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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
8/11/2019 Lesson 2 (Fuzzy Propositions)
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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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Unconditional and Unqualified Fuzzy
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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Unconditional and Unqualified Fuzzy
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