Linguistic variables and hedges in fuzzy logic

Linguistic variables are variables whose values are words or sentences in a natural or artificial language

– Zadeh

Linguistic Variables:

Linguistic variables and hedges are quite useful in representing values in fuzzy sets. In fuzzy systems, variable ranges over states – which are fuzzy numbers rather than real numbers. These fuzzy numbers are often represented as linguistic variables such as HOT, COLD, WARM etc.

Linguistic variables are defined in terms of base variables, whose values are real numbers. The base variable is the one that determines the linguistic variable’s value. Imprecision of linguistic variables makes them useful for reasoning.

That is, each linguistic variable consists of the following elements

• A name, which should capture the meaning of the base variable involved
• A base variable with its range of values (a closed interval of real numbers)
• A set of linguistic terms that refer to values of the base variable
• A semantic rule, which assigns to each linguistic term its meaning—an appropriate fuzzy number defined on the range of the base variable

An example of a linguistic variable is shown in the following diagram. Its name is “temperature,” which captures the meaning of the associated base variable—a variable that expresses the room temperature T (in degree) by real numbers in the interval [0,60], Linguistic values (states) of the linguistic variable are “Cold,” “warm”, “high” and “extreme.” Each of these linguistic terms is assigned one of the fuzzy values using the triangular function.

We can categorize linguistic variables as follow:

• Quantification terms:  all, most, many, about one-fourth, some
• Usuality terms: always, sometimes, seldom, never
• Likelihood terms: certain, likely, possible, certainly not

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Linguistic Hedges:

A linguistic hedge is an operation that modifies the meaning of a fuzzy set. In other words, hedges are the operators which modify the shapes of fuzzy sets by using adverbs such as more, high, less, and slightly.

For example, how high the temperature is? Depending upon we can have answers such as very high or slightly high etc. These are examples of hedges applied to the fuzzy set of the temperature.

Types of hedges:

• Intensify a fuzzy set (very, extremely)
• Dilute a fuzzy set (somewhat, sort of)
• Express probabilities (probably, not likely)
• Approximate a scalar or single number (exactly)
• Express vague quantities (most, seldom)

Example

A = cold climate with μ_A(x) as the MF

B = Hot climate with μ_B(x) as the MF

Not cold climate: A‘ =1 – μ_A(x)

Not hot climate: B‘ =1 – μ_B(x)

Extreme climate: A ∪ B = max⁡( μ_A(x), μ_B(x) )

Pleasant climate: A ∩ B = min⁡( μ_A(x), μ_B(x) )

Generalization of membership function:

Given a membership function, we can derive many more MFs representing several other linguistic hedges using the concept of Concentration and Dilation.

Concentration: A^k =[ μ_A(x) ]^k,     k > 1

Dilation: A^k =[ μ_A(x) ]^k,     k < 1

Example: Age = {Young, Middle-aged, Old}

• Thus corresponding to Young, we have Not young, Very young, Not very young and so on
• Similarly, with Old, we can have Not old, Very old, very very old, Extremely old etc.
• μ_ExtremlyOld (x)= (((μ_Old(x))²)²)²)
• μ_{MoreOrLessOld} (x)= (μ_old(x))^0.5

Example:

Modifier:

Let A be a fuzzy set in X. Then we can define the fuzzy sets ”very A” and ”more or less A”  by

Example:

Truth = {Absolutely false, Very false, False, Fairly true, True, Very true, Absolutely true}

One may define the membership function of linguistic terms of truth as

True(u)= u,  False(u)= 1 – u,  ∀u ∈ [0, 1]

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