The monograph is an examination of the fuzzy rational foundations of the structure of exact and inexact sciences over the epistemological space which is distinguished from the ontological space. It is thus concerned with the demarcation problem. It examines exact science and its critique of inexact science. The role of fuzzy rationality in these examinations is presented. The driving force of the discussions is the nature of the information that connects the cognitive relational structure of the epistemological space to the ontological space for knowing. The knowing action is undertaken by decision-choice agents who must process information to derive exact-inexact or true-false conclusions. The information processing is done with a paradigm and laws of thought that constitute the input-output machine. The nature of the paradigm selected depends on the nature of the information structure that is taken as input of the thought processing. Generally, the information structure received from the ontological space is defective from the simple principles of acquaintances and the limitations of cognitive agents operating in the epistemological space. How then do we arrive and claim exactness in our knowledge-production system? The general conclusion of this book is that the conditions of the fuzzy paradigm with its laws of thought and mathematics present a methodological unity of exact and inexact sciences where every zone of thought has fuzzy covering.

Fuzzy set and logic theory suggest that all natural language linguistic expressions are imprecise and must be assessed as a matter of degree. But in general membership degree is an imprecise notion which requires that Type 2 membership degrees be considered in most applications related to human decision making schemas. Even if the membership functions are restricted to be Type1, their combinations generate an interval – valued Type 2 membership. This is part of the general result that Classical equivalences breakdown in Fuzzy theory. Thus all classical formulas must be reassessed with an upper and lower expression that are generated by the breakdown of classical formulas.Key features:- Ontological grounding- Epistemological justification- Measurement of Membership- Breakdown of equivalences- FDCF is not equivalent to FCCF- Fuzzy Beliefs- Meta-Linguistic axioms- Ontological grounding- Epistemological justification- Measurement of Membership- Breakdown of equivalences- FDCF is not equivalent to FCCF- Fuzzy Beliefs- Meta-Linguistic axioms

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 signi- cance. The action which follows upon an opinion depends as much upon the amount of confidence in that opinion as it does upon fav- ableness 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.

Mathematics of Fuzziness – Basic Issues introduces a basic notion of ‘fuzziness’ and provides a conceptual mathematical framework to characterize such fuzzy phenomena in Studies in Fuzziness and Soft Computing. The book systematically presents a self-contained introduction to the essentials of mathematics of fuzziness ranging from fuzzy sets, fuzzy relations, fuzzy numbers, fuzzy algebra, fuzzy measures, fuzzy integrals, and fuzzy topology to fuzzy control in a strictly mathematical manner. It contains most of the authors’ research results in the field of fuzzy set theory and has evolved from the authors’ lecture notes to both undergraduate and graduate students over the last three decades. A lot of exercises in each chapter of the book are particularly suitable as a textbook for any undergraduate and graduate student in mathematics, computer science and engineering. The reading of the book will surely lay a solid foundation for further research on fuzzy set theory and its applications.

This potent area of technology allows us to formulate and solve a multitude of problems. Written by leading experts, this overview covers a number of aspects of fuzzy optimization, some related general issues, and various applications of this powerful tool.

In the Theory of Philosophical Consciencism, Professor Dompere establishes how Nkrumah used the theory of categorial conversion housing the necessary conditions of transformation to design strategies for creating the sufficient conditions for socio-political transformations. The theory of Philosophical Consciencism is about the institutionally destruction-creation process for socio-political transformation. The theory shows the scientific contributions of Nkrumah's thinking to the solution of the transformation problem in science and its application to social systemicity, where Nkrumah's analytical weapons were drawn from African conceptual system. The theory is developed as logico-mathematical foundations that guide the internal management of the command-control decision-choice systems for creative destruction of socio-political varieties in transformations through the development of qualitative mathematics making possible the construct of qualitative equations of motion for connecting varieties in qualitative transfers. The qualitative equations of motion through the Philosophical Consciencism constitute the sufficient conditions for transformations. The theory links rationality, intentionality, experiential information structure, defective-deceptive information structures in the control-dynamic games of power and dominance by the duals and poles under the principle of opposites with relational continua and unity relative to decision-choice systems that induce negation-negation transformations, where paths are established by the history of success-failure outcomes in the socially control-dynamic zero-sum games between the duals in any duality and between the poles of any actual-potential polarity. The main premise of the monograph is that there exists a set of sufficient conditions in support of the necessary conditions for internal transformation of socio-natural varieties. The theory is useful in understanding developmental processes and multi-polar-power zero-sum games for global dominance. The necessary conditions constitute the natural necessity that constrains cognitive freedom. The sufficient conditions constitute cognitive freedom that must overcome the necessity in socio-natural systems dynamics. Had this conceptual system been familiar to African leaders, the African transformation from colonialism to complete emancipation, rather than neocolonialism, would have been increasingly successful. This holds for those seeking triumph over injustices, oppression, imperialism and social change in all systems.

The monograph is about a meta-theory of knowledge-production process and the logical pathway that connects the epistemic possibility to the epistemic reality. It examines the general conditions of paradigms for information processing and isolates the classical and fuzzy paradigms for comparative analysis. The sets of conditions that give rise to them are defined, stated and analyzed to abstract the corresponding sets of laws of thought. The fuzzy paradigm with its corresponding logic and mathematics is related to inexact symbolism for the defective information structure where the results of the knowledge production must satisfy the epistemic conditionality, composed of fuzzy conditionality and fuzzy-stochastic conditionality under the principle of logical duality with continuum. The classical paradigm with its corresponding logic and mathematics is related to exact symbolism for exact information structure where the vagueness component of the defectiveness is assumed away, and where the results of the knowledge production must satisfy no epistemic conditionality or at the maximum only the stochastic conditionality under the principle of logical dualism with excluded middle. It is argued that the epistemic path that links ontological space to the epistemological space is information. The ontological space is taken as the primary category of reality while the epistemological space is shone to be a derivative. Such information is universally defective and together with assumptions imposed guides the development of paradigms with their laws of thought, logic of reasoning, mathematics and computational techniques. The relational structure is seen in terms of logical trinity with a given example as matter-information-energy transformational trinity which is supported by the time trinity of past-present-future relationality. The book is written for professionals, researchers and students working in philosophy of science, decision-choice theories, economies, sciences, computer science, engineering, cognitive psychology and researchers working on, or interested in fuzzy paradigm, fuzzy logic, fuzzy decisions, and phenomena of vagueness and ambiguities, fuzzy mathematics, fuzzy-stochastic processes and theory of knowledge. It is further aimed at research institutions and libraries. The subject matter belongs to extensive research and development taking place on fuzzy phenomena and the debate between the fuzzy paradigm and the classical paradigm relative to informatics, synergetic science and complexity theory. The book will have a global appeal and across disciplines. Its strength, besides the contents, is the special effort that is undertaken to make it relevant and accessible to different areas of sciences and knowledge production.

This important edited volume is the first such book ever published on fuzzy cognitive maps (FCMs). Professor Michael Glykas has done an exceptional job in bringing together and editing its seventeen chapters. The volume appears nearly a quarter century after my original article “Fuzzy Cognitive Maps” appeared in the International Journal of Man-Machine Studies in 1986. The volume accordingly reflects many years of research effort in the development of FCM theory and applications—and portends many more decades of FCM research and applications to come. FCMs are fuzzy feedback models of causality. They combine aspects of fuzzy logic, neural networks, semantic networks, expert systems, and nonlinear dynamical systems. That rich structure endows FCMs with their own complexity and lets them apply to a wide range of problems in engineering and in the soft and hard sciences. Their partial edge connections allow a user to directly represent causality as a matter of degree and to learn new edge strengths from training data. Their directed graph structure allows forward or what-if inferencing. FCM cycles or feedback paths allow for complex nonlinear dynamics. Control of FCM nonlinear dynamics can in many cases let the user encode and decode concept patterns as fixed-point attractors or limit cycles or perhaps as more exotic dynamical equilibria. These global equilibrium patterns are often “hidden” in the nonlinear dynamics. The user will not likely see these global patterns by simply inspecting the local causal edges or nodes of large FCMs.

The ‘Fuzzy Logic’ research group of the Microelectronics Institute of Seville is composed of researchers who have been doing research on fuzzy logic since the beginning of the 1990s. Mainly, this research has been focused on the microel- tronic design of fuzzy logic-based systems using implementation techniques which range from ASICs to FPGAs and DSPs. Another active line was the development of a CAD environment, named Xfuzzy, to ease such design. Several versions of Xfuzzy have been and are being currently developed by the group. The addressed applications had basically belonged to the control ?eld domain. In this sense, s- eral problems without a linear control solution had been studied thoroughly. Some examples are the navigation control of an autonomous mobile robot and the level control of a dosage system. The research group tackles a new activity with the work developed in this book: the application of fuzzy logic to video and image processing. We addressed our interest to problems related to pixel interpolation, with the aim of adapting such interpolation to the local features of the images. Our hypothesis was that measures and decisions to solve image interpolation, which traditionally had been done in a crisp way, could better be done in a fuzzy way. Validation of this general hypothesis has been done speci?cally in the interpolation problem of video de-interlacing. - interlacing is one of the main tasks in video processing.

This book studies optimized models with fuzzy quantities. It can be used by undergraduates in higher education, master graduates and doctor graduates. It also serves as a reference for researchers, particularly for those in the field of soft science.