1. Chapter 8 : Fuzzy Logic

2. 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 (true)
2 + 4 = (false)
For every x, if f(x) = sin x, then f(x) = cos x. (true)
It rains now. (true)

3. 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

4. 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”

5. 8.1.1 Proposition Logic Connectives - combine prepositional variables
Negation
Conjunction
Disjunction
Implication

6. 8.1.2 Logic function Logic function Logic formula
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

7. 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. 8.1.3 Tautology and inference rule
Example 8.5 (tautology) 1

9. 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 1

10. 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”

11. 8.1.5 Quantifier Universal quantifier Existential quantifier “for all”
Denoted symbolically by 
Existential quantifier
“there exists”
Denoted symbolically by 

12. 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]

13. 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

14. 8.2.2 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)

15. 8.2.2 Operators in fuzzy expression
The properties of fuzzy operators
Involution
Commutativity
Associativity
Distributivity
Idempotency
Absorption
Identity
De Morgan’s law

16. 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  =Min[a, ]=Min[a, 1 a]0 if a 0,1
Law of excluded middle
a  =Max[a, ]=Max[a, 1 a]1 if a 0,1

17. 8.2.3 Some examples of fuzzy logic operators
Example 8.12 : when a=1,b=0

18. 8.2.3 Some examples of fuzzy logic operators
Example 8.13 : when a=1,b=1

19. 8.2.3 Some examples of fuzzy logic operators
Example 8.14 : when a=0.6,b=0.7

20. 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

21. 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]}

22. 8.3.2 Fuzzy predicate Definition(Fuzzy predicate) Example
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

23. 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)

24. 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 25
100 50 75 u
very young
young
0.5 1 
base variable

25. 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}
undecided
Abs. true
Abs. false
fairly false
false
very false
fairly true
true
very true  1 

26. 8.4.2 Examples of fuzzy truth qualifier
Example 8.19 (Fuzzy set young and very young) 10 20 30 40 50 60
age
0.6
0.9 1 
young
very young

27. 8.4.2 Examples of fuzzy truth qualifier
P1=“20 is young is true” truth value : 0.9
P2=“20 is young is fairly true” truth value : 0.95
P3=“20 is young is very true” truth value : 0.81
P4=“20 is young is false” truth value : 0.1 1
0.9
0.1
0.81
0.95
t(young)
young(x)
false
fairly true
true
very true

28. 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.”

29. 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

30. 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)

31. 8.5.3 Representation of fuzzy rule
Fuzzy reasoning
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)
Generalized modus tollens (GMT)
Fact: y is B : R(y)
Result:x is A : R(x) = R(y)  R(x, y)

32. 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)