Notes on Linguistic variables from extracts of Zadeh
What are linguistic variables?
What is a fuzzy algorithm?
What is principle of incompatibility?
How is human intelligence different from machine intelligence?
Linguistic and fuzzy variables
In quantitative approach if we have a conditional statement like
“if x is 6 then y is 10”. The same statement in fuzzy logic approach can be given by
Basic operations of fuzzy sets(RECAP)
Example of these basic operators
Concentration and dilation of linguistic variables
CON(A) and DIL(A) are conventionally used to represent hedges very and more or
less.
combination of NOT(A), A AND B, A OR B, and CON(A) and DIL(A) can be used
in the construction of composite linguistic term.
The values of a linguistic variable are labels of fuzzy subsets of U which have the
form of phrases or sentences in a natural or artificial language. If U is the collection of
integers
and age is a linguistic variable labelled x, then the values of x might be young, not
young, very young, not very young, old and not old, not very old, not young and not
old.
x=hu where h is a hedge u is a term with a specified meaning.
x = very tall man,
Hedges serve the function of generating a larger set of values for a linguistic variable
from a small collection of primary terms. Hedge acts as an intensifier generating a
subset of the set on which it operates. As an example
or
Another example of hedges in a linguistic variable small
COMPUTATION OF THE MEANING OF VALUES OF A LINGUISTIC
VARIABLE
In addition to hu the terms not, or and and can also be used for the computation of
linguistic variables
where small is defined as
with the universe of discourse being
A linguistic variable is characterized by a quintuple(x,T(x), X,G,M)
x name of the variable
T(x) term set of x(set of linguistic variables or terms)
X universe of discourse
G syntactic rule which generates the terms in T(x)
Msemantic rule which associates with each linguistic value A its meaning M(A)
M(A) fuzzy set in X.
Example
If age is interpreted as a linguistic variable then its term set T(age)
T(age) = { young, not young, very young, not very young,….,
Middle aged, not middle aged………….
Old, not old, very old, not very old, more or less old,…..
Not very young and not very old,……………….}
X=[0,100] -- age of persons in the span of 0 and 100 years
“age is young” linguistic variable age is assigned to the linguistic value “young”
Age=20 assignment of numerical value 20 to the numerical variable age
Syntactic rule- refers to the way the linguistic values in the term set T(age) are
generated
Semantic rule- Defines the membership function of each linguistic value of the
term set
Primary terms(young, middle aged, old)
Modified by not, and, or
Hedges(very, more or less, quite, extremely )
Connectives – and, or, either and neither.
Try out these
µyoung(x) = bell(x,20,2,0) = [1+(x/40)4] -1
µold(x) = bell(x,30,3,100) = [1+(x-100/30)6] -1
Find
more or less old
not young and not old
young but not too young
extremely old
Example of contrast intensifier follow my worked out example in my notes
Orthogonality
A term set T=t1,t2, ….tn of a linguistic variable x on the universe X is orthogonal
if it fulfils the following property
n
  ( x)  1, x  X
ti
i 1
Where the ti’s are convex and normal fuzzy sets defined on X and these fuzzy
sets make up the term set T.
Fuzzy If-Then Rules
• Fuzzy rule, fuzzy implication, fuzzy
conditional statement
General format:
If x is A then y is B  A  B
x is A : antecedent (or premise)
y is B : consequence (or conclusion)
Examples:
• If pressure is high, then volume is
small.
 If the
• road is slippery, then driving is dangerous
 If a tomato is red, then it is ripe
• speed
If the
road
is slippery,
 If the
is high,
then apply
the brake a little then driving
is dangerous.
What do u mean by “if x is A then y is B”
• Ifdescribes
a tomato
red,twothen
itx is
The expression
a relationis
between
variables
and ripe.
y, which
suggests that a fuzzy if-then rule be defined as a binary fuzzy relation R on the
If the speed is high, then apply the
product of•X*Y
How can :If x brake
is A then Ya
is B”
be interpreted?
little.
A If-then rule is a relation between 2
variables x and y;
define as a binary fuzzy relation R on X  Y.
Two ways to interpret “If x is A then y is B”:
y
A coupled with
B
B
y
A entails
B
B
x
x
Incomplete------------------------------------------------------------------------------------------
The inference of the form
Where A,A’,B,B’ are fuzzy concepts is called as fuzzy conditional inference. As an
example
If A=A’and A and B are non-fuzzy then the above equation reduces to classical
modus ponens of a two valued logic.
In classical logic, modus ponendo ponens (Latin for the way that affirms by
affirming;[1] often abbreviated to MP or modus ponens) is a valid, simple argument
form sometimes referred to as affirming the antecedent or the law of detachment.
Modus ponens is a very common rule of inference, and takes the following form:
If P, then Q.
P.
Therefore, Q
But if A,A’ and B are fuzzy and A!=A’ then this form of inference cannot be made by
classical modus ponens. For this Zadeh formalized an inference rule named as
“compositional rule of inference”.