Lesson 2 (Fuzzy Propositions)

8/11/2019 Lesson 2 (Fuzzy Propositions)

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Models for Inexact Reasoning

Fuzzy Logic Lesson 2

Fuzzy Propositions

Master in Computational Logic

Department of Artificial Intelligence


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Fuzzy Propositions

Main difference between classical propositions and

fuzzy propositions:

The range of their truth values: [0, 1]

We will focus on the following types of propositions:

Unconditional and unqualified propositions

The temperature is high

Unconditional and qualified propositions

The temperature is high is very true

Conditional and unqualified propositions If the temperature is high, then it is hot

Conditional and qualified propositions

If the temperature is high, then it is hot is true


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Unconditional and Unqualified Fuzzy

Propositions

The canonical form pof fuzzy propositions ofthis type is:

:p V isF

V is a variable that takes values vfrom someuniversal set E

F is a fuzzy set on Ethat represents a given

imprecise predicate: tall, expensive, low, etc. Example:

p: temperature (V) is high (F)


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Propositions

Given a particular value V= v, this individualbelongs to Fwith membership grade F(v)

The membership grade is interpreted as the

degree of truth T(p) of proposition p

( ) ( )FT p v=

Note that T is also a fuzzy set on [0, 1]

It assigns truth value F(v) to each value vofvariable V


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Example

Let Vbe the air temperature (in OF) at someplace on the Earth

Let Fbe the fuzzy set that represents the

predicate high (temperature)


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Example

The degree of truth T(p) depends on: The actual value of the temperature

The given definition (meaning) of predicate high

Let us suppose that the actual temperature is 85 F


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Propositions

Role of functionT:

Provide us with a bridge between fuzzy sets and

fuzzy propositions

Note thatT is numerically trivial forunqualified propositions

The values are identical to those provided by the

fuzzy membership function

This does not happen when dealing with qualifiedpropositions (more complexT(p) functions)


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Unconditional and Qualified

Propositions

Two different canonical forms to represent

these propositions:

:p V isF isS

Vand Fhave the same meaning as in previousslides

Sis a fuzzy truth qualifier P is a fuzzy probability qualifier

: Pr( )p V isF isP


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Truth-qualified Propositions

There are different truth qualifiers

Unqualified propositions are special truth-

qualified propositions (Sis assumed to be true)


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Truth-qualified Propositions

In general, the degree of truth T(p) of any

truth-qualified proposition p is:

( ) ( ( )),S FT p v v E = The membership function G= s F can be

interpreted as the unqualified proposition

V is G


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Example

Proposition Tina is young is very true

Predicate: young

Qualifier: very true

Let us suppose that the age of Tina is 26


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Probability-qualified Proposition

There are different probability qualifiers:


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Probability-qualified Propositions

Given a probability distribution f on V, we candefine the probability of a fuzzy proposition:

Pr( ) ( ) ( )Fv V

V isF f v v

= We calculate the degreeT(p) to which a

proposition of the form [Pr(Vis F) is P] is trueas:

( ) (Pr( )) ( ) ( ))P p Fv V

T p V isF f v v

= =


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Example

p: Pro(temperature is around 75 F) is likely

The predicate around 75 F is represented by

the following membership function:


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Example

The probability distribution obtained from

relevant statistical data over many years is:

Then, we calculate Pr(temperature is around

75 F) as follows:


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Example

Now, we use the fuzzy qualifier to calculate

the truth value of the predicate:


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Conditional and Unqualified

Propositions

The canonical form pof fuzzy propositions ofthis type is:

: ,p if X is A then Y isB

X,Yare variables in universes E1 and E2 A, Bare fuzzy sets on X, Y

These propositions may also viewed as

propositions of the form:

,X Y is R


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Conditional and Unqualified

Propositions

R is a fuzzy set on XYdefined as:

( , ) [ ( ), ( )]A BR x y J x y=

J represents a suitable fuzzy implication There are many of them

In our examples we will use the Lukasiewicz

implication:

( ), min(1,1 )a b a b= +


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Example

A = .1/x1 + .8/x2 + 1/x3

B = .5/y1 + 1/y2

Lukasiewicz implication J = min(1, 1-a+b)

R?

T(p) = 1 when (X= x1 andY= y1) T(p) = .7 when (X= x2 andY= y1)

and so on


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Conditional and Qualified Propositions

Propositions of this type can be characterized

by two different canonical forms:

( ): ,p if X isA then Y isB is S

Sis a fuzzy truth qualifier

P is a fuzzy probability qualifier

Pr(Xis A|Y is B) is a conditional probability

Methods introduced before can be combined to

deal with propositions of this type

: Pr( | )p X is A Y isB isP


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Exercise

Couple Husband (height,

cm)

Wife (height,

cm)

1 150 150

2 190 160

3 180 150

4 170 150

5 160 160

6 180 150

7 170 160

8 150 150

9 170 170

10 150 190

Fuzzy Predicates tall= 0.5/160 + 0.75/170 + 1/180 + 1/190

short = 1/150 + 1/160 + 0.75/170 + 0.5/180

Statistics

Calculate the truth value

associated to the following

proposition:

Pr(husband is tall|wife is

short) is likely Use the Lukasiewicz

implication


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Homework

Exercise:

Describe the method to deal with conditional and

truth-qualified propositions

Provide a concrete example

Lesson 2 (Fuzzy Propositions)

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