1. Theory and Applications
FUZZY SETS AND
FUZZY LOGIC
Theory and Applications
PART 6 Fuzzy Logic
1. Classical logic
2. Multivalued logics
3. Fuzzy propositions
4. Fuzzy quantifiers
5. Linguistic hedges

2. Theory and Applications
FUZZY SETS AND
FUZZY LOGIC
Theory and Applications
6. Inference from conditional fuzzy propositions 7. Inference from conditional and qualified propositions 8. Inference from quantified propositions

3. Classical logic Inference rules
Various forms of tautologies can be used for making deductive inferences. They are referred to as inference rules. Examples : 3

4. Classical logic Existential quantifier
Existential quantification of a predicate P(x) is expressed by the form
"There exists an individual x (in the universal set X of the variable x) such that x is P". We have the following equality: 4

5. Classical logic Universal quantifier
Universal quantification of a predicate P(x) is expressed by the form.
“For every individual x (in the universal set) x is P". Clearly, the following equality holds: 5

6. Classical logic General quantifier Q
The quantifier Q applied to a predicate P(x),
x X, as a binary relation
where α, β specify the number of elements of X for which P(x) is true or false, respectively. Formally, 6

7. Multivalued logics 7

8. Multivalued logics n-valued logics
For any given n, the truth values in these generalized logics are usually labelled by rational numbers in the unit interval [0, 1]. The set Tn of truth values of an n-valued logic is thus defined as
These values can be interpreted as degrees of truth. 8

9. Multivalued logics
Lukasiewicz uses truth values in Tn and defines the primitives by the following equations: 9

10. Multivalued logics
Lukasiewicz, in fact, used only negation and implication as primitives and defined the other logic operations in terms of these two primitives as follows: 10

11. Fuzzy propositions Unconditional and unqualified proposition
The canonical form of fuzzy propositions of this type, p, is expressed by the sentence
where V is a variable that takes values v from some universal set V, and F is a fuzzy set on V that represents a fuzzy predicate, such as tall, expensive, low, normal, and so on. 11

12. Fuzzy propositions
Given a particular value of V (say, v), this value belongs to F with membership grade F(v). This membership grade is then interpreted as the degree of truth, T(p), of proposition p. That is,
for each given particular value v of variable V in proposition p. This means that T is in effect a fuzzy set on [0,1], which assigns the membership grade F(v) to each value v of variable V. 12

13. Fuzzy propositions 13

14. Fuzzy propositions
In some fuzzy propositions, values of variable V are assigned to individuals in a given set / . That is, variable V becomes a function V : / → V, where V ( i ) is the value of V for individual i in V. The canonical form must then be modified to the form 14

15. Fuzzy propositions Unconditional and qualified proposition
Propositions p of this type are characterized by either the canonical form
or the canonical form 15

16. Fuzzy propositions
In general, the degree of truth, T(p), of any truth-qualified proposition p is given for each v V by the equation
An example of a truth-qualified proposition is the proposition "Tina is young is very true." 16

17. Fuzzy propositions 17

18. Fuzzy propositions
Let us discuss now probability-qualified propositions of the form (8.8). For any given probability distribution f on V, we have
and, then, the degree T(p) to which proposition p of the form (8.8) is true is given by the formula 18

19. Fuzzy propositions
As an example, let variable V be the average daily temperature t in °F at some place on the Earth during a certain month. Then, the probability-qualified proposition
p : Pro { temperature t (at given place and time) is around 75 °F } is likely
may provide us with a meaningful characterization of one aspect of climate at the given place and time. 19

20. Fuzzy propositions 20

21. Fuzzy propositions Conditional and unqualified proposition
Propositions p of this type are expressed by the canonical form
where X, Y are variables whose values are in sets X, Y, respectively, and A, B are fuzzy sets on X, Y, respectively. 21

22. Fuzzy propositions
These propositions may also be viewed as propositions of the form
where R is a fuzzy set on X x Y that is determined for each x X and each y Y by the formula
where J denotes a binary operation on [0, 1] representing a suitable fuzzy implication. 22

23. Fuzzy propositions
Here, let us only illustrate the connection for one particular fuzzy implication, the Lukasiewicz implication
This means, for example, that T(p) = 1 when X = x1 and Y = y1; T(p) = .7 when X = x2 and Y = y1 and so on. 23

24. Fuzzy propositions Conditional and unqualified proposition
Propositions of this type can be characterized by either the canonical form
or the canonical form
where Pro {X is A | Y is B} is a conditional probability. 24

25. Fuzzy quantifiers First Kind - Ⅰ
There are two basic forms of propositions that contain fuzzy quantifiers of the first kind. One of them is the form
where V is a variable that for each individual i in a given set / assumes a value V(i), F is a fuzzy set defined on the set of values of variable V, and Q is a fuzzy number on R. 25

26. Fuzzy quantifiers
Any proposition p of this form can be converted into another proposition, p', of a simplified form,
where E is a fuzzy set on a given set / that is defined by the composition 26

27. Fuzzy quantifiers For example, the proposition
p : "There are about 10 students in a given class whose fluency in English is high“
can be replaced with the proposition
p’ : "There are about 10 high-fluency English-speaking students in a given class."
Here, E is the fuzzy set of "high-fluency English-speaking students in a given class." 27

28. Fuzzy quantifiers Proposition p' may be rewritten in the form
where W is a variable taking values in R that represents the scalar cardinality, W = |E|,
and, 28

29. Fuzzy quantifiers Example :
p : There are about three students in / whose fluency in English, V( i ), is high.
Assume that / = {Adam, Bob, Cathy, David, Eve}, and V is a variable with values in the interval [0, 100] that express degrees of fluency in English. 29

30. Fuzzy quantifiers 30

31. Fuzzy quantifiers First Kind - Ⅱ
Fuzzy quantifiers of the first kind may also appear in fuzzy propositions of the form
where V1, V2 are variables that take values from sets V1, V2, respectively, / is an index set by which distinct measurements of variables V1,V2 are identified (e.g., measurements on a set of individuals or measurements at distinct time instants), Q is a fuzzy number on R, and F1, F2 are fuzzy sets on V1, V2 respectively. 31

32. Fuzzy quantifiers
Any proposition p of this form can be expressed in a simplified form,
where E1, E2 are 32

33. Fuzzy quantifiers Moreover, p’ may be interpreted as
we may rewrite it in the form
where W is a variable taking values in R and W = | E1 ∩ E2|. 33

34. Fuzzy quantifiers Using the standard fuzzy intersection, we have
Now, for any given sets E1 and E2 , 34

35. Fuzzy quantifiers Second Kind
These are quantifiers such as "almost all," "about half," "most," and so on. They are represented by fuzzy numbers on the unit interval [0, 1].
Examples of some quantifiers of this kind are shown in Fig. 8.5. 35

36. Fuzzy quantifiers 36

37. Fuzzy quantifiers
Fuzzy propositions with quantifiers of the second kind have the general form
where Q is a fuzzy number on [0, 1], and the meaning of the remaining symbols is the same as previously defined. 37

38. Fuzzy quantifiers
Any proposition of the this form may be written in a simplified form,
where E1, E2 are fuzzy sets on X defined by 38

39. Fuzzy quantifiers we may rewrite p’ in the form where
for any given sets E1 and E2. 39

40. Linguistic hedges Linguistic hedges
Given a fuzzy predicate F on X and a modifier h that represents a linguistic hedge H, the modified fuzzy predicate HF is determined for each x X by the equation
This means that properties of linguistic hedges can be studied by studying properties of the associated modifiers. 40

41. Linguistic hedges Every modifier h satisfies the following conditions:41

42. Linguistic hedges
A convenient class of functions that satisfy these conditions is the class
where α R+ is a parameter by which individual modifiers in this class are distinguished and a [0, 1]. When α < 1, hα is a weak modifier; when α > 1, hα is a strong modifier; h1 is the identity modifier. 42

43. Inference from conditional propositions
For classical logic
Assume that the variables are related by an arbitrary relation on X × Y, not necessarily a function.
Given X = u and a relation R, we can infer that Y B, where B = { y Y |<x, y> R } (Fig. 8.7a).
Similarly, given X A, we can infer that Y B, where B = { y Y |<x, y> R, x A } (Fig. 8.7b).

44. Inference from conditional propositions
Observe that this inference may be expressed equally well in terms of characteristic functions XA, XB, XR of sets A, B, R respectively, by the equation

45. Inference from conditional propositions

46. Inference from conditional propositions
For fuzzy logic
Assume that R is a fuzzy relation on X x Y, and A', B' are fuzzy sets on X and Y, respectively. Then, if R and A' are given, we can obtain B' by the equation
which is a generalization by replacing the

47. Inference from conditional propositions
characteristic functions with the corresponding membership functions.
It can also be written in the matrix form as
called the compositional rule of inference.

48. Inference from conditional propositions

49. Inference from conditional propositions
Viewing proposition p as a rule and proposition q as a fact, the generalized modus ponens is expressed by the following schema:

50. Inference from conditional propositions
Example 8.1

51. Inference from conditional propositions

52. Inference from conditional propositions
Another inference rule in fuzzy logic, which is a generalized modus tollens, is expressed by the following schema:
In this case, the compositional rule of inference has the form

53. Inference from conditional propositions
Example 8.2

54. Inference from conditional propositions
The generalized hypothetical syllogism is expressed by the following schema:
X, Y, Z are variables taking values in sets X, Y, Z, respectively, and A, B,C are fuzzy sets on sets X, Y, Z, respectively.

55. Inference from conditional propositions
Given R1, R2, R3, obtained by these equations, we say that the generalized hypothetical syllogism holds if
which again expresses the compositional rule of inference. This equation may also be written in the matrix form

56. Inference from conditional propositions
Example 8.3

57. Inference from conditional and qualified propositions
Given a conditional and qualified fuzzy proposition p of the form
p : If X is A, then Y is B is S, (8.46)
where S is a fuzzy truth qualifier, and a fact is in the form "X is A’," we want to make an inference in the form “Y is B’."

58. Inference from conditional and qualified propositions
The method of truth-value restrictions is based on a manipulation of linguistic truth values. It involves the following four steps.

59. Inference from conditional and qualified propositions

60. Inference from conditional and qualified propositions
Example 8.4

61. Inference from conditional and qualified propositions

62. Inference from conditional and qualified propositions

63. Inference from conditional and qualified propositions
Theorem 8.1
Let a fuzzy proposition of the form (8.46) be given, where S is the identity function (i.e., S stands for true), and let a fact be given in the form "X is A'," where
for all and some x0 such that A(x0)= . Then, the inference “Y is B‘ " obtained by the method of truth-value restrictions is equal to the one obtained by the generalized modus ponens, provided that we use the same fuzzy implication in both inference methods.

64. Inference from qualified propositions
Given n quantified fuzzy propositions of the form
where Qi is either an absolute quantifier or a relative quantifier, and Wi is a variable compatible with the quantifier Qi, what can we infer from these propositions?

65. Inference from qualified propositions
Quantifier extension principle
Assume that the prospective inference is expressed in terms of a quantified fuzzy proposition of the form
The principle states the following: if there exists a function f : Rn →R such that W = f (W1, W2, …, Wn) and Q = f (Q1, Q2, …, Qn), where the meaning of f (Q1, Q2, …, Qn) is defined by the extension principle, then we may conclude that p follows from p1, p2, …, pn.

66. Inference from qualified propositions
Intersection / product syllogism
where E, F, G are are fuzzy sets on a universal set X, Q1 and Q2 are relative quantifiers (fuzzy numbers on [0, 1]), and Q1 • Q2 is the arithmetic product of the quantifiers.

67. Inference from qualified propositions
Propositions p1, p2, and p may be expressed in the form
where W1 = Prop(F / E), W2 = Prop(G / E∩F), and W = Prop(F∩G / E).

68. Inference from qualified propositions
Consequent conjunction syllogism
where E, F, G are are fuzzy sets on a universal set X, Q1 and Q2 are relative quantifiers and Q is a relative quantifier given by

69. Inference from qualified propositions
that is, Q is at least MAX(0, Q1 + Q2 - 1) and at most MIN(Q1, Q2). Here, MIN and MAX are extensions of min and max operations on real numbers to fuzzy numbers (Sec. 4.5).

70. Exercise 6
6.4
6.8
6.9 70