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Lot Zadeh [946] introduced the concept of linguistic variable (or fuzzy variable) in 1973, which allows computation with words in stead of numbers Linguistic variables are variables with values that are words or sentences from natural language For example, referring again to the set of tall people, tall is a linguistic variable Sensory inputs are linguistic variables, or nouns in a natural language, for example, temperature, pressure, displacement, etc Linguistic variables (and hedges, explained below) allow the translation of natural language into logical, or numerical statements, which provide the tools for approximate reasoning (refer to Section 212) Linguistic variables can be divided into di erent categories: Quanti cation variables, eg all, most, many, none, etc Usuality variables, eg sometimes, frequently, always, seldom, etc Likelihood variables, eg possible, likely, certain, etc In natural language, nouns are frequently combined with adjectives for quanti cations of these nouns For example, in the phrase very tall, the noun tall is quanti ed by the adjective very, indicating a person who is taller than tall In fuzzy systems theory, these adjectives are referred to as hedges A hedge serves as a modi er of fuzzy values In other words, the hedge very changes the membership of elements of the set tall to di erent membership values in the set very tall Hedges are implemented through subjective de nitions of mathematical functions, to transform membership values in a systematic manner To illustrate the implementation of hedges, consider again the set of tall people, and assume the membership function tall characterizes the degree of membership of elements to the set tall Our task is to create a new set, very tall of people that are very tall In this case, the hedge very can be implemented as the square function That is, very tall (x) = tall (x)2 Hence, if Peter belongs to the set tall with certainty 09, then he also belongs to the set very tall with certainty 081 This makes sense according to our natural understanding of the phrase very tall: Degree of membership to the set very tall should be less than membership to the set tall Alternatively, consider the set sort of tall to represent all people that are sort of tall, ie people that are shorter than tall In this case, the hedge sort of can be implemented as the square root function, sort of tall (x) = tall (x) So, if Peter belongs to the set tall with degree 081, he belongs to the set sort of tall with degree 09 Di erent kinds of hedges can be de ned, as listed below: Concentration hedges (eg very), where the membership values get relatively smaller That is, the membership values get more concentrated around points.

with higher membership degrees Concentration hedges can be de ned, in general terms, as (211) A (x) = A (x)p , f or p > 1 where A is the concentration of set A Dilation hedges (eg somewhat, sort of, generally), where membership values increases Dilation hedges are de ned, in general, as A (x) = A (x)1/p f or p > 1 (212)

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