4.1 Complement Concept and Complement Set
For an entityEcomprised of two parts,AandB, partAis commonly referred as the complement of part B. In other words, the “complement” concept is associated with two points. One is its way of composition , and the other is that two parts viabecomes one entity, i.e.,E=AB. More precisely,
Ais said to be the complement ofB with respect toE under .
The “complement” concept has been widely adopted in mathematical sys- tems. A simple example is that 6 is a complement number of 4 with respect to 10 under the addition +. Generally, we say a number 0≤n≤b is a com- plement number of 0≤m≤bwith respect tobunder the addition + if there is a unique number nsatisfying n+m= b. For simplicity, one usually says
that 6 is a complement number of 4, without explicitly mentioning 10 and +. Such simplification, however, may cause some confusion about the “comple- ment” concept. Choosing (9) as the complement set of Ain the Z-system is an example of such confusion.
For two sets,AandB, we have∪asandU asE; that is,U =A∪B. However, we are still unable to say thatA is the complement set of B with respect toU under ∪, because there are manyAthat satisfy U =A∪B for a given B. To make a uniqueA, we need to addA∩B =∅. This leads to the complement set definition given by (3). For a characteristic function by (5), the complement set definition is given by (8). It says thatµ¬A(a) is the
complement ofµA(a) with respect to the constant functionµU(a) = 1 under
the operation in (7), subject to 0 = min{µA(a), µ¬A(a)}for its uniqueness. For
the binary valued function by (5), the complementµ¬A(a) by (8) is equivalent
to the direct expression given by (9). Therefore, in such a case, either (8) or (9) can be used as the definition of the complementµ¬A(a).
For a real valued case by (10), the complementµ¬A(a) by (8) is no longer
equivalent to that given by (9). In this case, the correct way is to choose (8) to define the complementµ¬A(a). However, except for thoseµA(a) in the
degenerated case by (5), there is no solution forµ¬A(a) by (8) for a real-valued µA(a) by (10), i.e., the complement set does not exist.
4.2 Avoiding a Controversial Definition in Zadeh’s Complement Set
Unfortunately, Zadeh mistakenly selected (9) as the definition of the comple- mentµ¬A(a) in the Z-system, though theµ¬A(a) given by (9) is the comple-
ment of µA(a) with respect toµT(a) = 1 under +. In fact, the operation +
has been excluded from (7).
More precisely, 1−µA(a) can be regarded as the “mirror” or “conjugate”
function with respect toµT(a) = 1. To distinguish it fromµ¬A(a), we denote
this conjugate function as
µΞA(a) = 1−µA(a). (21)
The confusion of this concept with the commonly adopted complement con- cept discussed in Sect. 4.1 has caused bad consequences at least in two aspects. First, it produces unnecessary mistakes in applications of the Z-system. Though it maybe clear to certain senior fuzzy researchers that the “comple- ment” concept is different from the commonly adopted complement concept, it may be not clear to many new comers or those who simply apply the Z-system for practical uses. In their applications, a classical logical reasoning problem is extended into a fuzzy logic problem via simply turning a binary valued characteristic function into a fuzzy membership. This type of practice may cause mistakes and lead to unsuccessful applications, unless the original logical problem does not involve logical negation either directly or indirectly
(e.g., via the logical implicationA→B). This type of failures can be avoided by renaming the Zadeh’s “complement” definition. Also, it is a well adopted convention in the scientific community that one should not use duplicately a name or terminology, that has already been well adopted, on a different concept for unnecessary confusions.
Second, it contradicts classical logic as well as our common sense. If we follow our common sense to define the truth µT(a) by 1 or even a constant c, it follows the MAX operation by (7)(a) that c = µT(a) = µA∪¬A(a) =
max{µA(a),1−µA(a)}, which is only possible when µA(a) takes a binary
value of 0 or 1. Thus, the Z-system collapses back to classical logic. To be consistent, we can only letµT(a) = max{µA(a),1−µA(a)}, which is actually
a variable that varies between 0.5 and 1 asavaries. Similarly, it follows from (7)(b) thatµF(a) =µA∩¬A(a) = min{µA(a),1−µA(a)}, which forces us either
revert to classical logic or to accept a false membership function µF(a) =
min{µA(a),1−µA(a)} that varies between 0 and 0.5 as a varies. In other
words, a truth in the Z-system is no different to any fuzzy membership function 0.5 ≤ µA(a) ≤ 1, while a false in the Z-system is no different to any fuzzy
membership function 0≤µA(a)≤0.5.
The above confusion and conflicts incur a long-running debate in the liter- ature [3, 8]. Facing the challenges raised by critics, various kinds of reactions have arisen from the “pro-fuzzy” community. There are some “pro-fuzzy” people who worship the Z-system and simply ignore any controversy or over- react to challenges, which will not be further discussed here. There are also some researchers who attempt to resolve the controversy [1, 5]. To avoid the awkwardness, one remedy without explicitly discarding away (9) is the bold operations in (11) that modifies (7) such that 1−µA(a) becomes equivalent
to the complement set in a similar way to (8). As a result, µA∪¬A(a) = 1
andµA∩¬A(a) = 0 still hold, and the above mentioned confusion and conflicts
have been removed.
Moreover, there are also “pro-fuzzy” researchers who choose to defense the Z-system. One key argument is that the above so-called conflict repre- sents naturally a feature for handling partial membership or a multi-valued truth due to missing information. For example, one argument maybe illus- trated via a scenario that an agent knows exactly “John is 55 years old”, and is asked the question “Is John old?”. Considering partial membership, the agent’s answer may be “Well, John is a little old, but not quite so”. That is, it is not necessary to assign 1 toold(55)∨ ¬old(55). It appears naturally from (7) that µold∪¬old(55) = max{µold(55),1−µold(55)} <1 has no prob-
lem. However, there is also a hidden confusion. We fell acceptable that the value of old(55)∨ ¬old(55) is smaller than 1 does not mean that we can ac- cept µold∪¬old(55) = max{µold(55),1−µold(55)} < 1. For 0< µold(a) <1,
the value of old(55) ∨ ¬old(55) is not equal to µold∪¬old(55) = 1 with old∪¬old=U. That is, there is a confusion between the membershipµp∨q(a)
The definition µp∨q(a) = max{µp(a), µq(a)} can not automatically lead to
that the value ofp(a)∨q(a) is given by max{µp(a), µq(a)}.
Also, missing information can not be a real excuse too. It would be one if the conflicts and confusion could not be avoided by any other operation based on the same information. Actually, the bold operations in (11) refute this argument. The success of (11) on resolving the above discussed conflicts and confusion actually comes from honoring the additive principle behind both the set theory and the P-theory. Still, one may further argue that (11) brings other problems, e.g., making the idempotent law (i.e.A∪A=Aand
A∩A=A) invalid. Actually, the reason behind this problem is just what has been discussed at the end of Sect. 3.3, and can be solved if the dependence type ofPB|A(a), PA|B(a) in (16) are also considered [4]. Further ahead along this line, we are finally lead to the P-theory.
If one wants all the axioms of Boolean algebra to remain hold, the P-theory is a best choice for the development of models of uncertainty or partial truth. If one does not want to use the P-theory due to higher computation costs and requiring extra information for handlingPB|A(a), PA|B(a), one has to abandon
some axioms of Boolean algebra. Of course, depending on applications, one may choose to abandon the idempotent law or the exclude-middle law or other. However, one can not introduce a conceptual confusion by duplicately using a terminology with a well known meaning to name a new and different concept. If one abandons the complement definition 1−µA(a) or just simply
renaming 1−µA(a) by (21), everything is fine with the min–max operations,
which may be worth some further study, especially on the joint role ofµA(a)
andµΞA(a) in the Z-system.