**Logical Interpretation of Fuzzy Control**

In spite of the appearance there are several difficulties to give a rigorous logical interpretation of the *IF-THEN* rules. As an example, interpret a rule as *IF (temperature is "cold") THEN (heater is "high")* by the first order formula *Cold(x)→High(y)* and assume that r is an input such that *Cold(r)* is false. Then the formula *Cold(r)→High(t)* is true for any *t* and therefore any *t* gives a correct control given *r*. A rigorous logical justification of fuzzy control is given in Hájek's book (see Chapter 7) where fuzzy control is represented as a theory of Hájek's basic logic. Also in Gerla 2005 a logical approach to fuzzy control is proposed based on fuzzy logic programming. Indeed, denote by *f* the fuzzy function arising of a IF-THEN systems of rules. Then we can translate this system into fuzzy program in such a way that *f* is the interpretation of a vague predicate *Good(x,y)* in the least fuzzy Herbrand model of this program. This gives further useful tools to fuzzy control.

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