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 A c ( 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 = {x 1, ……, x n } to the range [0, 1]. That is X  [0, 1]. The sets-as-points theory: the fuzzy power set F(2 X )  I n [0, 1] n (unit hypercube) a fuzzy set  a point in the cube I n 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 x 1 (2nd, x 2 )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. 1 A 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, I n, 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 l p distance between fuzzy set A & B as: Then M is the l 1 between A &