"Applying this approach, systems belonging to different scientific disciplines are investigated in their natural forms. On the basis of experimental results, isomorphic relations between different systems are studied and, finally, some general principles applicable for all systems of a certain class are formulated." (George Klir, "An approach to general systems theory", 1969)
"In mathematics, logic, linguistics, and other abstract disciplines, the systems are not assigned to objects. They are defined by an enumeration of the variables, their admissible values, and their algebraic, topological, grammatical, and other properties which, in the given case, determine the relations between the variables under consideration." (George Klir, "An approach to general systems theory", 1969)
"Systems science is a science whose domain of inquiry consists of those properties of systems and associated problems that emanate from the general notion of systemhood." (George Klir, "Facets of Systems Science", 1991)
"The term system is unquestionably one of the most widely used terms not only in science, but in other areas of human endeavor as well. It is a highly overworked term, which enjoys different meanings under different circumstances and for different people. However, when separated from its specific connotations and uses, the term "system" is almost never explicitly defined." (George Klir, "Facets of Systems Science", 1991)
"What is systems science? This question, which I have been asked on countless occasions, can basically be answered either in terms of activities associated with systems science or in terms of the domain of its inquiry. The most natural answers to the question are, almost inevitably, the following definitions: Systems science is what systems scientists do when they claim they do science. Systems science is that field of scientific inquiry whose objects of study are systems." (George Klir, "Facets of Systems Science", 1991)
"A fuzzy set can be defined mathematically by assigning to each possible individual in the universe of discourse a value representing its grade of membership in the fuzzy set. This grade corresponds to the degree to which that individual is similar or compatible with the concept represented by the fuzzy set. Thus, individuals may belong in the fuzzy act to a greater or lesser degree as indicated by a larger or smaller membership grade. As already mentioned, these membership grades are very often represented by real-number values ranging in the closed interval between 0 and 1." (George J Klir & Bo Yuan, "Fuzzy Sets and Fuzzy Logic: Theory and Applications", 1995)
"Among the various paradigmatic changes in science and mathematics in this century, one such change concerns the concept of uncertainty. In science, this change has been manifested by a gradual transition from the traditional view, which insists that uncertainty is undesirable in science and should be avoided by all possible means, to an alternative view, which is tolerant of uncertainty and insists that science cannot avoid it. According to the traditional view, science should strive for certainty in all its manifestations (precision, specificity, sharpness, consistency, etc.); hence, uncertainty (imprecision, nonspecificity, vagueness, inconsistency, etc.) is regarded as unscientific. According to the alternative (or modem) view, uncertainty is considered essential to science; it is not only an unavoidable plague, but it has, in fact, a great utility." (George Klir, "Fuzzy sets and fuzzy logic", 1995)
"In constructing a model, we always attempt to maximize its usefulness. This aim is closely connected with the relationship among three key characteristics of every systems model: complexity, credibility, and uncertainty. This relationship is not as yet fully understood. We only know that uncertainty (predictive, prescriptive, etc.) has a pivotal role in any efforts to maximize the usefulness of systems models. Although usually (but not always) undesirable when considered alone, uncertainty becomes very valuable when considered in connection to the other characteristics of systems models: in general, allowing more uncertainty tends to reduce complexity and increase credibility of the resulting model. Our challenge in systems modelling is to develop methods by which an optimal level of allowable uncertainty can be estimated for each modelling problem." (George J Klir & Bo Yuan, "Fuzzy Sets and Fuzzy Logic: Theory and Applications", 1995)
"In spite of the insurmountable computational limits, we continue to pursue the many problems that possess the characteristics of organized complexity. These problems are too important for our well being to give up on them. The main challenge in pursuing these problems narrows down fundamentally to one question: how to deal with systems and associated problems whose complexities are beyond our information processing limits? That is, how can we deal with these problems if no computational power alone is sufficient?" (George Klir, "Fuzzy sets and fuzzy logic", 1995)
"Probability theory is an ideal tool for formalizing uncertainty in situations where class frequencies are known or where evidence is based on outcomes of a sufficiently long series of independent random experiments. Possibility theory, on the other hand, is ideal for formalizing incomplete information expressed in terms of fuzzy propositions." (George Klir, "Fuzzy sets and fuzzy logic", 1995)
"To select an appropriate fuzzy implication for approximate reasoning under each particular situation is a difficult problem. Although some theoretically supported guidelines are now available for some situations, we are still far from a general solution to this problem." (George Klir, "Fuzzy sets and fuzzy logic", 1995)
"Intuition, insight, and the ability of global comprehension
are possibly the most valuable assets of the human mind, particularly one that
is appropriately trained."
"Mathematics can roughly be divided into pure and applied.
Pure mathematics is basically oriented to the development of various axiomatic
theories, regardless of whether or not they have any real-world meaning. The
proper activity of the pure mathematician is thus to derive theorems from
postulated assumptions (axioms), and it is not his or her concern to determine
whether there is some interpretation of the theory in which the assumptions are
true. […] The role of applied mathematics is to search for practical
interpretations of the various mathematical theories and, when such
interpretations are found, to further develop the theories into useful
methodological tools for dealing with the interpreted systems and associated
problems. As such, applied mathematics is oriented to the development of
methods based on specific mathematical theories and their use in as many
interpreted areas as possible."
"Systems whose variables are classified into input and output variables are called directed systems; those for which no such classification is given are called neutral systems. A number of additional distinctions are recognized for state sets associated with the involved variables (basic or supporting) and provide a basis for further methodological classification of source systems. They include, for instance, the distinctions between crisp and fuzzy variables, discrete and continuous variables, and variables of different scales." (George J Klir & Doug Elias, "Architecture of Systems Problem Solving" 2nd Ed, 2003)
"The aim of architectural design is to prepare overall specifications, derived from the needs and desires of the user, for subsequent design and construction stages. The first task for the architect in each design project is thus to determine what the real needs and desires of the user are […]" (George J Klir & Doug Elias, "Architecture of Systems Problem Solving" 2nd Ed, 2003)
"The domain of systems science consists thus of all kinds of
relational properties which are valid for particular classes of systems, or, in
some rare instances, are valid for all systems. The chosen relational
classification of systems determines the way in which the domain of systems is
divided into subdomains, in a similar fashion as the domain of traditional
science has been divided into subdomains of the various disciplines and
specializations." (George J Klir & Doug Elias, "Architecture of Systems Problem
Solving" 2nd Ed, 2003)
"The principle of maximum entropy is employed for estimating
unknown probabilities (which cannot be derived deductively) on the basis of the
available information. According to this principle, the estimated probability
distribution should be such that its entropy reaches maximum within the
constraints of the situation, i.e., constraints that represent the available
information. This principle thus guarantees that no more information is used in
estimating the probabilities than available."
"The principle of minimum entropy is employed in the
formulation of resolution forms and related problems. According to this
principle, the entropy of the estimated probability distribution, conditioned
by a particular classification of the given events (e.g., states of the
variable involved), is minimum subject to the constraints of the situation.
This principle thus guarantees that all available information is used, as much
as possible within the given constraints (e.g., required number of states), in
the estimation of the unknown probabilities."
"The reconstruction problem can be stated as follows: Given a behavior system, viewed as an overall system, determine which sets of its subsystems, each viewed as a reconstruction hypothesis, are adequate for reconstructing the given system with an acceptable degree of approximation, solely from the information contained in the subsystems." (George J Klir & Doug Elias, "Architecture of Systems Problem Solving" 2nd Ed, 2003)
"Nevertheless, the use of fuzzy logic is supported by at least the following three arguments. First, fuzzy logic is rooted in the intuitively appealing idea that the truths of propositions used by humans are a matter of degree. An important consequence is that the basic principles and concepts of fuzzy logic are easily understood. Second, fuzzy logic has led to many successful applications, including many commercial products, in which the crucial part relies on representing and dealing with statements in natural language that involve vague terms. Third, fuzzy logic is a proper generalization of classical logic, follows an agenda similar to that of classical logic, and has already been highly developed. An important consequence is that fuzzy logic extends the rich realm of applications of classical logic to applications in which the bivalent character of classical logic is a limiting factor." (Radim Belohlavek & George J Klir, "Concepts and Fuzzy Logic", 2011)
"The principal idea employed by fuzzy logic is to allow for a partially ordered scale of truth-values, called also truth degrees, which contains the values representing false and true , but also some additional, intermediary truth degrees. That is, the set {0,1} of truth-values of classical logic, where 0 and 1 represent false and true , respectively, is replaced in fuzzy logic by a partially ordered scale of truth degrees with the smallest degree being 0 and the largest one being 1." (Radim Belohlavek & George J Klir, "Concepts and Fuzzy Logic", 2011)
"We use the term fuzzy logic to refer to all aspects of representing and manipulating knowledge that employ intermediary truth-values. This general, commonsense meaning of the term fuzzy logic encompasses, in particular, fuzzy sets, fuzzy relations, and formal deductive systems that admit intermediary truth-values, as well as the various methods based on them." (Radim Belohlavek & George J Klir, "Concepts and Fuzzy Logic", 2011)
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