Wladyslaw Homenda
Faculty of Mathematics and Information Science
Warsaw University of Technology, pl. Politechniki 1, 00-661 Warsaw, Poland
Decision making, fuzzy sets, balanced fuzzy sets, negative information.
In the paper aspects of negative information and of information symmetry in context of uncertain information
processing is considered. Both aspects are presented in frames of fuzzy sets theory involved in data aggrega-
tion and decision making process. Asymmetry of classical fuzziness and its orientation to positive information
are pointed out. The direct dependence of symmetry of uncertain information on negative information mainte-
nance is indicated. The symmetrical, so called balanced, extension of classical fuzzy sets integrating positive
and negative information an paralleling positiveness/negativeness with symmetry of fuzziness is presented.
Balanced counterparts of classical fuzzy connectives are introduced.
In the paper a discussion on aspects of negative infor-
mation and information symmetry is presented. The
discussion is based on an observation of asymmetry
of operators in classical theories of uncertain infor-
mation and in theories with focus turned on fuzzy
sets. Preliminary discussion on asymmetry of clas-
sical fuzzy sets is presented in Section 2. It shows
an inclination of classical fuzzy connectives to posi-
tive information and incompatibility with negative in-
formation. Issues related to negative information are
outlined in Section 3. Assumed symmetry of negative
and positive information integrates both types of in-
formation. The integrated approach to parallelism of
negativeness/positiveness and symmetry of informa-
tion is introduced in Section 4. The integration of neg-
ativeness and symmetry is inherently drawn in an idea
of so called balanced extension of fuzzy sets. This
idea was introduced in (Homenda, 2001) and then
discussed in several papers (Homenda, 2004; Home-
nda, 2003; Homenda and Pedrycz, 2002). Finally, an
application of the classical approach to uncertainty
versus its balanced model is compared in the exam-
ple 4.1. The example outlines importance of negative
information in decision making process in real envi-
Fuzzy set theory is often used to partition a universe
into two subsets if partition criteria are not crisp. If
partition criterion is uncertain, definition of subsets as
fuzzy sets over the universe is a natural way to model
uncertainty. However, neither crisp, nor fuzzy mod-
elling avoids problems with law of excluded middle.
Partitioning the universe into two complementary
sets suggests comparable significance of both sets un-
less additional principle is given. In such partition-
ing elements of the universe can be classified as true
and false, like and dislike, good and bad, etc. with-
out any emotional evaluation of these terms. We will
simply talk about positive and negative information,
again - without emotional evaluation of both terms.
Using several criteria in partitioning we may classify
elements of the universe with regard to every criterion
separately. Having a number of pairs of complemen-
tary sets it is necessary to aggregate these results in
order to get final two sets separation of the universe.
Using classical aggregators we can choose between
all good criteria or one good criterion. The first one,
where the elements classified as good one must have
all criteria good, is implemented by conjunction. The
second one, where the element classified as good can
have only one good criterion, is implemented by dis-
Homenda W. (2007).
In Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics, pages 40-46
junction. Both aggregation connectives, conjunction
and disjunction, raise clear asymmetry under comple-
ment. If elements of one set are classified as having
all criteria good, elements of the complementary set
must have at least one criterion bad instead of ex-
pected the same condition of all criteria bad. Keep-
ing the same condition (either all criteria, or at least
one criterion) in definition of both sets raises troubles
with law of excluded middle mentioned above. Fol-
lowing this way of thinking we need other connec-
tives that will balance aggregation of decisions based
on singular criterion. The above discussion leads to
the conclusion that classical fuzzy set theory is asym-
metrical with regard to processing opposite values of
given attributes.
2.1 Symmetrization of the Scale
A classical fuzzy set A in the universe X can be de-
fined in terms of its membership function µ : X
[0,1], where the value 0 means exclusion of the el-
ement from the set while the values greater than 0 ex-
press the grade of inclusion of the element into the
set. However, membership function does not define a
grade of exclusion, the grade of negative information.
Therefore, fuzzy sets theory distinguishes grades of
inclusion and reserves only one value - 0 - for exclu-
sion. This raises asymmetry of this interpretation.
Membership function defines fuzzy connectives:
union, intersection and complement. The definitions
are expressed by max, min and complement to 1, i.e.
d(x,y) = max{x,y}, c(x,y) = min{x,y} and n(x) =
1 x. Classical connectives are asymmetrical. Union
gets its value from the greater argument, despite of
the values of both arguments. Similarly, intersection
gets its value from the smallers argument only.
We can split values of a given criterion in the spirit
of good and bad allocating the values of the interval
[0,0.5) as pieces of negative information relevant to
bad values and the values of the interval (0.5.1] as
pieces of positive information relevant to good val-
ues. The value 0.5, the center of the unit interval
[0,1], is a numerical representation of the state of no
negative/positive information. Being compatible with
common meaning of membership function let us as-
sume that the greater the value of positive informa-
tion, the stronger the good value of the criterion. By
symmetry, the smaller the value of negative informa-
tion, the stronger the bad value of the criterion.
This interpretation is well-matched with the com-
mon sense of ordering of the negative/positive values.
The ordering could be seen as monotonicity of nega-
tive/positive information mapping: it starts from the
left end of the unit interval representing strong nega-
tive information, then goes toward middle of the unit
interval diminishing strength of negative information,
then crosses the middle point of the unit interval and
then goes towards the right end of the unit interval
increasing strength of positive information.
This interpretation is also well-matched with the
common sense of symmetry of the negative/positive
values with the symmetry center in the value 0.5. The
linear transformation f(x) = 2x 1 of the unit inter-
val [0,1] into the symmetrical interval [1,1] points
out the symmetry. In this transformation negative in-
formation is mapped to the interval [1,0), positive
information - to the interval (0, 1] and the state of no
information - to the value 0.
2.2 Connectives Asymmetry
The classical fuzzy connectives stay asymmetrical
even with the symmetrical bipolar scale of the inter-
val [1,1] applied. Both classical fuzzy connectives
get their values from the maximal argument (union,
maximum) and the minimal argument (intersection,
minimum). Classical fuzzy connectives were gen-
eralized to triangular norms: maximum is an exam-
ple of t-conorms, minimum is an example of t-norm,
c.f. (Schweizer and Sklar, 1983). Strong t-norms
and t-conorms, the special cases of triangular norms,
have an interesting property: if both arguments are
greater than 0 and smaller than 1, the result of strong
t-conorm exceeds the greater argument while the re-
sult of strong t-norm is less than smaller argument,
c.f. (Klement et al., 2000). This property might be in-
terpreted that union tends to positive information de-
spite of the values of its arguments while intersection
tends to negative information despite of the values of
its arguments. In other words, symmetrical interpre-
tation of the unipolar scale makes that strong t-norm
increases certainty of negative information and de-
creases certainty of positive information. And vice
versa, strong t-conorm decreases certainty of negative
information and increases certainty of positive infor-
mation. This observation emphasizes the asymmetry
of fuzzy connectives, c.f. Figure 1.
The problem of asymmetry of fuzzy connec-
tives was discussed in number of papers, e.g. (De-
tyniecki and Bouchon-Meunier, 2000b; Homenda and
Pedrycz, 1991; Homenda and Pedrycz, 2002; Silvert,
1979; Yager, 1988; Yager, 1993; Zhang W. R., 1989).
In these papers discussion on asymmetry of fuzzy sets
and uncertain information processing was undertaken
for different reasons, though common conclusions led
to importance of the symmetry problem in fuzziness
and uncertainty.
Figure 1: Asymmetry of classical triangular norms.
The mapping of negative and positive information in
the scale of unit interval [0, 1] as well as in the sym-
metrical interval [1,1] bring incompatibility with
connectives, so the question is raised if negative in-
formation can be considered as a subject of uncer-
tainty. The question seems justified since negative in-
formation is hardly interpretable in classical set the-
ory and classical fuzzy sets theory. However, nega-
tive information, as explained in the introductory re-
marks to this section, play important role in different
fields. From psychological studies it is known that
human beings convey symmetry in their behavior, c.f.
(Grabisch M., 2002). One can be faced with positive
(gain, satisfaction, etc.) or negative (loss, dissatisfac-
tion, etc.) quantities, but also with a kind of disin-
terest (does not matter, not interested in, etc.). For
instance, one either likes to listen to the music while
reading an interesting novel or does not like to listen
to the music then or even music is only a background
not affecting him at all. These quantities could be in-
terpreted in context of positive/negative/neutral infor-
mation. On the other hand, in economy psychologi-
cal attempt to decision making process with uncertain
premises overheads traditional models of customers
behavior. The pseudocertainty effect is a concept
from prospect theory. It refers to people’s tendency
to make risk-averse choices if the expected outcome
is positive, but risk-seeking choices to avoid nega-
tive outcomes. Their choices can be affected by sim-
ply reframing the descriptions of the outcomes with-
out changing the actual utility, c.f. (Kahneman and
Tversky, 2004). Aggregation of positive and negative
premises leads to implementation of a crisp decision.
Modelling of such an attempt requires processing of
positive/neutral/negative information.
An interesting contribution to positive/negative in-
formation maintaining could be found in the theory of
intuitionistic fuzzy sets (Atanassov, 1986) and in very
similar theory of vague sets (Gau and Buehrer, 1993).
Another approach to positive/negative information is
discussed in twofold fuzzy sets, c.f. (Dubois and
Prade, 1983). In these theories, uncertain informa-
tion is represented as a pair of positive/negative com-
ponents numerically described by membership values
from the unit interval [0,1]. Both components are tied
with degree of indeterminacy which stays that sum
of membership values of both components cannot ex-
ceed the value 1. However, no tool to combine both
components is provided in these theories. Since infor-
mation aggregation leading to non ambiguous result is
a clue issue in decision making process, these theories
must be supported by information aggregators in such
a process.
The very early medical expert system MYCIN,
c.f. (Buchanan and Shortliffe, 1984), combine posi-
tive and negative information by somewhat ad hock
invented aggregation operator. In (Detyniecki, 2000)
it was shown that MYCIN aggregation operator is
a particular case in a formal study on aggregation
of truth and falsity values, c.f. (Detyniecki and
Bouchon-Meunier, 2000a) for further discussion on
aggregation of positive and negative information.
Having many premisses, usually uncertain, we
need to produce nonambiguous information that
yields a unique decision. Therefore aggregation of in-
formation is crucial in decision making process. The
topic of information aggregating has been studied in
number of papers. An interesting considerations on
information aggregation could be found in - for in-
stance - (Calvo T., 2001; Detyniecki and Bouchon-
Meunier, 2000a; Silvert, 1979; Ovchinnikov, 1998;
Yager and Rybalov, 1998; Yager and Rybalov, 1996;
Zhang W. R., 1989).
Fuzzy connectives stay asymmetrical with sym-
metrized scale. The incompatibility of symmetrical
interpretation of the scale and asymmetrical behav-
ior of fuzzy connectives suggest incorrectness of scale
symmetrization. This discussion leads to the hypoth-
esis that Zadeh’s extension of crisp sets to fuzzy sets,
c.f. (Zadeh, 1965), relied on dispersion of positive in-
formation of the crisp point {1} into the interval (0,1].
However, negative information of the point {0} was
still bunched in this point, c.f. Figure 2. This hy-
pothesis can be supported by similarity of balanced
Figure 2: Extension of fuzzy sets to balanced fuzzy sets.
triangular norms and uninorms and nullnorms - the
products of different approaches to fuzzy connectives
extension, c.f. (Homenda, 2003)
4.1 Balanced Symmetrization of the
Now, the process of information dispersion is applied
again to information concentrated in the point {0}.
This operation extends classical fuzzy sets to bal-
anced fuzzy sets, c.f. (Homenda, 2004; Homenda,
2003). The extension is being done by dispersion
of crisp negative information bunched in the point
{0} into the interval [1.0) without affecting clas-
sical fuzzy sets based on the unit interval [0,1] c.f.
Figure 2. Thus, classical fuzzy sets will be immersed
in a new space of balanced fuzzy sets. Since both
types of information - positive and negative - are as-
sumed to be equally important, it would be reasonable
to expect that such an extension will provide a kind of
symmetry of positive/negative information.
Concluding, the following symmetry principle can
be formulated: the extension of fuzzy sets to balanced
fuzzy sets relies on spreading negative information
(information about exclusion) that fit the crisp point
{0} of fuzzy set into the interval [1, 0). The ex-
tension will preserve properties of classical operators
for positive information. It will provide the symme-
try of positive/negative information with the center of
symmetry placed in the point 0, c.f. Figure 2. It is
worth to underline that this operation is entirely dif-
ferent than simple linear rescaling of the unit inter-
val [0,1] into the interval [1,1]. The linear function
f(x) = 2x 1 is replaced by the transformation that is
not a function: it allocates the whole interval [1,0)
as a ”value” in the point 0.
Figure 3: Balanced extension of fuzzy operators.
4.2 Symmetry of Balanced Connectives
Triangular norms generalize the concept of set op-
erations: union and intersection, c.f. (Schweizer
and Sklar, 1983). Triangular norms, t-norms and t-
conorms, together with negation, the basic fuzzy con-
nectives are the subject of the discussion of connec-
tives symmetrization.
Definition 4.1 Triangular norms: t-norm t and t-
conorm s, are mappings t : [0,1] × [0,1] [0, 1] and
s : [0, 1] × [0,1] [0,1] satisfying the following ax-
1. t(a,t(b,c)) = t(t(a,b),c)
s(a,s(b,c)) = s(s(a, b), c) associativity
2. t(a,b) = t(b, a)
s(a,b) = s(b, a) commutativity
3. t(a,b) t(c, d) if a c & b d
s(a,b) s(c, d) if a c & b d monotonicity
4. t(1,a) = a for a [0,1] boundary
s(0,a) = a for a [0,1] conditions
t-norms and t-conorms are dual operations in the
sense that for any given t-norm t and given negation
operator assumed here to be complement to one, we
have the dual t-conorm s defined by the De Morgan
formula s(a,b) = 1t(1 a,1 b). And vice-versa,
for any given t-conorm s, we have the dual t-norm t
defined by the De Morgan formula t(a,b) = 1s(1
a,1 b). Duality of triangular norms causes duality
of their properties. Note that the max/min is a pair of
dual t-norm and t-conorm.
The idea of balanced extension of classical fuzzy
connectives must be compatible with the concept of
balanced extension of the unipolar scale and with the
symmetry principle formulated in Section 4.1. This
requirements and symmetry of the balanced fuzzy
scale of the interval [1,1] determines the domain of
symmetrized balanced connectives to be the square
[1,1] × [1,1]. Preservation of classical fuzzy sets
properties requires preservation of properties of clas-
sical fuzzy connectives on the unit square [0,1] ×
[0,1]. Conversely, expected symmetry of positive and
negative information puts strict restrictions on bal-
anced extension on the square [1,0] × [1,0]. The
same factors determine the co-domain of symmetrical
fuzzy connectives to the interval [1,1]. This idea of
the balanced extension of classical fuzzy connectives
is outlined in Figure 3. It is clear that balanced con-
nectives are simple reflection of respective classical
connectives on the square [1,0] × [1,0]. The re-
maining parts of the domain of balanced connectives
are not explicitly constrained. However, some con-
strains will be put when other properties of connec-
tives are considered. These properties come from nat-
ural extension of the axioms of the triangular norms
definition onto the whole domain of balanced opera-
This discussion leads to the definition of balanced
negation, balanced t-norms and balanced t-conorms.
Definition 4.2 of balanced connectives.
The mapping N : [1,1] [1, 1], N(x) = x is the
balanced negation.
The mappings T : [1,1] × [1,1] [1,1] and
S : [1, 1] × [1, 1] [1,1] are balanced t-norm
and balanced t-conorm, respectively, assuming that
they satisfy the following axioms in the whole domain
[1,1] × [1,1] unless defined explicitly:
1., 2., 3.
associativity, commutativity and monotonicity
4. T(1, a) = a for a [0, 1] boundary
S(0,a) = a for a [0,1] conditions
5. T(x, y) = N(T(N(x), N(y)))
S(x,y) = N(S(N(x),N(y))) symmetry
Conclusion 4.1 The definitions of balanced t-norm
and balanced t-conorm restricted to the unit square
[0,1] × [0,1] are equivalent to the classical t-norm
and classical t-conorm, respectively.
Conclusion 4.2 Balanced t-norm and balanced t-
conorm restricted to the square [1, 0] × [1, 0] are
isomorphic with the classical t-conorm and classical
t-norm, respectively.
Conclusion 4.3 Balanced t-norm vanishes on the
squares [1,0] × [0,1] and [0, 1] × [1, 0].
Example 4.1 Let us consider the strong t-norm gen-
erated by the additive generator f(x) = (1x)/x and
the strong t-conorm generated by the additive gener-
ator f(x) = x/(1 |x|), c.f. (Klement et al., 2000;
Schweizer and Sklar, 1983). The formula p(x, y) =
( f(x) + f(y)) defines the respective strong trian-
gular norms. The extension of this t-norm to balanced
Figure 4: Balanced t-conorm and balanced t-norm.
t-norm comes from the monotonicity and symmetry
axioms directly. Alternatively, the additive generator
f(x) = (1 x)/x of this classical t-norm could be ex-
tended to the interval [1, 1] with the formula f(x) =
(1 |x|)/x. Of course, this function is undefined in
the point 0. The formula p(x,y) = f
( f(x) + f(y))
defines the balanced t-norm for both arguments being
nonnegative or nonpositive.
The balanced counterpart of the strong t-conorm
generated by the additive generator f(x) = x/(1
|x|) is determined in the squares [0,1] × [0, 1] and
[1,0] × [1, 0]. The values in remaining parts of
the domain are unconstrained besides that they must
satisfy axioms of the definition. In this case we can
extend the additive generator to the whole interval
[1,1] assuming that in points 1 and 1 it gets the
values and , respectively. The balanced t-
conorm could be defined by the formula S(x,y) =
( f(x)+ f(y)) in its whole domain [1,1]×[1,1]
except the points (1,1) and (1,1), where this bal-
anced t-conorm is undefined, c.f. (Homenda, 2003;
Klement et al., 2000; Schweizer and Sklar, 1983) for
details. Contour graphs of these two balanced norms
are presented in Figure 4.
Let us consider simple decision making process in
real economical environment, i.e. with uncertain
premises. Assume that in the first set of six premises
ve have the numerical value 0.6 and the last one has
the value 0.4. In the second set of six premises the
one has numerical value 0.6 and five other have nu-
merical values 0.4. Classical fuzzy connectives em-
ployed as aggregators of premises do not distinguish
between these two sets. Max and min operators pro-
duce the same values for both sets: 0.6 and 0.4, re-
spectively. Employing the strong t-conorm based on
the additive operator f(x) = x/(1 x) we get the val-
ues 0.89 and 0.83 for both sets, respectively. In the
case of dual t-norm we get the respective values equal
to 0.17 and 0.11. The linear mapping f (x) = 2x 1
to the interval [1,1] gives the transformed values of
premises equal to 0.2 and 0.2 instead of 0.6 and
0.4. Classical triangular norms produce the following
the respective values: 0.78, 0.66 and 0.66, 0.78.
Therefore, besides small quantitative differences be-
tween aggregated numerical results no qualitative in-
dication is given with regard to the decision. Employ-
ing balanced modelling based on the additive opera-
tor f (x) = x/(1|x|) balanced t-conorm produces the
values 0.50 and 0.50 for both sets of premises, re-
spectively. Dual balanced t-norm produces the values
0.06 and 0.06, respectively. In the case of balanced
modelling clear indication is given with regard to the
The balanced extension of fuzzy sets discussed in
this paper is a contribution to the discussion on sub-
jects of negative information and symmetry of nega-
tive/positive types of information. These aspects of
information processing, though controversial in clas-
sical and traditional fields of information processing,
become useful and necessary in some important areas
of research and practice, as indicated in Section 3 and
have been studied in number of papers.
Negative and positive types information play im-
portant role in information aggregation. Multicriteria
decision making process could be seen as a kind of in-
formation aggregation leading to a synthetical result
applicable in unique choice between given options.
The synthesis must consider pros and contras a deci-
sion, must consider positive and negative premisses of
the decision. The concept of balanced fuzzy sets deals
with positiveness and negativeness assuming symme-
try of both types of information.
In (Homenda and Pedrycz, 2005) the concept of
negativeness and symmetry was applied in construc-
tion of the balanced computing unit, a variation of
fuzzy neuron. The concept of balanced computing
unit involves a generalization of balanced t-norms, so
called t-norms in weak form. The balanced comput-
ing unit based on weak form of fuzzy connectives may
exemplify decision making process with positive and
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