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## Fuzziness vs. Probability

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**Fuzziness vs. Probability**JIN Yan Nov. 17, 2004**The outline of Chapter 7**Part I Fuzziness vs. probability Part II Fuzzy sets & relevant theories**Part I**In general, Kosko's position is: • Probability is a subset of Fuzzy Logic. • Probability still works. It is just the binary subset of fuzzy theory.**Fuzziness in a Probabilistic World**Uncertainty—how to describe this notion in math? • Probability is the only answer ( probabilists hold this.) • Fuzziness. (probability describes randomness, but randomness doesn’t equal uncertainty. Fuzziness can describe those ambiguous phenomena which are also uncertainty.)**Randomness vs. Fuzziness**• Similarities: 1. both describe uncertainty. 2. both using the same unit interval [0,1]. 3. both combine sets & propositions associatively, commutatively, distributively.**Randomness vs. Fuzziness**• Distinctions: • Fuzziness event ambiguity. Randomness whether an event occurs. • How to treat a set A & its opposite Ac ( key one). • Probability is an objective characteristic Fuzziness is subjective. ( The last viewpoint comes from “Fuzziness versus probability again”, F. Jurkovič )**Randomness vs. Fuzziness**• The 1st distinction and the 3rd one: Example1: • There is a 20% chance to rain. ( probability & objectiveness ) • It’s a light rain. ( fuzziness & subjectiveness) Example 2: • Next figure will be an ellipse or a circle, a 50% chance for every occasion. ( probability & objectiveness ) • Next figure will be an inexact ellipse. ( fuzziness & subjectiveness)**or**An inexact ellipse**Randomness vs. Fuzziness**• The key distinction: Is always true ? The formula describes the law of excluded middle --- white or black. Yet how about those “a bit white & a bit black” events? ---- Fuzziness!!**Part II**• Fuzzy sets & relevant theories: • the geometry of fuzzy sets (√) • the fuzzy entropy theorem • the subsethood theorem**the Geometry of Fuzzy Sets: Sets as Points**• Membership function maps the domain X = {x1, ……, xn} to the range [0, 1]. That is X [0, 1]. • The sets-as-points theory: the fuzzy power set F(2X) In [0, 1]n (unit hypercube) a fuzzy set a point in the cube In nonfuzzy set the vertices of the cube midpoint maximally fuzzy point (if , A is the midpoint)**the Geometry of Fuzzy Sets: Sets as Points**Sets as points.The fuzzy subset A is a point in the unit 2-cube with coordinates (1/3, 3/4). The 1st element x1(2nd, x2)belongs to A to degree 1/3(3/4).**the Geometry of Fuzzy Sets: Sets as Points**• Combine set elements with the operators of Lukasiewicz continuous logic: Proposition: A is properly fuzzy iff , iff**the Geometry of Fuzzy Sets: Sets as Points**The fuzzier A is, the closer A is to the midpoint (black hole ) of the fuzzy cube. The less fuzzy A is, the closer A is to the nearest vertex.**the Geometry of Fuzzy Sets: Sets as Points**• The midpoint is full of paradox!! Bivalent paradox examples: e.g.1A bewhiskered barber states that he shaves a man iff he doesn’t shave himself. a very interesting question rise: who shaves the barber? let S the proposition that the barber shaves himself, not-S he does not, then the S & not-S are logically equivalent t( S ) = t (not-S ) = 1 - t( S ) t( S ) = 1/2**the Geometry of Fuzzy Sets: Sets as Points**e.g. 2 One man said: I’m lying. Does he lie when he say that he’s lying? e.g. 3 God is omnipotent and he can do anything. If God can invent a large stone that he himself cannot lift up?**Counting with Fuzzy Sets**The size or cardinality of A, i.e., M (A) M (A) of A equals the fuzzy Hamming norm of the vector drawn from the origin to A. (X, In, M) defines the fundamental measure space of fuzzy theory. A = ( 1/3,3/4) M (A) =1/3+3/4 = 13/12**Counting with Fuzzy Sets**Define the lp distance between fuzzy set A & B as: Then M is the l1 between A &