Reasoning and Fuzzy Logic - Fuzzy Logic: An Introductory Course for Engineering Students - page 98

Remark 3.3.1 It should be pointed out that the Modus Ponendo Tollens (MPT) can
be reduced, in the case of duality, to the disjunctive mode by means of the change
μ = ʱ , ˃ = ʲ , in which case since (μ · ˃) = μ + ˃ it follows μ · (μ · ˃)

=

μ · (μ + ˃ ) = ʱ · (ʱ + ʲ) ʲ = ˃ . Hence, it holds with the triplet W ˕ )

( W

˕ ,

N

˕ ,

.

3.4 Inference with Fuzzy Rules

A central topic fuzzy logic deals with are non-rigid, dynamic systems involving
'variables' x 1 ,..., x n , y taking values in, respectively, universes X 1 ,..., X n , Y ,

and constrained by imprecise rules r i of the type

If x 1 is P 1 i , and x 2 is P 2 i ,…, and x n is P ni , then y is Q i ( 1 i n ) ,

with predicates P ji ( 1 j n ) in X j , and Q i in Y .

Let's consider the simplest case with two variables x ∈ X , y ∈ Y , constrained by

a single rule IfxisP,thenyisQ .
When observing the system , the variable x in the rule's antecedent not

always will show ' xisP ', but ' x is P ∗ ' with P ∗ some predicate slightly modificate
from P . For example, if P

( x , y )

short , it could be P ∗ = v = er y shor t , almost short ,

etc. A concrete example is

Rule: If tomatoes are red, they are ripe
Observation: Tomatoes are very red
Conclusion: Tomatoes are very ripe,

ripe , P ∗ er y r ed , and Q ∗ where P = red , Q = = v = v er y r i pe . Hence, the

corresponding scheme of forwards reasoning is

Rule: If x is P , then y is Q
Observation: x is P ∗
Consequence: y is Q ∗

P ∗ are known, and Q ∗ is unknown. This scheme is the Generalized
Modus Ponens (GMP), and to find Q ∗ through fuzzy artillery it is needed to have the
representations
• μ P ∈[

where P , Q ,

X ,of P .

, ]

0 1

Y ,of Q .

• μ Q ∈[ 0 , 1 ]

X ,of P ∗ .

• μ P ∗ ∈[ 0 , 1 ]

• (μ P ₒ μ Q )( x , y ) = J (μ P ( x ), μ Q ( y )) , with a convenient T -conditional J ,for

the rule.