Epistemic Foundations of Fuzziness: Unified Theories on by Kofi Kissi Dompere

By Kofi Kissi Dompere

This monograph is a remedy on optimum fuzzy rationality as an enveloping of decision-choice rationalities the place restricted details, vagueness, ambiguities and inexactness are crucial features of our wisdom constitution and reasoning strategies. the quantity is dedicated to a unified method of epistemic versions and theories of decision-choice habit less than overall uncertainties composed of fuzzy and stochastic kinds. The unified epistemic research of decision-choice types and theories starts off with the query of ways top to combine vagueness, ambiguities, constrained info, subjectivity and approximation into the decision-choice technique. the reply to the query results in the moving of the classical paradigm of reasoning to fuzzy paradigm. this can be by way of discussions and institution of the epistemic foundations of fuzzy arithmetic the place the character and position of data and information are explicated and represented.

The epistemic beginning permits overall uncertainties that constrain decision-choice actions, wisdom company, good judgment and mathematical buildings as our cognitive tools to be mentioned in connection with the phenomena of fuzzification, defuzzification and fuzzy good judgment. The discussions on those phenomena lead us to research and current versions and theories on decision-choice rationality and the wanted arithmetic for challenge formula, reasoning and computations. The epistemic constructions of 2 quantity platforms made of classical numbers and fuzzy numbers are mentioned in terms of their variations, similarities and relative relevance to decision-choice rationality. The houses of the 2 quantity platforms bring about the epistemic research of 2 mathematical platforms that permit the partition of the mathematical house in aid of decision-choice area of data and non-knowledge creation into 4 cognitively separate yet interdependent cohorts whose homes are analyzed by way of the tools and strategies of class concept. The 4 cohorts are pointed out as non-fuzzy and non-stochastic, non-fuzzy and stochastic either one of which belong to the classical paradigm and classical mathematical area; and fuzzy and non-stochastic, and fuzzy and stochastic cohorts either one of which belong to the bushy paradigm and fuzzy mathematical house. the variations within the epistemic foundations of the 2 mathematical platforms are mentioned. The dialogue results in the institution of the necessity for fuzzy arithmetic and computing as a brand new approach of reasoning in either precise and inexact sciences.

The mathematical buildings of the cohorts are imposed at the decision-choice strategy to permit a grouping of decision-choice types and theories. The corresponding sessions of decision-choice theories have an analogous features because the logico-mathematical cohorts relative to the assumed information-knowledge constructions. The 4 groupings of versions and theories on decision-choice actions are then categorized as: 1) non-fuzzy and non-stochastic category with precise and entire information-knowledge constitution (no uncertainty), 2) non-fuzzy and stochastic category with specific and restricted information-knowledge constitution (stochastic uncertainty), three) fuzzy and non-stochastic type with complete and fuzzy information-knowledge constitution (fuzzy uncertainty) and four) Fuzzy and stochastic type with fuzzy and constrained information-knowledge constitution (fuzzy and stochastic uncertainties). some of these diversified periods of choice selection difficulties have their corresponding rationalities that are absolutely mentioned to provide a unified logical approach of theories on decision-choice approach.

The quantity is concluded with epistemic discussions at the nature of contradictions and paradoxes considered as logical decision-choice difficulties within the classical paradigm, and the way those contradictions and paradoxes can be resolved via fuzzy paradigm and the tools and methods of optimum fuzzy decision-choice rationality. The logical challenge of sorites paradox with its answer is given to illustrate. viewers contains these operating within the components of economies, decision-choice theories, philosophy of sciences, epistemology, arithmetic, machine technological know-how, engineering, cognitive psychology, fuzzy arithmetic and arithmetic of fuzzy-stochastic processes.

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2 The Rationality and Two Number Systems One important element in fuzzy mathematical development is the notion that the classical number system can be replaced by the fuzzy number system through the replacement of the concept of exact numbers by inexact numbers called fuzzy numbers. In this respect, every classical number has a corresponding fuzzy number that constitutes a fuzzy covering. For any classical number we may think of the fuzzy covering as a set of interval values where the closeness of every interval within the set is defined by a membership function.

Definition Let N ε ( x0 ) be an ε-neighborhood set of classical number x0 ∈ R . The set N ε ( x0 ) is said to be a fuzzy number if it is equipped with a membership characteristic function µ Nε ( x0 ) ( x ) ∈ [ 0,1] that shows the distribution of degree of belonging of any number x ∈ R such that ⎧∈ ( α ,1] if ⎪⎪ if µ Nε ( x0 ) ( x ) ⎨= 1 ⎪ ⎪⎩∈ [1,0 ) if dµ dx ≥ 0, x ≤ x0 dµ dx = 0, x = x0 dµ dx ≤ 0 , x ≥ x0 where ε is any number defining an interval around x0 ∈ R . The fuzzy number may then be specified as X = N∈ ( x0 ) ,µ N∈ ( x0 ) ( x0 ) and the collection of all these numbers together constitutes the fuzzy number system e .

In fact reality is another way of viewing the space of potential and perceptive reality is another way of viewing our knowledge structure. Our knowledge structure and the process of its construction are plagued with problems of uncertainties due to information incompleteness and ambiguities in concepts, reasoning and communications to which our mathematics is called upon to help. Let us look closer at these problem characteristics. 2 Uncertainty, Risk and Mathematics Our knowledge structure as a perceptive reality that provides us with the understanding and awareness of the universe is a subset of space of actual reality.

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