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

depth0pt width6cm

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

depth0pt width6cm

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:

- For each of the antecedents find the minimum of the
membership function for the input data. Apply this to the
consequent.
- For all rules construct a fuzzy set that is a truncated set using the maximum of the membership values obtained.

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.

Fri Oct 25 14:41:29 BST 1996