FUZZY SETS
AND
FUZZY LOGIC
PART 6
Fuzzy Logic
Theory and
Applications
1. Classical logic
2. Multivalued logics
3. Fuzzy propositions
4. Fuzzy quantifiers
5. Linguistic hedges
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
Classical logic
• Inference rules
Various forms of tautologies can be used for
making deductive inferences. They are
referred to as inference rules. Examples :
(a  (a  b))  b (modus ponens),
(b  (a  b))  a (modus tollens),
((a  b)  (b  c))  (a  c) (hypotheti
cal syllogism).
3
Classical logic
• Existential quantifier 
Existential quantification of a predicate P(x) is
expressed by the form
(x) P( x),
"There exists an individual x (in the universal set
X of the variable x) such that x is P". We have
the following equality:
(x) P ( x)  V P ( x)
x X
4
Classical logic
• Universal quantifier 
Universal quantification of a predicate P(x) is
expressed by the form.
(x) P( x),
“For every individual x (in the universal set) x is
P". Clearly, the following equality holds:
(x) P( x)   P( x)
xX
5
Classical logic
• General quantifier Q
The quantifier Q applied to a predicate P(x),
x  X, as a binary relation
Q  {(α, β) | α, β  N, α  β | X |},
where α, β specify the number of elements of X
for which P(x) is true or false, respectively.
Formally,
 | {x  X | P( x) is true} |,
 | {x  X | P( x) is false} | .
6
Multivalued logics
7
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
0
1
2
n  2 n 1 

Tn  0 
,
,
,
,
 1.
n 1 n 1 n 1
n 1 n 1 

These values can be interpreted as degrees of
truth.
8
Multivalued logics
Lukasiewicz uses truth values in Tn and defines
the primitives by the following equations:
a  1  a,
a  b  min(a, b),
a  b  max(a, b),
a  b  min(1, 1  b  a),
a  b  1 | a  b | .
9
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:
a  b  ( a  b )  b,
a b  a b,
a  b  (a  b)  (b  a ).
10
Fuzzy propositions
• Unconditional and unqualified proposition
The canonical form of fuzzy propositions of this
type, p, is expressed by the sentence
p : V is F ,
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
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,
T ( p )  F (v )
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
Fuzzy propositions
13
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 p : V is F must then be
modified to the form
p : V(i) is F
, where i  I .
14
Fuzzy propositions
• Unconditional and qualified proposition
Propositions p of this type are characterized by
either the canonical form
p : V is F is S ,
or the canonical form
p : Pro{V is F} is P, (8.8)
15
Fuzzy propositions
In general, the degree of truth, T(p), of any truthqualified proposition p is given for each v  V by
the equation
T ( p)  S ( F (v)).
An example of a truth-qualified proposition is the
proposition "Tina is young is very true."
16
Fuzzy propositions
17
Fuzzy propositions
Let us discuss now probability-qualified
propositions of the form (8.8). For any given
probability distribution f on V, we have
Pro{V is F}   f (v)  F (v);
vV
and, then, the degree T(p) to which proposition p
of the form (8.8) is true is given by the formula
T ( p)  P( f (v)  F (v))
vV
18
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
Fuzzy propositions
20
Fuzzy propositions
• Conditional and unqualified proposition
Propositions p of this type are expressed by the
canonical form
p : If X is A, thenY is B,
where X, Y are variables whose values are in
sets X, Y, respectively, and A, B are fuzzy sets
on X, Y, respectively.
21
Fuzzy propositions
These propositions may also be viewed as
propositions of the form
X , Y is R,
where R is a fuzzy set on X x Y that is
determined for each x X and each y Y by the
formula
R( x, y)  J[ A( x), B( y)],
where J denotes a binary operation on [0, 1]
representing a suitable fuzzy implication.
22
Fuzzy propositions
Here, let us only illustrate the connection for one
particular fuzzy implication, the Lukasiewicz
implication
J(a, b)  min(1, 1  a  b).
Let A  .1 x1  .8 x2  1 x3 and B  .5 y1  1 y2 .
T henR  1 x1 , y1  1 x1 , y2  .7 x2 , y1  1 x2 , y2
 .5 x3 , y1  1 x3 , y2 .
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
Fuzzy propositions
• Conditional and qualified proposition
Propositions of this type can be characterized by
either the canonical form
p : If X is A, thenY is B is S ,
or the canonical form
p : Pro{X is A | Y is B } is P,
where Pro {X is A | Y is B} is a conditional
probability.
24
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
p : T hereare Q i's in I such thatV(i) is F ,
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
Fuzzy quantifiers
Any proposition p of this form can be converted
into another proposition, p', of a simplified form,
p': T hereare Q E's,
where E is a fuzzy set on a given set / that is
defined by the composition
E (i)  F (V(i)) for all i  I .
26
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 Englishspeaking students in a given class."
Here, E is the fuzzy set of "high-fluency Englishspeaking students in a given class."
27
Fuzzy quantifiers
Proposition p' may be rewritten in the form
p': W is Q,
where W is a variable taking values in R that
represents the scalar cardinality, W = |E|,
| E |  E (i)   F (V(i))
iI
iI
and,
T ( p)  T ( p' )  Q(| E |).
28
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
Fuzzy quantifiers
30
Fuzzy quantifiers
• First Kind - Ⅱ
Fuzzy quantifiers of the first kind may also
appear in fuzzy propositions of the form
p : Thereare Q i's in I such thatV1 (i) is F1 and V2 (i) is F2 ,
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
Fuzzy quantifiers
Any proposition p of this form can be expressed
in a simplified form,
p': Q E1's E2's ,
where E1, E2 are
E1 (i)  F1 (V1 (i))
E2 (i)  F2 (V2 (i)) for all i  I .
32
Fuzzy quantifiers
Moreover, p’ may be interpreted as
p': T hereare Q( E1 and E2 )'s.
we may rewrite it in the form
p': W is Q,
where W is a variable taking values in R and W
= | E1 ∩ E2|.
33
Fuzzy quantifiers
Using the standard fuzzy intersection, we have
W   min[F1 (V1 (i)), F2 (V2 (i))],
iI
Now, for any given sets E1 and E2 ,
T ( p)  T ( p' )  Q(W).
34
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
Fuzzy quantifiers
36
Fuzzy quantifiers
Fuzzy propositions with quantifiers of the second
kind have the general form
p : Amongi's in I such thatV1 (i) is F1 thereare
Qi's in I such thatV2 (i) is F2 ,
where Q is a fuzzy number on [0, 1], and the
meaning of the remaining symbols is the same
as previously defined.
37
Fuzzy quantifiers
Any proposition of the this form may be written
in a simplified form,
p': QE1's are QE2's,
where E1, E2 are fuzzy sets on X defined by
E1 (i)  F1 (V1 (i)),
E2 (i)  F2 (V2 (i)) for all i  I .
38
Fuzzy quantifiers
we may rewrite p’ in the form
p': W is Q,
where
min[F (V (i)), F (V (i))]

| E1  E2 |
W
. W
| E1 |
 F (V (i))
iI
1
iI
1
1
2
2
1
for any given sets E1 and E2.
39
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
HF ( x)  h( F ( x)).
This means that properties of linguistic hedges
can be studied by studying properties of the
associated modifiers.
40
Linguistic hedges
Every modifier
conditions:
h
satisfies
the
following
1. h(0)  0 and h(1)  1.
2. h is a continuous function;
3. if h is strong,thenh 1 is weak and vice versa;
4. given another modifier g , composit ions of g with h
and h with g are also modifiersand, moreover,if
both h and g are strong(weak), thenso are composit ions.
41
Linguistic hedges
A convenient class of functions that satisfy these
conditions is the class

h (a)  a ,
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
Exercise 6
• 8.4
• 8.8
• 8.9
43