Welcome to lecture number 41 of Fuzzy Sets
Logic and Systems and Applications. In this
lecture, we will discuss Linguistic Hedges.
So, before we finally discuss linguistic hedges
let us go through the term linguistic variables.
So, a linguistic variable is characterised
by a quintuple and this quintuple basically
has five variables 
and here these five variables are x, T x.
So, x is generic variable and T x here is
the term set, capital X is the universe of
discourse G here is this syntactic rule which
generates the term set T x. And we have M
which is nothing but, the semantic rule which
associates with each linguistic value.
So, that is how we have these five variables
and these five variables are put together
to be called as quintuple. So, if here let
us say we take an example of a linguistic
variable.
Let say a linguistic variable here is age.
So, if we have age as the linguistic variable.
So, let’s see how these five variables look
like. So, we see that if we have age, a g
e age. So, what is the generic variable in
this case? So, here the linguistic variable
or the generic variable basically, the linguistic
variable is the age, but we have x as the
generic variable, generic variable. And then
here we have the term set T x.
So, for age, we can have term set and this
term set can be a collection of multiple linguistic
values. When we say linguistic values it means
fuzzy sets. So, every linguistic value is
represented by a particular fuzzy set and
here the term set T x and since here age is
a linguistic variable. So our T of age or
I would say the term set for age can be here
in this case young, middle aged, old. So,
this is just for example, and we can have
multiples segregations multiple fuzzy sets
for age. So, term set can be basically a collection
of the linguistic values for a particular
linguistic variable.
So, if we have a linguistic term like age
is young, this will denote the assignment
of the linguistic value young to the linguistic
variable age. So, the linguistic variable
as I have mentioned here if we have age as
the linguistic variable. So, linguistic variable
is normally represented in terms of the generic
variable x.
So, as I mentioned that this linguistic variable
can be segregated can be divided into multiple
linguistic values. Here in this case, we have
divided the whole region of age in 3. So,
these three regions these three regions basically
are young, middle aged, old like that, but
we can have multiple such regions which could
be represented by the linguistic values and
these linguistic values are normally represented
by fuzzy sets.
So, here we have this fuzzy set, this linguistic
value young and this young is nothing but,
the lower side of the age representation.
So, young is a fuzzy set here is a left open
fuzzy set and we have this the middle one
is middle age and here this middle age is
represented by another fuzzy set. Similarly,
we have the old age here which is represented
by a fuzzy set which is right open.
So, as I mentioned that when we are dealing
with the term set we can suitably divide a
particular linguistic variable into multiple
fuzzy regions and every fuzzy region is represented
by a particular linguistic value which is
represented by a fuzzy set. So this way a
linguistic variable is normally represented
and as in this case we have seen that age
if we have taken age as the linguistic variable
age has been divided into various fuzzy regions.
And all these regions are nothing but, is
the collection of all these regions is term
set.
So, here we have a generic variable x which
is the measure of age, for example we have
1 year, 2 years, 3 year like that and it can
further go up to here in this case we are
we have 90. So, these values are actually
the values of the generic variable x and x
here is the age.
Now, what is the universe of discourse? So,
the universe of discourse can be all you know
the possible limit of age where the all possible
values of the generic variable age, linguistic
variable age could be settle. So, within the
total a space basically is here termed as
universe of discourse capital X. So, in the
term set T age each term can be characterized
by a fuzzy set of a universe of discourse.
So, the capital X here is the universe of
discourse and this universe of discourse basically
says that whatever value that small x can
take or age can take will be belonging into
the limit of the universe of discourse. Then
comes the syntactic rule here, so, the syntactic
rule is G and it is symbolically defined by
G. So, the syntactic rule refers to the way
the linguistic values in the term set T are
generated.
So, in the figure that we have just discussed
for age where we have created multiple fuzzy
regions for age; young, middle age etcetera
are the syntactic levels. So, based on this
syntactic rule, we create young, middle aged,
old etcetera. Similarly, semantic rule M.
So, M basically helps us in specifying the
procedure for computing the meaning of any
linguistic value through specified membership
function. So, we will have couple of examples
and with this we will be able to understand
all these parameters.
Now, comes the composite linguistic term.
So, here for linguistic variables we use word
as values of linguistic variables in cases
of linguistics we often use more than one
word to describe a variable. So, for example,
here the intensity of light, if we choose
the intensity of light as a linguistic variable.
So, let us now see how these five parameters
as we have seen those parameters which were
as a quintuple. How actually are these coming
up for this linguistic variable.
So, intensity of the light basically here,
if we are taking this as the linguistic variable.
So, what is the generic variable? How we are
going to measure the intensity of light? How
we are going to represent the intensity of
light, let us say it is x, so, x is the generic
variable and then what is the universe of
discourse capital X. So, that the limit the
range within which we are supposed to take
the generic variable values and then the term
set.
So, the term set here for this case could
be if it is the intensity of light, it could
be simply either low or high or medium or
like that or may be dim, bright. So, these
kinds of fuzzy regions or the linguistic values
can be included in the term set. And similarly,
the syntactic and semantic rules could be
created for these the linguistic variable
the intensity of light. Here we have written
very bright, slightly dim, more or less bright,
all these are little further if we discussed
the hedges. So, these comes under that.
So, we will understand these terms when we
move little ahead. So, the linguistic variables
we see here the term set which we have for
linguistic variables. So, this can further
we generated and let us now understand first
that the linguistic variable may be a composite
term and can be classified into three groups.
So, as a whole linguistic variable can have
primary terms and then linguistic hedges and
then we have the negation and complement or
and connectives. For example, if we take intensity
of light so, what are the primary terms for
the intensity of light?
So, primary terms could be for intensity of
light, it could be as I already mentioned
that could be low intensity, medium intensity
or may be high intensity. Similarly, the linguistic
hedges for the intensity of light could be
maybe we if we add an adjective here, so we
could simply write very low intensity or may
be more or less, medium or whatever.
So, similar linguistic values could be included.
So, the linguistic hedges basically are with
the adjectives in along with the primary terms.
And then comes the negation class here we
have you know the linguistic fuzzy regions
where the primary terms are taken as either
the complement or negation or with some other
connectives like and or, or. So, for example,
here we can write we can use a term not low
or may be not medium. Similarly, we can have
another fuzzy value which could be like this
low, but not very low.
So, here we see that, but is the, but is a
connective. So, this way we see that we have
some composite terms and these are basically
divided into three groups; first group is
primary terms, second group is the linguistic
hedges and the third group is the complement
and connectives. So, let us further understand
this by taking some examples and these primary
terms and then the linguistic hedges negation
complements and connectives can be understood
further.
So, if we talk of the primary terms let us
say the age which we have already taken in
the beginning as the linguistic variable,
then it is primary term could be the young
as I have already mentioned and in this primary
terms, the middle aged can also be one of
the linguistic value, similarly old. So, we
have already seen that we have three fuzzy
regions. This could be like this.
So, we have let’s say this is the universe
of discourse 
here and within this we have the generic variable
x which is nothing but the age the measure
of age let’s say years. So, let us say we
have created three primary terms and these
three primary terms here are middle age or
middle aged and then we have the old and here
we have let’s say young. So, these three
are the primary terms which have been crated
here and further these primary terms can be
used with some adjectives are connectives
to get the linguistic hedges.
So, we will discuss this in the coming slides.
So, here the primary terms basically are the
fuzzy regions, primary fuzzy regions which
are being created just to represent the basic
building regions basic regions. So, for example,
here the age as the generic variable, age
as the linguistic variable here is divided
into three region, three basic regions and
hence these young middle aged old are called
as basically the primary terms.
Now, these primary terms as I have already
mentioned can be used as hedges by using the
adjectives along with the primary terms.
So, here as I mentioned linguistic hedges.
So, in linguistic, fundamental atomic terms.
Basically, atomic terms are often modified
with adjectives. So, here with adjectives
and adverbs such as very, slight, more or
less, fairly, slightly, almost, barely, mostly,
roughly, approximately etcetera. So, we see
that everywhere we have some adjectives or
adverbs and when this is being used along
with the primary terms or atomic terms then
this becomes linguistic hedges.
So, let us understand this further with some
mathematical expressions of linguistic hedges.
So, if we have here a fuzzy set let say A
and if it is continuous fuzzy set then we
represent a fuzzy set, continuous fuzzy set
like this. And similarly, we represent discrete
fuzzy set like this. So, the membership values
which we use here for representing any fuzzy
set let say A whether it is a continuous fuzzy
set or discrete fuzzy set.
So, if we have a membership function mu x
which has been used in the basic fuzzy set
in the primary term or the atomic term that
the symbol that has been used. So, if we have
mu x, then how can we represent a linguistic
hedge using the membership function of the
primary term. So, here if we have the mu x
which is here. The mu x is nothing but the
membership function of primary fuzzy value
So, if we have mu x as the primary fuzzy value
when we say fuzzy set, primary fuzzy set it
means it is the primary term like in case
of age we have seen the young. So, young is
represented by a primary set, this is a primary
fuzzy set these also sometimes termed as a
fuzzy value or the linguistic value because
all these the linguistic value, fuzzy value
etcetera are represented by a suitable fuzzy
set
So, now let us make use of mu x which is used
in the primary fuzzy set and let us make linguistic
hedge out of it. So, if we have mu x for a
primary fuzzy set then if we have to let say
use very for example, if we have let say this
mu x, I will write mu x here and this mu x
has been used for let say young. This mu x
has been used for young. It means, we have
a fuzzy region or fuzzy value which is represented
by a fuzzy set A and this A is nothing but,
this is for young. So, this mu x is for mu
x here is a membership function and this is
for the primary fuzzy set primary region primary
term.
So, if let say we would like to say it like
this. We would like to modify the fuzzy set
like this like very young. So, how can we
make use of this mu x and we convert it into
very young. So, very young we have used very
before young, young we already know. So, how
can we get the a fuzzy set let say which is
A and this is for very young. It is very easy
and we simply make use of the mu x that was
given to us we have to is square the mu x.
So, the mu x that was given to us means simply
take it and then we will square it and that
is it.
So, mu A can be converted into mu very A very
easily by just taking square of the membership
functions. So, this way we are able to convert
this into a linguistic hedge.
Similarly, a membership value if it is given
let’s say again this is the basic membership
value mu A and we are interested in more or
less as the linguistic hedge, we have to simply
dilate it. When we say dilate means we are
reducing the power, means we are taking these
square root of it here.
So, more or less can be represented by mu,
the membership function for more or less can
be simply the mu more or less of x is equal
to mu A of x and then and the root of it.
Similarly, when we want to find the membership
function of the extremely something extremely
if we want to use the hedge is extremely,
then this is equal to basically mu very very
3 times and then that is how it become mu
a of x and then we write here this raise to
the power 8. This means we have applied very
3 times. If we take very one time it is squares
the membership function means we write mu
a of x raise to the power 2.
So, this way we are able to make use of the
hedge and we can applied the adjective over
the primary term to make the linguistic hedge.
So, with this I would like to stop here.
And in the next lecture we will continue with
some examples on linguistic hedges and negations
complements and connectives.
Thank you.
