Table Of ContentKofiKissiDompere
EpistemicFoundationsofFuzziness
Studiesin Fuzzinessand Soft Computing,Volume236
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FuzzySetsandTheirExtensions: ISBN978-3-540-78483-8
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inEpidemiologyandBeyond,2008
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TheoriesandTheirApplications,2008
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FuzzyEngineeringEconomicswith
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JohnN.Mordeson,JoshuaD.Potter, ISBN978-3-540-70809-4
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ApplyingFuzzyMathematicstoFormal
Knowledge-BasedNeurocomputing:
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AFuzzyLogicApproach,2009
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SoftComputingApplicationsinIndustry,2008 Vol.235.KofiKissiDompere
ISBN978-3-540-77464-8 FuzzyRationality,2009
ISBN978-3-540-88082-0
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ManagementofKnowledgeImperfectionin Vol.236.KofiKissiDompere
BuildingIntelligentSystems,2008 EpistemicFoundationsofFuzziness,2009
ISBN978-3-540-77462-4 ISBN978-3-540-88084-4
Kofi Kissi Dompere
Epistemic Foundations of
Fuzziness
Unified Theories on Decision-Choice Processes
ABC
Author
Prof.KofiKissiDompere
DepartmentofEconomics
HowardUniversity
Washington,D.C.,20059
USA
E-Mail:[email protected]
ISBN978-3-540-88084-4 e-ISBN978-3-540-88085-1
DOI10.1007/978-3-540-88085-1
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To Arnold Kaufman, for his continual dedication to
the development of foundations of mathematics of
fuzzy processes and decision sciences.
To H. J. Zimmermann, for his dedication to fuzzy sys-
tems research and editorial management of interna-
tional journal of Fuzzy Sets and Systems.
To all scholars working on the frontier of fuzzy mathe-
matics, new mathematics, algorithms of soft and natu-
ral computing and fuzzy decisions.
To all researchers who are working against logical
anchoring that the prevailing intellectual order offers
in order to energize the spirit of free and unlimited
inquiries that are required for the progress of our
knowledge enterprise.
In memory of my parents
father, Kwasi Kissi and mother, Adwoa Fosua,
parents, Mr. and Mrs. Herbert Isaac Somuah,
uncles, Kwabena Aware, Kwabena Atta,
Kwame Amakye
Preface
It is necessary to practice methodological doubt, like Descartes, in or-
der to loosen the hold of mental habits; and it is necessary to cultivate
logical imagination, in order to have a number of hypotheses at com-
mand, and not to be the slave of the one which common sense has ren-
dered easy to imagine. These two processes, of doubting the familiar
and imagining the unfamiliar, are corrective, and form the chief part of
the mental training required for a philosopher.
Bertrand Russell
At every stage and in all circumstances knowledge is incomplete and
provisional, conditioned and limited by the historical circumstances
under which it was acquired, including the means and methods used for
gaining it and the historically conditioned assumptions and categories
used in the formulation of ideas and conclusions.
Maurice Cornforth
This monograph is the second in the series of meta-theoretic analysis of fuzzy
paradigm and its contribution and possible contribution to formal reasoning in
order to free the knowledge production process from the ridge frame of the
classical paradigm that makes its application to soft and inexact sciences dif-
ficult or irrelevant. The work in the previous monograph was strictly devoted
to problems of theory of knowledge and critique of classical, bounded and
other rationalities in decision-choice processes regarding the principles of
verification, falsification or corroboration in knowledge production.
This monograph deals mostly with epistemic decision-choice models and
theories and how they are related to both the classical and fuzzy paradigms.
Four categories of decision-choice models and theories are presented. They
are distinguished by the nature of assumptions made about the information-
VIII Preface
knowledge structure that imposes restrictions on the choice of type of para-
digm which then gives rise to a particular rationality. The four categories of
decision-choice theories correspond to four categories of information-
knowledge structure. The categories are 1) non-fuzzy (exact) and non-
stochastic (complete) information-knowledge structure (Cohort I), 2) non-
fuzzy (exact) and stochastic (incomplete) information-knowledge structure
(Cohort II), 3) fuzzy (inexact) and non-stochastic (complete) information-
knowledge structure (Cohort III) and 4) fuzzy (inexact) and stochastic (in-
complete) information-knowledge structure. The partition of the decision-
choice space allows us to present the uniqueness of theories in each cohort as
well as present a unified epistemic analysis of models and theories of deci-
sion-choice process that may encounter deficient and non-deficient informa-
tion-knowledge structure.
The imposed partition provides us with channels to examine the founda-
tions of logic and mathematical reasoning that may be utilized in modeling
and theory construction of decision-choice processes in general, and econom-
ics and other social sciences in particular. How do the logic and mathematical
reasoning relate to rationality that may be induced from the defined cohorts?
To answer this question we draw on the meta-theoretic gains from the previ-
ous monograph as they relate to classical and fuzzy paradigms and specify the
particular cohort that their developments occur. The classical paradigm has its
complete system of mathematical reasoning with its defined variables for ma-
nipulations; and the fuzzy paradigm has its mathematical reasoning with its
defined variables for manipulation. From the viewpoint of mathematical con-
struct in support of modeling and theoretical construct of the decision-choice
processes, these cohorts are associated with specific analytical variables that
may be quantitative or linguistic in nature on which logical operations are per-
formed. The variable under logical operations in cohort I is described as non-
fuzzy and non-stochastic variable whose meaning is given by one exact mag-
nitude (e.g. X-value). It may also be referred to as perfect or exact variable.
The variable under logical operation in cohort II is described by non-fuzzy
and stochastic variable. It is also referred to as random variable whose mean-
ing is given by two exact magnitudes of classical quantity (x) and exact prob-
ability value (p) [e.g., (x, p)]. The random variable only accounts for limited
information-knowledge deficiency. The variable under logical operation in
cohort III, on the other hand, is described as fuzzy and non-stochastic (com-
plete) variable. It is also called fuzzy variable whose meaning is given by two
magnitudes. One is inexact or linguistic magnitude which is represented by a
set and the other is a functional expression of relationally sujective values of
Preface IX
the degree of inexactness in the set magnitudes. It takes account only of
fuzzy-information deficiency. The variable under logical operation in cohort
IV is described as either fuzzy-random variable or random-fuzzy variable
whose meaning finds expressions in either three or four magnitudes in com-
plex relations where the space of decision-choice actions is described by
fuzzy-random-interactive processes. These are new variables that are being
introduced into our formal analytics by the fuzzy paradigm. They take account
of both limited and fuzzy information-knowledge deficiencies. These entire
variables are either quantitative or linguistic or qualitative depending on the
problem and the analytical structure. The variables in cohort IV are the most
complete, complex and difficult under logical operations and yet it is their
manipulations that capture the true essence of cognitive rationality in human
decision-choice actions.
The characteristics of these variables are such that we can classify them as
classical and fuzzy variables that relate to logical operations and operators of
classical and fuzzy paradigms with their logics and mathematics. We then
treat the analytically epistemic conditions of the cohort of classical numbers
and mathematics as applied to the classical paradigm. The set of classical
numbers constitute the classical number system with defined properties and
behavior. From the foundation of the classical numbers and mathematics, a
move is made to deal with the epistemic foundation of fuzzy numbers and
mathematics. The set of the fuzzy numbers constitutes the fuzzy number sys-
tem with its own properties and behavior as different logical operators are
acted upon. We then present their epistemic similarities, differences and
points of commonness. Both the classical and fuzzy numbers are translated
into symbolic systems that contain either total uncertainty or partial uncer-
tainty composed of fuzziness and randomness. The properties of the two
number systems lead to the presentation of two relative foundations of classi-
cal and fuzzy number and mathematical systems whose toolboxes become
available to examine human behavior over decision-choice variables.
The important point here is that it is an illusion of the mind to think of
symbolic representation of thought that is free of uncertainty, accident and
necessity in human decision-choice processes as it is assumed in the classical
system. There is also no logically and mathematically perfect language. If
there is one, the nature of humanity will cease to exist. Thus the more exact-
ness we seek of symbolic representation of human thought, the more irrele-
vance we create about human understanding of his or her complex social and
natural environments. This is another way of looking at the problem of
X Preface
Zadeh’s incompatibility principle which is a challenge that must be overcome
in symbolic and mathematical reasoning as well as explication and exact sci-
ence. The fuzzy paradigm is an attempt to overcome this problem. The rele-
vance of the fuzzy paradigm in human thought is to admit this imperfection.
The strength of fuzzy logic is that it has corresponding mathematical system
with different rules of operations where symbolic representation of thought is
not free of vagueness, ambiguities and subjectivity. This strength of fuzzy
logic is not found in multi-valued logic. The classical exact variables are ob-
tained by assuming away all factors in human cognitive behavior that will
force them into inexactness. The fuzzy variables on the other hand are ob-
tained by admitting the presence of cognitive limitations that render vague-
ness, ambiguities, linguistic deformities, defective explication that character-
ize human reasoning. A search for epistemic foundation of fuzzy mathematics
is appropriate and useful for further development.
The motivation behind the presentation of the epistemic foundation of
fuzzy mathematics is derived from certain analytical reasons in classical
mathematics. The nature of epsilon-delta (ε-δ) reasoning in theories of limits
in mathematics, and stability problems in engineering, economics and physics,
presents some epistemic concerns if these areas of knowledge are claimed to
belong to exact science. It may be noted that both epsilon (ε)and delta
(δ)are arbitrary constants since one of them depends on the other as given.
Examples are definitions of continuity, or Cauchy sequence in mathematics or
different definitions of stability of equilibrium in engineering dynamical and
optimal control systems and economic dynamics and evolution. The point
here is simple. In self-contained logical system the given parameters for
analysis or definition should not be arbitrary. They must be optimally derived
within the logical system and in relation to the environment of the analysis if
such logical system is not opened. The given parameter may be abstracted
from an analytical structure of methods and techniques of fuzzy decision where
subjectivity is part of the logical process. The (ε-δ) reasoning is basically an
approximate reasoning that is not different from epsilon-neighborhood (N (i))
ε
reasoning in the classical number system. Such approximation can be found in
all areas claimed to belong to exact sciences. This epsilon-neighborhood rea-
soning is used to define the fuzzy numbers as sets that allow us to present the
relative epistemic foundation of fuzzy arithmetic and mathematical reasoning.
The basic foundation of classical mathematics cannot be separated from the
foundation of its logic and conditions of truth acceptance. The reasoning
based on Aristotelian logic is suggested to be exact and hence applicable to
exact sciences and all other knowledge forms that submit themselves to this
Preface XI
exactness of classical mathematical reasoning. The accepted fundamental no-
tion in the classical paradigm in arriving at conditions of exactness is the Aris-
totelian logic that “every statement is either true or false but not both”. This
principle also forms the foundational notion of classical mathematical reason-
ing in the acceptance of true and false statements.
The implication here is that in our knowledge construction process, the set
of true statements and the set of false statements are mutually exclusive and
collectively exhaustive for all known and potential statements. The field of
application of such mathematical reasoning is, therefore, itself and other areas
of sciences that want to claim the properties of exactness. “For mathematics is
the study of all structures whose form can be expressed in symbols, it is the
grammar of all symbolic systems and, as such, its methods are peculiarly ap-
propriate to the investigation of its own internal structure” [R20.4, p.4].
By replacing the fundamental classical logical proposition that “every
statement is either true or false but not both” with fundamental fuzzy proposi-
tion that “every statement is a set of true-false characteristics in varying pro-
portion and that the acceptance of a proposition to be true or false is done on
the basis of subjective decision-choice action in reconciling the conflict in the
true-false proportion” a new system of thinking is born. This new system of
thinking is the fuzzy logic and mathematics that allow computable systems to
be constructed towards solution and conclusion. The fundamental fuzzy
proposition allows the analysts or decision-choice agents to account for
vagueness, imprecision, ambiguities, ill-definedness and inexactness and still
maintain symbolic logical operations and mathematical rigor in them in the
analytical structure of the fuzzy paradigm that not only rivals the classical
paradigm but enhances our cognitive abilities to use it in dealing with exact
and inexact decision-choice constructs in all areas of human endeavor. Fuzzy
logic empowers human thinking with the use of rigorous symbolic reasoning
to arrive at meaningful conclusions whose degree of exactness is part of rea-
soning through decision-choice action. In development of the fuzzy paradigm,
a number of meta-theoretic problems have to be overcome at the onset. The
first is to define the nature of the symbols and their concept representations
that include vagueness, ambiguities, imprecision and others that rob the con-
cepts of their exactness and deprive them of exact symbolic representations.
This is achieved by accepting grades, degrees and shades of meaning and
truth and designing their symbolic representation. The second is to construct
appropriate logical and mathematical operators to act on the symbols to derive
conclusions with qualified exact meanings and solutions that follow fuzzy
