.2.4.1. The conditions of separability of total uncertainties will also constitute the conditions for separability of total risk into fuzzy risk and stochastic risk. The total uncertainty will come to us in combination of: 1) interaction of prior fuzzy and posterior stochastic; and 2) interaction of prior stochastic and posterior fuzzy in representation of uncertainties. At the level of non-separability we have to consider two new variables of fuzzy random variable and random fuzzy variable that correspond to fuzzy-random process and random-fuzzy process. The epistemic order in which these uncertainties appear will be determined in terms of assumptions imposed by the analyst on the basis of his or

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her understanding of the problem at issue. This statement also holds for fuzzyrandom and random-fuzzy variables. Here emerges a notion of fuzzy epistemic duality in the uncertainty and risk spaces. The differences between fuzzy-random and random-fuzzy in case of non-separarability arise in the order of fuzzification-defuzzification process to obtain exact-value equivalence and randomization-derandomization process to obtain certainty equivalence. At the level of epistemics, fuzzy analysis is an important toolbox to provide an explication for vagueness, errors, illdefindness, imprecision, subjective phenomenon, inexactness and others in our knowledge structure that give rise to penumbral regions and indeterminacies in human decision-choice processes. At the level of knowledge applications, fuzzy logic defines an important toolbox for logical reasoning with linguistic variables that may or may not contain fuzzy characteristics in such a manner that allows us to overcome the reasoning constraint imposed on human thought by the classical principle of non-acceptance of contradiction. Here, the invention of fuzzy membership characteristic functions is an attempt to provide subjective measures of grades of these phenomena in a way that allows interpretation of combined interaction of subjective and objective measures. It further allows us to make sense of elements in the penumbral region in our communication and decisionchoice processes. A point of epistemic reference of the perceptive knowledge space of decision-choice processes in relations to analysis and synthesis for logical and mathematical reasoning will be useful. In the classical non-stochastic and non-fuzzy knowledge environment, we speak of ordinary analytical variables as objects of logical manipulations to derive hypothesis and conclusions. In the classical non-fuzzy and stochastic environment, we speak of random analytical variables as objects under logical manipulations. In the non-classical fuzzy and non-stochastic knowledge environment, we speak of fuzzy analytical variables as representations of objects under logical manipulations. In the case of non-classical and fuzzy-stochastic knowledge environment, we speak of a) fuzzy random analytical environment, or b) random fuzzy analytical variables which are applicable to Cohort IV. A complexity in theoretical construct emerges in dealing with decisionchoice behavior in an environment that comes to us as either fuzzy-stochastic or stochastic-fuzzy. The cognitive construct that assumes the environment of decision-choice process, as characterized by prior fuzzy and posterior stochastic, leads to decision-choice theory whose main propositions are description of optimal fuzzy-stochastic rationality. Similarly, the theory that emerges from assuming an environment characterized by prior stochastic and posterior fuzzy

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in the cognitive construct of decision-choice behavior yields propositions about decision-choice intelligence of optimal stochastic-fuzzy rationality. An epistemic question arises as to whether the optimal fuzzy-stochastic rationality and optimal stochastic-fuzzy rationality constitute a duality. In other words, does the cognitive process in the environment of prior fuzzy and posterior stochastic yield the same decision-choice intelligence as the cognitive process in the environment that assumes prior stochastic and posterior fuzzy? Similarly, will the theory constructed by using fuzzy-random variable yield the same result as the one constructed in using random-fuzzy variable? If the answers are yes then the order in which fuzzy and stochastic appear is epistemically irrelevant and the choice will be just a matter of logical convenience and analytical simplicity to the theorist to deal with the theoretical construct of decision-choice behavior under total uncertainty and risk. We, however, must examine the conditions of isomorphism and explain why this is the case. If the answer is no, in the sense that they do not yield the same decision-choice intelligence, then the two approaches must be cognitively used to construct two theories and examine their symmetric differences. A further theoretical task remains in terms whether there are conditions, no matter how remote they may be, for them to yield the same result. Furthermore, we must investigate the cognitive sources of the differences and how they may affect theories of risk-taking behavior either in terms of explanatory or prescriptive science. A further note of clarification on the concepts of prior and posterior as they relate to fuzzy and stochastic characteristics will be useful. The decision-choice environment is said to be prior fuzzy and posterior stochastic if the fuzziness is first to be cleaned up (defuzzified) and then the decision-choice problem is sent to non-fuzzy and stochastic environment for de-randomization. An alternative way to view the problem is to first obtain fuzzy optimal solution to the fuzzystochastic decision-choice problem and then establish stochastic process over the optimal fuzzy rationality whose solution must yield propositions that describe optimal fuzzy-stochastic rationality. Similarly the decision-choice environment is said to be prior stochastic and posterior fuzzy if the stochastic characteristics are first cleaned up (de-randomize) to obtain certainty-value equivalence before dealing analytically with the fuzzy characteristics (defuzzify) to obtain exact-value equivalences. Alternatively, the problem may be viewed in terms of first obtaining optimal stochastic solution and then establish fuzzy process over the optimal stochastic rationality to arrive at optimal stochastic fuzzy rationality. The epistemic nature and the analysis of the paths of rationality are presented in Figure 3.2.4.2.

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RATIONALITY AS AN IDEAL DECISION-CHOICE STATE CLASSICAL

FUZZY

Non-Stochastic Rationality

Classical NonStochastic Rationality

No Risk

Fuzzy NonStochastic Rationality

Fuzzy Risk

Methodological Toolbox

Stochastic Rationality

Classical Stochastic Rationality

Stochastic Risk

Fuzzy Stochastic Rationality

Fuzzy-Stochastic Risk

Methodological Toolbox

CLASSICAL LOGIC AND MATHEMATICS

FUZZY LOGIC AND MATHEMATICS

DECISION-CHOICE PROCESSES

RATIONALITY AS AN ATTRIBUTE OF DECISION-CHOICE AGENTS

Fig. 3.2.4.2. An Epistemic Nature of Rationality as an Ideal State

The theoretical environment for fuzzy non-stochastic environment stays the same while the specification of the problem with random characteristics must be restructured to retain the use of the fuzzy paradigm and its mathematics in order to abstract solution to the decision-choice problem that allows the construct of either the optimal fuzzy stochastic rationality or optimal stochastic fuzzy rationality. Both types of rationality must receive axiomatic descriptions

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that allow decision-choice behavior under both fuzzy uncertainty and stochastic uncertainty with combined fuzzy and stochastic risks of decision-choice alternatives. The analytical process involves the establishment of the logical framework that fixes the boundaries of acceptable theories that either explain or prescribe the path to the ideal decision-choice process under the fundamental assumptions of fuzziness and incompleteness of knowledge space for decision-choice activities. The basic characterizations of the decision-choice environment to provide an epistemic model of rationality as an ideal are established by the following assumptions that are meant to capture consistency between optimal rationality as ideal behavior and optimal rationality as attribute of decision agents in Cohort IV: I.

Knowledge Structure or Nature of Uncertainty

1. Characterization of membership set that defines the fuzzy decision space (postulate of fuzzy knowledge and penumbral region); 2. Characterization of fuzzy measures or possibility measures a) probability measure of fuzzy event b) fuzzy measure on random event (postulate of inexact measures on quantity and quality characteristics); II. Cost-Benefit Structure or Goal-Constraint Structure

3. Characterization of fuzzy stochastic reward (fuzzy objective) space (postulate of fuzzy goal space); 4. Characterization of fuzzy stochastic cost (fuzzy constraint) (postulate of fuzzy constraint space); III. Motivation and Action Structure or Preference Structure

5. Characterization of space of decision-choice action (postulate of benefit-choice conflict or contradiction); 6. Characterization of fuzzy preference ordering consistent with ordered fuzzy set that contains minimal fuzzy element and maximal fuzzy element with membership values (postulate of optimal process) These six characterizations together, and operating under fuzzy logic and mathematics lead to the optimal rationality that combines fuzzy uncertainty

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and stochastic uncertainty in examining decision-choice behavior with either explanatory or prescriptive science. The abstracted types of optimal decisionchoice rationality are complex that allows us to account for the disparities between the path of actual decision-choice outcomes and the path of the ideal outcomes enveloped by optimal rationality. The fuzzy stochastic optimal rationality and the stochastic fuzzy optimal rationality are described by three elements of optimal decision-choice value, optimal degree of acceptance of optimal rationality and optimal level of fuzziness or stochasticity as obtained from the membership characteristic functions and probabilistic distribution functions from subjectively perceived. The optimal degree of acceptance must be interpreted in relation to subjective classification of vagueness, ambiguity and other related concepts as they give rise to penumbral region of knowledge and decision-choice activities in addition to information incompleteness that give rise to randomness of the penumbral region of knowledge and decision-choice actions. Again an important epistemic lesson from the decision-choice theories in the fuzzy environment is that we may observe certain decision-choice behaviors to go counter to the established axioms and theories in Cohorts I and II simply because either the full knowledge sector or incomplete knowledge and linguistic reasoning that support decision-choice processes contain fuzzy characteristics that lead to contradictions and incoherence as are observed in the experimental results on decision-choice studies in psychology and economics under classical logic and mathematics. These contradictions and incoherence generates fuzzy risk that is assumed away in the classical paradigm. Of course the disparity may be the result of inappropriate characterization of the theoretical environment that is constructed to abstract the path to the ideal.

3.3 The Organic Paradigm of Decision-Choice Theories The four frameworks that we have presented follow an organic epistemic paradigm of theoretical and empirical analysis. The organic paradigm is initialized by existence of decision-choice agents who makes decisions and choices for many reasons including self-preservation. The decision-choice agents are linked to decision-choice activities by decision-choice processes. The decision-choice agents are endowed with optimal rationality as attribute to deal with conditions in the universal object set. This endowment forces the decision-choice agents to pursue the path of best decision-choice action in

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OPTIMAL RATIONALITY

COST-BENEFIT STRUCTURE/ GOALCONSTRAINT STRUCTURE

MOTIVATION STRUCTURE

DECISIONCHOICE AGENTS IDEAL STATE OF DECISIONCHOICE PROCESS

ATTRIBUTES OF DECISIONCHOICE AGENTS

KNOWLEDGE STRUCTURE

Fig. 3.3.1. Epistemic Geometry of the Organic Paradigm of Decision-Choice Theories

every situation. This best path defines the optimal rationality as ideal state of decision-choice process which is a surrogate representation of optimal rationality as an attribute of decision-choice agents. All theoretical constructs about decision-choice agents conform to the organic paradigm that follows the pyramidal logic of interactions which allows the examination of the degree of isomorphism between optimal rationality as attribute and optimal rationality as the path of an ideal state of the decision-choice process. The nature of the pyramidal logic presents the relational interactions that are shown in Figure 3.3.1. The organic paradigm is mutually defined by two components of primary category of logical reality with its pyramidal relations of three elements and a derived category of logical reality with its pyramidal relations of three elements. The primary category is composed of a) nature of humans as decisionchoice agents b) the specification of this nature and c) the surrogate representation of this nature that forms a relational pyramid. The human nature of decision-choice agents is taken to be optimal rationality as indisputable attribute.

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This intelligence is engrained in decision-choice agents by natural process that goes through refinements. This is fundamental assumption that provides the specification of the endowment in b). The construction of the surrogate representation as pointed out in c) presents theoretical and empirical challenges for human thought process in philosophical and scientific knowledge. It is the search for this surrogate representation that gives rise to another relational pyramid that must be superimposed on the first pyramid. The first pyramid is the primary category of logical reality as we have stated and the second pyramid is a logical derivative of the primary. The derived pyramid presents an epistemic framework that allows us to examine the isomorphism of correspondences between the primary category of optimal rationality as attribute of decision-choice agents and the derived category of optimal rationality as an ideal decision-choice process. The derived category is a cognitive construct that may or may not be good surrogate representations of the primary category depending on the set of assumptions that may characterize the key components of the logical pyramid. Given the primary category of logical reality two challenges are faced by knowledge construction. One challenge defines the path of the validity of the assumed optimal rationality as an attribute of decision-choice agents. It is on this path of understanding human nature that cognitive sciences can bring their powers of analysis to clarify the issue. So far, cognitive psychology and psychiatry and other related cognitive studies have little to offer. The fundamental assumption of optimal rationality as attribute to human nature have not been disputed even though some critics of theories about optimal decision-choice rationality reject it or at least are uncomfortable with it [R18], [R18.28], [R18.30]. It is here that criticisms of optimality hypothesis in decision theories show its maximum futility. We shall return to this point. The second epistemic challenge defines a path of knowledge construction that accept the validity of optimal rationality as attribute of decision-choice agents and then such for optimal rationality as an ideal state of decisionchoice activities. In other words, the theories are geared to construct or search for the best surrogate representation of optimal rationality. One may view the optimal rationality as an attribute in terms of primary category of reality while the optimal rationality as an ideal state is viewed in terms of derived category of reality. To deal with the second cognitive challenge, the organic paradigm requires the derived pyramid to be such that all theoretical constructs on decisionchoice process must specify 1) the nature of the knowledge structure or representation of knowledge, 2) the nature of the real cost-benefit structure or the

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goal-constraint structure, and 3) The motivation structure or incentive schemes. These three elements constitute the derived relational pyramid. The specification of the knowledge structure will reveal the nature of uncertainty and hence risk surrounding the decision-choice alternatives. It is the specification of this knowledge structure that also partitions the decision space into different cohorts. The real cost-benefit structure allows the goals and constraints to be specified in terms of conflicts in the transformation-substitution processes in the dynamics of actual-potential duality induced by decisionchoice process. The motivation structure propels the direction of the decisionchoice activities and hence the resolution of the conflicts that bring about the substitutions and transformations where the actual fades into the potentials and a potential replaces the actual by the decision-choice processes. The three elements constitute a derived relational pyramid that allows the analysis of decision-choice intelligence of decision-choice agent by establishing the ideal optimal process, examine deviations from the ideal and find explanations to the deviations. The organic paradigm for the study of decision-choice process is all encompassing. It covers all human decision-choice activities including knowledge construction itself. Within this organic paradigm arise sub-paradigms in different knowledge sectors. These sector include political science, sociology, engineering, economics, law, finance, all physical, chemical, medical, biological and cognitive sciences in addition to others sub-fields that are to many to mention. In fact, this is one way of viewing the unity of science. The nature of the knowledge structure and its specification, the nature of cost-benefit structure and its representation and the nature of the motivation structure and its representation will vary from discipline to discipline and even from problem to problem in the same discipline. The structure of the representation of the theoretical or empirical model of decision-choice process of either explanatory or prescriptive type is defined and fixed by the parametric characterization of the knowledge structure that provides the linkage between the cost-benefit (goal-constraint) structure and the motivation structure in the organic paradigm. Nonetheless the organic paradigm remains the same for research on decision-choice activities in all areas. Even the critics of hypothesis of optimal decision-choice rationality find themselves working within this organic paradigm. As an illustrative example of the use of organic paradigm, let us take a look at the concept of rationality in economics and psychology.

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3.4 A Brief Reflection on Optimal Rationality in Economics and Psychology At this point a brief reflection on decision-choice rationality as applied in economics and psychological studies at theoretical and empirical levels is necessary [R12], [R12.8], [R12.10], [R12.12], [R12.16], [R12.18]. This brief reflection may also be extended into different scientific areas where the concept of optimal rationality is important. In broad scientific research, where best decision-choice actions are the focus of investigation, optimal decisionchoice rationality may be viewed as fixing the paradigmic boundaries of acceptable and competing theories for explaining or prescribing the best path of decision choice actions. The best path of decision-choice action is the path of optimal decision-choice rationality and the epistemic framework for theoretical and empirical works is referred to as the rational decision-choice paradigm in economics and political science. In the study of economic decisions, the specification of the knowledge structure has taken many different forms that include perfect and imperfect knowledge about the relevant variables and parameters. The cost-benefit structure or goal-constraint structure is specified in terms of conflicts in substitution-transformation process in the dynamics of input-output duality of resource space. The motivation structure is specified in terms of satisfaction, represented by direct or indirect utility function and measured in subjective or ordinal units. This may be viewed also as the incentive structure. The analytical structure allows the study of economic decisions to be posed as constrained optimization problem in other to abstract the path of optimal rationality that is a surrogate of optimal rationality as an attribute of decision-choice agents. Here, the optimization index is satisfaction defined by utility function whose arguments are real benefits, where real cost characteristics define the allowable region optimally rational decision-choice activities. This is the paradigm of optimal rationality in economic decision-choice process that conforms to substitution-transformation process with categorial conversion acting on the mutual negation of actual-potential duality. So far, this paradigm for the study of economic decisions and choices has been restricted to Cohort I with exact and perfect knowledge structure and Cohort II with exact and stochastic knowledge structure. The criticisms by psychologists against this economic paradigm are often misplaced. At motivation structure, the psychological concepts of satisficing or aspiration levels and others, to replace satisfaction as measured by utility are limited and have not been fruitful in general scientific research. This stems

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from the basic epistemic difficulty in specifying the exact or inexact conditions of individual and collective motivation such as satisficing or aspiration level in the decision-choice processes. As we have pointed out in the previous chapters, the psychological concept of aspiration or satisficing level is vague and brings into surface the conditions of sorities paradox or paradox of heap. The conditions of what is relevant or irrelevant in explanatory or prescriptive decision-choice theories must be viewed in terms of cognitive abstractions that will allow the construct of the derived category of optimal decisionchoice rationality as an ideal state of the path of decision-choice action. At the level of cost-benefit structure in decision-choice processes, in relation to substitution-transformation process, and in relation to mutual negation of actualpotential duality, under the action of categorial conversion, the psychological studies of the decision-choice activities have little to say if any. Substitution-transformation in the actual potential space is evolution or what the psychologists may call the manner in which decisions are made and brought into being by cost-benefit balances. The introduction of time as an important element in the decision-choice process introduces important complications about the nature of cost-benefit evaluations and assessment of the motivation structure. Here, we must distinguish between cost time and benefit time whose comparative impact changes the motivation structure. The experimental studies of decision-choice activities by psychologists fail to capture this basic reality. It would seem that the psychological studies are geared to examine the attribute of decision-choice agents in terms of the type of intelligence of rationality in the decision-choice process. This also may be the focus of cognitive sciences when it relates to decision-choice process. The economists assume decision-choice agents to be endowed with rationality as attribute in terms of Euler’s mini-man principle in the universal object set (that is: Nothing happens in the universe that does not have a sense of either certain maximum or minimum.). Given optimal rationality as an attribute of decision-choice agents, economic and other decision-choice theories are designed to explain or prescribe the ideal state of decision-choice process. A basic dichotomy emerges. The psychologists and cognitive scientists may be viewed as researching to find conditions of rationality in the brains and minds of decision-choice agents it seems. The economists and other decision-choice theorists assume the existence of optimal rationality as an attribute of decision-choice agents and research on the conditions of rationality as an ideal state as well as examine the possible deviations from the actual. The paths of dual studies are shown in Figure 3.4.1.

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GLOBAL OPTIMAL RATIONALITY

STUDY RATIONALITY AS AN ATTRIBUTE OF DECISIONCHOICE AGENTS

STUDY RATIONALITY AS AN IDEAL STATE OF DECISION-CHOICE PROCESS

ASSUMPTIONS FOR THE STUDIES

EXISTENCE OF DECISIONCHOICE AGENTS

PSYCHOLOGY AND OTHER COGNITIVE SCIENCES

DECISION-CHOICE AGENTS ENDOWED WITH THE ATTRIBUTE OF OPTIMAL RATIONALITY

ECONOMICS AND OTHER AREAS OF DECISION SCIENCES

THEORIES ABOUT DECISIONCHOICE ACTIVITIES

Fig. 3.4.1. Epistemic Paths of Studies on Decision-Choice Activities

Here the cognitive sciences can enlighten us and provide illumination about the intellectual attribute of decision-choice agents and whether the assumption of optimal rationality as attribute must be retained or altered and if it is to be altered then how do we contend with Eulerian principle of mini-max (that is: Nothing happens in the universe that does not have a sense of either certain maximum or minimum.) in the universal object set. The epistemic point here is that the concept formation, knowledge representation, the goal-constraint structure, the motivation and incentive structure are covered by the characteristics of vagueness and inexactness that generate philosophical paradoxes

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such as sorites paradox, in analytical deductions or penumbral regions of decision-choice activities when the classical paradigm of logic and corresponding mathematics are the available toolbox for problem formulation, solution, analysis and synthesis. The resolution of sorites paradox by the toolbox of fuzzy paradigm and analytical procedures of fuzzy rationality is provided in a companion volume [R17.24].

4

Epistemics of Risk and Optimal Decision-Choice Rationality

4.1 Introduction The concept and idea of risk are introduced and related to the decision-choice process in a companion monograph where the focus is on the character of risk as associated in applications of rationality to the construct of social information-knowledge structure [R17.23]. In Chapter One of this monograph, the concept of uncertainty and expectations are examined and indirectly related to risk and brought into the analysis of events. The concept of risk in these analyses, like that used in most theories on the subject as well as its linguistic use, is not defined. The definitions in dictionaries and encyclopedia are short of usefulness for scientific work. In general, it is taken for granted that its meaning is understood. The phenomenon of risk is everywhere around us and yet its meaning lives in obscurity. It influences behavior in decision-choice space by either providing positive or negative motivation for choice and yet it seems conceptually amorphous that presents us with some dilemmas and at times helps to generate phantom problems in general enterprise of knowledge production. The theories on risk are mostly based on probabilistic measures and reasoning, and hence related to information constraints on human perceptive understanding of events which is then linked to uncertainty as a reflection of lack of information fullness. As has been discussed, in a companion volume referenced above, probability in the classical sense views information as opposite of uncertainty in terms of that which we do not know. The phenomenon of risk is then viewed in terms of that which is not known (uncertainty) about decision-choice actions. By viewing the relationships of information, probability, uncertainty and risk in this way a framework is established for defining quantity of information as increase of degree of awareness (consciousness) and hence as the degree of reduction of uncertainty through the information processes. K.K. Dompere: Fuzzy Rationality, STUDFUZZ 237, pp. 83–104. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

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4.2 The Conceptual-Measurement Problem of Information, Uncertainty and Risk Since risk is associated with uncertainty in terms of the classical view, the degree of risk-lessness may be associated with quantity of information and uncertainty reduction. Thus the process of measuring quantity of information through probabilistic reasoning can be transferred to the measurement of level and degree of risk. In an epistemic sense, theories of risk then become theories of probability in some defined structure and in terms of theories of uncertainty that assume special form of statistical theory of information. Viewed in terms of theories of probability, and statistical theory of information, the theories about risk are merely concerned with risk measurements while theories of decision under risk are concerned with optimal decision-choice behavior relative to the presence of risk. In general, the two are integrated. The same analytical structure is found in theories of information where the treatment of the concept of information is neglected in the favor of a search of measurement of quantity of information. The efficient measurement of information becomes the central problem of information theory. The phenomenon of information is then viewed and defined in terms of quantity of information which becomes separated from the intrinsic meaning, content, relevance and other important qualitative characteristics of information which tend to impose usability structure on the acquired information. The central focus of risk theory is not different from that of information theory in that in the risk theory, the concern is about the measurement of the level and degree of risk that must influence the decision-choice process, the boundaries of acceptable risk as well as decision on quantitative risk aversion. Viewed in a broad general context of information-decision-interactive processes, risk theory is information theory that must relate to decision-choice theories. Basically, the statistical approach to information is analytically limiting and its extension to risk analysis carries with it the inherent logical limitations and difficulties. This framework of conceptual analysis applies to risk and information theories based on either subjective or objective probability with probabilistic reasoning to stochastic processes in decision or other science where judgment may be called upon for verification, conformation or corroboration. Science proceeds from perception, concept formation, meaning of concepts through explication, measurement of concept, abstraction of properties of the measurement, and defining the uses to which the measured concept may be put into the services of decision-choice actions. The concept of utility of in-

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formation, possesses meaning only to the user or group of users relative to the object of decision-choice action where uncertainty, risk and rationality may be defined. The utility of information proceeds a) from the viewpoint of the user (subjective), b) from the viewpoint of information content and, c) from the viewpoint of decision-choice process relative to goals and objectives. These three viewpoints allow us to examine the nature of quantitative risk and its effect on the decision-choice action. Just as the existing information theory defines information in terms of its measurability and the properties of information measures, so also the existing theories of risk define it in terms of measurability of risk and the properties of the measures of risk. The question, however, is: what is information and what is being measured? Similarly, what is risk and what is being measured? Related to these two questions is another question of what uncertainty is, and how the concept of uncertainty is perceived, explicated and measured? All these relate to probability with the question as to what is the concept of probability and what does probability

SOURCE

UNCERTAINTY/ CERTAINTY

INFORMATION

CENTER OF DECISION

RISK

PROBABILITY

OBJECT

Fig. 4.2.1. Relationship among Information, Uncertainty, Probability and Risk

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measure. The probe into these questions and the search for answers find meaning in the logical relationship among object information, source, uncertainty, risk, probability and center of decision choice action as represented in Figure 4.2.1. In view of this, we are confronted with a question: how do the concept of probability, its measurement and use relate to those of information, uncertainty and risk? We have discussed and defined the concept of information where we introduced the principle of variety in terms of characteristics in which information was viewed as an abstract reflection of properties in the universal object set; and that such abstract reflection is nothing but surrogate representation of the object presenting itself in the energy field [R17.23]. The definition of uncertainty was provided where uncertainty was divided into stochastic and fuzzy uncertainties. These may be further divided but the

SOURCE

UNCERTAINTY/ CERTAINTY

INFORMATION

CENTER OF DECISION

RISK

FUZZINESS/ POSSIBILITY

OBJECT

Fig. 4.2.2. Relationship among Information, Uncertainty, Fuzziness, Possibility and Risk

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two broad characterizations are sufficient for the point of discussion and the epistemic conception and clarity that we want pursue. Corresponding to the con-cepts of stochastic uncertainty and fuzzy uncertainty, we introduced the concepts of stochastic risk and fuzzy risk respectively in the previous chapters of this monograph. The relational roles of fuzziness, possibility, information, risk, source and object also take a similar structure as that of Figure 4.2.1. This is illustrated in Figure 4.2.2. The possibility, fuzziness and corresponding measures may be related to qualitative and subjective characteristics of uncertainty, risk and the center of decision that are missing in probability. The conceptual and analytical process presents an epistemic hexagon around the center of decision in which decision-choice agents must operate from information source to the object of decision-choice action. It is within this transformation process that uncertainties and risk tend to arise. Information is transformed to knowledge which appears as subjective or objective in terms of collective acceptance. In terms of a particular decision-choice element the information-knowledge structure may be full in terms of sufficiency of limited in terms wanting. Let us coin the terms full information-knowledge structure and limited information-knowledge structure. Both full and limited information-knowledge structures may appear to the decision-choice agent as exact or fuzzy (broadly defined) The two together presents us with information-knowledge deficiency which reveal itself as total uncertainty of stochastic and fuzzy types. The stochastic uncertainty leads to stochastic belief system that reflects stochastic risk while the fuzzy uncertainty leads to possibilistic belief system that reflects fuzzy risk. In this respect, what are the central foci of theory of decision under risk and the theory of decision under uncertainty? How both are related to the risk theory and under what set of conditions are they similar or different?

4.3 Epistemic Definitions of Uncertainty and Risk To answer these questions we may note that uncertainty and risk just like information are both part of descriptive language of commonsense and sciences. They also form part of the reporting language of knowledge production. Just like information and knowledge they are used without being defined or explicated. The meanings are always implicit and assumed to be understood in scientific analysis, thus leading to varying subjective interpretations of their implied results and their relational effects on decision-choice rationality. The

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search for relationship among uncertainty, risk and decision-choice rationality requires explications and explicitly epistemic definitions of both uncertainty and risk. Definition 4.3.1: Pure Uncertainty Pure uncertainty arises when the information available to the decision-choice agent is such that he is unable to answer any of the relevant questions that may be formulated regarding a prospect and decision-choice action. Partial uncertainty arises when only some of the questions can be answered. Definition 4.3.2: Risk A prospect in uncertain conditions is said to curry potential risk if at least one of the outcomes has undesirable effect that far outweighs the inherent benefit. Definition 4.3.3: Pure Risk Pure risk regarding a decision-choice action arises in those situations in which the available knowledge is such that the decision-choice agent cannot formulate any further questions from the information, if when answered would lead to alterations of decision-choice agent’s subjective weights that currently capture his or her possibilistic and probabilistic beliefs which are assigned to possible and probable outcome in a prospect that carries damaging consequences. Partial risk arises when only some of the questions can be answered. Some observations may be made about the two definitions in terms of their similarity and difference. As defined, uncertainty and certainty relate to degrees of information limitation (information deficiency). Pure uncertainty relates to no information and complete ignorance. Certainty relates to full information where pure uncertainty is assigned zero degree and pure certainty assigned a degree value of one. Between zero and one, we have deferent degrees of uncertainty and certainty that depends on the size of information relative to a prospect. Uncertainty always presents itself at the level of the potential and vanishes at the level of the actual with the substitution-transformational dynamics of actual-potential duality. The definition of risk needs some clarification. In every decision-choice action on a prospect we have risk, benefit and cost (damage). The risk, benefit and damage appear as potential ex-ante decision. The benefit appears as desirable outcome and cost (damage) appears as unwanted outcome ex-post decision. All risks are potential but not actual and all costs (damages) and benefits are both potential and actual. A prospect is said to contain risk if it is associated with undesirable outcome ex-ante decision-choice action in such a way

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that actual resources ex-ante decision-choice action will be completely at stake. The point here is, risk is associated with potential cost and damage in substitution-transformation process under the dynamics of actual-potential duality. Cost and benefit, while potential ex-ante, are actual ex-post decisionchoice action. Risk resides at the level of potential, and disappears by categorial conversion to cost, or to benefit at the level of the actual. Optimal decision-choice rationality must reconcile uncertainty, risk, cost and benefit in all decision-choice action from the potential to the actual. The complex natures of the concepts of uncertainty and risk in relation to decision-choice rationality require further epistemic examination. It must be noted that the definition of uncertainty is embedded in the framework of information under the principle of information sufficiency while that of risk is embedded in framework of knowledge in that uncertainty is an information process and risk is a knowledge process under the principle of knowledge sufficiency. This is an important distinguishing feature that also unites them by logical transformation. It is within this epistemic framework that theories of decision under uncertainty must be distinguished from theories of decision under risk. Within this framework, there are two important epistemic components of total uncertainty that present itself to a decision-choice agent where the degree of certainty is greater than zero but between zero and one. They are stochastic uncertainty that gives rise to probabilistic belief and fuzzy uncertainty that also gives rise to possibilistic belief. The differences and similarities have been discussed in Chapter One of this monograph. We shall solidify them by giving their epistemic definitions and contents. Definition 4.3.4: Stochastic Uncertainty Stochastic uncertainty arises when the information is limited in such a way that only exact answers can be given to some of the relevant questions that have been formulated by the decision-choice agent regarding the outcomes of a prospect. The prospect and its outcome are said to be exact information-limited and the decision-choice rationality is said to be exact information constrained. Definition 4.3.5: Fuzzy Uncertainty Fuzzy uncertainty arises when the available information, whether full or incomplete, is such that some answers to relevant questions that have been formulated regarding a prospect are vague and ambiguous. Pure fuzzy uncertainty arises when all answers provided to relevant questions are vague, ambiguous, approximate, and inexact and others. The prospect and its outcome

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are said to be fuzzy information-limited and the corresponding decisionchoice rationality is said to be fuzzy information constrained. Definition 4.3.6: Stochastic Risk Stochastic risk regarding a prospect in decision-choice action is that situation in which the available knowledge is such that the decision-choice agent can formulate some essential questions which when answered would cause him or her to change the distribution of weights that reflect his or here current probabilistic belief regarding stochastic outcomes of a given prospect. Definition 4.3.7: Fuzzy Risk Fuzzy risk arises in decision-choice action regarding a prospect in a situation in which the available knowledge is such that the decision-choice agent cannot formulate any further questions from the available information if when answered would make him or her change the distribution of weights that reflect his or her possibilistic belief regarding the outcomes of a given prospect. Observations 4.3.1 The definition of fuzzy uncertainty relates to all levels of stochastic uncertainty except pure uncertainty where there is nothing to be vague about. Stochastic uncertainty relates to limited exact information from the source while stochastic risk relates to limited exact knowledge regarding outcomes in a given prospect. Fuzzy uncertainty relates to ambiguity, vagueness and inexactness of information from a source while fuzzy risk relates not to limited knowledge but rather to ambiguities, vagueness and other elements in the full or limited knowledge that creates penumbral regions in relations to the outcomes of a prospect. The whole of the concept of risk may be seen in terms of the risk square that is necessary for full risk analysis as shown in Figure 4.3.1. In the theories on risk, decision under uncertainty and decision under risk, the tradition has been to go from potential to the possible and then to the actual in triangular mode by bypassing the possible and neglecting the fuzzy risk. In the final analysis of decision-choice actions, subjective judgment becomes the driving force of an action. Such subjective judgment is embedded in vagueness and ambiguities that place the decision-choice agent in a penumbral region that generates fuzzy risk. The relative dominance of stochastic and fuzzy risks is decision-choice specific. In some decision-choice situations, fuzzy risk dominates stochastic risk and hence requires fuzzy logic in reasoning.

4.3 Epistemic Definitions of Uncertainty and Risk

POTENTIAL THE POTENTIAL Complete Risk

Total Actual Risk ACTUAL Realized Belief

91

POSSIBILITY POSSIBILITYPossibilistic Possibilistic Belief Fuzzy Risk

Stochastic Risk PROBABILITY Probabilistic Belief

Fig. 4.3.1. The Uncertainty-Risk Square Relative to the Knowledge Square

Uncertainty and risk are cognitive characteristics of human decision-choice rationality. Both of them vanish where the decision-choice process is absent. Can we speak of uncertainty and risk without decision? Questions may be raised regarding the concept and the role of accident, so we shall discuss them and risk at a proper logical linkage. Alternatively, how relevant are information and knowledge at an absence of decision-choice process. Let us keep in mind that not deciding is also a decision of a Bernoulli type. In general, information and knowledge are sought to assist in improving decision-choice action in oder to set benefits against costs and minimize risk as defined. Every disaster or an unwanted happening has at least a potential evasive decisionchoice strategy and every risk is potential while every disaster is both potential and actual. The potential disaster is associated with risk ex-ante decisionchoice action. The actual disaster is associated with outcome ex-post decisionchoice action.

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4.4 Accident, Necessity and the Knowledge-Risk Squares Uncertainty and risk may be related to the information-knowledge structure in decision-choice process that generates uncertainty-risk squire. This uncertainty risk square may be related to accident-necessity construct of human experience and decision-choice rationality. As it has been argued in [R20.21], accidents are not characteristics of nature. Necessity and pure substitutiontransformations are the fundamental characteristics of the universal system. Accident appears as cognitive phenomenon due to various levels of ignorance that proceed from complete ignorance to a complete knowledge about an element in the universal object set as defined in [R17.23]. Necessity resides in every process in nature and human social formation which for all practical purposes is part of nature where necessity determines the outcome of the process in which it resides. Necessity relates to conditions of outcomes while accident relates to conditions of human knowledge. The search for information is an attempt to expose the necessity to perception while the search for knowledge is an attempt to understand the necessity through the receptor-processor structure and hence to explain the accident as well as reduce it through the applications of the set of optimal rules of decision-choice rationality regarding decision-choice action on a prospect. Necessity reveals itself to human cognition through series of accidents that may harbor possible risks of various kinds and in various degrees of intensity. The conditions of accidents lead to risk-benefit configuration where risk may be viewed as a component of cost. Information deficiency about the necessity hides from the decision-choice agent the distribution of possible outcomes through the creation of uncertainties about the substitution-transformation process that creates outcomes. The information deficiency also create knowledge deficiency about the range of possible outcomes and hence the accidents which are then translated into potential risks within the process. Necessity relates to uncertainty due to the principle of information deficiency, and accident relates to risk due to the principle of knowledge deficiency. At all circumstances of substitution-transformation process in the dynamics of actual-potential duality, the received information by decision-choice agents is incomplete and fuzzy, conditioned by the dynamics of past-present duality that has generated and transmitted it, and constrained by the means, methods and the logic of acquisition, processing and ideas formed to create and accept knowledge. The limited nature of information generates stochastic uncertainty while the nature of ambiguity and vagueness in the received information gen-

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erates fuzzy uncertainty. The two together generate total uncertainty in the system’s dynamics. The processing of the limited and fuzzy information with incomplete logical tools and faulty reasoning leads to limited and fuzzy knowledge. The conditions of fuzziness rests on receiving some information without which one will be unable to even assess the nature of fuzziness contained in it. If one receives information about substitution-transformation process and logically processes this information into acceptable knowledge about the working mechanism of a particular substitution-transformation process and, able to state not what the observed facts from the information are, but to provide explanation to them so one can confirm the necessity and accidents that will motivate applications of decision-choice rationality in order to avoid or minimize the risks associated with the distribution of potential outcomes. Risk resides in accident, accident resides in necessity and necessity resides in substitution-transformation process which in turn resides in potential actual polarity. It is human limitations in understanding the conditions of necessity, and how necessity envelopes series of accidents, that define the profile of distribution of risks in the decision-choice process regarding possible outcomes of prospects. So far, we have provided definitions for information, knowledge, uncertainty, and risk. The epistemic question within the process of linguistic explication is simply: What is necessity and what is accident? We may now define the concepts of necessity and accident. Definition 4.4.1: Necessity Necessity is a set of characteristics in substitution-transformation process that generates conditions which result in actualizing a particular potential but not other wise in the dynamics of actual-potential duality. The outcome is a necessity of conditions of categorial conversion Definition 4.4.2: Accident Accident is composed of characteristics in substitution-transformation process as well as in the organic distribution of possible outcomes of dynamics of an actual-potential duality, generating condition whose knowledge is obscured from the decision-choice agent regarding the operations of the necessity. It may be noted that necessity is a condition of categorial convertibility. It is an enveloping of a series of accidents inherent in the process unity and trajectory. Not every accident is an undesirable event. Every potential outcome of an event has potential risk, benefit and cost. The outcome is said to be undesirable if the combined risk and cost outweigh the benefits. The definitions

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of necessity and accident suggest that while necessity is an enveloping of a series of accidents they are mutually determined. In categorial conversion process the necessity forms the primary category while the accidents constitute derived categories of the transformation process. In this way, necessity and accident are sufficient condition to ensure transformation to particular outcome that actualizes the inherent potential. It is from conditions of accident and necessity that cognitive expectation expresses its impact on the structure of decision-choice rationality. We have discussed the cognitive role of expectations on the construct of decision-choice rationality [R17.24] In the view of the structure of the analysis, categories of risk are defined relative to categories of lack of levels of knowledge which relates to internal accidents that are established by information-knowledge conversion process under categories of uncertainty, ex-ante the decision-choice action, to asset the necessity in events. The understandings of necessity, as it is related to information-transmission process leading to uncertainty, and the understanding of accident as it relates to knowledge-creation process leading to risk, are related to decision-choice rationality. This decision-choice rationality propels us to seek information and process it to develop a knowledge structure on the basis of which expectations are formed to reduce the chance of a particular outcome from emerging. In this respect, risk arises from the general operations of causality of the dynamics of actual potential transformation process of nature and society in terms of their conflicts and arrangements.

4.5 Risk, Freedom and Decision-Choice Rationality Rationality implies freedom of action in the decision-choice space constrained only by resource availability. Resource availability is defined to include capacity to process information to create knowledge in addition to other relevant elements in the universal object set. However, necessity implies a process in nature and society where conditions are such that a particular outcome is inherent in any substitution-transformation process through a series of accidents to effect categorial conversion. Every substitution-transformation process in nature and society function through a set of conditions that guide its outcome as a necessity but nothing else. The outcome is maintained by a set of conditions that propels the necessity. There are many possible outcomes associated with a particular substitution-transformation process, but there is only one possible outcome associated with a necessity. The distribution of possible outcomes from the substitution-transformation process presents a distribution

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of cost-benefit-risk configurations. Some of these possible outcomes may be associated with cases where the risk plus cost outweighs the inherent benefit where some may have the inherent benefit to outweigh the cost plus risk. The act of knowing the distribution and hence the accident and risk requires knowledge. A particular outcome can only be changed by changing the set of the underlying conditions to a different set. This will alter the distributions of accidents in order to bring into place the desired outcome. Changing the conditions of substitution-transformation process merely alters one necessity to another with different inherent outcome. Changing the conditions require a search for information about the existing necessity, the necessity corresponding to the desired outcome; and a search for knowledge about the underlying accidents that will lead one to understand the distribution of risks associated with the distribution of the possible outcomes. A search for knowledge, therefore, is a search for information about the conditions of the necessity and the understanding of the underlying accidents where the obtained information is cognitively transformed into knowledge and understanding of a particular substitution-transformation process with a particular necessity and the inherent outcome. The question that arises is: can this understanding from the obtained knowledge be used to produce a set of conditions that will propel the existing substitution-transformation process to change course and bring about a different necessity that will in turn bring about an actualization of the desired potential? Let us keep in mind that every possibility is a potential but not every potential is a possibility relative to a prospect. The question may be directly stated in terms of necessity, freedom and decision-choice rationality. In substitution-transformation processes under the dynamics of actual-potential dualities, do humans, acting on decision-choice actions, have freedom under necessity in determining the outcomes since the conditions of the necessities assert the inherent outcomes? How free are decision-choice agents in exercising the rules of optimal decision-choice rationality? 4.5.1

Freedom under Uncertainty and Risk

To answer this question, regarding freedom of decision-choice agents to exercise decision-choice rationality under conditions of uncertainty and risk, we must relate the concepts of necessity, accident and outcome to causality. The substitution-transformation process harbors the laws of causal relations that depend on interrelationships of series of accidents to connect the necessity to the outcome of the possible potential through categorial conversion. The cau-

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sality is generated by the internal conflicts or instabilities in the substitutiontransformation process that give rise to the conversion moment essential to assert the necessity and accidents that will actualize the possible potential. Such a possible outcome may be desirable or undesirable, and if it is undesirable, in the sense of producing potential disaster where total cost outweighs the total benefit, then the risk is defined. Can such risk be avoided through decision-choice action of decision agents? The avoidance or aversion of any risk on the basis of rules of decisionchoice rationality depends on knowledge of both internal necessity and accidents in the substitution-transformation process. By acquiring information about the necessity and process it into knowledge of the working mechanism one gains profound understanding about the accidents and hence possible risk. This understanding may then be utilized through the principle of decisionchoice rationality to avoid or minimize risk associated with accidents. Similarly, the knowledge obtained offers a possibility and probability to help alter the conditions that a particular necessity operates. It is here that freedom resides in the necessity. It is also here that decision-choice rationality presents itself in freedom. In this process, freedom is not independent of necessity in the substitution-transformation processes. In this respect, freedom in nature and society is a derived category from necessity that envelops the accidents in the substitution-transformation process. In reference to freedom and applications of optimal decision-choice rationality, necessity is a primary category while accidents and risk and human will are derived categories of reality. Risk is always a potential that resides in necessity and accident. One can only understand risk by acquiring information to reduce uncertainty about the necessity and acquire knowledge to understand accidents and hence risk. Decision-choice agents are always exposed to risk as a potential phenomenon and further exposed to risk as a possible phenomenon to the extent to which they lack information about the necessity and knowledge about the accident. The acquisition of the relevant knowledge offers decision-choice agents freedom to deal with accident as a by product of necessity and its inherent outcome. Within the freedom to exercise the logic of decision-choice rationality on the basis of knowledge, three possibilities are opened to the decisionchoice agent in dealing with risk in the potential-actual spaces. The decisionchoice may: 1. do nothing and face the consequences of the potential risk when it is actualizes within the dynamics of cost-benefit duality. This is nothingness strategy in dealing with risk;

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2. intervene in the conditions that maintain the inherent necessity of the substitution-transformation process in order to produce a different necessity whose outcome is of minimum risk in the sense that the benefit outweighs the risk plus cost. This is intervention strategy in dealing with risk. 3. avoid the risk by avoiding the encounter with the potential outcome and hence the associated the potential risk. This is avoidance decisionchoice strategy. Just as every decision-choice action carries with it both costs and benefits so also it carries with it risk which may be subsumed under cost in general. This allows us to view decision-choice actions in general, not only in terms of costbenefit frontiers, but in terms of risk-benefit configurations that instruct the direction of the decision-choice rationality in the information-knowledge restricted domain. The nothingness strategy in dealing with risk is uncharacteristic of human action even in the case where the constraints are over-binding in such a way that all options are closed and where the potential risk outweigh the potential benefit and the consequences must be faced. Even here, the decision-choice agent will act under rationality to minimize the consequences. This situation may be referred to as over-binding necessity, but makes freedom meaningless in optimal decision-choice rationality relative to accident and necessity. The intervention strategy is to use the information about the necessity, process it to obtain knowledge about the accident in the possible outcome and then intervene if the distribution of the risk-outcome structure is undesirable. The essential logic is to intervene in the conditions that create the sequence of accidents by consciously creating a new set of conditions with judicious application of the knowledge acquired under demands of decision-choice rationality. Intervention can only take place to change the conditions of the necessity which will then generate a new series of accidents through internal conflicts. The new conditions and accidents are to spin a new possibility space whose desired risk-outcome element will be the most probable. It may also be kept in mind that the intervention process, by changing the conditions of the necessity mainly switches one necessity to another necessity whose probability of the possible is increased to asset the desired potential against the unwanted actual. It is here that human freedom defines itself within the necessity and the corresponding series of accidents and the risk-outcome element through the information-knowledge structure and decision-choice rationality.

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Here, arises the principle of compatibility between necessity and human freedom where the necessity is the essential precondition for human freedom in dealing with uncertainty, accidents, risk, benefit and costs that constitute the essential ingredients of categorial conversion from the potential to the actual through the principle of causality. Here, information becomes the basic ingredient of perception of the structure of the necessity and uncertainty that it generates; and knowledge about its internal conflict dynamics provides the ingredients for the understanding of accidents, and risk-outcome configuration within the uncertainty that provides powerful instrument of intervention through the use of conditions and rules of optimal decision-choice rationality. The third available approach is the decision-choice avoidance strategy. In this case, the decision-choice action is to avoid risk and forgo the distribution of its cost-benefit outcomes. This situation corresponds to the case where the decision-choice agent has information about the necessity, but does not understand its causal relations with the accidents and risk-outcome configuration. The decision-choice agent is unable to intervene in the structure of the necessity because there are serious cognitive limitations that restrict the understanding of the accidents and the causal relations to the risk-outcome structure. Here, the decision is not to participate and to avoid the risk-outcome element when the potential is actualize. In economic analysis of cost-benefit distribution, this is called with-and-without decision-choice problem. The ranking of conditions of certainty of the actual is given a greater weight than the conditions of uncertainty even if the potential has risk-benefit configuration that may be judged to exceed the certainty structure of ex-ante decision-choice action. 4.5.2

The Principle of Compatibility of Necessity and Freedom

To examine the principle of compatibility of freedom and necessity, and the decision-choice rationality that they may induce in the categorial conversion from potential to the actual in ex-ante and ex-post behavior respectively, let us examine the three cases a little closer. Case one and three involve nothingness strategy and risk-avoidance strategy which characterize attitude to risk-taking and risk-avoidance in conditions of decision-choice principle. Case two, involving intervention strategy, characterizes risk engineering or risk planning. The risk taking and avoidance principles epistemically assume that necessity and accident are non-interventionable in the sense that they are external to the cognitive operations of the decision-choice agent and hence must be accepted

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as they are. The distribution of risk-outcome configurations are mandated by forces completely outside the human will. In this case, the objectives of decision-choice agents are simply to decide on whether to take risk or not, and when risk taken is selected, then to decide on what risk taking and what to avoid. Here, human freedom in necessity is expressed not in terms of ability to change the distribution of risk-outcome elements in the possibility set but, rather in terms of organizing the subjective preferences over the risk-outcome distribution that is associated with a particular necessity. The usefulness of search for information about the necessity and the search for knowledge about the inherent accidents is merely to assist the decisionchoice agent to shape subjective preferences that may be expressed over the risk-outcome distribution for a particular necessity and for the application of conditions and optimal decision-choice rationality. In this respect, given a decision-choice agent’s subjective preferences, risk-taking and risk-avoidance depend on information-knowledge structure and behavioral rules established by the conditions of optimal decision-choice rationality. Here, rationality is defined over probabilities, given the possible potential risk-outcome structure and hence in the probability space. The explanation of the decision-choice process in this respect, will depend on how well behavior conforms to the

Rationality

Decision

Choice Action System

Freedom

Necessity

Optimality

Fig. 4.5.2.1. Epistemic Structure of Decision and Choice in Necessity and Freedom

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conditions of the decision-choice rationality. Freedom in necessity is thus expressed only at the level of the probability space. The cognitive process is to reconcile the conflicts in rationality-necessity-freedom structure with optimality-decision-choice structure to define decision-choice action on the riskoutcome configuration. The interactions are shown in Figure 4.5.2.1. The decision finds expression in necessity, and choice finds expression in freedom while freedom is expressed within the internal accidents that define the structure of risk-outcome configuration. Necessity and accidents reside in the substitution-transformation process while freedom and rationality resides in decision-choice activity, all of which define the action system toward the actualization of the possible potential in the risk-outcome space. Let us turn our attention to risk engineering. The nature of the case of risk engineering is extremely interesting in its epistemic sense. Here, freedom in necessity is taken to an important level of cognition where it is not only about information-knowledge acquisition to define preferences over risk-outcome distribution but to intervene in the conditions of necessity in order to alter the distributions of the accidents in the particular necessity and thus to create a different necessity whose possible potential risk-outcome is desirable. Here, decision-choice rationality is first expressed in the possibility space where the possible potential risk-outcome elements are rearranged to tilt the distribution of the internal accidents in favor of a necessity whose inherent outcome is that which is most preferred. In risk engineering process, there is a rejection of a particular necessity by human freedom in exercising the decision-choice action to rearrange the possible potential risk-outcome distribution. This is the first principle of freedom in necessity in risk engineering. The usefulness of a search for information about the necessity and a search for knowledge about the inherent series of the accidents is not only to assist the process of defining the preferences over the possible potential riskoutcome configuration in the probability space but rather to intervene in the possibility distribution over the potential elements in order to alter the distribution over risk-outcomes in the probability space. In this respect, given human innate endowment of optimal decision-choice behavior in the sense of Euler’s mini-max principle, risk engineering depends on information about the necessity, knowledge about accidents and behavioral rules about both the possibility and probability spaces. Subjective preferences are first defined over the potential elements that will allow the construct of the possibility space. From the set of possible potential risk-outcomes another preferences are then established over the set of probable possibilities for a decision-choice action over the probable risk-outcome distribution. In risk-engineering, free-

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dom of human decision-choice rationality is exercised in the possibility space that define the fist principle of freedom in necessity. The freedom in necessity is again exercised in probability space defining the second principle of freedom in necessity. In all these, it is observed that information-knowledge process shapes human freedom in necessity leading to a defined decision-choice principle in human actions under conditions of uncertainty and risk. The principle of compatibility of freedom and necessity in human action is a process whereby the development and use of information-knowledge structure present a process where the potential is set against the actual in accordance with human will and decision-choice rationality through substitution-transformation processes in the dynamics of the actual-potential duality (this is the necessityfreedom compatibility principle). In the three action cases that we have presented, the decision-choice agent has a role and action to undertake. The nature of the role and action are performed with a belief system that may be justified or unjustified in the sense that has been specified and discussed in a companion volume [R17.24]. The belief system that supports the role and action appears as a set of assumptions that helps to define the decision-choice environment that humans operate. There are two belief systems that define human vision in decision-choice action under Euler mini-max principle of human intelligence in the substitutiontransformation process in the dynamics of actual-potential duality whether one proceeds from methodology of logical constructionism of that of reductionism. They are possibilistic belief and probabilistic belief of the riskoutcome distribution in the necessity and accidents that are the basic characteristics of the internal structure of the substitution-transformation processes. The two belief systems are embedded in two types of uncertainty of fuzzy uncertainty and stochastic uncertainty. The relational structure among the belief systems, uncertainty systems and the human will to alter some potential elements to be actualized by the use of information-knowledge process defines human freedom in the substitution-transformation process. In both do-nothingness and avoidance decision-choice strategies that decision-choice agent is assumed to know the possibility distribution of the riskoutcome configuration in accordance with information on the necessity and knowledge of the accidents which are taken as immutable in the sense that the necessary possible potential risk-outcome will be actualized. The potential space is taken to exist and unimportant for decision-choice analysis. The concern is on what the possibility space presents from the potential space. The possible potential risk-outcomes are under stochastically uncertain conditions that shape the belief system under knowledge constraint to define the cognitive weights assigned to the various elements in the possibility space as to the

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probabilistic belief of their individual actualization. The knowledge of the accidents is then used to construct probability distribution over the assumed elements in the possibility space that contains the necessary risk-outcome. Here, only one belief system is of relevance regarding the actualization of the possible potential risk-outcome. The lack of certainty about the element to be actualized is stochastic uncertainty which is due to incomplete knowledge that hides from the decision-choice agent the risk-outcome’s necessity. The same knowledge provides the decision-choice agent the freedom under conditions of uncertainty for implementing decision-choice action in accordance with decision-choice rationality. Freedom in necessity is only defined in the stochastic decision-choice space. This is the necessity-freedom compatibility principle II in the substitution-transformation process in the dynamics of actual-potential duality. In the intervention decision-choice strategy to deal with risk-outcome configuration, however, the decision-choice agent assumes the existence of the potential space or the universal object set to be outside his or her control. The acceptance of the universal object set as given and beyond human intervention is based on the belief that the elements in the universal object set or the potential space is construction outside human cognitive abilities and that the decision-choice agents are part of the universal object set but not external to it. However, there is the belief system that project notion that the possible potential risk-outcome elements that affect human life can be constructed from the universal object set, that is from the potential space. This belief is the possibilistic belief that creates the vision of freedom to intervene in the possibility space to create new uncertain conditions that will lead to the restructuring of the potential risk-outcome distribution. The corresponding freedom is the necessity-freedom compatibility principle I in substitution-transformation process in the dynamics of actual-potential duality. The concern here is on what potential elements must be included in the possibility space when human freedom is being exercised. The decision is clouded with uncertainty. The uncertainty in this space is the fuzzy uncertainty that is due, among other things, to the cognitive vagueness of the universal characteristic set, human linguistic structure and explication that may be required of knowledge production. Thus, the fist human freedom in necessity is defined in the possibility space under the limitations and applications of the conditions of decision-choice rationality. The decision-choice agents, in this respect, externalize themselves from the possibility space to which they belief they can exercise some control and assert their freedom in the necessity. From the information about the existing necessity and the knowledge of the corresponding accidents, the possibility space is intervened to change the conditions so as to redefine the risk-outcome

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elements in the possibility space that will produce a new necessity with its information structure and a corresponding set of accidents with its knowledge structure. All what human will can do, in risk engineering, is to change the conditions of the necessity in order to affect the internal conflicts that generate accidents on the basis of which a preferred potential risk-output outcome may be actualized. The information about the new necessity is used to produce knowledge about the corresponding sequence of accidents and the formation of a corresponding probabilistic belief structure. The use of the information on the new necessity and the use of knowledge about the accidents in order to implement decision-choice action under rationality is the necessity-freedom compatibility II in the substitution-transformation process. The logical paths of the theories of risk taking and decision-choice rationality and their relation to necessity-freedom compatibility principles are presented in Figure 4.5.2.2.

THEORIES OF RISK TAKING

Fuzzy Risk

Stochastic Risk Uncertainty

Possibilistic Necessity-Freedom Compatibility I

Probabilistic Action System

Necessity-Freedom Compatibility II

Stochastic

Fuzzy

Belief Fuzzy Decision-Choice Rationality

Stochastic DecisionChoice Rationality

DCISION-CHOICE RATIONALITY

Fig. 4.5.2.2. Epistemic Structure of Rationality and Necessity-Freedom Compatibility Principle in Substitution-Transformation Process

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The nature of the Figure 4.5.2.2 is that at the center is the action system by decision-choice agents dealing with two superimposed pyramidal structures of uncertainty-fuzziness-stochasticity pyramid on one hand, and belief-possibilistic-probabilistic pyramid on the other hand that define the range of environment of decision-choice action as well as the structural interactions between decision-choice rationality and the class of theories on risk taking that is constructible. Out of the pyramidal interactions and decision-choice action, emerges the use of the information-knowledge structure to establish freedom in necessity.

5

Reflections on Some Decision-Choice Theories on Uncertainty and Risk

Some reflections on theories of decision-choice activities under uncertainty and risk will be useful at this point. The theories may be grouped into those that are concerned with probabilistic belief, stochastic uncertainty and stochastic risk on one hand and those that are concerned with possibilistic belief, fuzzy uncertainty and fuzzy risk on the other. Our reflections will mostly focus on the former since it encompasses the bulk of the analysis. In each case, however, we shall suggest the entry point of the theories and the possible contribution on the aggregate and how the theories deal with uncertainty and risk in the substitution-transformation process in the dynamics of actual-potential duality. Let us keep in mind that the knowledge enterprise or the search for knowledge is about the understanding of the accidental processes in the behavior of the necessity as well as about what is not known in universal object set in order to find an explanation to what there is (the actual) and how what there is comes to be. This reflection on risk and uncertainty apply to economic and financial risks as well as general risk viewed in terms of cost-benefit balances. The usefulness of acquired information-knowledge structure to decisionchoice agents, lies in the understanding of some elements in the potential space, how some potential elements become possible elements whose underlying conflicts give rise to the necessity and corresponding elements for the actualization of a potential that gives rise to the problem of the understanding of the behavior of the probable and actual. This is a complex and complicated process. Our information-knowledge structure for dealing with the task of understanding the actual-potential transformations is incomplete, expanding and even shrinking sometimes leading to formation of probabilistic belief and the creation of stochastic uncertainty and risk associated with the necessity. Our information-knowledge structure in handling the task of understanding, is not only incomplete but in many cases vague, ambiguous and inexact leading to K.K. Dompere: Fuzzy Rationality, STUDFUZZ 237, pp. 105–132. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

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possibilistic belief and the creation of fuzzy uncertainty and risk that are in simultaneous association with the necessity. The two situations are further made difficult by cognitive limitations of decision-choice agents regarding information processing and appropriate logical calculus. All these lead to the basic notion that necessities are not fully comprehended and associate risks elude decision agents. Under this epistemic frame, how do theories about decision-agents agents help us to understand their behavior under uncertainty and risk? We shall turn our attention to the classical theories.

5.1 The Classical Decision-Choice Theories under Stochastic Uncertainty and Risk The classical theories of decision-choice behavior under uncertainty and risk are merely about stochastic uncertainty and stochastic risk. The general epistemic framework is presented in Chapter Four of this monograph where the decision-choice intelligence under uncertainty and probabilistic belief is referred to as exact classical stochastic rationality. While there are many versions and variations in this group of theories, all of them take their entry points from the epistemic structure of classical decision-choice theories with exact classical non-stochastic rationality. Complete information-knowledge structure that provides certainty in the decision-choice process is replaced with defective information-knowledge structure that creates uncertain conditions which then affect the distribution of stochastic beliefs. The epistemic logical structure of decision-choice action under certainty and exactness is then transformed to decision-choice action under uncertainty where behavioral conclusions are drawn with probabilistic logic. From this entry point, the classical theories proceed from a fundamental assumption of a given possibility space on the basis of which a probability space is constructed and an acceptance is made of the probabilities that represent the weights of probabilistic belief. The conclusions follow probabilistic logical inference. By some modifications, this classical approach is extended to classical theories of risktaking. Technically, they are the same or there are very little differences between the classical theories of decision-choice action under uncertainty and those under risk. The probability space is viewed as both uncertain and risky. It is uncertain because of deficient information structure and risk of variations in cost-benefit distributions of the outcomes associated with events. The main concern of these classical theories is not about how necessity and accidents are formed in substitution-transformation processes from the space

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of potential to the space of the actual, and how they may be understood as well as influence human decision-choice actions or how human decisionchoice action may influence the substitution-transformation process. The concern is about the formation of stochastic belief system that will support decision-choice action on the postulated elements in the possibility space. The belief system is constructed on the postulate of deficiency in the informationknowledge structure for decision-choice action. Given the belief system, the classical theories under uncertainty and risk seek as well as examine the optimal rationality whose rules of behavior provide conditions of optimal decision-choice action and how such conditions relate to risk-taking or riskavoidance behavior. Two important epistemic problem elements enter into the examination of the decision-choice actions on the distribution of the possible risk-outcome elements in the possibility space. The first one is the problem of quantification of the distribution of the stochastic beliefs associated with different possibilities. The second problem involves the use of the quantified stochastic beliefs in addition to the possible risk-outcome elements to examine the motivation behind the decision-choice action of decision-choice agents given the relational structure of necessity and accident. It is here that goals and objectives of decision-choice agents enter to determine the best decision-choice action on the probable elements. The solution of the first problem leads to a logical transformation of the possible to the probable; in a sense, a transformation of the possibility space to a probability space. The solution to the second problem requires a logical linkage between objectives and possible potential outcomes before the event is actualized. It is observed that goals and objectives of decision-choice agents are shredded in utility function or some index of satisfaction that allows the assessment of degrees of preferences associated with the various possible potential outcomes of an event. These units of utility assessments are then weighted by their quantified values of the associated stochastic belief. A question immediately arises as to how to use the weighted utility values to project decision-choice behavior. The answer to this question is constrained by the mathematical properties of the probability space as well as the logical boundaries imposed by the Aristotelian logical inference for both methodologies of constructionism and reductionism. In respect of this, many suggested criteria have been made including maximum weighted utility value, maximum expected utility, minimum variance, mean variance and others, all of which have their roots in the probability space and the probabilistic logic where mathematics of expectations takes on a prominent stage leading to expected

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utility hypothesis where the expectations are defined in terms of weighted average of the utility values and where the decision-choice action is to select the outcome that optimizes the expected value or some risk index. A closer look at the classical approach to the theory of decision under risk reveals that the decision-choice agent is assumed to have deficient information-knowledge structure regarding the necessity and accident in the possibility space. The possible risk-outcome configuration is, however, fully known. Even though all the possible outcomes are known and the probabilities are defined, the necessity is unknown and hence every possible outcome is a candidate for actualization. There is an implication that the decision-choice agent can select either risk or an outcome in exercising freedom in necessity. The interrelationship is such that risk and outcome reside in mutual unity in terms of their conceptual relation to decision and choice. In the deciding process the relevance of each outcome to a decision-choice agent is captured by a satisfaction or utility value. The accompanying risk is indirectly captured by the belief structure as measured by probability. The concepts of event, outcome and risk have been introduced as if we have a clear understanding as to what they mean. This is also true of their uses in the literature on uncertainty and risk in decision-choice process. The literature has limited their implied meaning and uses to economic and financial decision-choice processes to the neglect of its general meaning defined in terms of possible potential cost-benefit balances. To understand theories on risk and decision-choice actions, we must ask a question as to what risk is. To answer the question as to what risk is, we must understand what an outcome is, and to understand the concept of outcome we must have a clear definition of the concept of an event. We shall define all these concepts in the broad general framework of substitution-transformation process and cognitive interpretations of categorial conversions of potential to the actual. The cognitive interpretations must be viewed in terms of human enterprise of information acquisition and knowledge production and deficiencies that may come to characterize the information-knowledge structure. Definition 5.1.1: Deficient Information-Knowledge Structure An information-knowledge structure is said to be deficient if it is incomplete, or fuzzy, or both, in relation to a phenomenon where incompleteness implies limited information about the necessity and limited knowledge about the accidents within the phenomenon; and fuzziness encompasses all characteristics that deprive some elements in the information-knowledge structure from exactness. It is said to be non-deficient or perfect if it is complete and exact.

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Definition 5.1.2: Event An event is a process encompassing of the interplay of necessity and accident of actualizing the potential element in the possibility space whose outcome distribution is assumed to be known or given to, and whose necessity is unknown to the decision-choice agent due to deficiencies in the informationknowledge structure intended to support decision-choice action. The essential defining factor of event is a composition of many possible outcomes and a deficient information-knowledge structure about the necessity and accidents that create conditions of decision-choice ignorance. The deficient information-knowledge structure may be due to limited information-knowledge structure or fuzzy information-knowledge structure or both where fuzziness encompasses many elements including vagueness, ambiguities, hedging and many others that create inexactness in the information-knowledge structure that supports a degree of belief in undertaking decision-choice action. Definition 5.1.3: Potential Outcome A potential outcome in the possibility space is an element of the set of accidents as well as a candidate of the necessity. It presents itself to the decision-choice agent in terms of potential cost-benefit configuration where the characteristics of costs are undesirable and the characteristics of benefits are desirable. The decision-choice agents with their limited and vague informationknowledge structure form beliefs regarding the actualization of each possible potential outcome. This belief in classical theories of decision-choice behavior under both risk and uncertainty is viewed in terms of exact and limited information-knowledge structure which becomes represented as distribution of probability values over possible potential outcomes. The defining elements of potential outcome are event and relative benefit-cost content. Much of the theories of decision-choice action under uncertainty and risk are simply about the study of the behavior of decision-choice agents in stochastic space which is another way of saying that the theories are about choice under deficient information-knowledge structure. The fundamental characteristics of the probability space are: a) deficient information-knowledge structure in support of probabilistic belief, b) exactness of belief values and, c) clarity in concepts and exactness in probabilistic reasoning. Decision-choice actions are specified around an event which is merely defined in a stochastic space under a fundamental assumption of a given and known possibility space. How does risk-taking appear in this decision-choice setting? In order to understand the risk-taking and risk-avoiding phenomena we must explicate

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the concept of risk in relation to the concepts of event and outcomes as may be seen by the decision-choice agents. Every possible potential outcome of an event presents itself in the possibility space as a triplet that may be written as E = C ⊗ B ⊗ U = {e = ( c, b,β ) | c ∈ C, b ∈ B, β ∈ U

}

(5.1.1)

where E = A set of possible events with a generic element, e, that represents an outcome C = A set of possible potential costs with generic element, c, associated with each outcome B = A set of potential benefits with a generic element, b, associated with each outcome U = A set of probabilistic values with a generic element, β, associated with each outcome

The cost and benefit characteristics may be quantitatively or qualitatively defined or both. The epistemic analysis should be indifferent to the nature of the measurements used. Equation 5.1.1 has no meaning in nature without cognitive agents. The characteristics of benefits, costs, and belief acquire no meaning in substitution- transformation processes without cognitive agents. In the natural order of the universal object set, in which cognitive agents are elements, there are only substitution-transformation processes in the dynamics of the actualpotential duality operating under categorial conversions. There is no uncertainty, no benefits and no costs. Every categorial conversion is a transformation with either intra-categorial substitution or inter-categorial substitution in the universal object set that is infinitely closed under transformations with continual self-restructuring with the disappearance of the old and emergence of the new. Keep in mind that there is no risk but continual substitution-transformation processes in the economy of nature and in the universal object set. Whether a substitution-transformation process is risky or not depends on its interpretive impact as assessed by cognitive agents in terms of their cost-benefit balances. It is only at the presence of cognitive agents that some of these transformations and substitutions induce conceptual categories of benefits and costs where information-knowledge deficiencies present uncertainty and risk. An

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event E is then seen as a set of possible potential cost-benefit outcomes under uncertainty. Substitution-transformation activities in nature and universal object set are processes where one or more elements transformed to another element or elements in natural and social processes of what there is, what there is not, what would be and what ought to be. We may use the concept of the structure of information deficiency to define an uncertain event and the cost-benefit structure under uncertainty to define a general concept of risk. Definition 5.1.4: Uncertain Event An event E is a collection of outcomes and it is said to be uncertain if the information-knowledge structure in support of decision-choice agents’ beliefs of its necessity and accidents is deficient in such a way that the necessity (necessary outcome) is obscured from the decision-choice agent. Since the deficiency in the information-knowledge structure may be due to limitationality or fuzziness we have random events and probabilistic beliefs that correspond to limited information deficiency and fuzzy events and possibilistic beliefs that correspond to fuzzy information deficiency or randomfuzzy and fuzzy-random events with combined possibilistic-probabilistic beliefs that correspond to complex deficiency in information-knowledge structure due to simultaneous presence of limitationality and fuzziness. The classical theories work with conditions of exactness thus assuming away fuzziness that encompasses all kinds of deficiencies in the information-knowledge structure. As defined every event is seen in terms of degrees of deficiency in the information-knowledge structure. Definition 5.1.5: Risk A risk is a phenomenon in transformation-substitution processes whose possible potential outcome is cognitively viewed to have greater costs than benefits in the categorial dynamics of actual-potential duality as viewed within the universal object set. Definition 5.1.6: Risk Outcome An element, e of an event E , ( e ∈ E ) that is a cost-benefit outcome is said to be risk-producing if its costs outweigh its benefits for a given probability distribution that captures the stochastic beliefs of the decision-choice agents, otherwise it is said to be non-risk producing. Definition 5.1.7: Risky Event An event E , in a substitution-transformation process, is said to be risky to decision-choice agents if the possible potential outcomes are composed of a

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mixture of risk-producing and non-risk-producing elements of necessity and accidents in the categorial dynamics of actual-potential duality. Any decision-choice agent acting on a risky event is said to behave under risk or in risk-taking behavior. The decision-choice agent is said to be in riskaversion behavior if there are action tendencies to avoid the risky event E . The phenomenon of risk, its concept and its explication are cognitive that derive their meanings from cognition and the existence of cognitive agents. It is this observation that gives rise to the study of decision-choice behavior under uncertainty and risk in terms of decision-choice theories and experiments on cognitive agents. The decision-choice theories under risk must work on the combined relationships of the composite elements of costs, benefits, information-knowledge deficiency as have been discussed in order to design a logical system of understanding. Risk appears as either stochastic or fuzzy or both in nature from the given definitions and specifications. The stochastic risk is cognitively derived from probabilistic belief structure while fuzzy risk is derived from possibilistic belief structure. In relation to these discussions, let us examine the epistemic structure of classical theories of risk-bearing or theories of decision-choice action under risk. Given the event E , the classical theories of decision-choice action first assume the existence of exact probability measures without vagueness that capture the probabilistic beliefs attached to the possible potential outcomes in the possibility space. It is at this point that theories of probability and probabilistic logic become essential in cognition, decision-choice action and knowledge production. The supporting approach is to deal with the distribution of the degree of preferences associated with possible potential costbenefit outcomes where the cost and benefit characteristics may be viewed in terms of net or gross relative values. The degrees of preference is then defined by relative satisfactions and measured by utilities. It is also at this point that theories on utility, its existence, properties and applied and theoretical uses acquire epistemic relevance in examining decision-choice rationality under risk. At this juncture, all the discussions provided under non-fuzzy stochastic rationality hold for behavior under risk. The definition of the concept of risk that has been offered in this discussion is general. This general definition will acquire a modification under specific decision-choice situation and defined environment of action. Within the classical theories of decision-choice behavior under risk, the non-fuzzy stochastic optimal decision-choice rationality, as developed to explain or prescribe optimal rules of behavioral action in the non-fuzzy stochastic space, acquires different modifications of the maximum expected utility

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criterion. This criterion may be replaced by maximum expected value criterion or minimum risk criterion where risk is measured in terms of statistical variance. The statistical variance may be connected to mean variance criterion or maximum weighted utility where the weights are corresponding probabilities. Other variations of the maximum expected utility criterion may be developed depending on the requirement of the theory and the application to which it may be put. For theoretical works on theories under risk, or risk bearing, see [R21.1], [R21.5], [R21.16], [R21.17], [R21.11]. We may add that the epistemic framework of analysis of risk-taking behavior is equivalent to that developed under decision-choice theories of uncertain behavior. Any other claims are mere modifications within the classical paradigm and defined logical boundaries. It will become clear that the classical class of decision-choice theories, under uncertainty and risk, is a special class of all possible theories that may be constructed for decision-choice behavior under uncertainty and risk.

5.2 Principle of Insufficient Reason, Principle of Sufficient Reason and Uncertainty-Risk Decision-Choice Theories A note of epistemic relevance will be useful here in other to further elucidate the nature of the theories of behavior under risk and uncertainty. The guiding principles of these theories are to be found in the classical paradigm. Here there is interplay of a number of epistemic principles that establish the path of analysis to the derived conclusions. The core of these principles may be stated as: 1. Principle of sufficient reason; 2. Principle of sufficient belief in support of the principle of sufficient reason; 3. Principle of sufficient justification in support of the principle of sufficient belief; 4. Principle of sufficient knowledge in support of the principle of sufficient justification; 5. Principle of analytical sufficiency in support of the principle of the principle of knowledge sufficiency; 6. Principle of sufficient information in support of the principle of analytical sufficiency;

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7. Principle of sufficient cause that relates the conditions of accidents and links them to the necessity as their enveloping in substitution-transformation processes; 8. Principle of insufficient reason or indifference in support of both the principle of insufficient information; 9. Principle of excluded middle or Aristotelian logic for truth verification in support of the logical relationships of the above principles regarding conclusions; All these principles relate to the questions of what there is (the actual) and its knowability and what there is not (the potential) and its process to what ought to be (possible potential) and its process to what would be (probable) and its actualization. The epistemic problem is how these principles help in the understanding of risk and decision-choice rationality under risk. In this understanding, all the principles must constitute a logical unity in the enterprise of knowledge production and the decision-choice process. Let us observe that all axioms of probability and classical decision-choice behavior under uncertainty and risk relate to deficient information-knowledge structure and decision-choice rationality in human decision-choice activities in society and nature. The decision-choice activities are part of general substitution-transformation process where, from the primary category of reality, human decision-choice activities become elements of various derived categories. The degree of success of humans in accomplishing the various activities through the decision-choice rationality depends on the constructed information-knowledge structure in support of the understanding and hence of the belief structure of the behaviors of the primary and derived categories. The links between the primary category and derived categories are provided by intra-categorial conversions and inter-categorial conversions, both of which take place through the logics of constructionism and reductionism in the substitution-transformation process. Recall that in constructionism, given intra-categorial and inter-categorial conversions, we take as our starting point in the analytical construct, the postulate that the propositions that are formed to be valid about the derived categories must have their roots in the primary logical category. In reductionism, however, a logical stand in the analytical process is taken where we operate with a postulate that propositions that are accepted as valid about the derived categories must be completely reducible to propositions that are validly rooted in the primary category. How do these provide understanding of risk and the

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relationship between risk and decision-choice rationality? In both intracategorial and inter-categorial conversions in dynamics of actual-potential duality, there is a recognition of the existence of a difference between necessity and outcome on one hand and between damage and risk on the other. Risk finds expression in the potential damage while actual damage finds expressions in actualized possible potential, and realized risk finds expressions in cost-benefit configuration that is associated with actualized possible outcome. The categorial differences among the pairs find expressions in the substitution-transformation processes in the dynamics of the actual-potential duality. It may be observed that propositions about risk acquire no cognitive sense if the propositions about damage or cost are not meaningful. Alternatively, propositions about risk are valid to the extent to which propositions about potential damage are valid. Given the information-knowledge structure, the belief of existence of risk in decision-choice action is logically caused by the belief in potential damage which takes its causation from a belief in potential outcome whose cause is rooted in the belief surrounding the interplay of necessity, accidents and chance as generated by the categorial dynamics of potential-actual transformations. From the viewpoint of risk analysis and decision-choice rationality, the existence of actual-potential duality is taken as primary category of reality from which possible potential outcome is a cognitive derivative. The possible potential event is a set of possible potential outcomes from which one is a necessity that defines the enveloping of its accidents. The possible potential outcome becomes the primary logical category from which risk becomes a derived category of logical reality on the basis of which decision-choice rationality is formed. Notice that risk is a potential and will always remain so waiting to be actualized. It is a set whose membership size is the same as the size of the set of outcomes. The actual outcome is a derive category from the potential outcome which constitutes its primary category. The actual outcome is a singleton set that represents the necessity in the possible potential outcomes. The actualized potential outcome becomes the primary category of the actualized damage (cost) and benefit. The actualized cost-benefit relationship becomes the supporting justification of the belief in the risk in the possible potential. The transmission process of the sequential links of the primary to the interconnected derived categories is the substitution-transformation processes that provide enveloping paths from the primary category to the final derivative. All the categorial conversions derive their logical link from all or some of the nine principles stated above.

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For example in the decision-choice rationality there is always a sufficient reason why there are valid propositions that leads to some contingent existence of risk. This involves the principle of sufficient reason. The principle of sufficient reason is basically a derivative of principle of sufficient belief that constitutes a logical posterior to the principle of sufficient justification. The principle of sufficient justification is a derivative from the principle of knowledge sufficiency which is a derived logical category from the principle of analytical sufficiency which arises from the principle of sufficient information that leads us to the principle of sufficient cause. The principle of sufficient cause allows one to relate information on necessity and knowledge on accidents in the substitution-transformation processes, in the construct of the possibility space where by the principle of insufficient reason, the possible potentials, may be given equal possibilistic belief under conditions of vagueness and ambiguity. Finally, the truth verification process may proceed on the principles of either excluded middle (Aristotelian logic) or non-excluded middle (fuzzy logic or non-Aristotelian logic). All these are justifiable through the methodologies of constructionism and reductionism in substitutiontransformation processes that we have presented (see also [R17.22], [R20.21]. Whether any of the above principles may be restricted, or not, in terms of their structures and uses will depend on the nature of the both the actual and logical problem selected for analysis.

5.3 Conceptual Foundations of Fuzzy Risk and the Decision-Choice Rationality We now turn our attention to the concept of fuzzy risk in relation to fuzzy uncertainty. So far, there has not been explicit theoretical works on fuzzy risk and measures of fuzzy risk in general decision-choice processes even though substantial literature exists on fuzzy decision theories [R5], [R5.7], [R5.40], [R5.75], [R5.78], [R6]. Nonetheless, the theories of decision-making, under fuzzy uncertainty, may be related indirectly to decision-choice process under fuzzy risk. This relation is not different from that of theories of decision under stochastic uncertainty and stochastic risk. Our concern here is not on the development of the mathematical structure of fuzzy risk, but rather on providing an epistemic framework of the measurements of fuzzy risk and the theory of behavior under fuzzy risk. In the above definitions of uncertain event, outcome and risk, we argued that they appear as either stochastic, or fuzzy, or both. The classical theories on risk are mostly defined in limited but exact

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information-knowledge support for decision-choice actions. The logic of analysis is that of Aristotelian type within the classical paradigm. This logic fails, or experiences some inferential difficulties, when the deficiency in the information-knowledge structure is the result of fuzziness alone or both fuzziness and stochasticity that give rise to fuzzy uncertainty or fuzzy-stochastic uncertainty or stochastic-fuzzy uncertainty. To deal with the nature of fuzziness and its risk conditions we must exit from the classical paradigm whose reasoning process is governed by Aristotelian logical inference and exact information-knowledge structure that supports the classical decision-choice rationality. Fuzziness gives rise to penumbral region of decision-choice action where rational judgment is called upon. It is in this penumbral region that fuzzy risk is encountered by decision-choice agents. Thus, a decision-choice theory, which deals with choice under fuzzy risk, must involve a process of clarifying the conditions that hide the cognition of the decision-choice agent from complete illumination. The difficulty in seeing through the penumbral region is resolved through the manipulation of possibilistic belief. When we relax the classical assumption of exact probabilities, we encounter vagueness in thought process of the decision-choice agents in forming their probabilistic belief. Furthermore, the probabilistic thought processes based on the fundamental principle of classical law of reasoning that, there are only true or false propositions but not both must be replaced by the fuzzy logical principle that, propositions about nature and society have relative standing in truth and falsity, where this relative standing is a cognitive measure in degrees. The principle of duality is thus maintained. This is the framework that motivated a whole debate on the theory of decision making under vagueness, particularly when the probabilistic belief is purely subjective [R1I.4], [R16], [R16.10], [R16.11], [R16.36], [R16.43]. Here, the probabilities used in specifying some measures of risk acquire added problem in that we have to deal with vague or inexact probabilities that violate the classical postulate of exactness required by its logical system. The presence of penumbral regions in either full information-knowledge structure or less-than-full information-knowledge structure leads to three different types of fuzzy risks. These risks are: non-stochastic fuzzy risk that originates from non-stochastic fuzzy uncertainty; fuzzy-stochastic risk that corresponds to fuzzy-stochastic uncertainty and; stochastic-fuzzy risk that is the result of stochastic-fuzzy uncertainty. The information relational structure and the fuzzy risk-uncertainty structure may be seen in terms of pyramidal relations in Figure 5.3.1. The measure of a non-stochastic fuzzy risk may be constructed by the use of the membership characteristic functions. The meas-

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FUZZY UNCERTAINTY

STOCASTICFUZZY RISK

FUZZYSTOCHASTIC RISK PENUMBRAL REGIONS

STOCHASTICFUZZY UNCERTAINTY

FUZZYSTOCHASTIC UNCERTAINTY FUZZY RISK

Fig. 5.3.1. Fuzzy Risk-Uncertainty Relational Structure

ures of stochastic-fuzzy and fuzzy stochastic risks are complicated, due also to complications of their corresponding uncertainties that define overlapping penumbral regions, which have a tendency to give cognitive denseness in terms of intelligibility of applications of decision-choice rationality. Fuzzy analyses require an examination of whether the fuzzy-stochastic on stochastic-fuzzy uncertainties are additively or multiplicatively separable. The conditions of separability are discussed in another monograph [R17.24]. If the total uncertainties are separable then total risk is simply an indirect sum of non-stochastic fuzzy risk and stochastic risk. The indirectness refers to the notion that the summation operation cannot be done unless either the fuzzy risk is transformed into stochastic risk or the reverse. This requires that the fuzzy variable is either transformed to random variable or the random variable is transformed to a fuzzy variable. If the total uncertainties are not separable then we must deal with two complicated decision-choice spaces whose topological structures and corresponding properties have been discussed in[R10.54], [R17.24]. The decision-choice variables appear as random-fuzzy variable or fuzzy-random variable that complicates the penumbral region of decision-choice action and hence increases the risk. The fuzzy-random variable in the choice process leads to

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vague probabilities or fuzzy probabilities while the random-fuzzy variable leads to random membership functions. The theories of decision-choice actions under these types of risk require different methods of reasoning and risk computations. In general, there are two distinct categories of total risk. They are stochastic risk and fuzzy risk. Both of them require different analytical methods and principles of reasoning. The methods of analysis of stochastic risk and decision-choice behavior under it are provided by the classical paradigm whose decision-choice intelligence is the classical optimal decision-choice rationality in risk analyses. The fuzzy risk comes to us as three types of non-stochastic fuzzy risk, fuzzy-stochastic risk and stochastic-fuzzy risk. The method of analysis of all these types of fuzzy risk and the theories of decision-choice behavior under them are provided by the fuzzy paradigm whose decision choice intelligence is fuzzy optimal decision-choice rationality. The fuzzy optimal decision-choice rationality appears as non-stochastic fuzzy optimal rationality, stochastic-fuzzy optimal rationality and fuzzy-stochastic optimal rationality in risk analyses. The fuzzy logical approach in dealing with risk analysis and decision-choice process under risk where stochastic-risk analysis through probabilistic reasoning is a part of total risk analyses requires an integrated fuzzy logic, probabilistic reason and possibilistic laws of thought in substitution-transformation process under categorial dynamics of actualpotential duality. The epistemic framework provides us with an integrated approach to deal with total risk measures and decision-choice behavior under total risk. The philosophical and mathematical problems of decision-choice behavior under all kinds of fuzzy uncertainties are the challenges and also a frontier of theory of decision-choice behavior under uncertainty and risk.

5.4 Risk Engineering, Risk Bearing and Decision-Choice Rationality The above epistemic framework for risk analysis and behavior under risk takes risk as given and outside the control of decision-choice agents. The decision-choice agents can, however, reduce the conditions of risk and perhaps the size of risk measure by reducing uncertainty through reductions in incomplete and vague information-knowledge structure surrounding the necessity and accidents that cloud the nature of risk. The concept of risk engineering is, however different. It is introduced here for the understanding of two related phenomena that involve general understanding of risks associated with gam-

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bling or lottery and the functioning of social institutions, and risks associated with natural processes. We shall refer to the former as social risk and the latter as natural risk. Definition 5.4.1: Social Risk Social risk is simply that which is human made and may be associated with social institutions and their management, and where the necessity and accidents are generated by the interplay of human actions and design, in evolving social institutions, and where the decision-choice agents are confronted with a deficient information structure about the necessity and deficient knowledge structure about the accidents in social substitution-transformation processes of categorial dynamics of social actual-potential duality. Definition 5.4.2: Natural Risk Natural risk is that which results from natural processes due to deficiency in the information structure about the inherent necessity and deficiency in the knowledge structure about the internal accidents of a substitution-transformation process of categorial dynamics of natural actual-potential duality in the universal object set. 5.4.1

Social Risk Engineering

It may be said that much of the theory on risk and of behavior under risk and uncertainty is about social risks. These risks are engineered through social institutions that emerge through social formations. Some are intentional and, others are unintentional that emerge from interactions and interplay of social forces outside the control of individuals and groups. For example, lottery and games of social gambling belong to intentional social risk engineering the outcome of which is hidden from the gamers that must design decision-choice strategies to overcome the risk. Important example of unintentional social risk engineering is that associated with market system of resource allocation, production and distribution. The nature and the size of the unintentional social risk depend essentially on the institutionally structural relationships and the crafted rules that help the protective belt of the society. Since the set of the institutions, their structural interactions and the rules that govern and restrict individual behavior to conformity are human made, the associated risks of decision-choice actions are indirectly engineered. In both cases the nature and structure of the risks can be changed by social intervention through institutional restructuring and changes in social rules of the protective belt and internal interaction of social relations. In the intentional social risk engineering,

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the risk engineer creates the possibility space and hence has an access to relevant information about the necessity and knowledge about the internal accidents of the process. Given the engineered possibility space, the corresponding probability space is constructed. The information-knowledge structure in support of the necessity is not fully available to the gamers or lottery participants. Participation therefore is risk-taking by the individual in the hope that the internal necessity will conform to the decision-choice agent’s action. The decision-choice actions by an individual or a group either to participate or not in intentionally engineered risk system, involves risk-taking and riskaversion when there is no social rule to enforce participation. Here participation implies risk-taking and non-participation implies risk-aversion. These views on risk-taking and risk-aversion are different from the traditional uses but have similarity with them. Compulsory rule of participation deprives all decision-choice agents of risk-aversion decision leaving them with decisionchoice on strategies, the action of which will depend essentially on individual intuitive reflection on and belief in the supporting information-knowledge structure of the process regarding the necessity and chance of the gaming and gambling process. In our contemporary gaming and gambling, compulsory participation is not socially not mandated. This does not mean that we can rule out a rise of social situation where the bearing of intentionally engineered risk is legally mandated. The intentionally engineered risk system has no secondary risk market where one can insure oneself against possible losses at unfavorable outcome. The individual risks in this subsystem are mutually exclusive and collectively exhaustive in some sense. It may be reflected that no participation is also a decision-choice action. The choice of individual not to participate ends further decision contemplation on the necessity and accident. A choice to participate creates a follow up decision-choice problem of a choice of optimal strategy for optimizing cost-benefit outcome (risk-benefit outcome). It may immediately be observed that the whole democratic electoral process belongs to this un-mandated intentionally risk-taking system. The individual rationality must be viewed in terms of subjective utility in gambling and expectations of possible reward that may outweigh the cost of participation. The decision-choice rationality is following the best course of action as seen by the decision-choice agent in terms of optimization of risk. The same information-knowledge structure is available to all individuals and the public. However, their subjective interpretation in support of their beliefs may be asymmetric even if their satisfaction values are the same. The implication of this statement is that differences in behavior toward risk are explainable by asymmetry of subjective belief and asymmetry in satisfaction values given the

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information-knowledge structure. Since the possibility space has been given and the probabilities of outcome has been artificially constructed, the measure of risk can easily be done with the probability values and expectations where such probability values are exact and the probabilistic logic is of Aristotelian type. This leads to a lost function of stochastic type that helps to define quantitative risk. The structure of the concept of risk acquires a different dimension in unintentionally engineered risk system that emerges out of social formation and interplay of corresponding institutions. The possibility space here emerges out of substitution-transformation processes of categorial dynamics of social actual-potential duality of the institutional arrangements. Even though the setup institutional arrangements are human engineering design, the outcome possibilities are beyond human control making the possibility space an unintentional construct as well as external to decision-choice agents operating in the social setup. Another important characteristic of the unintentionally engineered risk system is that the information-knowledge structure in support of the corresponding social risk is incomplete and fuzzy. The fuzziness may be due to vagueness in the possibilities, methods of information acquisition, and techniques of knowledge construct. The social substitution-transformation process is such that information about the necessity is not only limited but also exhibits properties of multiasymmetry to decision-choice agents. Such multi-asymmetry is also the case for knowledge about its accidents. Decision-choice agents are conditioned, in one way or the other, to participate in this risk-taking social enterprise. Such risk-taking may be divided into individual risk-taking and collective risktaking. The collective risk-taking behavior depends on collective decisionchoice rationality that is permissible within the social formation. The costbenefit outcomes of the collective risk-taking process affect all decisionchoice agents in the social setup. Thus risk-taking is said to be collective if the effects of the actualized possible potential impact on all members and such risk is uninsurable (for example national war actions) such as collective decision to elect a decision-making group or political leader. The individual risktaking behavior depends on individual decision-choice rationality. The costbenefit outcomes of the individual risk-taking process impact only on the individual(s) on whose behave the decision-choice actions are taken. Thus risktaking in decision-choice process is said to be individual if the effects of the actualized potential have no impact on other members of the society except the individual risk-taker and such risk may have property of insurability (for

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INFORMATION-KNOWLEDGE STRUCTURE

FUZZY UNCERTAINTY

STOCHASTIC UNCERTAINTY

RISK-ENGENEERING SYSTEMS

Fuzzy Rationalities

NATURAL RISKENGINEERING

SOCIAL RISK-ENGINEERING

Unintentionally Risk-Engineered

Social Adaptation

Collective RiskTaking Collective RiskAdaptation

Stochastic Rationalities

Intentionally RiskEngineered

Individual Risk-Taking

Individual riskAdaptation Uninsurable Risks

Insurable Risks

RiskIntervention Projects INSURANCE AND RISK MARKETS

DECISIONCHOICE AGENTS

Fig. 5.4.1.1. The Typology of Risk Structure, Risk-Engineering and Decision-Choice Agents

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example all market risks). The structure of the risk-engineering system is presented in Figure 5.4.1. The individual component of the unintentionally engineered risk, such as market risks, has similarities and differences with intentionally engineered risk, such as lottery. Both of them are individually structured as well as have common theoretical structure regarding decision-choice behavior. The former, however, has a given possibility space and known and exact probabilities of the possible potential outcomes with hidden information-knowledge structure about the necessity and accidents in the risk-engineered system. Generally, the possibility space of the latter is not given and must be constructed from the information signals of the potential. The corresponding probability space must also be constructed from the available information on the necessity and the probabilities of the possible potential outcomes are to be estimated on the basis of the knowledge that individuals hold on the accidents. The process of moving from the potential to the possibility space and then to the probability space is complicated. This complication introduces into the decision-choice process some conditions of fuzziness where the probabilities may contain elements of vagueness and the reasoning process may be clouded with ambiguities and approximations. The intentionally engineered risks (such as those associated with gambling) are not transferable through any system of exchange. In other words, no market exists for insurance. The same elements of individual component of unintentionally engineered risk system may have actual and potential transferability through a medium of exchange such as insurance markets. Even here transferability is limited depending on the allowable domain of the institutional setup. The existence of transferability of risks has substantial effects on risk-taking behavior and the corresponding decisionchoice rationality. The theory of risk-taking behavior must take into account the element of transferability of risk and existences of institutions of risk transferability (that is an insurance scheme). The important epistemic point here is a question: can the framework for studying risk-taking behavior of gambling that falls under intentionally engineered risk system be used in the study of risk-taking behavior under conditions of unintentionally engineered risk, such as those of market system? If so, are there some amendments that need to be taken, and if not, what kind of changes need to be undertaken, keeping in mind that the former has exact probabilities by engineering design and the latter may contain elements of fuzziness, vague probabilities and reasoning approximations? These questions apply not only to theories of decision-choice actions under uncertainty and risk, but also to those of game theory that may be used in the study of the strategies of risk-taking and risk-

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avoidance in market systems as well as international resource games where institutions of risk transfers through exchange may not be available. These discussions allow us to provide definition for risk engineering. Definition 5.4.3: Social Risk-Engineering Social risk engineering is a process whereby decision-choice agents operating individually or collectively and with information on necessity and knowledge about accidents of social process act to alter either the quantitative or qualitative or both elements of social risk through changes in the social institutional configuration in order to intervene in the possibility space to influence the necessity and the accidents inherent in the potential with the objective of redefining the probability space and the corresponding probabilities of outcome possibilities in the substitution-transformation activities of the categorial dynamics of the social actual-potential duality. 5.4.2

Natural Risk Engineering

Some epistemic reflections have been given to social risk engineering where two types social risk-engineering system are identified. Let us turn our attention to natural risk-engineering. The thing we need to establish is the explication of the concept of natural risk-engineering. The natural risk is defined in relation to outcomes of natural processes while the social risk is defined in relations to outcomes of social processes. In both natural and social process, risks arise as a result of deficiencies in the information about the inherent necessities and deficiencies in the knowledge structures about the corresponding accidents. The differences of the social and natural processes that generate risks are due to conditions of controllability of decision-choice agents. The social risk can be altered in qualitative and quantitative terms by actively tempering with the possibility space through changes in the social institutional configuration whose dynamics of the substitution-transformation process and the conditions of categorial convertibility in the behavior of social actual-potential duality alters the necessities and corresponding accidents. There are many events where the social risk is an interventionable in the social risk system. This situation is not available to decision-choice agents when the natural risk is confronted. The necessity and accidents in the substitutiontransformation activities in natural processes where the laws of categorial conversion in the dynamics of natural actual-potential dualities are such that many times they are unknown or vaguely or approximately known by either individual or collective claims. In this respect, the decision-choice cannot in-

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terfere in the qualitative and quantitative natural risk. What then do we mean by natural risk-engineering? A definition will be appropriate before some epistemic discussions. Definition 5.4.4: Natural Risk-Engineering Natural risk engineering is a process whereby decision-choice agents operating individually or collectively and by the use of information on the necessity and knowledge about the accidents of natural process act to adapt themselves to quantitative or qualitative or both elements of natural risks through the design of social or and artificial responses in order to intervene in the impact of risk when the possible necessary potential outcome is actualized in the substitution-transformation processes of the categorial dynamics of the natural potential-actual duality. In natural risk-engineering, the objective of the design is centered on the use of the available information-knowledge structure at the guidance of the adaptation principle in order to cope with natural disasters when the possible potential is actualized. The decision-choice agents with available informationknowledge structure can not intervene in the necessity and accidents of natural processes but they can intervene in their impact though risk-engineering responses to outcome possibilities. The possibility space is outside the control of human engineering abilities and hence the possible potential outcomes with corresponding risks cannot be humanly engineered. The possibility space can, however, be estimated from the available information on the potential and the inherent necessity of the natural process. The probability space may be projected while the probabilities of the outcome potential may be estimated on the basis of available knowledge of the accidents that correspond to the internal necessity. The natural risk-engineering is, therefore, about natural-risk response engineering to possible potential outcomes of substitution-transformation processes under the forces of categorial dynamics of natural actual-potential duality. The typological relationships of types of risk-engineering phenomenon in terms of social and natural is provided in Figure 5.5.4.1 where comparative structures between social and natural risk-engineering are provided. The natural risk may be individual of collective. The collective natural risks are those that affect general society or a significant portion. There is no medium of risk- transfers through institutional exchange for such collective natural risks and where the risk-engineering is a collective responsibility. Some of the individual natural risks are transferable through institutions of exchange, such as insurance institutions depending on the nature of the risk. The possibilities of risk transferability provide the reason for the construction of the

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theory of insurance and market system of risk transfers that may be viewed in terms of supply of risk and demand for risk. Here the business of insurance must also be seen as risk-taking on the part of the firms and risk-avoidance on the part of the insured, irrespective of the nature of the risk. Thus, the theory of insurance, and the theory of behavior under risk, and the theory of behavior under uncertainty, have common epistemic structure and analytical foundations. The understanding of one must reinforce the understanding of the other and how the principal-agent problem arises in the theory of transfers of possible potential. The phenomenon of risk-engineering belongs to prescriptive science where optimal prescriptive decision-choice rules are constructed for practice in order to deal with future possibilities and their actualization. These optimal decision-choice rules will fall under an appropriate type of decision-choice rationality depending on the type of the set of assumptions that one makes about the information-knowledge structure as has been discussed in Chapter Five of this monograph. The techniques and methods for risk engineering will be drawn from different knowledge sectors and the logic and mathematical reasoning will be defined by the type of topological space that is consistent with the assumed uncertainty structure. From the view point of total uncertainty and total risk it will be analytically useful if the risk engineering process accounts for risks associated with both stochastic and fuzzy uncertainties. This requires that the problem of the risk-engineering process be formulated in either fuzzystochastic space or stochastic-fuzzy space. The problem formulation and the search for answer and analysis require the usage of both probabilistic and fuzzy logical reasoning and corresponding mathematics that will allow us to construct rules of either fuzzy-stochastic rationality or stochastic-fuzzy rationality.

5.5 Costs-Benefit Rationality, Risk-Taking and Risk-Engineering It is useful at this point to make some epistemic reflections on the relationship between cost-benefit rationality and risk-taking on one hand and cost-benefit rationality and risk-engineering on the other hand. The structure of costbenefit rationality and its relationship to fuzzy rationality in decision-choice behavior has been discussed in [R.7.35], [R7.36]. The approach offers us a different understanding of studying decision-choice behavior with conflicts in the dynamics of cost-benefit duality. Risk and uncertainty received indirect analysis. Cost-benefit rationality was, however, presented as a general ap-

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proach to the study and analysis of decision-choice trajectory. In the discussions in [R17.23], and the current monograph, an explicit postulate of costbenefit rationality is made. The postulate puts forward the fundamental idea that all human decisions, irrespective of the object and decision-choice agent, are cost-benefit driven in the sense that every decision-choice action that decision-choice agents confront has costs and benefits whose balances for decision-choice action lead to optimal cost-benefit rationality. In the general framework of information-knowledge structure as we have discussed, the cost-benefit rationality will present itself as either fuzzy-stochastic optimal cost-benefit rationality or stochastic-fuzzy optimal cost-benefit rationality. Here arises a sub-paradigm of analysis and the study of human decisionchoice actions where such actions are undertaken within some conditions of different available alternatives and under some defined comparative consistency. To meet this comparative consistency, the cost-benefit rationality requires that there exists a relative comparison of available alternatives and that such comparison be based on relative values that order all the possible potential alternatives available to the decision-choice agent. For decision-choice processes to be based on cost-benefit rationality, cost and benefit characteristics must be identified, computed and organized into data sets to provide the required information-knowledge structure for decision-choice action. The provision for the information has led to the development of the theory of computable cost-benefit identification matrixes [R7.36]. From the viewpoint of human cognition, every event is a set of cost-benefit outcomes where each outcome is distinguished by its own informationknowledge support that allows the possible potential to be distinguished from the impossible potential thus creating the possibility space for possible potential elements. The set of cost-benefit information-knowledge structures that define outcome identities is deficient in all its components due to limited information vagueness, ambiguities, human limitations and default linguistic explication. The deficient information-knowledge structures place the possible potential cost-benefit outcomes into uncertainty conditions that logically connect the possibility space to the probability space that indicates the distribution of the chances that each cost-benefit outcome may be actualized. The cost-benefit outcomes are the results of substitution-transformation processes in nature and society. View in this light, every possible potential outcome has its possible potential benefit and its possible potential cost support where the potential cost is maintained by the possible potential benefit support. Events are risky by cognitive interpretation in terms of their impacts on human existences in all forms when they occur. Risk is thus defined in terms

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of relative cost-benefit values. This is a generalized view that captures the effects of natural and social substitution-transformation processes. This is another broad interpretation of the economic concept of opportunity cost or alternative cost where the universal substitution-transformation processes are seen in terms of infinite production of continual change within itself. It is on the basis of the benefit-cost effects, that possible potential outcomes give meaning to risk where risk is defined in terms of possible potential costs and benefits of events. Risk is nothing but interpretive conditions of possible potential outcomes of an event where at least one of the outcomes is perceived to have possible potential costs that outweigh the possible potential benefits given the available defective information-knowledge structure that hides from the decision agents the necessity or the outcome that will be actualized. In fact, the loss and risk functions in statistical decision theory may be viewed in this light. The definition and measurements of benefits and costs will vary over decision-choice items and subject areas including, justification, verification and corroboration principles in acceptance of theories and hypotheses. Since each event in nature and society is composed of different possible potential outcomes, where benefits may outweigh costs for some outcomes and costs of some outcomes may outweigh their corresponding benefits and the cognition of that which may be actualized depends on defective informationknowledge structure, we may define risky events in terms of their cost-benefit balances and the measure of distribution of the degrees of belief attached to probable actualization of any of the possible potential outcomes as cognitively revealed by a defective information-knowledge structure. The relationships among costs, benefits and risk are shown in Figure 5.5.1. Definition 5.5.1: Risk event, Risk-free event, Pure risk Event and Riskneutral event An event, therefore, is said to be risky if the total costs outweighs the corresponding total benefits for at least one of the possible potential outcomes and the possible potential outcome to be actualized is not clearly known to decision-choice agents due to defective information-knowledge structure about the inherent necessity and accidents. Definition 5.5.2: Risk-free Event An event is said to be risk-free if the total benefits outweigh the total costs for all the possible potential outcomes and possible potential outcome to be actualized is unknown to decision-choice agents due to information-knowledge deficiency.

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DECISION-CHOICE AGENT

INFORMATION-KNOWLEDGE STRUCTURE

EVENT NECESSITY

ACCIDENTS POSSIBLE POTENTIAL OUTCOMES

BENEFITS

COSTS

CONDITIONS OF RISK

CATEGORIES OF RISK

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RISK-FREE OUTCOME

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RISK-ENGINEERING ACTUALIZED OUTCOME

Fig. 5.5.1. The Relational Structure of Costs, Benefits, Risk-Taking and Risk-Engineering

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Definition 5.5.3: Pure Risk An event is said to be pure risk event if the total costs outweighs total benefits for each and all the possible potential outcomes and the possible potential outcome to be actualized is unknown due to information deficiency about the necessity and knowledge deficiency about the internal accidents. Definition 5.5.4: Risk Neutral Any event is said to be risk neutral if the total costs is just equal to total benefits for each and all possible potential outcomes given a deficient informationknowledge structure about the substitution-transformation processes. The definitions that have been offered involve the interplay of cost-benefit configurations and deficient information-knowledge structure where such information- knowledge deficiency is expressed through possibility measures and probability measures that capture possibilistic and probabilistic beliefs of decision-choice agents respectively. For individual events, the distribution of probabilistic belief is relevant to risk-taking and risk-engineering decisions for risk minimization under pure risk and net benefit maximization under riskfree events respectively. Such distribution is irrelevant for risk-neutral events in the sense that all possible outcomes produce the same cost-benefit effect. It is on the basis of differential cost-benefit configuration of different possible potential outcomes that possibility and probability measures become useful computational definition of risk through the relationship between expected value and variance of the event. It is also through the accepted computational definition that the variance is used to order events on the basis of the content of risk and the concept of minimum-risk rationality arises when the decisionchoice process involves multiple events where each event has multiple possible potential cost-benefit configurations. The acceptable minimum-risk decision-choice rationality must utilize risk measures whose constructs combine fuzzy risk and stochastic risk. The combined risk measure with the minimumrisk decision-choice rationality is then used to implement risk-taking or riskengineering. Since the defining structure of risk relates to possibility and probability measures, it indirectly incorporates the subjective intensity effects of costs and benefit characteristics. The time structure of index of optimal decision choice rationality will depend essentially on the time framework of the effects of the possible potential outcome. The cost-benefit framework for defining risk given the measures of possibilistic and probabilistic beliefs imply that a reduction in cost or increase in benefit is a reduction in risk. Similarly a reduction in benefit or an increase in cost given the existing deficient informa-

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tion-knowledge structure implies an increase in risk. It is on these bases that a reduction in risk is benefit and an increase in risk is a cost and hence we can speak of benefit-risk analysis.

6

Fuzzy Decision-Choice Rationality and Paradoxes in Decision-Choice Theories

We have presented a notion that paradoxes in sciences, mathematics and logic come to us as temporary and permanent within the logical environment of their creation. The resolution of temporary paradox is intra-paradigm in nature while the resolution of permanent paradox is an inter-paradigm process. The former is an internal movement in terms of sub-paradigm changes while the latter is a revolution in paradigm with inter-paradigm changes. In this respect, it is useful to introduce the concept of global paradigm into the analysis. Every global paradigm is a family of sub-paradigms. A resolution of a temporary paradox may result in intra-paradigm changes and hence inter-subparadigm shifting. Intra-paradigm changes are merely refinements of existing paradigm. A resolution of a permanent paradox results in inter-paradigm shifting from one global paradigm to another global paradigm. The new global paradigm may have the old paradigm as its sub-paradigm in the sense that it offers solutions to solved and unsolved problems in the old paradigm as well as providing new epistemic look and rationality in knowledge acceptance process. An example of a permanent paradox in logic and philosophy is that of sorites (the paradox of heap), the resolution of which we have provided through a paradigm shifting from classical paradigm to fuzzy paradigm in a companion volume [R17.24].

6.1 The Four Structures of Decision-Choice Rationality Let us now relate fuzzy decision-choice rationality to paradoxes in decisionchoice theories. To have an epistemic clarity, appreciate complexities of the paradoxes, and understand computational difficulties as well as logical confusions associated with paradoxes in decision-choice theories, it is useful to parK.K. Dompere: Fuzzy Rationality, STUDFUZZ 237, pp. 133–170. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

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tition actual and potential theories in the general decision-choice space into sub-categories with distinguishing characteristics. These sub-categories are defined by assumed information-knowledge structures for decision-choice processes and the nature of supporting mathematics and logic of reasoning. The categories of actual and possible decision-choice theories and the corresponding rationalities are presented in Figure 6.1.1. These categories have corresponding logic and mathematics of reasoning that allows a decisionchoice rationality to be abstracted. The categories of decision-choice theories center on the conceptual interactions of fuzzy and stochastic uncertainties. The required categories of logic and mathematics are placed in Cohort I, II, III and IV in Chapter Three in this monograph. From the structure of the family of the categories of decision-choice theories we can examine the suitability of the use of the available logical and mathematical toolbox of the classical paradigm in either explanatory or pre-

CATEGORIES OF DECISION – CHOICE THEORIES

NON-STOCHASTIC THEORIES

Cohort I

CLASSICAL DETERMINISTIC

CLASSICAL RATIONALITY

Cohort II

Cohort III

STOCHASTIC THEORIES

NON-FUZZY STOCHASTIC THEORIES

FEZZY DETERMINISTIC

FUZZY THEORIES

Cohort IVA

FUZZY RATIONALITY

CLASSICAL RATIONALITY

FUZZY RISK

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RISK-FREE

Cohort IV

Cohort IVB

FUZZY STOCHASTIC

STOCHASTIC FUZZY

FUZZY STOCHASTIC RATIONALITY

STOCHASTIC FUZZY RATIONALITY

FUZZY STOCHASTIC RISK

STOCHASTIC FUZZY RISK

CATEGORIES OF RISK THEORIES

Fig. 6.1.1. Categories of Decision-Choice and Risk Theories

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scriptive theories or both. Let the category of possible theories in a particular cohort be the constructed theories and axioms. Are these theories and axioms directed to either explain or to define the character of decision-choice rationality as an attribute of decision agents? Alternatively, are these theories and corresponding axioms directed to define and explain an ideal path of decision-choice rationality that will provide an isomorphism between the attribute and the ideal state given the attribute as an endowment of decisionchoice agents? The differences and similarities between optimal rationality as an attribute and optimal rationality as an ideal state of decision-choice process are discussed in a companion volume [R17.23]. In that volume, it is pointed out that most experimental research on decision-choice behavior is directed to the former while the theories and axioms in economics and other decision-choice sciences are directed to the latter. The answers to these two questions are important to understanding some disagreements in research on decision-choice theories. Our epistemic position is that all decision-choice agents are endowed with the attribute of optimal decision-choice rationality in the sense of possessing a calculating ability to seek optimum by minimizing cost and maximizing benefit both of which are broadly defined. The description of optimal sequential steps that define the enveloping path to the optimum by the method of constructionism is where disagreements tend to arise given optimal rationality as an attribute. It is precisely on the verification process of the axioms relating to the descriptive notion of this enveloping path of optimal rationality and its relationship to optimal rationality as attribute that paradoxes tend to arise. Two questions tend to arise. Are the experimental verifications directed to dismiss or affirm the optimal rationality as an attribute? Alternatively, are they directed toward the test of the degree of validity of the theories and contained axioms, which are irreducible primaries, as reasonable descriptions of decision-choice behavior of decision-choice agents? If the experiments on decision-choice theories are directed towards the verification of optimal rationality as an attribute then we are inclined to question the adequacies of the environments of the experiments as to whether they are sufficiently structured to account for information-knowledge incompleteness and information-knowledge vagueness, or inexactness, or in general, the deficiencies in the information-knowledge structure. What do we mean here regarding information and knowledge? We simply mean that the environment of the experiments must be structured to account for two limitations of knowledge support for the decision-choice action in the experimental design. These limitations are fuzziness and incompleteness in the supporting information-

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knowledge structures of decision-choice activities. Furthermore, we must examine the differences and similarities of the environment of the experimental objects and the environment of the theoretical objects and answer a question as to how the experimental tests related to the theories and axioms of optimal rationality as an ideal state of decision-choice behavior. It is through the process of relating the results of the decision-choice experiments to decisionchoice theories and axioms that paradoxes and contradictions tend to emerge. The paradoxes may arise if either the experimental decision-choice agent is descriptively different from the theoretical decision-choice agent; or the experimental decision-choice environment is different from the theoretical decision-choice environment. Similarly, paradoxes may arise if the conditions of an experiment neglect the logical category of decision-choice theories that the results of the experiments are related given that the logic of reasoning within the paradigm is correct. In reference to categories of decision-choice theories and the corresponding mathematical space, every decision-choice agent actually functions in either fuzzy-stochastic space or stochastic-fuzzy space in the sense that our knowledge is incomplete and some parts are vague and ambiguous. These two spaces are both similar and yet different. They are similar in terms of containing fuzzy and stochastic uncertainties and they are different in terms of the order in which the uncertainties appear and the nature of the required logico-mathematical toolbox for analyses and syntheses as has been presented in Chapter Two of this monograph. These spaces seem to be isomorphic characterization of actual world that decision-choice agents function. The experiments on decision-choice behavior are conducted on decisionchoice agents who are functioning in these spaces containing stochastic and fuzzy uncertainties. Stochasticity means limited knowledge and fuzziness means unclear knowledge. Thus ambiguity and vagueness are not the result of having a limited knowledge but the result of having unclear knowledge which may result from complexities in receiving and processing of information signals. The theories and axioms of decision-choice behavior, on the other hand, are constructed in either non-stochastic and non-fuzzy space or non-fuzzy and stochastic space that are free from two or one type of uncertainties. Examples of the former are the axioms and theories of collective choice [R5], [R5.7], [R12], [R12.6 [R12.9], [R12.27]; while the examples of the latter include those of von Neumann-Morgenstern [R22.5], [R22.11], Savage [R16.43], [R22.5], Alias [R21.2], [R22.1] Arrow [R21.5], Marschak [R14.45] and stochastic game theories [R19.2], [R19.6], [19.24]. The theories and axioms of the latter are constructed in a decision-choice space free of fuzzy uncertainties but contain stochastic uncertainties. The actual or thought experiments on the

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other hand are conducted in decision-choice spaces that contain both stochastic and fuzzy uncertainties that introduce vagueness, ambiguities, inexactness and others which we have classified under a general umbrella of fuzziness. The presence of fuzzy uncertainty gives rise to penumbral region of decisionchoice activities with resulting fuzzy risks. The conditions of penumbral region are not included in the constructions of decision-choice theories and axioms but seem to be included in the experimental process. The same experimental process and theoretical process on decision-choice space have one thing in common in that they use logic and mathematical toolbox of classical paradigm in deriving their result. Such toolbox will be inappropriate for processing the experimental data if the fuzzy uncertainties are not abstracted out before its use. The presence of fuzzy uncertainties and fuzzy risk between the experimental process and theoretical construct is an epistemic difference that constitutes an important source of observable paradoxes in decision-choice theories. The same vagueness, ambiguities and others are responsible for, and provide explanation to many paradoxes in science, logic, mathematics and other knowledge sectors whose methods of reasoning are governed by the classical paradigm. With these epistemic views in mind, in addition to the structure of categories of possible decision-choice theories and optimal rationality as provided in Figure 6.1 and corresponding logico-mathematical spaces of Chapter Three and let us examine some of these paradoxes in decision-choice theories which we shall attribute to inability of the classical paradigm to deal with the presence of fuzziness in logico-mathematical spaces.

6.2 Fuzzy Decision-Choice Rationality and the Arrow's Paradox (Impossibility Theorem) The Arrow’s paradox is generally referred to as Arrow’s impossibility theorem in economics and other social sciences. This paradox shows itself in all collective decision-choice situations if the Arrow’s conditions are to be met. These conditions, sometimes referred to as axioms, are composed of a) unrestricted domain, b) non-imposition, c) non-dictatorship, d) monotonicity and e) independence of irrelevant alternatives at the level of the social collectivity. These conditions are supported by the postulates of rationality of individual decision-choice agents functioning in non-stochastic and non-fuzzy decisionchoice environment. The paradox flows from Arrow’s description of the environment of decision-choice process of the individuals and the conditions re-

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quired for their aggregation. In other words, it is an examination of democratic decision-choice process as a resolution of conflicts in the collective decision-choice activities where dictatorship is not allowed and all individual preferences must equally count. The Arrow’s theoretical and axiomatic structure was built in non-stochastic and non-fuzzy decision-choice space that allows the use of the toolbox of the classical paradigm composed of its logic and mathematics of exactness without ambiguities and logical contradictions. The problem of Arrow belongs to the class of collective decision-choice action that must be expressed over a set of alternatives whose cardinality is generally greater than, or equal to three. The cardinality of the set of individuals in the social collectivity is generally greater than or equal to two. The individuals in the collective have differential preferences that are subjectively defined. The individual subjective preferences are to be harmonized through some form of aggregation into collective preferences for collective decision-choice action. The collective decisionchoice problem whose solutions lead to a paradox may be stated. Given the individual preference orderings and Arrow’s interpretation of democratic social organization: is there any method of aggregation that will create an isomorphism between individual optimal decision-choice rationality and the collective optimal decision-choice rationality that satisfies conditions of comparability, transitivity and acyclicity, where all individual preferences equally count and without a dictator? The answer is simply that such a channel of aggregation is impossible leading to the impossibility theorem; and the paradox is that the isomorphic structures of the individual and collective decisionchoice rationalities of democratic decision-choice actions in a democratic social organization require a dictator contrary to the requirement that all individual preferences must count in the social decision-choice enterprise. Let us take a closer look as to the nature of the paradox, how it arises, and where it arises. The Arrow’s problem with subjective phenomenon, even though formulated in non-fuzzy and non-stochastic decision space, belongs to fuzzy and non-stochastic decision-choice space. The logical methods and the techniques of analysis are drawn from the toolbox of the classical paradigm that disallows contradictions, ambiguities degrees of truth and falsity, and vagueness. It is not, therefore, surprising to arrive at a paradox and an impossibility theorem. The known mathematical methods of the classical paradigm do not provide us with channels of aggregation of elements with subjective phenomenon and presence of vagueness that create conflicts in preferences in the social decision-choice space. This is true of all collective or social decision-choice problems that seek universality of representation and solution, and

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where the preferences are defined in terms of Aristotelian logical values. The acceptable solutions with the use of the toolbox of classical paradigm are the rationality of either unanimity or dictatorship if some of Arrow’s axioms are not to be violated. The difficulty arises from the simple idea that the toolbox of the classical paradigm is being applied in an environment that is inconsistent with the fundamental assumption of the paradigm. The axioms of individual optimal decision-choice rationality composed of completeness (comparability), transitivity and or acyclicity are maintained by Arrow. Given the axioms of individual optimal decision-choice behavior, Arrow imposes extra axioms on social decision-choice behavior that capture conditions of democratic social decisionchoice action that is fair and just to individual preferences in the collective decision-choice space in order to create an isomorphism between individual optimal decision-choice rationality and collective optimal decision-choice rationality. In order to understand how the paradox emerges, it is important to realize that even though the axioms of individual decision-choice behavior are created in non-fuzzy and non-stochastic decision-choice space the subjective phenomenon is suppressed and silent in a way that does not affect the use of the classical toolbox where interpersonal aggregation is not required. The individual constitutes both the dictator and the collective in all cases of individual decision-choice action over alternatives without conflict in ranking, thus providing isomorphic decision-choice structures with both unanimity and dictatorship where harmony exists between the dictator and the collective with the same preferences without a need for of the use of weighted preferences. This is not the case in the study of collective decision-choice action. We must note the Arrows paradox vanishes if the individuals have the same preferences. Things are, however, different when Arrow requires the collective optimal rationality of social collectivity with more than one individual to be isomorphic to that of the individual decision-choice rationality. This harmony of conflict in decision-choice among alternatives is broken and conflict and tension are created under different processes of aggregation. The subjective phenomenon that is dormant in the individual decision-choice space becomes active, moving the collective decision problem to the active zone of fuzzy nonstochastic space where a weakness of the classical paradigm is revealed in terms of Aristotelian logic. In the case of the individual decision-choice space, the comparison is self-interpersonal of classical identity. Such classical identity is not constructible without violating at least one of the Arrow’s axiomatic interpretations of democratic social decision-choice actions. The Arrow’s

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paradox may be viewed as temporary or permanent. If it is viewed as temporary paradox then the resolution must be sought by intra-paradigm changes through sub-paradigm shifting within the organic classical paradigm by altering the axiomatic interpretation of democratic decision-choice action in order to obtain the isomorphism between individual optimal decision-choice rationality and collective optimal decision-choice rationality. On the other hand, if it is viewed as permanent paradox in the classical paradigm of its creation, then inter-paradigm shifting is required to bring about its resolution under the same set of conditions. It is the view here that the Arrow’s paradox is a permanent one in the classical paradigm. Its resolution lies in the use of the toolbox of fuzzy paradigm leading to fuzzy and non-stochastic rationality. We have argued that the fuzzy non-stochastic optimal decision-choice rationality needs not coincide with non-stochastic and non-fuzzy optimal decision-choice rationality. The point here is that the nature of democratic decision-choice process by social collectivity is such that the solution set contains unanimity, dictatorship, majority-minority points with differential weights and by insisting on the idea that all preferences in the collective should equally count Arrow imposes Aristotelian principle of excluded middle where only two polar cases of unanimity and dictatorship are left in the solution set. Unanimity requires similarity in individual preferences in the collective which violates the assumed differences in individual preferences. Thus, unanimity is ruled out as a solution leaving dictatorship as the optimal solution. The solution of dictatorship to Arrows collective choice problem is the result of avoiding vagueness, and grades of individual participation in the democratic choice process in addition to the use of the toolbox of classical paradigm that disallows contradiction and conflicts.

6.3 Fuzzy Optimal Decision-Choice Rationality, Utility Theory, Probability and Paradoxes From our discussions on Arrow’s paradox, we would like to reflect on other paradoxes in decision-choice theories that involve expectations. We pointed out that Arrow’s paradox is a creation within the classical paradigm in nonfuzzy and non-stochastic information-knowledge space. In this space there are no uncertainties and risks. The set of Arrow’s axioms involves defining relationships among utilities, actions, incentive structure (rewards), and exact and certain information-knowledge structure. The axioms that we are going to examine in this section, however involve interrelationships among utilities, un-

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The nature of the paradoxes and the conditions that give rise to them may be understood by examining two environments: a) the environments in which the actual or thought experiments are conducted (that is, the experimental environment) and b) the environment in which the theory and axioms are constructed (that is, the theoretical environment). Let us visit the discussions in Chapter Four of the companion volume [R17.23]. Four important decisionchoice spaces are identified in the discussions. They are a) space of potentials, b) space of actual, c) possibility space, and d) probability space. Given these spaces, the epistemic analysis of the nature of paradoxes in decision-choice theories requires us to critically examine the concept of possibility and probability, how they relate to each other and how to explicate them in relation to decision-choice rationality. The possibility space performs a linkage function between the space of potentials and the probability space in the decision-choice processes and the rationality that it may induce. The probability space in decision-choice process involves in establishing a relation between the present decision-choice actions to unknown or partially known knowledge about current and future elements in terms of ranking the possible outcomes conditional on a given gambling hypothesis. A choice among alternative gambling elements (possible outcomes) is based on subjective assessment of knowledge about probabilities of these possible outcomes. The manner, in which the knowledge about probabilities of alternative outcomes is obtained, relative to decision and choice, will affect the structural understanding of what is known, uncertainties and the degree of belief associated with the expected outcome. The uncertainties surrounding decision-choice process is of two types. On one hand, we have fuzzy uncertainties that are associated with possibility space. On the other hand we have stochastic uncertainties that are associated with probability space. The degree of belief of any outcome in an uncertain environment is thus dual in nature. On the side of fuzzy uncertainty, we have possibilistic belief that is associated with possibility space, while on the side of stochastic uncertainty, we have probabilistic belief that is also associated with probability space. The conceptual differences and their analytical usefulness are discussed in [R17.23]. Both possibilistic and probabilistic beliefs in decision-choice actions on gambling alternatives are based on subjective knowledge structures that satisfy their individual justification principle as their cognitive supports in what is believed. The logical and computational complexities of information-knowledge support in belief structures are generally simplified in theory to the point of triviality in theoretical representation of actual phenomenon. The accepted point of triviality in each theoretical

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structure depends on the degree to which personal judgment and justifications are allowed to enter into the process of probability assessment relative to the possibilities. There is a general intuitive notion concerning human judgment and action in assessment and selection processes which we take as immutable assumption or primitive postulate of natural order in construction of possibilities, probabilities and decision-choice actions. The intuitive notion is simply that decision-choice agents are endowed with capacity to rank objects by comparing and contrasting elements in the decision-choice spaces by some index. The assumed innate ability and cognitive capacity flow from an assumed optimal rationality as an unquestionable attribute of decision-choice agents. In an uncertain world of decision-choice processes, the postulate of innate ability is expressed in terms of capacity to rank outcomes by degrees of belief over potential elements that allow the development of the possibility space. The same postulate also allows a second ranking of beliefs over the elements in the possibility space to be developed leading to the construction of probability space. The same postulate allows a third ranking of the elements in the decision-choice space for action. All these involve collection, processing and interpretations of information signals to obtain knowledge support as justifications in ranking degrees of beliefs surrounding alternative outcomes in an uncertain decision-choice space. The potential space is thus linked to the space of the actual through the relational interactions of possibility, probability and decision-choice spaces that are induced by cognitive transformation functions. Given the potential space, the ranking process is to reconcile the supporting degrees of beliefs from possibility space to the probability space so the actual outcome can be realized by the optimal selection process. It is in the process of epistemic reconciliations between degrees of possibilistic belief and probabilistic belief through the theoretical and experimental processes on decision-choice structures that paradoxes tend to arise. In understanding the paradoxes in decision-choice theories and experiments we must examine the underlying assumptions that link the possibility space to the probability space in the transformations from the potential to the actual. The examinations of the underlying assumptions suggest that the theories and axioms may be constructed around the probable set of outcomes H given the possibility space Π and the knowledge support, K that justifies the belief index. In other words, what is the best decision-choice action that the knowledge K supports? Alternatively, we may ask a question as to what knowledge structure, K , would be needed to support a degree of belief in particular deci-

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sion-choice action on a gambling alternative H1 ∈ H ? These questions may be posed without assuming that the possibility space is given. The process is such that given the possibility space, we can derive or compute the index of probabilistic belief from the observed decision-choice action of uncertain outcomes given the innate endowment of decision-choice agents. With these conceptual backgrounds, let us examine the three paradoxes. We have so far not said anything about paradigm of analysis. However, the current scientific culture is such that much of our analytical work and development of computable systems are mostly based on the use of the toolbox of organic classical paradigm under the principles of exactness and non-contradiction with intersub-paradigm changes depending on types of anomalies that are encountered in our cognitive pursuits. In other words, our formal cognitive systems operate on the principle of classical optimal rationality. The world, in which we function, however, is such that our informal cognitive systems operate under conditions of inexactness, vagueness and many types of ambiguities that create dichotomy between our formal and informal cognitive systems in the decision-choice space. The need to integrate the two has lead to the development of organic fuzzy paradigm and the toolbox of fuzzy optimal rationality and epistemic analysis of which this monograph is about. 6.3.1

The Ellsberg Paradox and Savage Axioms

The Savage problem flows from the Bayesian view of probability as opposed to the classical view of probability. Both views are constructed around the concept of degree of belief that one holds regarding uncertain outcomes. The former relies on subjective information while the latter is claimed to be based on objective information. The essential difference, therefore, is that the computations of probability values in the Bayesian analytical framework are done on the basis of subjective belief system. In the classical objective probability, the justified belief system on obtained values is the supporting knowledge of the relative frequency. The probability value is derived from relative frequency which then becomes the measure of the degree of belief in one’s expectations on outcomes of events associated with a particular uncertainty. Both classical and Bayesian belief systems are probabilistic that take the possibilistic belief as given. By connecting relative frequency measure to probabilistic belief, the assumption of a given possibilistic belief is always maintained in the classical framework that becomes classical paradigm-preserving in the sense of principles of exactness and non-contradiction. While the classical measure of probability affects the decision-choice process, it is outside

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the domain of decision-choice action as well as impinges on it as an external factor of the decision-choice system under uncertainty. The knowledge in support of decision-choice action is externalized. It is this externality of the classical concept of probability that introduces linearity into the expected utility hypothesis [R22.1], [R22.8] in terms of monotonic linear transformation. Things are different in the case of the Bayesian framework which is based on subjective phenomena where the probabilistic belief is internalized as part of knowledge construction and decision-choice process. A problem thus arises in the Bayesian framework as to whether one can have a justified belief in supporting the probabilistic degree of belief that will constitute an acceptable measure of subjective probability and is this probability viewed as external or internal to the decision-choice system. To this problem we have Savage response that holds on to the Bayesian view that the subjective probability measure can be operationally abstracted from decision-choice process among gambles in a defined uncertain condition. The degrees of belief in addition to decision-choice action are internal to the agent of decision and choice. By introducing subjective phenomena into the decision-choice system the Bayesian framework becomes involved with ambiguity and vagueness and yet the same toolbox of the classical paradigm is call into action. It is here that violation of Savage axioms become possible and Ellsberg paradox begins to emerge. The nature of the Ellsberg paradox reveals itself as the experimental violations of axioms of completeness and monotonicity of ranking in an exact and uncertain decision-choice space as advanced by Savage from which probability estimates are to be inferred from individual decision-choice behavior in gambling situations. First we must observe that the Savage axioms were constructed in non-fuzzy (exact) and stochastic (limited information) environment. This environment contains stochastic uncertainties whose measurements are exact without vagueness or ambiguity as acknowledged by Ellsberg as well as pointed out by Savage [R16.43]. Prelude to his thought experimental on Savage axioms Ellsberg reminded us of Frank Knight’s distinction between measurable uncertainty or risk and unmeasurable uncertainty [R21.13], [R4.34] which we have called stochastic uncertainty and fuzzy uncertainty respectively that partition the uncertainty space. Furthermore, the stochastic uncertainty is associated with stochastic risk while the fuzzy uncertainty is associated with fuzzy risk [R7.35]. The stochastic uncertainty is associated with probability space while fuzzy uncertainty is associated with possibility space. Thus unmeasurable uncertainty comes with risk of ambiguity in the decision-choice process (see Figure 6.1.1).

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6 Paradoxes in Decision-Choice Theories

In order to use the toolbox of the classical paradigm with its Aristotelian logic where principle of exactness is required it becomes necessary to keep separate the possibility space from the probability space. This is exactly what Savage does to obtain complete stochastic environment for his theory and axioms on the basis of which a probability measure may be abstracted as a measure of degree of belief associated with probabilistic uncertainty surrounding decision-choice action. In other words, the set of axioms assumes the prior existence of possibility space and hence we can take the possibilistic belief as given. By separating the two spaces, and implicitly assuming the existence of the possibility space, Savage has done away with vagueness and ambiguity that may cause theoretical decision-choice agents to violate his axioms due to decision-choice actions in the penumbral regions. In this respect, all expected utility analyses with postulates, axioms and hypotheses are direct extensions of the classical decision-choice representation from non-fuzzy and non-stochastic information-knowledge environment to non-fuzzy and stochastic environment by neglecting fuzzy uncertainties and possibilistic degrees of belief. This allows Savage to directly extend the use of the toolbox of the classical paradigm in studying non-fuzzy and non-stochastic optimal decisionchoice rationality to non-fuzzy and stochastic optimal decision-choice rationality The Savage axioms tend to generate optimal non-fuzzy optimal stochastic decision-choice rationality which is tested by Ellsberg. In testing the validity of the Savage axioms through thought experiment, Ellsberg introduces ambiguity-vagueness complications into the information-knowledge space that supports stochastic decision-choice actions under exactness thus uniting the stochastic space with fuzzy space to obtain fuzzy-stochastic space or stochastic-fuzzy space, and corresponding to them, an optimal fuzzy-stochastic decision-choice rationality or optimal stochastic-fuzzy decision-choice rationality. The categories of the decision choice theories are given in Figure 6.1.1. We must observe that the non-fuzzy stochastic decision-choice variable in Savage analytical framework is transformed to either fuzzy-stochastic or stochastic-fuzzy variable. The result is the emergence of computational complexities where the decision-choice agents are confronted with either fuzzy events in stochastic uncertainties or random events in fuzzy uncertainties. The computational complexities place the decision agents in penumbral region of choice as well as create decision-choice indifference between black and red choices as pointed out by Einhorn and Hogarth [R16.10], [R21.12]. These computational complexities may be illustrated with decision tree of Figure 6.3.1.1.

6.3 Utility Theory, Probability and Paradoxes

147

The fuzzy uncertainty was imposed on the stochastic uncertainty through the experimental design of decision-choice elements in the two Urns where the proportion of red and black balls are unknown in Urn I but with known quantity of balls and Urn II contains the same quantity of balls with equal proportions of black and red balls. The probability of red and black balls in Urn II is known but that of Urn I is not known. Can we infer from the decision-choice actions by decision-choice agents the subjective probabilities to red and black balls that are assigned? The answer to this question would be affirmative if the decision-choice behaviors of the agents in the experiment follow the Savage axioms. The answer would, however, be negative if the behaviors violate any of the Savage axioms. Ellsberg arrives at the latter answer where the decision-choice agents violate the completeness and monotonicity axioms. This is the paradox [R21.13], [R21.28]. This conclusion is also agreed to by Fellner [R16.10]. The Ellsberg’s explanation is anchored on the presence of vagueness and ambiguities in subjective judgment associated with assessing the probabilities associated with the degree of belief in the decision-choice process. How do the ambiguities tend to enter the decision-choice process? Let us suppose that we begin with both Urn I and II with n balls with unknown black-red proportions. Furthermore, let U1 and U 2 represent Urn I and II respectively. Let Θ1 = {θ1,θ2 , θ i ,θ n−1}1 be all possible red-black proportions in U1 and Θ2 = {θ1,θ2 , θ i ,θ n−1}2 in U 2 both of which define the possibility space. In Ellsberg’s experiment Θ2 = {θ1 }2 = {5B, 5R}2 which is one state of known nature while Θ1 = {θ1,θ2 , θ i ,θ n−1}1 contains ( n − 1) unknown state of nature. The computational complexities and decision indifference is illustrated in Figure 6.3.2.11. The individual decision-choice agent must first construct the possibility space since the space of potentials has been given by n balls of black-red combinations. Given Urn I, the decision-choice agent must compute ( n − 1) set of conditional statements of the form Φ1 = {P ( B|θ1 ), P ( B|θ1 ), , P ( B|θ n−1 )}1

(6.3.1.1)

By the method of Bayesian logic each one of the terms may be written as: P ( B|θi ) =

P ( B ∩ θi ) P ( θi )

(6.3.1.2)

No computational problem arises with the numerator P ( B ∩ θi ) to the extent to which the decision-choice agent’s possibilistic belief leads to identify θi .

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6 Paradoxes in Decision-Choice Theories

1B ⇒ P (1B ) = 1 n θ1 Unknown

θ's

( n − 1) R ⇒ P ( (1 − n ) R ) = ( n − 1) θ2

n

2B ⇒ P ( 2B ) = 2 n

( n − 2) R ⇒ P (( n − 2) R ) = ( n − 2)

n

U1 θ n− 2

( n − 2) B ⇒ P ( ( n − 2) B) = ( n − 2)

n

2R ⇒ P ( 2R ) = 2 n

( n − 1) B ⇒ P ( ( n − 1) B) = ( n − 1)

n

θ n−1 1R ⇒ P (1B ) = 1 n

Known θ

5B ⇒ P ( B ) = 0.5

U2 5R ⇒ P ( R ) = 0.5

Fig. 6.3.1.1. Possibility and Probability Interactions in Decision

The denominator cannot be computed at the available information-knowledge structure. It is here that vagueness and ambiguities tend to arise to create a penumbral decision-choice region that seems to lead to inconsistencies in probability assessments. In Urn II all the relevant spaces of a) potential is restricted by number of balls (n), b) possibility defined by one element of red-black proportion (5B,

6.3 Utility Theory, Probability and Paradoxes

149

5R) that allows the probability space to be specified for the decision-choice action on red or black. In Urn I, however, the space of potential is given to the decision-choice agent through the restriction of number of balls of black and red. The possibility space composed of (n-1) elements must be constructed from the potential without the probability space is undefined. The epistemic relevance of this relative analysis of Urn I and II is that the Savage axioms are created under assumed conditions of given potential and possibility spaces. The Ellsberg’s experimental test denies the foundational assumptions on which Savage’s axiomatic construct is developed. The problem of Ellsberg’s experiment is simply that there is no logical way that allows the decisionchoice agent to connect the potential to the probability. Epistemic discussions on the relationships of spaces of potential, possibility, probability and actual are provided in Chapters One and Two of this monograph while their relationship to knowledge construct is provided in a companion volume [R17.23]. The inconsistencies of probability assessments, under Ellsberg’s experiment and Savage axioms, are illustrated by Einhorn and Hogarth [R12.12], [R21.12] where individuals are indifferent between black or red in Urn I which implies P ( B1 ) = P ( R1 ) just as the probability distribution in Urn II is P ( B2 ) = P ( R 2 ) which implies that the decision-choice agents must be indifferent between selecting from Urn I and Urn II. The response however is such that the decision-agents prefer selecting from Urn II rather than Urn I which implies that P ( R 2 ) >P ( R1 ) = 0.5 and P ( B2 ) >P ( B1 ) = 0.5⎫⎪ ⎬ P ( R 2 ) = 0.5>P ( R1 ) and P ( B2 ) = 0.5>P ( B1 ) ⎪⎭ Thus P ( R 2 ) + P ( B2 ) > 1in the first case and P ( R1 ) + P ( B1 ) <1 in the second case that produce inconsistencies in the probability assessment in that we expect P ( R 2 ) +P ( B2 ) = P ( R1 ) +P ( B1 ) = 1. This inconsistencies are paradoxical that are explained by ambiguities and vagueness in the informationknowledge structure [R21.13], [R16.10], [R21.12].The use of ambiguities and vagueness as explanation of possible violation of Savage’s axioms is acknowledged by Savage as it is also pointed out by Ellsberg that Savage acknowledges “that the aura of vagueness attached to many judgments of personal probability might lead to systematic violations of his axioms.” [R16.10, p.660]. Ellsberg suggests that ambiguity is subjective variable. It has been argued in a companion volume that the process of our knowledge construct is riddled with subjective phenomena and the acceptance of a claim to an item as

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6 Paradoxes in Decision-Choice Theories

knowledge is transformed from subjective knowledge to objective knowledge by collective decision. The transformation process is an enveloping of decision-choice action in four corners that may be referred to as knowledge square [R17.23, Chapter One Figures. 1.1.1A&B]. This knowledge structure that flows from the knowledge square contains stochastic and fuzzy uncertainties that influence the decision-choice process. Attempts have been made to rehabilitate the Savage axioms when an aura of ambiguities is introduced into the stochastic uncertainty space by method of artificial grafting without recognizing that the paradox arises as a result of simultaneous presence of stochastic uncertainty and fuzzy uncertainty that renders the logic and mathematics of the classical paradigm ineffective in reasoning. The Ellsberg paradox is a permanent one whose resolution requires us to exist from the classical paradigm that imposes the principle of nonacceptance of contradiction and duality in thinking. As we have explained above, the space of the potential has been disconnected from the probability space by conditions of Ellsberg’s experiment. The probabilities of Θ1 = {θ1,θ2 , θi ,θ n −1}1 must be constructed from the subjective assessment of the possibilistic degrees of belief of the decision-choice agent. The appropriate logico-mathematical space for studying such decision-choice behavior that allows the possibility space to be developed to provide a continuous link between the potential space and the probability space as well as to resolve the Ellsberg’s paradox is in stochastic-fuzzy space and with methods of optimal fuzzy rationality whose toolbox is to be found in fuzzy paradigm, but not in classical paradigm. Nonetheless, some researchers on decision-choice theories under stochastic uncertainties and theories on subjective probability seem to think that the Ellsberg’s paradox is a temporary one in the classical paradigm and can be resolved by adjustment factor within the paradigm of its creation. For example Fellner states “The use of a correction factor seems to me suitable for restoring the comparability which is essential to the Savage type concept of subjective probability” [R10.10pp. 672]. Similarly Einhorn and Hogarth state “Our model postulates an anchoring-and-adjustment strategy for assessing probabilities” [R21.12, pp 436]. In both cases the adjustment factor or correction factor is to measure ambiguity which is then used as a correction factor in restoring comparability and monotonicity in Savage axioms using the same toolbox of classical paradigm. Ambiguity, which carries with it fuzzy uncertainty, is measured in probability space rather than in the possibility space. As it has been remarked in many places, fuzziness or ambiguity is not stochastic uncertainty and hence cannot and should not be meas-

6.3 Utility Theory, Probability and Paradoxes

151

ured and computed in probability space without acceptable logical transformation. Ambiguity and vagueness are part of fuzzy uncertainties and hence must be handled with fuzzy logic rather than probabilistic reasoning. Furthermore, the adjustment factors introduced by Fellner [R16.10] Einhorn and Hogarth [R21.12] do not tell us where these adjustments tend to arise in the logical space and how they relate to the possibility that connects the potential to the probability and then to the actual. 6.3.2

A Fuzzy Reflection on Some Suggested Resolutions of Ellsberg Paradox

In the framework of fuzzy paradigm, fuzzy uncertainty and stochastic uncertainty are considered to be in the same space that may be called fuzzystochastic space with corresponding fuzzy-stochastic variable or stochasticfuzzy space with corresponding stochastic-fuzzy variable as discussed in previous chapters. Both variables are made up of varying proportions of fuzzy variable and stochastic variable depending on the nature of the problem of interest. The separability and non-separability of the fuzzy-stochastic and stochastic-fuzzy variables into fuzzy variable and stochastic variable require analytical work with toolbox of fuzzy paradigm that has been explained in [R17.23], [R17.24]. The manners through which Fellner, Einhorn and Hogarth present the adjustment factor of vagueness indirectly assume separability of fuzzy-stochastic variable into fuzzy variable and stochastic variable without presenting the supporting conditions. Furthermore, the addition and subtraction of the fuzzy variable to or from the random variable is ad hock and even the adjustment factors may not be in probability units. An important problem here is simply the fuzzy variableS and the random variables are heterogeneous entities measured in different units and hence not directly summable. Both required conditions of separability and aggregation are discussed and presented in chapter Five of this monograph. The fuzzy and stochastic variables cannot be added without transforming one to the other so as to define them with the same common unit. Such unit transformation processes is also discussed in [R10.54] as well as in a companion monograph [R17.24]. The paradox arises because Savage is constructing his axioms in classical stochastic space with given possibility space using classical toolbox while Ellsberg is conducting his experiment in non-classical topological space using classical toolbox. The resolution centers on the concept of degrees of belonging as measured by membership functions and translated into possi-

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6 Paradoxes in Decision-Choice Theories

bilistic belief index. We may consider the Urn I with 100 black and red balls with unknown proportion. For the benefit of generality the number of balls are taken to be (n) where B = black and R = red. We assume that P ( B ) ∈ [ 0,1] = PB and P ( R ) ∈ [ 0,1] = PR where P ’s are sets of probability measures that capture stochastic uncertainty as defined over red and black balls in Urn I such that P ( B ) + P ( R ) = 1 . Let the set of black balls be B and that of red balls be R . To deal with the introduction of ambiguity and vagueness into the probability estimate require us to fuzzify and defuzzify the ambiguities and subjective weighting of possible probabilities associate with black-red differential proportions in Urn I. As shown in eqn. (6.3.1.2) and revealed in Figure (6.3.1.1) the resolution of the paradox requires us to develop the possibility space that will show the subjective assessment of conditions of vagueness and provide a logical connection to the probability space. All these must find expressions in acceptable logical spaces. Urn I presents us with possibility distribution of blackred proportions for finite possibility space Π . Given the possibility space, a probability space may be abstracted with probability distribution from which cumulative probability may be computed for black and red balls. From the postulate of equally likely outcome, each of the n balls has equal chance of ( 1 n ) in the selection process. It will become clear in this discussion why P ( B1 ) = P ( R 1 ) in Urn I by decision-choice agents’ indifference between black and red balls and why Urn II is preferred to Urn I in black-red selection process. Let us consider Urn I with n black (B) and red (R) balls with unknown proportions ( n = 100 in Ellsberg thought experiment). We assume that P ( B ) ∈ [ 0,1] and P ( R ) ∈ [ 0,1] . Let i be the possible number of black balls and hence ( n − i ) is the possible number of red balls that Urn I may contain. Since the proportion is not known i is a variable. Furthermore, let the set of possible number of black balls be B and that of red balls be R . These are fuzzy numbers whose values depend on subjective possibilistic belief. From the above information, we may specify black-red combinations to obtain the potential space as: Π = {( i, n − i ) | i ∈ B and ( n − i ) ∈ R ∋ ( i + n − i ) = n}

(6.3.2.1)

Thus from Θ1 = {θ1,θ2 , θi ,θ n −1}1 we have θ1 = (1, ( n − 1) ) and θi = ( i, ( n − i ) ) where i is the number of black balls and (n-i) is the number of red balls. The sets B and R contain elements that are made up of ambiguous black-red num-

6.3 Utility Theory, Probability and Paradoxes

153

ber proportions and the set Π contains both stochastic and fuzzy uncertainties thus defining either fuzzy-stochastic or stochastic-fuzzy space. To deal with decision-choice action in this space or the Ellsberg introduction of ambiguity and vagueness into the probability estimates from decision-choice action requires fuzzification to construct possibility distribution and defuzzification of ambiguities and subjective weighting that captures the possibilistic belief in order to connect the potential to the probability space. Thus equation (6.3.1.1) may be fuzzified as:

{

Π = ( i, µ B ( i ) ) , ( ( n − i ),µ R ( n − i ) ) |

(6.3.2.2)

i ∈ B, µ B ( i ) , µ R ( n − i ) ∈ [ 0,1] ∋ ( i + n − i ) = n}

Some observations may be useful at this point. Equation (6.3.2.2) defines a possibility or fuzzy space with a membership distribution function where n −1

n −1

i =1

i =1

µ R ( n − i ) = ⎡⎣1− µ B ( i ) ⎤⎦ ∀i , Π = n − 1 , B = ∑ µ B ( i ) R = ∑ (1−µ R ( i ) ) . It

may be noticed that the elements in the set Π appear as black-red pairs and the elements in the set, Π also appear as black-red pairs but equipped with a fuzzy operator. To link the possibility space that defines the logical path to the probability space eqn.(6.3.2.2.2) of the possibility space may be transformed into equivalent probability units in the form that define another probability space:

{

}

P = ( i, P ( i ) ) , ( ( n − i ), P ( n − i ) ) | i ∈ B, P ( i ) , P ( n − i ) ∈ [ 0,1] ∋ ( i + n − i ) = n

(6.3.2.3) Given the possibility space and the measure of possibilistic belief as expressed by membership characteristic function, we may now impose on the possibility space on the probability space and define the measure of probabilistic belief. The process is to logically combine eqns. (6.3.2.2) and (6.3.2.3) by constructing Φ = Π ∪ P as fuzzy-stochastic or stochastic-fuzzy set to obtain:

{

Φ = ( i, µ B ( i ) , P ( i ) ) , ( ( n − i ),µ R ( n − i ) , P ( n − i ) ) | i ∈ B, µ B ( i ) , µ R ( n − i ) ∈ [ 0,1]}

(6.3.2.4)

where + P ( i ) + P ( n − i ) = 1, ∀i which implies the condition that P ( B ) + P ( R ) = 1 at any black-red combination. An illustrative table of eqn. (6.3.2.4) may be set up with n= 10 as:

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6 Paradoxes in Decision-Choice Theories

Table 6.3.2.1. Measures of Fuzzy and Stochastic Uncertainties of Black-Red Ball Combinations

{i, µ ( i ) , P ( i )}

{n − i, µ ( n − i ) , P ( n − i )}

B

B

RED

BLACK i 1

µ B (i )

µ B (1)

P (i )

PB (1)

U B ( i ) = µ B ( i ) + PB ( i )

U B (1) = µ B (1) + PB (1)

n − i µ R ( n − i ) P ( n − i ) U R ( n − i ) = µ R ( n − 1) + PR ( n − 1) 9

2 µ B ( 2 ) PB ( 2 ) U B ( 2 ) = µ B ( 2 ) + PB ( 2 ) 3 µ B ( 3) PB ( 3) U B ( 3) = µ B ( 3) + PB ( 3)

8 7

5 µ B ( 5 ) PB ( 5 ) U B ( 5 ) = µ B ( 4 ) + PB ( 4 )

5

4 µ B ( 4 ) PB ( 4 ) U B ( 4 ) = µ B ( 4 ) + PB ( 4 )

6 µ B ( 6 ) PB ( 6 ) U B ( 6 ) = µ B ( 4 ) + PB ( 4 )

7 µ B ( 7 ) PB ( 7 ) U B ( 7 ) = µ B ( 4 ) + PB ( 4 ) 8 µ B ( 8 ) PB ( 8 ) U B ( 8 ) = µ B ( 4 ) + PB ( 4 ) 9 µ B ( 9 ) PB ( 9 ) U B ( 9 ) = µ B ( 4 ) + PB ( 4 )

6 4 3 2 1

µ R (9)

PR ( 9 )

µ R (8) µR (7)

PR ( 8 ) PR ( 7 )

µ R (5)

PR ( 5 )

µR (6)

PR ( 6 )

U R ( 9 ) = µ R ( 9 ) + PR ( 9 )

U R ( 8 ) = µ R ( 8 ) + PR ( 8 ) U R ( 7 ) = µ R ( 7 ) + PR ( 7 ) U R ( 6 ) = µ R ( 6 ) + PR ( 6 ) U R ( 5 ) = µ R ( 5 ) + PR ( 5 )

µR ( 4)

PR ( 4 )

U R ( 4 ) = µ R ( 4 ) + PR ( 4 )

µR ( 2) µ R (1)

PR ( 2 ) PR (1)

U R ( 2 ) = µ R ( 2 ) + PR ( 2 ) U R (1) = µ R (1) + PR (1)

µ R ( 3)

PR ( 3)

U R ( 3) = µ R ( 3) + PR ( 3)

The index of total uncertainty that must be processed by the decision-choice agent for decision-choice action to select is the sum U B = µ B ( i ) + P ( i ) for black balls and U R = µ R ( n − i ) + P ( n − i ) where U R = ⎡⎣1 − µ B ( i ) ⎤⎦ + ⎡⎣1 − P ( i ) ⎤⎦ = 2 − ⎡⎣µ B ( i ) + P ( i ) ⎤⎦ for the red balls. This total uncertainty is not computable in their present form. We may now construct the subjective probability assessment from the membership characteristic functions. If i is the number of black balls believed to be possibly contained in Urn I and hence ( n − i ) is the number of possible red balls. In a discrete case like the balls there are ( n − 1) − values of membership characteristic functions and ( n − 1) − values of probabilities that may be assessed for each of B and R . For black and red balls we may specify M B = {µ B (1) , µ B ( 2 ) , , µ B ( i ) , , µB ( n − 1)} as a set of membership degrees of belonging to B and M R = {µ R ( n − 1) , µ R ( n − 2 ) , , µ R ( n − i ) , , µR (1)} that of belonging to R as shown in Tables (6.3.2.1) and (6.3.2.2). The fuzzy complementary conditions by construct require that µ R ( i ) = ⎡⎣1 − µ B ( n-i ) ⎤⎦ where for example µ R ( 2 ) = ⎡⎣1 − µ B ( n − 2 ) ⎤⎦ . The sum U B = µ B ( i ) + P ( i ) as a sum of possibilistic belief on fuzzy uncertainty and probabilistic belief on fuzzy-stochastic uncertainty is a composite aggregate since they are in two different units. The components must be transformed to one common unit for aggregation. In this respect, the measures of fuzzy uncertainty may be reduced to units of stochastic uncertainty by a transformation

{

}

6.3 Utility Theory, Probability and Paradoxes

⎫ ⎪ ⎪ ⎪ ⎬ ⎧ ⎫⎪ ⎡1− µ ( i ) ⎤ ⎪ ⎪ Π B =PF ( R ) = ⎨ piF | piF = n−⎣1 B ⎦ ⎬⎪ ∑ ⎡⎣1− µB ( i )⎤⎦ ⎪⎪ ⎪⎩ i =1 ⎭⎭ ⎧ ⎫ ⎪ µ (i ) ⎪ Π B =PF ( B ) = ⎨ piF | piF = n−1 B ⎬ ∑ µB ( i ) ⎪ ⎪⎩ i =1 ⎭

155

(6.3.2.5)

For conditions of equal likelihood for each ball irrespective of whether black or red, each ball has ( 1 n ) probability of being selected and hence P ( i ) = ( i n ) and P ( n − i ) = ( n −i n ) . The measure of total uncertainty in terms of probability is ⎛ ⎞ ⎛i⎞ µ (i ) U B = ⎡⎣Π B ( i ) +PB ( i ) ⎤⎦ = ⎜ n−1 B ⎟ + ⎜ ⎟ ⎜ ∑ µB ( i ) ⎟ ⎝ n ⎠ ⎝ i=1 ⎠ U R = ⎡⎣Π R ( n − i ) +PB ( n − i ) ⎤⎦ =

⎡⎣1− µB ( i ) ⎤⎦ n −1

∑ ⎡⎣1− µB ( i )⎤⎦ i =1

⎫ ⎪ ⎪ ⎪ ⎬,i ≥ 1 − n i ⎛ ⎞⎪ +⎜ ⎟⎪ ⎝ n ⎠⎪ ⎭

(6.3.2.6)

Note that at each outcome under fuzzy and stochastic uncertainties that capture both possibilistic and probabilistic beliefs (ambiguity, vagueness, subjectivity and randomness) we have a measure of total uncertainties defined in probability units to exceed unity as seen in eqn. (6.3.2.7).

⎡ ⎤ ⎡⎛ i ⎞ n − i ⎤ ⎡1− µ ( i ) ⎤ µ (i ) ≥ 1, ∀i = 1, P ( B ) + P ( R ) = ⎢ n−1 B + n−⎣1 B ⎦ ⎥ + ⎢⎜ ⎟ + ⎢ ∑ µB ( i ) ∑ ⎣⎡1− µB ( i )⎦⎤ ⎥ ⎣⎝ n ⎠ n ⎥⎦ i =1 ⎣ i=1 ⎦

n (6.3.2.7)

Equation (6.3.2.7) may explicitly be written as P ( B ) + P ( R ) = ⎡⎣Π B ( i ) +Π R ( n − i ) ⎤⎦ + ⎡⎣ P ( i ) + P ( n − i ) ⎤⎦ ≥ 1, ∀i = 1,

n (6.3.2.8)

Equations (6.3.2.7) and (6.3.2.8) hold when fuzzy and stochastic uncertainties are separable in the fuzzy-random space or random-fuzzy space with defined topologies (see Figure 3.1 in Chapter Three of this monograph) and for conditions of separability and other related topics see Chapter Two and discussions in [R17.24]. It suggests superadditivity conditions of probability and hence inconsistent with known probability theory if probability is viewed as the only

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6 Paradoxes in Decision-Choice Theories

and complete measure of total uncertainty and if fuzzy risk is assumed away. In this case, we have a contradiction. This is not surprising since Savage axioms are defined in the classical stochastic space without vagueness and Ellsberg analysis is conducted in fuzzy-stochastic space whose unity measure of total uncertainty requires a different topological space and logical framework for decision-choice analyses. There are two approaches to overcome the problem of superadditivity.Under conditions of seperaility of fuzzy-stochastic space with fuzzy random variable Eqn. (6.3.2.7) can be normalized by redefining the units respectively as: n −1 ⎛ ⎞ ⎡ ⎤ ⎜ nµ B ( i ) + ∑ i µ B ( i ) ⎟ i ⎛ ⎞ µ (i ) i =1 ⎟ , ∀i = 1, U B =P ( B ) = ⎢ n−1 B + ⎜ ⎟ ⎥ = ⎜ n −1 ⎢ ∑ µB ( i ) ⎝ n ⎠ ⎥ ⎜ ⎟ n∑ µB ( i ) ⎣ i=1 ⎦ ⎜ ⎟ i =1 ⎝ ⎠

n

(6.3.2.9)

n −1 ⎛ ⎞ ⎡ ⎤ ⎜ n ⎡⎣1 − µB ( i ) ⎤⎦ + ( n − i ) ∑ ⎡⎣1 − µB ( i ) ⎤⎦ ⎟ − n i ⎡⎣1− µB ( i ) ⎤⎦ ⎛ ⎞⎥ ⎜ i =1 ⎟ + = U R =P ( R ) = ⎢ n−1 n −1 ⎢ ∑ ⎡⎣1− µB ( i )⎤⎦ ⎜⎝ n ⎟⎠ ⎥ ⎜ ⎟ n∑ ⎡⎣1 − µB ( i ) ⎤⎦ ⎣ i=1 ⎦ ⎜ ⎟ i =1 ⎝ ⎠ (6.3.2.10)

The total measure of stochastic and fussy uncertainties may then be specified in the units of probability as U T = ( U B + U R ) ∈ [ 0, 2] since we are adding values of two probability spaces. The components are in the same common unit and hence we can normalize them with the measure of total fuzzy and stochastic uncertainties to obtain

(

UB

UT

)+(

UR

UT

) =1

(6.3.2.11)

It is obvious that 0 ≤ ( UB UT ) ≤ 1 and 0 ≤ ( UR UT ) ≤ 1 where the probability value in fuzzy- stochastic space is ( UR UT ) = 1 − ( UB UT ) . The concept of subadditivity as described in [R12.12], [R21.4] is not obvious. Over valuation of vagueness which implies high possibilistic belief with Π T = ( Π B + Π R ) > 1 will lead to subadditivity while under valuation which implies Π T = ( Π B + Π R ) < 1 will lead to superadditivity. An alternative way presents itself to us through the method of Bayesian probabilistic reasoning using conditional logic. In this framework, the fuzzystochastic space and fuzzy random variable are taken as interactive with either

6.3 Utility Theory, Probability and Paradoxes

157

prior fuzzy and posterior stochastic or prior stochastic and posterior fuzzy. In the case of Ellsberg’s decision-choice experiment, we have prior fuzzy and posterior stochastic that allows us to use the conditional probability statement of equation (6.3.1.2) which is reproduced here for quick reference. P ( B|θi ) =

P ( B ∩ θi )

(6.3.1.2)B

P ( θi )

Equation (6.3.1.2B) is fuzzy conditional probability composed of fuzzy joint and fuzzy marginal probabilities. From eqn. (6.3.1.5), we know the fuzzy marginal as well as the fuzzy conditional and hence the probabilities of black and red outcomes may be computed as ⎛ ⎞ ⎛ ⎞ ⎛i⎞ µ (i ) iµ (i ) P ( B|θi ) P ( θi ) =P ( B ∩ θ i ) = ⎜ n−1 B ⎟ ⎜ ⎟ = ⎜ n−1B ⎟ ∀i = 1, ⎜ ∑ µB ( i ) ⎟ ⎝ n ⎠ ⎜ n ∑ µB ( i ) ⎟ ⎝ i=1 ⎠ ⎝ i=1 ⎠

n

⎛ ⎞ ⎛ ⎞ ⎛ n − i ⎞ ⎜ ( n − i ) µB ( i ) ⎟ µ (i ) ∀i = 1, P ( R|θ i ) P ( θi ) =P ( R ∩ θi ) = ⎜ n−1 B ⎟ ⎜ = n −1 ⎜ ∑ µB ( i ) ⎟ ⎝ n ⎟⎠ ⎜ n ∑ µB ( i ) ⎟ ⎝ i=1 ⎠ ⎝ i=1 ⎠

(6.3.2.12)

n (6.3.2.13)

Proposition 6.6.2.2.1

P ( B ) +P ( R ) =1 Proof ⎛ ⎞ ⎜ iµ ( i ) ⎟ ⎟ P ( B ) = ∑ P ( B ∩ θi ) = ∑ ⎡⎣ P ( B|θ i ) P ( θi ) ⎤⎦ = ∑ ⎜ n −1 B ⎟ i =1 i =1 i =1 ⎜ ⎜ n∑ µ B ( i ) ⎟ ⎝ i =1 ⎠ n −1

n −1

n −1

⎛ ⎞ ⎜ ( n − i ) µ (i ) ⎟ and P ( R ) = ∑ P ( R ∩ θi ) = ∑ ⎡⎣ P ( R|θi ) P ( θi ) ⎤⎦ = ∑ ⎜ n −1 B ⎟ ⎟ i =1 i =1 i =1 ⎜ ⎜ n∑ µ B ( i ) ⎟ ⎝ i =1 ⎠ n −1

n −1

n −1

158

6 Paradoxes in Decision-Choice Theories n −1

n −1

n −1

n −1

i =1

i =1

i =1

i =1

P ( B) +P ( R ) = ∑ P ( B ∩ θi ) + ∑ P ( R ∩ θi ) = ∑ P ( B|θi ) + ∑ ⎡⎣ P ( R|θi ) P ( θi ) ⎤⎦ ⎛ ⎞ ⎛ ⎞ ⎛ n −1 ⎞ n∑ µ B ( i ) ⎟ ⎟ ⎜ ⎟ ⎜ 1 n −1 ⎜ n − iµ ( i ) ( n − i ) µ (i ) ⎟ + ∑ ⎜ n −1 B ⎟ = ⎜ ni =−11 ⎟ =1 = ∑ ⎜ n −1B ⎟ i =1 ⎜ ⎟ ⎜ ⎟ i =1 ⎜ ⎜ n∑ µ B ( i ) ⎟ ⎜ n∑ µ B ( i ) ⎟ ⎜ n ∑ µ B ( i ) ⎟ ⎝ i =1 ⎠ ⎝ i =1 ⎠ ⎝ i =1 ⎠ Hence P ( B ) +P ( R ) =1 when vagueness and randomness are properly handled contrary to the result arrived in Ellsberg’s decision-choice experiment. In this structure, Einhorn and Hogarth’s method of anchoring-and-adjustment strategy for assessing probabilities is not necessary since we are concerned with total uncertainties surrounding the decision-choice action. This total uncertainty is the interactive sum of fuzzy uncertainty and stochastic uncertainty. It may be pointed out that the analytical structure will remain the same if instead of using µ B ( i ) for θi we use µ R ( n − i ) for θ n −i ∀i since B and R are fuzzy complements. 6.3.3

Fuzzy Optimal Rationality and Ellsberg Decision-Choice Experiment

The nature of the decision-choice action in the Ellsberg’s experiment and paradox require further analytical work in the fuzzy decision-choice space. The question is: under what set of conditions will the decision-choice agent select black or red from Urn I? Central to the judgment under ambiguity in this fuzzy logical framework is the degree of confidence attached to the probability assessments where µB ( i ) is the degree of confidence attached to pi with i specifying the number of black balls and hence pi = P ( B ) = ( i n ) and [1 − pi ] =P ( n − i ) = P ( R ) attached to red balls with confidence coefficient, µ R ( n − i ) = ⎡⎣1 − µ B ( i ) ⎤⎦ . A question arises as to how are these probabilities connected to Urn I to Urn II for decision to bet on black or red in either Urn I relative to Urn II. The answer to this question requires us to utilize the toolbox of optimal fuzzy decision-choice rationality. The logic involves the idea of duality where the degree of confidence attached to the assessment of the number ( i ) of black balls in the Urn I is constrained by the degree of confidence attached to the number of red balls such that the total of black and red is always to, n. This idea leads us to the fuzzy decision-choice space and a formulation of fuzzy decision-choice problem of the form: opt µ∆ ( i ) = opt ⎡⎣ µB ( i ) ∧ µR ( n − i ) ⎤⎦ i

i

(6.3.3.1)

6.3 Utility Theory, Probability and Paradoxes

159

The solution to this fuzzy free optimization problem may be transformed into constrained optimization problem as: ⎧opt µB ( i ) ⎪ opt µ∆ ( i ) = ⎨ i i ⎪⎩s.t ⎡⎣ µB( i ) − µR ( n − i ) ⎤⎦

(6.3.3.2)

0

The formulation of the problem presented in eqn. (6.3.3.2) is a combined maximum-minimum formulation. The method of solution to the problem provides us with a defuzzification process. The optimal solution to this type of fuzzy mathematical programming problem can be found in [R7.40], [R9.16], [R9.25], [R9.41 [R9.42], [R10.52], [R10.7517]. The solution yields an optimal value of the form ( n 2 ,α *) where µR ( n 2 ) * = µB ( n 2 ) = α * = µ ∆ ( n 2 ) defines a confidence coefficient at ( n 2 ) black balls and ( n 2 ) red balls and the corresponding probability is ⎛ n ⎞ ⎟ = 0.5 = P ( B ) = P ( R ) Π ( n 2 ) =P ( n 2 ) = ⎜ µ∆ ( 2 ) n−1 ⎜ ∑ µB ( i ) ⎟ i =1 ⎝ ⎠

(6.3.3.3)

The decision-choice indifference between black and red as reported in [24], [25] is basically fuzzy indifference that constitutes a solution to an optimal fuzzy decision problem for the decision-choice agent. We may now use the method of α − level set as the support of decisionchoice action to define two mutually exclusive sets for the black balls, H and H′ as

H = {i | µB ( i ) > α * and P ( i ) > 0.5} ⇒ P ( B ) > 0.5,

H′ = {i | µB ( i ) ≤ α * and P ( i ) ≤ 0.5} ⇒ P ( B ) ≤ 0.5,

( a ) ⎫⎪ ⎬ ( b )⎪⎭

(6.3.3.4)

Similarly two mutually exclusive and collectively exhaustive sets G and G′ may also be defined for the red ball in the Urn I as

G = {i | µ R ( n − i ) > α * and P ( n − i ) > 0.5} ⇒ P ( R ) > 0.5,

G′ = {i | µ R ( n − i ) ≤ α * and P ( n − i ) ≤ 0.5} ⇒ P ( R ) ≤ 0.5,

( a ) ⎫⎪ ⎬ ( b )⎪⎭

(6.3.3.5)

Very interesting interpretations emerge when these are connected to the Ellsberg’s thought experiment. Equations (6.3.3.4a) and (6.3.3.5a) imply that the decision-choice agent chooses Urn I over Urn II in selecting Black or Red ball.

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6 Paradoxes in Decision-Choice Theories

Equations (6.3.3.4b) and (6.3.3.5b) imply that the decision-choice agent selects Black or Red ball from Urn II. The nature of the optimal fuzzy decision-choice solution is provided in Figure (6.3.2.3.1). The decision-choice agent will always select from Urn II if the assessment is such that at α * -level of confidence coefficient P ( B ) = P ( R ) in Urn I for reasons of surety of Urn I.

1

µR ( n − i ) µ∆ ( i ) = µB ( i ) ∧ µR ( n − i ) µB ( i )

0

i* = ( n2 )

i

1

µ ∆ ( i* ) = α *

α*

0

i* = ( n 2 ) Fig. 6.3.3.1. A Resolution to Ellsberg Thought Experiment and Paradox by the Method of Optimal Fuzzy Rationality

6.4 A Numerical Example for the Ellsberg’s Experiment

161

From the stand point of general uncertainties the Ellsberg’s paradox may be viewed in terms of probability of ambiguous (fuzzy) event or ambiguous probability of exact event. The former involves fuzzy events such as heavy or light rain while the latter involves fuzzy probabilities defined over exact events. In the former case, the fuzzification takes place over the values of the events whose subjective value interpretation provides a fuzzy dataset for assessing the probabilities that corresponds to random fuzzy variable. In the latter case, the fuzzification takes place over the values of the probabilities to obtain fuzzy probabilities made up of classical probabilities and confidence weights. Here, the probabilities become fuzzy numbers where the assessment is through a logical manipulation of fuzzy random variable. The Ellsberg’s paradox belongs to the former (for discussions on methods of fuzzification and defuzzification see [R9.64], [R17.24] and [R10.132]). Savage produces his axioms of classical optimal decision-choice rationality under probabilistic belief system in deals with stochastic uncertainty and hence stochastic risk without the complications of fuzzy uncertainty and fuzzy risk as basic requirements of utilizing the toolbox of the classical paradigm. Ellsberg’s attempt to basically reconcile optimal decision-choice rationality as attribute of decision-choice agents with Savage’s axioms of classical optimal decisionchoice rationality as an ideal state of decision-choice process introduces fuzzy uncertainty and hence possibilistic belief system in addition to the stochastic uncertainty with probabilistic belief system. The processing of these two types of uncertainties for decision-choice actions requires paradigm shifting from known probabilistic reasoning to posibilistic-probabilistic reasoning that will allow axioms of behavior in fuzzy-stochastic space to be fully understood.

6.4 A Numerical Example for the Ellsberg’s Experiment A numerical example will be useful in the fuzzy analytical framework. Let us consider the Urn I to contain ten (10) balls of Black and Red of unknown proportion and Urn II the same 10 black and red ball equal proportion. The reference set is Q = {1,2,3,4,5,6,7,8,9,10} around which the possibility space may be defined as: Π = ( B,R ) = {(1,9 )( 2,8 ) , ( 3,7 )( 4,6 ) , ( 5,5 ) , ( 6, 4 ) , ( 7,3)( 8, 2 )( 9,1)} . From this we fuzzify Π by introducing the membership characteristic function as the degree of black-red proportions in Urn I. Let B be the fuzzy collection of

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6 Paradoxes in Decision-Choice Theories

Table 6.4.1. Measures of Fuzzy and Stochastic Uncertainties of Black-Red Ball Combinations

{i, µ ( i ) , P ( i ) , Π ( θ )} B

{n − i, µ ( n − i ) , P ( n − i )} B

i

RED

BLACK i 1

µ B ( i ) P ( i ) Π ( θ i ) P ( B i ∩ θ i ) = P ( θ i ) P ( Bi | θ i ) 0.1 0.05 0.005 0.2

n − i µ R ( n − i ) P ( n − i ) P ( R n − i ∩θ i ) = P ( θ i ) P ( R n −i | θ i ) 9 0.8 0.9 0.045

2

0.5

0.2

0.13

0.026

8

0.5

0.8

0.104

3 4

0.8 0.6

0.3 0.4

0.21 0.15

0.063 0.060

7 6

0.2 0.4

0.7 0.6

0.147 0.090

5 6

0.7 0.4

0.5 0.6

0.18 0.10

0.090 0.06

5 4

0.3 0.6

0.5 0.4

0.09 0.040

7

0.3

0.7

0.08

0.056

3

0.7

0.3

0.024

8 9

0.2 0.2

0.8 0.9

0.05 0.05

0.04 0.045

2 1

0.8 0.8

0.2 0.1

0.01 0.005

1

0.445

0.555

various quantities of backs and R that of red from Π each of which is equipped fuzzy operator of the form µ B ( i ) and µ R ( n − 1) respectively. Corresponding to black-red combination we have probabilities of the form P ( i ) and P ( n − 1) respectively. The table (6.3.4.1) is consistent with Table (6.3.2.1). As we have pointed out column four is a composite value in different fuzzy and random units that needs to be transformed to one unit of measurement. We have decided to transform the fuzzy units into probability 9

units. From the table it is easily seen that P ( B ) = ∑ P ( Bi ∩ θi ) = 0.445 and i=1 9 a P ( R ) = ∑ P ( R n −i ∩ θi ) = 0.555 and hence P ( B ) + P ( R ) = 0.445 + 0.555 = 1 i =1 9 ⎛ ⎞ where P ( θ i ) = ⎜ µ B ( i ) ∑ µ R ( i ) ⎟ ∀i are defined in terms of number of posi =1 ⎝ ⎠ sible blacks contained in the Urn I. The P ( θ i ) may be defined in terms of membership distribution around the number of red balls instead of the black balls. In this example with the defined membership distribution function, the individual will bet on red ball ® from Urn I and black ball from Urn II. One epistemic importance of this example is that vagueness or ambiguity as part of fuzzy uncertainty measured by an appropriate index of possibilistic belief can be transformed into stochastic uncertainty measured by an appropriate index of probabilistic belief for comparability and summability. Another important outcome is that the total uncertainty space preserves the unity as demanded by the axioms of probability. Furthermore, given the probabilistic space the measure of total uncertainty will be influenced by individual perception of ambiguity couched in possibilistic belief.

6.5 Other Decision Paradoxes and Paradoxes in General

163

6.5 Fuzzy Optimal Decision-Choice Rationality, Other Decision Paradoxes and Paradoxes in General Let us have a short reflection on other decision-choice paradoxes such as St. Petersburg’s paradox, Allais paradox, Newcomb’s paradox and paradoxes in general. It has already been pointed out that epistemically any paradox is either temporary or permanent in the paradigm of its creation. We have argued that the St. Petersburg’s paradox is a temporary one whose resolution is obtained in the classical paradigm through the restriction of the behavior of the utility function by decreasing marginal utility of money. This restriction allows upper bound to be imposed on the reward-utility space. Things are different for Newcomb’s paradox and Allais paradox. If these paradoxes are temporary then their resolutions can be obtained within the ambit of the organic classical paradigm in which their creation occurred. The resolutions within it may, however, require inter-sub-paradigm shifting in terms of intraorganic paradigm changes as we have explained previously. If it is a permanent one in the sense that resolution cannot be found within the paradigm of its creation then inter-organic paradigm shifting may be required. How do we find out whether a paradox is a temporary or a permanent one? The process of epistemic discovery of the temporariness or permanence of a paradox requires us to logically examine the corners of the knowledge square which is produced here for easy reference. In particular, one must examine the assumptions that allow cognitive connections among the potential, the possible, the probable and the actual as well as the culture of knowledge production that is defined by the ruling paradigm, logical structures that define the linkages of the knowledge square to provide a complete system of thought, production of knowledge elements, verification of true-false validity and acceptance principles of knowledge items. Given the organic paradigm of knowledge production, each block of the knowledge square has its own selfcontained system of logical reasoning that must be consistent with selfcontained systems of logical reasoning of other blocks. These self-contained systems of logical reasoning are dependent on each other as well as together define the structure of the organic paradigm and the nature of the enveloping of the organic path of knowledge enterprise. Most paradoxes in science, philosophy, and other knowledge sectors arise from two major sources. They may arise in the process of linking the possibility space to the probability space and then to the space of the actual. This is a paradox of inter-family categorial conversions of the cognitive transformation processes that may be due to the deficiencies in the organic paradigm and the

164

6 Paradoxes in Decision-Choice Theories

U =FC

ϕ1 :FC

ϕ1 ϕ2 :FC

ϕ4 : FC

FA

ϕ3 : FB

FP ϕ2 : FP

FB

Fig. 6.5.1. Substitution-Transformation Process in Terms of Categorial Relations and Cognitive Functions in the Knowledge Square Where U = Potential Space P = Possibility Space B = Probability Space and A = the Space of Actual; FP = Family of Category of Possibilities FB = Family of Category of Probables, FA = Family of Category of Actual U = FC = family of categories of potential with an Epistemic Constraint Structured as ( A ⊂ B ⊆ P with FA ⊆ FB ⊆ FP ⊆ U ) . the ϕi ' s Are Cognitive Transformation Functions in the Knowledge Construction Process.

corresponding decision-choice rationality. The construction of the relevant elements of the possibility space from the potential space is a knowledgesector specific imbued with vagueness, ambiguities, subjectivity, contradictions, true-false duality, and ignorance (fuzzy characteristics) that create cognitive complexities. The logical movement from the possibility space to the probability space and then to the space of the actual through the classical paradigm and the toolbox of the classical optimal decision-choice rationality neglects or assumes away these essential characteristics that present complexities for the benefit of precision and simplicity. As it has been observed and remarked by Zadeh :“ As complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes until a threshold is reached beyond which precision and significance (or relevance) become almost mutually exclusive characteristics [R10.139 p.3]. Generally, precision

6.5 Other Decision Paradoxes and Paradoxes in General

165

and uncertainty constitute a duality that mutually defines themselves in both relative proportions in conceptual unity. The more precise we make our scientific knowledge item through explication and logic, the more we increase the content of uncertainty of the knowledge’s application through irrelevance. The epistemic point here is that precision and uncertainty are mutually dependent in our knowledge transformation process as well as decision-choice rationality that may be practiced. It is within this cognitive transformation from a family of categories of possibilities to the families of categories of probabilities and actual elements that permanent paradoxes tend to emerge in our knowledge production process. The presence of these permanent paradoxes in the organic classical paradigm may then be explained with the neglected essential characteristics in the possibility space. Their resolutions may find their tools from the shifting of the organic classical paradigm to an organic paradigm that accepts vagueness, contradictions, ambiguities, presence of true-false duality and others in our knowledge production. The known organic paradigm is the organic fuzzy paradigm with the toolbox of fuzzy optimal decision-choice rationality. Paradoxes may also arise within logical systems of the blocks of the knowledge square in the applications of the toolbox of organic classical paradigm. Such paradoxes are temporary and are the result of either faulty applications of logical rules within the organic classical paradigm or improper conception of intra-categorial conversions within families of categories of probability or actual. Temporary paradoxes are logical-system specific and are explainable by incorrect or faulty applications of the toolbox of the organic paradigm but not by the essential characteristics of possibility space. Temporary paradoxes may be resolve by correction in logic or by intra-sub-paradigm shifts with the organic classical paradigm. If a paradox in the organic classical paradigm cannot be resolved from within its logical system in the analyses and syntheses of the family of categories of probability and actual, then one must proceed to examine the assumptions, categorial conversions and cognitive transformations that connect the family of categories of possibility to those of categories of probability and actual. First, we notice that all the axioms of probability, expected utility and decision-choice action under uncertainty, take the possibility space as given. They, further, maintain the Euler’s max-min postulate of universal system where everything that happens in the universal system has a sense of either maximum or minimum to cognitive agents. All the vagueness, ambiguities and contradictions are replaced with exactness that functions under Aristotelian principles of non-contradiction. These are true with Allais paradox, New-

166

6 Paradoxes in Decision-Choice Theories

comb’s paradox and other paradoxes. Where then do ambiguities and vagueness enter into both Newcomb’s paradox and Allais paradox and how do paradoxes enter into the knowledge construction process under the governance of the classical paradigm and the use of the toolbox of classical optimal decision-choice rationality? It is an epistemic view here that most paradoxes in sciences and philosophy arise in the process of logically linking the space of potential to the possibility space than to the probability space and finally to the actual. The construction of the relevant elements in the possibility space from the space of potential is knowledge-sector specific where the properties of the constructed space include vagueness, ambiguities, conceptual approximations, and subjectivity that tend to influence decision-choice rationality. Let us look a little closer at the knowledge square and examine the possible logical connections that allow the universal knowledge bag to be constructed on the basis of acceptable decision-choice rationality. The path connections of the knowledge square may conceptually be partitioned into four knowledge pyramids, of Cases I, II, III, and IV as shown in Figure 6.5.2. Case I is a cog-

U = FC

FP

FP CASE II

CASE I

FB

U = FC

FP

FB

FA

U = FC CASE IV

CASE III

FA

FA

FB

Fig. 6.5.2. The Quadrangle Knowledge Pyramids as a Partition of the Knowledge Square

6.5 Other Decision Paradoxes and Paradoxes in General

167

nitive movement from the potential, U = F C to the possible F P and then ends with the probable F P without a connection to the actual. Case II is the cognitive movement from the possible FP to the probable FB to end up at the actual FA without any considerations of the potential. Case III is a cognitive movement from the potential U = F C to the possible FP and then ends up with the actual FA to the neglect of the probable FB and stochastic uncertainty (limited information). Case IV is a cognitive movement from the potential U = F C to the probable FB and to the actual FA while the possible FP is neglected or assumed away. These pyramidal logical structures obey the triangular logical laws of cognitive soft computing, we may add. Let us relate these knowledge pyramids to emergence of paradoxes in classical paradigm and the use of classical optimal decision-choice rationality. It may be noticed that Case I may be dismissed as logically inappropriate since the whole enterprise of knowledge production is about the actual (what there is) and hence the endpoint of probable is not cognitively acceptable in any paradigm of knowledge enterprise. Case II is cognitively inappropriate since there cannot be the possible without the potential except we are willing to initialize the knowledge production process with possible; in which case we have to explain how the possible is conceptualized. Furthermore, we must explain the structure of the characteristics of the elements of the possibility space. It seems that Case III has been the path structure of the classical paradigm and its rationality on the basis of Aristotelian logic and the principle of exactness and certainty (full information) in addition to full neglect of fuzzy characteristics of the possibility space. To rescue the classical knowledge construction from the defect of information limitation of human thought process in the classical paradigm the classical paradigm adopts the knowledge pyramid of Case IV where the cognitive movement is from the potential to the probable and to the actual to the neglect of conditions of the possible through probably an implicit assumption. Even though the assumption of certainty is relaxed by introducing the probability space in order to incorporate elements of human ignorance (limited information), the conditions of fuzzy characteristics including vagueness and ambiguities are assumed away since the cognitive process bypasses the possibility space and certain and uncertain variables are assumed to be exact. The nature of operations with the classical paradigm in Case III, and IV has been an important source of permanent paradoxes in the classical knowledge production process. All the temporary paradoxes are merely misapplications of the logical rules with the classical paradigm. The problem may be viewed as beginning from our number construction process that is accepted in the classical paradigm in the space of the actual

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6 Paradoxes in Decision-Choice Theories

that embraces conditions of exactness and certainty which is then extended to the principles of absolute knowledge and the use of Aristotelian logic in reasoning where a proposition is either true or false but not both in Case III. The use this classical paradigm results in non-fuzzy and non-stochastic rationality. The application of the classical paradigm in conditions of Case III then receives a backward logical path extension to conditions of Case IV to deal with the problems of uncertainty associated with limited information in human operations leading to optimal non-fuzzy stochastic decision-choice rationality. The cognitive structure of the classical paradigm and optimal decision-choice rationality that it engenders have influenced and continue to influence our thinking as well as guide our decision-choice process in all areas of human endeavors including knowledge production, acceptance and practice. The search for knowledge with the rules of classical optimal decisionchoice rationality runs into paradoxes as a result of circumventing the full use of sequential path of the knowledge square in favor of one of the knowledge pyramids. The integration of the knowledge pyramids of either Cases I and IV or Cases II and III produces transformation and cognitive unity that reinstate the knowledge square. The integrating factor is through the fuzzy paradigm with its rules of fuzzy optimal decision-choice rationality. From the structures and relationships of the knowledge square and pyramids, one can understand and begin to unravel the problems surrounding the Newcomb’s paradox and Allais paradox. The Newcomb’s thought experiment is created in fuzzy-stochastic space where the choice variable is a fuzzyrandom variable however much of the analysis has always been viewed and analyzed in the classical non-fuzzy stochastic space where vagueness as an element of the information set is excluded [R23], [R23.14], [R23.28], [23.38], [R23.51].The thought experiment places the decision-choice agent in both possibility and probability spaces such that the decision-choice variables are fuzzy-stochastic or stochastic-fuzzy. This is done by the introduction of a perfect predictor (Divine, may be) of the actions of the decision-choice agent who are confronted with fuzzy incomplete information leading to decisionchoice conflict between the use of the rules of expected utility hypothesis and the rules of dominance principles in ranking and decision-choice action. No logical allowance is made for computational limitations and vagueness in interpretations of the available information. The logical extensions of the interpretation of the conflict to the proposition that free will and determinism are incompatible is a direct application of the Aristotelian logic and the neglect of the principle of true-false duality where degrees of truth and falsity can exist in the same statement and action as the

6.5 Other Decision Paradoxes and Paradoxes in General

169

fuzzy paradigm offers us. Within the epistemics of fuzzy paradigm, the perfect predictor also allows us to reexamine the past-present-future interactions that exert preponderating effects on current decision-choice actions and the selection of the appropriate rules of optimal decision-choice rationality. It may be noted that while the rules of the classical optimal decision-choice rationality do not allow decisions on truth validity to be constrained by falsity since it is either one or the other in mutual exclusivity and vice versa the cognitive operations with the rules of fuzzy optimal rationality impose on the decision-choice acceptance of truth validity the epistemic constraint of falsity as a non-empty set. Thus the cognitive force behind the development of fuzzy paradigm and the fuzzy optimal decision-choice rationality is the realization of the inherent inadequacies and problems of decision-choice rationality in Aristotelian logical system supported by classical theories of crisp set, nonvague probability in such a way that the classical paradigm shows logical deficiencies when we are confronted with imprecision, vagueness, ambiguities, and complexities that are characteristics of our information-knowledge structure and decision-choice processes. The Allais paradox may also be viewed in the same light in terms of the knowledge square and the partitioned pyramids. It shares some of the epistemic problems with Ellsberg’s paradox and Newcomb’s paradox. The first question that must be asked is whether Allais, and von Neumann and Morgenstern are working in the same analytical space. The second question is whether von Neumann-Morgenstern axioms are interpreted in reference to either explanatory theory or prescriptive theory. If von Neumann-Morgenstern axioms with surrounding theory are viewed in reference to prescriptive theory, then the Allais paradox does not arise in that a claim can be made that the decision-choice agents are unable to follow the optimal rules derived under expected utility hypothesis. Alternatively, if the axioms and the surrounding theory are viewed in terms of explanatory science then we may raise a question as to what are the essential choice variables that the decision-choice agents must act on. Two situations present themselves: 1) the choice variable is probability given the sizes of the reward, 2) reward is the decision-choice variable given the probability. Action 1) and 2) must be undertaken: A) irrespective of the value of expected utility or B) to maximize the expected utility of the decision-choice agent. All these tend to be mixed in the experiments on decision-choice actions that tend to produce paradox in that, the actual observed decision-choice behavior takes place in the fuzzy-stochastic space with fuzzy-random variables while the expected utility hypothesis and supporting axioms take place in non-fuzzy-stochastic space which seem to suggest that

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6 Paradoxes in Decision-Choice Theories

Allais, von Neumann and Morgenstern are operating in two different analytical spaces. Is it possible that decision-choice agents see the probabilities in vague terms or in linguistic numbers rather than in crisp numbers as decision-choice actions are undertaken under uncertain conditions? Is it also possible that the decision-choice agents are using the rules of fuzzy optimal decision-choice rationality in uncertain conditions rather than using the classical optimal stochastic rationality? Additionally, it may be pointed out that social ideology creates anchoring behavior in the decision-choice process in such a way that the rules of classical optimal decision-choice may be violated. The reason is that the classical paradigm composed of its mathematics and logic presents optimal decision-choice behavior without the consideration of the preponderating effects of social ideology. On the other hand, the fuzzy paradigm suggests that decision-choice agents in uncertain decision-choice space are guided by the rules of fuzzy optimal decision-choice rationality. The rules of optimal fuzzy decision-choice rationality can be formulated to include important parameters that constrain the decision-choice behavior to many types of optimal behavior including bounded rationality and others which we have argued belong to fuzzy optimal decision-choice rationality.

References

R1

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[R2.5.1]

On Cost-Benefit Rationality and Discounting

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R19 [R19.1]

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R20

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R21

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R22

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Index

A Accident, 92 – 94 Accounting theory, 176 Action set, 141 – 144 Action structure, 57 – 75 Actual-potential duality, 7 – 22, 92 – 98, 122, 125 – 127 Adaptation principle, 126 – 127 Aggregation, 154, 171 – 173 Aggregation problem, 70 Aggregation process, 139 Allais’s paradox, 61 – 65, 141 – 142, 165 – 170 Almost false, 35 Almost true, 35 Ambiguity, 51 – 82, 92 – 93, 144 – 156 Ambiguous event, 141 – 158, 161 – 163 Analytical power, 33 – 34 Analytical sufficiency, 113 – 114 Anomalies, 30 – 31 Approximate reasoning, 34 – 39, 275 – 279 Aristotelian logic, 116, 167 – 169 Aristotelian logical inference, 107 – 108 Aristotelian principle, 140 Arrow’s axiomatic interpretations, 139 – 140 Arrow’s axioms, 140 – 141 Arrow’s collective choice problem, 140 Arrow’s paradox, 137 – 140

Arrow’s problem, 138 – 139 Aspiration level, 25 – 26, 29 – 34, 48, 79 – 82 Axiomatic system of expectation, 23 – 25

B Bayesian analytical framework, 144 – 145 Belief-possibilistic-probabilistic – pyramid, 103 – 104 Belief system, 167 – 168 Bernoullian principle, 21 Bernoulli’s resolution, 60 Best decision-choice action, 79 – 82 Best path, 7 Black, Max, 65 Bounded rationality hypothesis, 24 – 26, 28 – 49, 262 – 264

C Categorial conversion, 33 – 34, 79 – 82, 94 – 104, 110 – 116, 165 – 170 Categorial convertibility, 125 – 126 Categorial derivatives, 23 Categorial uncertainty, 6 Categories of decision-choice theories, 134 – 137 Categories of risk, 94 – 98 Categories of risk theories, 134 – 137 Category of reality, 77 – 78 Category theory, 253 – 254

282

Index

Center of decision, 85 – 87 Characteristics-based information, 2 Circumference of expectation formation, 51 – 53 Classical decision-choice theories, 106 – 113 Classical exact decision-choice space, 141 – 145 Classical law of reasoning, 117 Classical paradigm, 23 – 34 Classical paradigm preserving, 144 – 147 Classical rationality, 8 – 10 Classical optimal rationality, 28 – 34, 42 – 44 Classical rationality, 233 – 218 Classical theory of risk-bearing, 112 – 113 Classification of ambiguity, 75 Classification of vagueness, 75 Cognitive attribute, 28 – 34 Cognitive capacity, 32 – 33 Cognitive computation, 63, 69 Cognitive ignorance, 7 – 8 Cognitive limitation, 106 Cognitive systems, 144 – 147 Cognitive transformation function, 143 – 145 Collective acceptance, 3 Collective decision-choice action, 139 – 140 Collective motivation, 80 – 82 Collective risk-taking process, 122 – 124 Common justification support, 18 – 22 Comparative analysis, 12 Computable algorithms, 59 Concept of aspiration level, 79 – 82 Concept of risk, 111 Concept of satisficing, 79 – 82 Conceptual-measurement problem, 86 – 87 Conditions of justification, 4 – 5 Constrained optimal rationality, 25 – 26

Constrained optimization problem, 79 – 82 Constructionism, 107 – 108, 114 – 115 Contingent existence of risk, 116 Contingent valuation method, 186 – 189 Contradiction, 45 – 49 Corroboration, 32 Cost, 56 – 57 Cost-benefit balances, 110 Cost-benefit configuration, 109 Cost-benefit distribution, 98 Cost-benefit duality, 47 – 48, 96 – 98 Cost-benefit evaluation, 80 – 82 Cost-benefit rationality, 45 – 49, 127 – 132, 174 – 199 Cost-benefit-risk configuration, 95 Cost-benefit structure, 56 – 78 Crisp partition, 4 – 5 Crisp set theory, 30 – 34

D Decision-choice agent, 4, 8, 75 Decision-choice criteria, 180 – 182 Decision-choice conflict, 264 – 266 Decision-choice intelligence, 3, 58 – 60 Decision-choice modulus, 29 – 34 Decision-choice outcomes, 3 Decision-choice problems, 141 – 144 Decision-choice process, 174 – 176, 191 – 199, 245 – 248 Decision-choice rationality, 1 – 4, 83 – 104, 119 – 127, 133 – 137, 229 – 233 Decision-choice space, 3, 55 – 57, 138 – 140 Decision-choice theories, 105 – 132 Deficient information-knowledge – structure, 108 – 113 Definition of accident, 93 – 94 Definition of event, 109 Definition of fuzzy risk, 90 – 91 Definition of fuzzy uncertainty, 89 – 91 Definition of natural risk, 120 – 121

Index Definition of natural risk-engineering, 126 – 127 Definition of necessity, 93 – 94 Definition of potential outcome, 109 Definition of pure uncertainty, 88 – 89 Definition of pure risk, 129 – 132 Definition of risk, 87 – 91, 111 – 112 Definition of risk neutral, 131 – 132 Definition of risk outcome, 111 – 112 Definition of risk-free event, 129 – 130 Definition of risky event, 111 – 112, 129 – 130 Definition of social risk, 120 – 121 Definition of social risk engineering, 125 Definition of stochastic risk, 90 – 91 Definition of stochastic uncertainty, 89 – 91 Definition of uncertain event, 111 – 112 Definition of uncertainty, 86 – 91 Defuzzification, 34 – 38, 40 – 45, 153 – 160 Degree ob belief, 144 – 146 Degree of confidence, 158 Degree of convertibility, 14 – 15 Degree of fuzziness, 63 Degree of possibilistic belief, 21 – 22 Degree of preference, 111 – 112 Degree of probabilistic belief, 17 – 22 Degree of risk, 84 – 86 Degree of transformation, 14 – 15 Democratic choice process, 140 – 142 Democratic social organization, 138 Derived category, 23 – 26, 114 Descriptive language of commonsense, 87 – 88 Descriptive language of science, 87 – 88 Determinism, 168 – 170 Dictatorship, 140 Discounting, 184 – 185 Disequilibrium states, 38

283

E Economic costing, 176 – 180 Economics, 78 – 82 Einhorn, H., 146, 149 – 152 Ellsberg’s experiment, 149 – 158, 161 – 170 Ellsberg’s paradox, 61 – 65, 141 – 158 Engineered risk system, 121 Epistemic analyses, 55 – 75 Epistemic constraint construct, 164 Epistemic geometry, 8, 76 – 78 Epistemic model of rationality, 74 Epistemics of risk, 83 – 104 Epsilon neighborhood covering, 39 – 40 Euler’s mini-max principle, 1, 57, 80 – 82, 100 – 104 Euler’s statement, 48 Evidential support, 5 Exact information-knowledge structure, 117 – 119 Exact knowledge, 56 – 58 Exact stochastic preference ordering, 61 Exactness, 60 – 70, 109 – 113 Expectation formation, 11 – 14, 22 – 26 Expectation rationality, 23 – 24 Expectations, 1 – 26 Expected utility theory, 58 – 60, 107 – 108, 274 – 275 Expectations, 199 – 203 Experimental process, 143 – 144 Experimental design, 147 – 151 Explanatory hypothesis, 19 – 22 Explanatory science, 62 – 66 Explication of probability, 11 – 14, 53 – 55 Explication of possibility, 11 – 14, 53 – 55 Explicadum, 54 Explicatum, 54

284

Index

F Falsification, 68 Fellner, W., 151 – 152 Finite set of possibilities, 9 – 11 First principle of freedom, 1o1 – 104 Forecasted knowledge, 26 Formal cognitive system, 144 Foundations of fuzzy risk, 116 – 119 Four structures of rationality, 133 – 137 Free will, 168 – 170 Freedom, 94 – 104 Fundamental assumption, 56 – 59 Future, 4 – 5 Fuzzification, 34 – 38, 40 – 45 Fuzzification-defuzzification-process, 71 – 75 Fuzziness, 87, 113 – 119, 124 – 128, 203 – 213 Fuzzy alpha-level optimality, 48 – 49 Fuzzy category, 35 – 38 Fuzzy cost, 66 – 67 Fuzzy decision, 213 – 219 Fuzzy decision space, 74 Fuzzy economic theory, 33 Fuzzy event, 74 – 75 Fuzzy false statement, 36 – 38 Fuzzy group, 35 – 38 Fuzzy indifference, 159 Fuzzy logic, 225 – 229 Fuzzy logical category, 33 Fuzzy logical framework, 158 Fuzzy mathematics, 46, 233 – 242 Fuzzy measure, 74 – 75 Fuzzy neural network, 33 Fuzzy non-stochastic rationality, 64 Fuzzy number system, 38 – 40 Fuzzy optimal rationality, 25 – 26 Fuzzy optimization, 33 Fuzzy optimization, 229 – 233 Fuzzy paradigm, 31 – 49 Fuzzy partition, 4 – 5 Fuzzy preference ordering, 74 – 75 Fuzzy probabilities, 161 – 170, 242 – 245

Fuzzy probability theory, 33 Fuzzy-random event, 111 – 112 Fuzzy-random space, 155 – 157 Fuzzy-random variable, 70 – 75, 161, 242 – 245 Fuzzy rationality, 1 – 26, 27 – 49, 127 – 129, 134 – 144, 163 – 170 Fuzzy reward, 66 – 67 Fuzzy risk, 52 – 53, 62 – 79, 105 – 106, 145 – 147, 156 Fuzzy risk covering, 49 Fuzzy set theory, 30 – 34 Fuzzy-stochastic cost, 74 – 75 Fuzzy-stochastic knowledge, 71 – 75 Fuzzy-stochastic optimal rationality, 119 163 – 170 Fuzzy-stochastic process, 69 – 74 Fuzzy-stochastic reward, 74 – 75 Fuzzy-stochastic risk, 117 – 119 Fuzzy-stochastic space, 153 – 154 Fuzzy-stochastic uncertainty, 117 – 119 Fuzzy sub-categories, 31 – 34 Fuzzy toolbox, 46 Fuzzy truth statement, 36 – 38 Fuzzy uncertainty, 6 – 11, 52 – 53, 85 – 87, 101 – 106, 142 – 143, 145 – 147

G Game theory, 219 – 225’ 264 – 266 Goal-constraint structure, 56 – 82

H Hogarth, R. M., 146 – 147, 147 – 152 Human ignorance, 167 – 170

I Ideal category, 52 – 53 Ideal decision-choice state, 28 – 34, 135 – 137 Ideology, 247 – 248 Ill-defined explication, 6 Ill-defined problem, 54

Index Impossibility theorem, 137 – 140 Imprecise evidence, 6 Incentive structure, 79 – 82 Incomplete knowledge, 4 – 5 Individual risk-taking process, 122 – 124 Inexactness, 54 Information, 5 – 11, 248 – 253, 264 – 266 Information deficiency, 92 – 96 Information-knowledge deficiency, 87 Information-knowledge structure, 83 – 87, 105 – 113, 124 – 128, 134 – 136 Information-knowledge support, 142 – 143 Information processing capacity, 3, 29 – 30 Input-output duality, 79 – 82 Input-output process, 1 – 3 Insufficient reason, 114 – 116 Insurance, 124 – 129 Isomorphism, 72 Intentional social risk, 120 Inter-categorial conversion, 114 – 116 Inter-paradigm changes, 133 – 136 Intra-categorial conversion, 114 – 116 Intra-paradigm changes, 133 – 136

J Judgment under ambiguity, 158, 161 – 170 Isomorphism, 140 – 141 Justification, 13 Justification principle, 14 – 15 Justification set, 5 Justified belief, 19 – 22 Justified expectations, 23 – 25 Justified prediction, 26 Justified support, 20 – 22

K Keynesian principle of indifference, 21 – 22

285

Knight, Frank, 145 – 147 Knowledge, 5 – 11, 248 – 253 Knowledge accumulation process, 13 – 22 Knowledge category, 33 – 34 Knowledge construction, 32 – 34 Knowledge creation process, 94 Knowledge justification, 17 – 22 Knowledge production culture, 3 Knowledge-risk square, 92 – 94 Knowledge square, 10 – 11, 91 – 92, 163 – 170 Knowledge structure, 17 – 22, 56 – 78 Language formation, 68 Linguistic approximation, 62 – 66 Linguistic hedges, 63 – 65 Linguistic reasoning, 16 – 18, 32 – 38, 51 – 53 Logical categorial derivatives, 55 Logico-mathematical space, 150

M Majority-minority points, 140 Marginal utility of money, 60 Mathematics of expectation, 107 – 108 Maximum degree of belief, 20 Measurable uncertainty, 145 – 147 Measurement approximations, 68 Method of Bayesian logic, 147 – 151 Method of reductionism, 7, 11 Methodology of science, 30 Mini-max postulate, 57, 62 – 66 Minimum variance, 107 – 108 Motivation, 7, 57 – 79

N Nature of ambiguity, 92 – 93 Natural risk engineering, 125 – 127 Nature of uncertainty, 56 – 78 Necessity, 92 – 94 Necessity-freedom-compatibility – principle, 101 – 104 Newcomb’s paradox, 165 – 170

286

Index

Non-Aristotelian logic, 116 Non-existence of impossibility, 15 Non-fuzzy condition, 56 – 57 Non-humanistic systems, 69 – 74 Non-mathematical algorithms, 33 Non-optimal rationality, 22 – 26 Non-seperability, 151 Non-stochastic condition, 56 – 57 Non-stochastic decision-choice system, 64 – 65 Non-stochastic fuzzy risk, 63, 117 – 119 Non-stochastic rationality, 73, 106 – 113 Nothingness strategy, 96 – 98 Numerical example of resolution to – Ellsberg’s paradox; 161 – 170

O Optimal cost-benefit rationality, 128 – 132 Optimal decision-choice rationality, 71 – 74 Optimal fuzzy-stochastic rationality, 69 – 74 Optimal rationality, 22 – 26 Optimality, 257 – 262 Optimization index, 79 – 82 Optimization of risk, 121 – 125 Optimal risk-taking behavior, 64 Optimal stochastic-fuzzy rationality, 69 – 74 Organic paradigm, 75 – 78

P Paradigm shifting, 43 – 45 Paradoxes, 61 – 65, 133 – 170, 272 – 274 Paradox of heap, 80 Parrat’s statement, 68 Past, 4 – 5 Past-present duality, 92 – 95 Penumbral regions, 51 – 55, 63 – 67, 74 – 75, 117 – 119

Perception-to-reality set, 6 – 7 Perfect knowledge structure, 79 – 80 Permanent paradox, 133 – 135, 140 – 141 Phenomenon of risk, 83 – 91 Philosophy of science, 266 – 272 Possibilistic belief, 16 – 22, 101 – 106, 143 – 147 Possibilistic uncertainty, 51 – 53 Possibility index, 14 – 15 Possibility measure, 66 – 67 Possibility-probability interaction, 146 – 158 Possibility space, 5 – 14, 107 – 108, 142 Possible outcomes, 142 Postulate of benefit-cost conflict, 66 – 67 Postulate of exact measures, 60 Postulate of benefit-cost conflict, 74 – 75 Postulate of cost-benefit rationality, 128 – 129 Postulate of fuzzy goal, 66 – 6 Postulate of fuzzy knowledge, 74 – 75 Postulate of optimal process, 66 – 67 Potential-actual space, 96 – 98 Potential phenomenon, 96 – 98 Potential space, 5 – 11 Preference structure, 57 – 78 Prescriptive hypothesis, 19 – 22 Prescriptive science, 20 – 22 Prescriptive science, 62 – 66, 280 Present, 4 – 5 Pricing, 182 – 184 Primary category, 23 – 26, 76 – 78, 94 – 98, 114 – 116 Principle of compatibility, 98 – 104 Principle of decision-choice rationality, 21 Principle of duality, 117 Principle of exactness, 144 – 147, 167 – 170 Principle of excluded middle, 114 – 116, 140 Principle of fuzziness, 64

Index Principle of fuzzy optimal rationality, 21 Principle of information deficiency, 92 – 94 Principle of information sufficiency, 89 – 91 Principle of knowledge deficiency, 92 – 94 Principle of insufficient reason, 21, 113 – 116 Principle of knowledge sufficiency, 89 – 91 Principle of non-acceptance of – contradiction, 36 – 38 Principle of sufficient belief, 113 – 114 Principle of sufficient cause, 114 – 116 Principle of sufficient information, 113 – 114 Principle of sufficient reasoning, 21 – 22, 113 – 116 Probabilistic belief, 16 – 22, 101 – 104, 105 – 106, 143 – 147 Probabilistic belief formation, 22 – 26 Probabilistic hypothesis, 17 – 22 Probabilistic reasoning, 84, 109 – 111, 254 – 257 Probabilistic uncertainty, 6 – 11 Probability, 53 – 55, 140 – 144 Probability index, 15 – 22 Probability measure, 141 – 144 Probability set, 12 – 15 Probability space, 5 – 14, 107 – 108,142 Process unity, 93 – 98 Property of insurability, 122 – 123 Prospect, 4, 88 Psychology, 78 – 82 Pure fuzzy uncertainty, 89 – 90 Pyramidal logic, 18, 76 – 78 Pyramidal relation, 76 – 78

Q Quadrangle knowledge pyramids, 166 – 170 Qualitative-quantitative duality,63

287

Quantitative evidence, 4 – 5

R Random event, 74 – 75, 111 – 112 Random-fuzzy event, 111 – 112 Random-fuzzy space, 155 – 157 Random-fuzzy variable, 70 – 75, 242 – 245 Rational expectation hypothesis, 24 – 26 Rationality as an attribute, 135 – 137 Rationality-necessity-freedom structure, 100 – 104 Real number system, 38 – 40 Reasonable level, 25 – 26, 29 – 34 Receptor-processor structure, 92 – 94 Reduction of fuzziness, 64 – 66 Reductionism, 107 – 108, 114 – 115 Relations-based information, 2 Relative classical optimal rationality, 44 Relative fuzzy optimality ratio, 44 Reporting language of knowledge, 87 – 89 Resident dualities, 27 – 29 Resolutions of Ellsberg paradox, 151 – -158 Revealed preference approach, 189 – 191 Reward, 56 – 57 Reward space, 141 – 144 Risk, 51 – 82, 84 – 104 Risk analysis, 225 – 229 Risk-engineered system, 124 – 131 Risk engineering, 98 – 104, 119 – 127 Risk-engineering phenomena, 126 – 127 Risk engineering process, 127 Risk index, 108 Risk market, 121 – 125 Risk planning, 98 – 104 Risk theory, 84 – 94 Risk transferability, 126 – 127 Risk-avoidance behavior, 107 – 108

288

Index

Risk-avoidance strategy, 98 – 104 Risk-benefit configuration, 92 – 93 Risk-lessness, 84 – 86 Risk-outcome structure, 97 – 104 Risk-taking phenomena, 109 Riskness, 272 – 274 Russell, B., 13, 65

Subjective information, 5 – 7 Subjective knowledge, 2 – 4, 26 Sub-optimal rationality, 25 – 49 Substitution-transformation process, 1 – 22, 89 – 92, 92 – 113, 164 – 170 Sufficient justification, 113 – 114 Sufficient knowledge, 113

S

T

Satisficing level, 25 – 26, 29 – 34, 48, 79 – 82 Savage axioms, 141 – 158, Second principle of freedom, 101 – 104 Set of goals, 6 Simon, H. A., 24 Social actual-potential risk Social collectivity, 140 Social decision-choice space, 138 – 140 Social institutional configuration, 125 – 126 Social insurance, 124 – 129 Social risk engineering, 120 – 125 Sorities paradox, 80 – 82 Space of actual, 9 – 11, 142 Space of potential, 9 – 11, 142 St. Petersburg paradox, 60, 141 State of nature, 141 – 144 Statistical decision theory, 58 – 60 Statistical theory of information, 84 Stochastic optimal rationality, 119 Stochastic space, 153 – 154 Stochastic-fuzzy process, 69 – 74 Stochastic-fuzzy uncertainty, 117 – 119 Stochastic-fuzzy optimal cost-benefit – rationality, 128 – 132 Stochastic-fuzzy variable, 151 – 156 Stochastic rationality, 73 Stochastic risk, 58 – 60, 106 – 113, 145 – 147, 161 Stochastic risk-bearing, 58 – 60 Stochastic uncertainty, 6 – 11, 54 – 55, 85 – 87, 101 – 104, 106 – 113, 142 – 143, 145 – 147, 161 Stochasticity, 117 – 119

Theoretical knowledge, 59 Theory of contradiction, 45 – 46 Theory of insurance, 127 Theory of planning, 280 Theory of probability, 14 – 15 Theory of risk measurement, 49, 51 – 82 Time set 4 – 5 Toolbox of classical paradigm, 140 Tools of fuzzy paradigm, 82 Transferability of risk, 124 – 132 True-false duality, 168 – 170 Two-valued truth, 30 Typology of risk structure, 123

U Unanimity, 140 – 142 Uncertainty, 1 – 26, 161 – 170, 199 – 203 Uncertainty-fuzziness-stochastic – pyramid, 103 – 104 Uncertainty-risk decision-choice – theories, 113 – 116 Uncertainty-risk square, 91 Uncertainty space, 70 Unintentional social risk, 1200 Universal object set, 15 Universal system of knowledge, 32 Unmeasurable uncertainty, 145 – 147 Utility hypothesis, 58 – 60, 108, 169 – 170 Utility theory, 140 – 144, 274 – 275

Index

V Vague elements, 69 Vagueness, 30 – 32, 138 – 145, 275 – 279 Verification, 68 Verification principle,2 Violation of duality, 15 von Neumann-Morgenstern axioms, 141 – 144

W Weighted utility values, 107 – 111 What there is, 3 – 5, 111 – 114 What there is not, 111,

What would be, 111 – 114 What ought to be, 111 – 114 Worst-to-best possibility, 9

Z Zadeh, L. A., 164 Zonal analysis, 55 – 57 Zone of epistemic accessibility, 6 Zone of epistemic ignorance, 6 Zone of fuzzy non-stochastic space, 139 – 140 Zone of knowledge acquisition, 68

289