Reasoning and Fuzzy Logic - Fuzzy Logic: An Introductory Course for Engineering Students - page 92

1

, if a b

J min ( a , b ) = S ( N ( a ), b ) =

b , if a > b

it will result

1

, if N ( a ) b

S ( a , b ) =

b , if N ( a )> b

a function that is not a t-conorm, since S 0. Hence, J min is not

an S-implication. An analogous reasoning shows that J pr od ˕ are not S-implications.

( a , 0 ) = 0 = a ,if a >

μ ₒ ˃ = μ + μ · ˃ Example 3.2.9 The protoform (coming from the Sasaki hook),

gives

J 1 (

, ) = ( ( ), ( , )),

a b S N a T a b

with S a continuous t-conorm, T a continous t-norm, an N an strong negation. These
function are called Q -conditionals ( Q for Quantum). For example,

• S = max, T = min, N = N 0 ,is J 1 ( a , b ) = max ( 1 − a , min ( a , b )) , is the so-called

Early-Zadeh operator

• S = max, T = prod , N = N 0 ,is J 1 ( a , b ) = max ( 1 − a , ab )

prod ∗ , T a 2 b • S = = min, N = N 0 ,is J 1 ( a , b ) = 1 − a +

W ∗ , T • S = = W , N = N 0 ,is J 1 ( a , b ) = max ( 1 − a , b ) , that coincides with the

Kleene-Diennes implication

W ∗ , T

• S = = prod , N = N 0 ,is J 1 ( a , b ) = 1 − a + ab , that coincides with the

Reichenbach implication

W ∗ , T • S = = min, N = N 0 ,is J 1 ( a , b ) = min ( 1 , 1 − a + b ) , that coincides with

the Łukasiewicz implication

Withwhich t-norm T 0 do verify theMP inequality theseQ-operators? For instance,

• W ( a , max ( 1 − a , min ( a , b )) = max ( 0 , a + min ( a , b ) − 1 ) = W ( a , min ( a , b ))

min ( a , b ) b

• W ( a , max ( 1 − a , a · b ) = max ( 0 , a + a · b ) − 1 ) = W ( a , a · b ) a · b b

a 2 b a 2 a 2 b

• W ( a , 1 − a + ) = max ( 0 , , b ) = b

• W ( a , max ( 1 − a , b )) b (as it is proven before)

• ( , − + )

W a 1 a ab b (as it is proven before)

μ ₒ ˃ = ˃ + μ · ˃

Example 3.2.10 The protoform (coming from the Dishkant

hook), gives the D-operators:

J 2 ( a , b ) = S ( b , T ( N ( a ), N ( b ))),

with which J 2 ( N ( b ), N ( a )) = S ( N ( a ), T ( a , b )) = J 1 ( a , b ) or, equivalently,

: D-operators are the contrasymmetricals of Q-operators.
Hence, it can be repeated all that has been said for J 1 . For example,
If S

J 2 ( a , b ) = J 1 ( N ( b ), N ( a ))

= max, T = min, N = N 0 ,itis J 2 ( a , b ) = J 1 ( 1 − a , 1 − b ) =

max ( b , min ( 1 − b , 1 − a )) = max ( b , 1 − max ( a , b )) , that verifies