Once the membership functions have been described and the rules for a FIS generated then some method for combining the rules has to be found. Zadeh [36] extends the modus ponens of traditional logic and introduces the compositional rule of inference or the generalized modus ponens:
.5 true in
Ant1: If x is A then y is B
Ant2: x is 
Cons: y is 
where x and y are objects and 
are fuzzy
concepts represented by fuzzy sets. Note that this reduces to
modus ponens when 
and 
. Mizumoto and Zimmerman
[24] also describe generalized modus tollens:
.5 true in
Ant1: If x is A then y is B
Ant2: y is 
Cons: x is 
which, when 
and 
, reduces
to modus tollens. There are various fuzzy relations that allow
for the handling of ``If x is A then y is B''. In
[24] the many proposed approaches are all considered
with a view to their suitability for generalised modus ponens
and generalised modus tollens. This very detailed piece of
theoretical comparison and the similar work by Lee [20], although interesting in themselves, do not indicate
how a FIS developer might choose an implication function.
In practice the method commonly adopted is known as the min-max method [5]. Given a set of fuzzy rules of the form described in equation 13 the process is as follows:
and 
then 
IF 
and 
then 
where x and y are
inputs and A1, A2, B1, B2, C1 and C2 are fuzzy sets.
Fig. 4 shows how this method works.
This method is easy to compute and whilst other approaches are available and their theoretical properties are well known there is little reported in the literature on how a FIS developer may choose a composition method.
