Editing Fuzzy control system (section)

==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.<ref name = "Hájek" />

In Gerla 2005 <ref>{{cite journal|last1=Gerla|first1=Giangiacomo|title=Fuzzy logic programming and fuzzy control|journal=Studia Logica|date=2005|volume=79|issue=2|pages=231–254|doi=10.1007/s11225-005-2977-0|citeseerx=10.1.1.103.1143|s2cid=14958568}}</ref> another logical approach to fuzzy control is proposed based on fuzzy logic programming: Denote by ''f'' the fuzzy function arising of an IF-THEN systems of rules. Then this system can be translated into a fuzzy program P containing a series of rules whose head is "Good(x,y)". The interpretation of this predicate in the least fuzzy [[Herbrand model]] of P coincides with f. This gives further useful tools to fuzzy control.