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.
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Nov. 17, 2004
Part I Fuzziness vs. probability
Part II Fuzzy sets & relevant theories
In general, Kosko's position is:
Uncertainty—how to describe this notion in math?
(probability describes randomness, but randomness doesn’t equal uncertainty. Fuzziness can describe those ambiguous phenomena which are also uncertainty.)
1. both describe uncertainty.
2. both using the same unit interval [0,1].
3. both combine sets & propositions
associatively, commutatively, distributively.
Randomness whether an event occurs.
Fuzziness is subjective.
( The last viewpoint comes from “Fuzziness versus probability again”, F. Jurkovič )
every occasion. ( probability & objectiveness )
An inexact ellipse
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!!
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)
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).
Proposition: A is properly fuzzy
iff , iff
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.
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
One man said: I’m lying. Does he lie when he say that he’s lying?
God is omnipotent and he can do anything. If God can invent a large stone that he himself cannot lift up?
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
Define the lp distance between fuzzy set A & B as:
Then M is the l1 between A &