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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.

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fuzziness vs probability

Fuzziness vs. Probability

JIN Yan

Nov. 17, 2004

the outline of chapter 7
The outline of Chapter 7

Part I Fuzziness vs. probability

Part II Fuzzy sets & relevant theories

part i
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
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
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 fuzziness6
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 fuzziness7
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)
slide8

or

An inexact ellipse

randomness vs fuzziness9
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
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
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 points12
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 points13
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 points14
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 points15
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 points16
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
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 sets18
Counting with Fuzzy Sets

Define the lp distance between fuzzy set A & B as:

Then M is the l1 between A &