Chapter 8 : Fuzzy Logic
8.1.1 Proposition Logic
As in our ordinary informal language, “sentence”
is used in the logic. Especially, a sentence having
only “true (1)” or “false (0)” as its truth value is
called “proposition”.
Example 8.1




Smith hits 30 home runs in one season
2+4=7
For every x, if f(x) = sin x, then f(x) = cos x.
It rains now.
(true)
(false)
(true)
(true)
8.1.1 Proposition Logic
Example 8.2
The followings are not propositions
 Why are you interested in the fuzzy theory?
 He hits 5 home runs in one season.
 x + 5 = 0
 x + y = z
8.1.1 Proposition Logic
Definition(Logic variable)
If we represent a proposition as a
variable, the variable can have the value
true or false.
 This type of variable is called as a
“proposition variable” or “logic variable”

8.1.1 Proposition Logic
Connectives - combine prepositional
variables
Negation
 Conjunction
 Disjunction
 Implication

a
a b
ab
a b
8.1.2 Logic function
Logic function

a combination of propositional variables
by using connectives
Logic formula
Truth values 0 and 1 are logic formulas
 If is a logic variable, and are a logic
formulas
 If a and b represent a logic formulas,
a  b and a  b are also logic formulas

8.1.3 Tautology and inference rule
Definition(Tautology)
A “tautology” is a logic formula whose
value is always true regardless of its
logic variables.
 A “contradiction” is one which is always
false.

8.1.3 Tautology and inference rule
Example 8.5 (tautology)
a b a b
( a  b)
b
( a  b)  b
1 1
1
0
0
1
1 0
0 1
0 0
0
1
1
1
0
0
1
0
1
1
1
1
( a  b)  b
8.1.3 Tautology and inference rule
Example 8.7
(a  (a  b))  b
a b (a  b) (a  (a  b)) (a  (a  b))  b
11
1
1
1
10
0
0
1
01
1
0
1
00
1
0
1
8.1.4 Predicate logic
Definition(Predicate logic)
“Predicate logic” is a logic which
represents a proposition with the
predicate and individual
 Example
 “Socrates is a man”
 “Two is less than four”

8.1.5 Quantifier
Universal quantifier
“for all”
 Denoted symbolically by 

Existential quantifier
“there exists”
 Denoted symbolically by 

8.2.1 Fuzzy expression
Fuzzy expression(formula)

can have its truth value in the interval
[0,1]


f : [0,1]  [0,1]
Generalization

f : [0,1]n  [0,1]
8.2.1 Fuzzy Expression
Definition(Fuzzy logic)

a logic represented by the fuzzy
expression (formula) which satisfies the
followings
Truth values, 0 and 1, and variable xi ([0,1],
i = 1, 2, …, n) are fuzzy expressions
 If f is a fuzzy expression, ~f is also a fuzzy
expression
 If f and g are fuzzy expressions, f  g and f
 g are also fuzzy expressions

8.2.2 Operators in fuzzy expression
Operators in fuzzy expression
Negation
=1–a
 Conjunction a  b = Min (a, b)
 Disjunction a  b = Max (a, b)
 Implication a  b = Min (1, 1+ba)

8.2.2 Operators in fuzzy expression
The properties of fuzzy operators
a a
 Involution
a  b  b  a, a  b  b  a
 Commutativity
 Associativity
a  b  c  a  (b  c)
a  (b  c)  a  b  a  c
 Distributivity
a  a  a, a  a  a
 Idempotency
 Absorption
a  a  b  a
a  0  0, a  1  1
 Absorption
 Identity
a  1  a, a  0  a
 De Morgan’s law
a b  ab
8.2.2 Operators in fuzzy expression
“law of contradiction” and “law of
excluded middle” are not verified in
the fuzzy logic
Law of contradiction

a  a =Min[a, a ]=Min[a, 1 a]0 if a 0,1
Law of excluded middle

a  a =Max[a, a ]=Max[a, 1 a]1 if a 0,1
8.2.3 Some examples of fuzzy logic
operators
Example 8.12 : when a=1,b=0




a 0
a  b  min 1,0  0
a  b  max 1,0  1
a  b  min 1,1 1  0  0
8.2.3 Some examples of fuzzy logic
operators
Example 8.13 : when a=1,b=1




a 0
a  b  min 1,1  1
a  b  max 1,1  1
a  b  min 1,1 1  1  1
8.2.3 Some examples of fuzzy logic
operators
Example 8.14 : when a=0.6,b=0.7




a  0.4
a  b  min 0.6,0.7  0.7
a  b  max 0.6,0.7  0.7
a  b  min 1,1  0.6  0.7  1
8.3.1 Definition of linguistic variable
Linguistic variable = (x, T(x), U, G, M)





x: name of variable
T(x): set of linguistic terms which can be a
value of the variable
U: set of universe of discourse which defines
the characteristics of the variable
G: syntactic grammar which produces terms in
T(x)
M: semantic rules which map terms in T(x) to
fuzzy sets in U
8.3.1 Definition of linguistic variable
Example 8.15

X = (Age, T(Age), U, G, M)






Age: name of the variable X
T(Age): {young, very young, very very young, …}
U: [0,100] universe of discourse
G(Age): Ti+1 = {young}  {very Ti}
M(young) = {(u, young(u)) | u  [0,100]}
1 if u  0,25

 young u    u  25  2
if u  [25,100]

1 
5


8.3.2 Fuzzy predicate
Definition(Fuzzy predicate)

If the set defining the predicates of individual is
a fuzzy set, the predicate is called a fuzzy
predicate
Example



“z is expensive.”
“w is young.”
The terms “expensive” and “young” are fuzzy
terms. Therefore the sets “expensive(z)” and
“young(w)” are fuzzy sets
8.3.2 Fuzzy predicate
When a fuzzy predicate “x is P” is given,
we can interpret it in two ways


P(x) is a fuzzy set. The membership degree of
x in the set P is defined by the membership
function P(x)
P(x) is the satisfactory degree of x for the
property P. Therefore, the truth value of the
fuzzy predicate is defined by the membership
function

Truth value = P(x)
8.3.3 Fuzzy modifier
A new term can be obtained when we add a
modifier “very” to a primary term
 very young(u) = (young(u))2

1
young
0.5
very young
0
u
25
50
75
100
base variable
8.4 Fuzzy Qualifier
Baldwin defined fuzzy modifier terms in the
universal set V = { |   [0,1]} as follows
T = {true, very true, fairly true, absolutely true, … ,
absolutely false, fairly false,
false}

true v   v
very true v   true v 2
 fairlytrue v   true v 1/ 2
 falsev   1   true v 
very falsev    falsev 2
undecided
1
fairly
false
false
very
false
fairly
true
true
ver
y
true
Abs.
false
Abs.
true
 fairly falsev    falsev 1/ 2
0
1

8.4.2 Examples of fuzzy truth qualifier
Example 8.19 (Fuzzy set young and very young)

1
0.9
0.6
young
very young
0
10
20
30
40
50
60
age
8.4.2 Examples of fuzzy truth qualifier
P1=“20
P2=“20
P3=“20
P4=“20
is
is
is
is
young
young
young
young
is
is
is
is
true”
fairly true”
very true”
false”
truth
truth
truth
truth
value
value
value
value
t(young)
1
0.95
0.9
fairly
true
0.81
false
true
very
true
0.1
0
0.9
1
young(x)
:
:
:
:
0.9
0.95
0.81
0.1
8.5.1 Inference and knowledge representation
In general, the “inference” is a process to
obtain new information by using existing
knowledge
The representation of knowledge is an
important issue in the inference
When we consider the representation
methods, the following rule type “if-then”
is the most popular form

“If x is a, then y is b.”
8.5.1 Inference and knowledge representation
Modus ponens
Fact: x is a
 Rule: If x is a, then y is b
 Result: y is b

Modus tollens
Fact: y isb
 Rule: If x is a then y is b
 Result: x isa

8.5.3 Representation of fuzzy rule
General form of fuzzy rule
 If A(x), then B(y) ) (If x is A, then y is B)
This rule can be represented by a relation
R(x, y)
 R(x, y): A(x)  B(y)
Generalized modus ponens (GMP)



Fact: x is A
: R(x)
Rule: If x is A then y is B : R(x, y)
Result: y is B
: R(y) = R(x)  R(x, y)
8.5.3 Representation of fuzzy rule
Fuzzy reasoning

Generalized modus ponens (GMP)




Fact: x is A
Rule: If x is A then y is B
Result:y is B
: R(x)
: R(x, y)
: R(y) = R(x)  R(x, y)
Generalized modus tollens (GMT)



Fact: y is B
Rule: If x is A then y is B
Result:x is A
: R(y)
: R(x, y)
: R(x) = R(y)  R(x, y)
8.5.3 Representation of fuzzy rule
Example

We have knowledge such as



If x is A then y is B
x is A
From the above knowledge, how can we apply
the inference procedure to get new information
about z ?

We apply the implication operator to get implication
relation


R(x, y): A(x)  B(y)
We manipulate the fact into the form and the apply
the generalized modus ponens

R(x) = R(y)  R(x, y)