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In Rocha[1997a] I have presented all the measures of uncertainty needed to deal with the
presented next. In particular I developed measures of uncertainty for both discrete and
domains. Also introduced the notion of relative uncertainty as opposed to absolute uncertainty.
former relates the uncertainty present in a given situation, to the maximum uncertainty possible
universal set. Relative uncertainty requires measures which vary between 0 and 1 for no and
uncertainty respectively. In this paper I will simply utilize these measures, please refer to
for more detailed definitions.
IVFS offer, in addition to fuzziness, a nonspecific description of membership in a set. An
IVFS A, for
each x in X, captures two forms of uncertainty:
vagueness (as in the case of normal fuzzy set) and
nonspecificity. Vagueness, or fuzziness, is absolutely specific; when we create a
fuzzy set we have
perfect knowledge of the degree to which a certain element x of X
belongs to A. In contrast, when we
create an IVFS we have nonspecific knowledge of the degree of membership; hence the
utilization of an
interval to describe the membership of x in A.
Categories are bundles of somehow, in some context, associated concepts. Cognitive agents
survive in a
particular environment by categorizing their perceptions, feelings, thoughts, and language. The
evolutionary value of categorization skills is related to the ability cognitive agents have to
and group, relevant events in their environments which may demand reactions necessary for their
survival. If organisms can map a potentially infinite number of events in their environments to a
relatively small number of categories of events demanding a particular reaction, and if this
allows them to respond effectively to relevant aspects of their environment, then only a finite
memory is necessary for an organism to respond to a potentially infinitely complex environment.
words, only through effective categorization can knowledge exist in complicated environments.
Thus, knowledge is equated with the survival of organisms capable of using memories of
processes to choose suitable actions in different environmental contexts. It is not the purpose of
article to dwell into the interesting issues of evolutionary epistemology [Campbell, 1974; Lorenz,
I simply want to start this discussion by positioning categorization not only as a very important
the survival of memory empowered organisms, but in fact, a necessary aspect of such organisms
context of natural selection. Understanding categorization as an evolutionary (control)
between a memory empowered organism and its environment, implies the understanding of
not as an observer independent mapping of real world categories into an organism's memory, but
as the organism's, embodied, thus subjective, own construction of relevant to its survival
in its environment. This is the basis for the constructivist position of systems theory and second
cybernetics [Piaget 1971; Maturana and Varela, 1987; Glanville, 1988; Von Glasersfeld, 1990;
1991], which I have discussed elsewhere [Rocha, 1996a, 1997a; Henry and Rocha, 1996].
Since effective categorization of a potentially infinitely complex environment allows an
survive with limited amounts of memory, we can also see a connection between uncertainty and
categorization. Klir  has argued that the utilization of uncertainty is an important tool to
complexity. If the embodiment of an organism allows it to recognize (construct) relevant events
environment, but if all the recognizable events are still too complex to grasp by a limited memory
the establishment of one-to-many relations between tokens of these events and the events
might be advantageous for its survival. In other words, the introduction of uncertainty may be a
for systems with a limited amount of memory, in order to maintain relevant information about
environment. Thus, it is considered important for models of human categories to capture all
forms of uncertainty.
George Lakoff  has stressed the relevance of the idea of categories as subjective
any beings doing the categorizing, and how it is at odds with the traditional objectivist scientific
paradigms. In the following, I will address the historical relation between set theory and our
understanding of categories; in particular, I will discuss what kind of extensions we need to
fuzzy sets so that they may become better tools in the modeling of subjective, uncertain,
One other characteristic of the classical view of categorization has to with an observer
epistemology, or objectivism. Cognitive categories were thought to represent objective
distinctions in the
real world. Frequently, this objectivism is linked to the way classical categories are constructed
on all-or-nothing sets of objects: "if categories are defined only by properties inherent in the
categories should be independent of the peculiarities of any beings doing the categorizing"
page 7]. I do not subscribe to this point of view; we can use classical categories both in realist or
constructivist epistemologies. The properties of classical, all-or-nothing, categories, need not be
considered inherent in the members. The question is who or what is to establish the shared
properties of a
particular category. A model, where these shared properties are regarded as observer dependent,
established in reference to the particular physiology and cognition of the agent doing the
built under a constructivist epistemology. If on the other hand, these properties are considered to
one and ultimate truth of the real world, then the aim is the definition of an objectivist model of
Any modern theory of categorization will include classical categories as a special case of a
scheme, and that does not mean some categories are objective and others are subjective. Thus,
categories have to do with an all-or-nothing description of sets, based on a list of shared
defined in some model. The chosen structure of categories and the chosen model of knowledge
representation/manipulation, which can be objectivist or constructivist, may be independent
when modeling cognitive categorization.
Eleanor Rosch [1975, 1978] proposed a theory of category prototypes in which, basically,
elements are considered better representatives of a category than others. It was also shown that
categories cannot be defined by a mere listing of properties shared by all elements. Naturally,
became candidates for the simulation of prototype categories on two counts: (i) membership
could represent the degree of prototypicality of a concept regarding a particular category; (ii) a
could also be defined as the degree to which its elements observe a number of properties, in
these properties may represent relevant characteristics of the prototype -- the element that best
represents a category. These two points are distinct. The first one does not collide with the
concept of categories because it makes no claim whatsoever on the mechanisms of creation and
manipulation of categories. It may be challenged, as I will do in the sequel, on the grounds that
due to its
simplicity, models using it must be extremely complicated. Nonetheless, it does offer the
requirement a category must observe: a group (set) of elements with varying degrees of
representativeness of the category itself.
Now, the second point goes beyond the definition of a category and enters the domain of
creation of categories. As in the classic case, categories are seen as groups of elements observing
a list of
properties, the only difference is that elements are allowed to observe these properties to a
However, the so called radial categories [Lakoff, 1987] cannot be formed by a listing of
by all its elements, even if to a degree. They refer to categories possessing a central subcategory
defined by some coherent (to a model or context) listing of properties, plus some other elements
must be learned one by one once different contexts are introduced, but which are unpredictable
core's context and its listing of shared properties(1). Thus,
the second interpretation of fuzzy sets as
categories leads fuzzy logic to a corner which renders it uninteresting to the modeling of
Since fuzzy sets, at least to a degree, can be included in objectivist or constructivist
dismissal as good models of cognitive categories has to be made on different grounds. In the
will maintain that fuzzy sets are unsatisfactory because they (i) lead to very complicated models,
not capture all forms of uncertainty necessary to model mental behavior, and (iii) leave all the
considerations of a logic of subjective belief to the larger imbedding model, which makes them
tools in true constructivist approaches. A formal extension based on evidence theory is proposed
A complex category is assumed to be formed by the connection of several other categories;
reasoning defines the sort of operations that can be used to instantiate this association. Smith and
Osherson's  results, showed that a single fuzzy connective cannot model the
association of entire
categories into more complex ones. Their analysis centered on the traditional fuzzy set
(max-min) union and intersection. They observed that max-min rules cannot account for the
degrees of elements of a complex category which may be lower than the minimum or higher than
maximum of their membership degrees in the constituent categories. Their analysis is very
regarding the full-scope of fuzzy set connectives, since we can use other operators [see Dubois
Prade, 1985], to obtain any desired value of membership in the [0, 1] interval of membership.
their basic criticism remains extremely valid: even if we find an appropriate fuzzy set connective
particular element, this connective will not yield an accurate value of membership for other
the same category. Hence, a model of cognitive categorization which uses fuzzy sets as
need several fuzzy set connectives to associate two categories into a more complex one (in the
for each element). Such model will have to define the mechanisms which choose an appropriate
connective for each element of a category. Therefore, a model of cognitive
categorization based solely on
fuzzy sets and their connectives will be very complicated and cumbersome. No single fuzzy set
connective can account for the exceptions of different contexts, thus the necessity of a complex
which recognizes these several contexts before applying a particular connective to a particular
Turksen's model simplifies the pure fuzzy set approach since we will find more categories
which can be
combined into complex categories with a single connective used for all elements of the universal
though it does not apply to all categories. The problem is that categories demand membership
which more than nonspecific can be conflicting. That is, the contextual effects may need more
interval of variance to be accurately represented. Also, even though IVFS use nonspecific
thus allowing a certain amount of contextual variance, the several contexts are not explicitly
for in the categorical representation.
A(x): X B[0,
B[0, 1] is the set of all possible bodies of evidence
(Fx, mx) on
I[0, 1]. Such bodies of evidence
are defined by a basic probability assignment mx on
I([0, 1]), for every x in X (focal elements must be
intervals). Notice that [0, 1] is an infinite, uncountable, set, while X can be
countable or uncountable.
Thus, evidence sets are set structures which provide interval degrees of membership, weighted by
probability constraint of DST. They are defined by two complementary dimensions: membership
belief. The first represents a fuzzy, nonspecific, degree of membership, and the second a
degree of belief on that membership, which introduces
conflict of evidence as several, subjectively defined,
competing membership intervals weighted by the
basic probability constraint are created.
forms of uncertainty define a 3
dimensional uncertainty space for set
structures, where crisp sets occupy the origin,
fuzzy sets the fuzziness axis, IVFS the
fuzziness-nonspecificity plane, and evidence
sets most of the rest of this space. The total
uncertainty, U, of an evidence set A is defined
U(A) = (IF(A),
IN(A),IS(A)). The three
indices of uncertainty, which vary between 0
and 1, IF (fuzziness), IN (nonspecificity), and
IS (conflict) where introduced in
Rocha[1996a, 1997a], where it was also
proven that IN and IS possess good
mathematical properties, wanted of
information measures. These indices are identified with a different way of measuring
referred to as relative uncertainty, better suited for nondiscrete domains. For a complete
these issues, please refer to [Rocha, 1996a, 1997a; Rocha et al 1996]. The uncertainty situation of
several set structures known is summarized in table I.
None of the fuzzy set and IVFS approaches to cognitive categorization consider, explicitly,
the notion of
subjectivity. This is so because fuzzy sets do not offer an explicit account of belief in evidence;
words, we have degrees of prototypicality and not judgements of degrees of
prototypicality as Eleanor
Rosch required in the previous quote. The interpretation I suggest [Rocha, 1994] for the multiple
intervals of evidence sets, in light of the problem of human categorization processes, considers
of membership Ijx, with its
correspondent evidential weight mx(
Ijx), as the representation of the
prototypicality of a particular element x of X , in category A
according to a particular perspective. In
other words, each interval Ijx
represents a particular perspective of the element x of a category
represented by an evidence set A. Thus, each element x of our evidence
set A will have its membership
varying within several intervals representing different, possibly conflicting, perspectives. An
instance, represents the case where we have a single perspective on the category in question, even
admits a nonspecific representation (an interval)(2). The
ability to maintain several of these perspectives,
which may conflict at times, in representations of categories such as evidence sets, allows a
cognitive categorization or knowledge representation to directly access particular
contexts affecting the
definition of a particular category, essential for radial categories. In other words, the several
membership of evidence sets refer to different perspectives which explicitly point to
In so doing, evidence sets facilitate the inclusion of subjectivity in models of cognitive
addition to the inclusion of the several forms of uncertainty.
"Whenever I write in this essay 'degree of support' that given evidence provides for a
proposition or the
'degree of belief' that an individual accords the proposition, I picture in my mind an act of
judgment. I do
not pretend that there exists an objective relation between given evidence and a given proposition
determines a precise numerical degree of support. Nor do I pretend that an actual human being's
mind with respect to a proposition can ever be described by a precise real number called his
belief, nor even that it can ever determine such a number. Rather, I merely suppose that an
make a judgement. Having surveyed the sometimes vague and sometimes confused
understanding that constitutes a given body of evidence, he can announce a number that
degree to which he judges that evidence to support a given proposition and, hence, the degree of
wishes to accord the proposition." [Shafer, 1976, p. 21, italics added]
Shafer's intent captured in the previous quotation seems to follow Rosch's earlier quotation in
context of cognitive categorization. The degrees of belief on which evidence theory is based do
to be objective claims about some real evidence, they are rather proposed as
judgements, formalized in
the form of a degree. Likewise, Rosch's prototypes are not assumed to be an objective grading of
concepts in a category, but rather judgements of some uncertain, highly context-dependent,
Evidence sets offer a way to model these ideas since an independent membership grading of
(concepts) in a category is offered together with an explicit formalization of the belief posited on
membership. In a sense, in evidence sets, membership in a category and judgments over
different, complementary, qualities of the same phenomenon. None of the other structures so far
presented is able to offer both this independent characterization of membership and a
judgments imposed on this membership: traditional set structures (crisp, fuzzy, or
offer only an independent degree of membership, while evidence theory by itself offers
formalization of belief which constrains the elements of a universal set with a probability
The operations of complementation, interesection, and union are the most basic connectives
in a theory
of approximate reasoning. All other connectives can be easily construed from these, therefore, I
discuss these three operators here. Naturally, complementation, intersection, and union as defined
for evidence sets, subsume, as special cases, the same operations for IVFS and fuzzy sets.
complement of an evidence set [Rocha, 1995b] is defined as the complement of each of its
focal elements with the preservation of their respective evidential strengths:
Ii and Jj are intervals of [0,1]. Their intersection is
an evidence set C(x) = A(x) SV‹t$W‹|$U…öŒª B(x), whose
intervals of membership Kk and respective basic probability
assignment mC(Kk) are defined by:
Dempster's rule of combination eliminates all focal elements which do not coincide (or
intersect) in both
bodies of evidence being combined, while the operations of section 7.1 maintain some evidential
for these, though enhancing those that do intersect.
The uncertainty decreasing operation can be used when we have coherent evidence of
combining evidence sets, and when we wish to reduce dramatically the amount of uncertainty
some simulation of human reasoning processes. In an artificial system, this operation might be
we fast decision-making processes. Say, if we possess two categories which must be combined
to make a fast decision, then uncertainty must be reduced and the most coherent result chosen.
other hand, if we do not have coherent membership evidence, or if we do not need to engage in
decision making, but instead desire to search for more conflictuous, far-fetched, associations
wildly different contexts), then the uncertainty increasing operations should be chosen. These
by enabling the maintenance of many different contexts of the same category, which may be later
reduced by the uncertainty decreasing operation, may open the way for the modeling of metaphor
models of human categorization, as previously uncorrelated contexts, considered incoherent by
Dempster's rule, may be bridged together.
Nakamura and Iwai's data-retrieval system is based on a structure with two different kinds of
Concepts (e.g. Key-words) and Properties (e.g. records like books). Each concept is associated
number of properties which may be shared with other concepts. Based on the amount of
with one another, a measure of similarity is
established between concepts (figure 3).
The inverse of similarity creates a (usually
non-Euclidean) distance measure.
Users input an initial concept of interest.
The system creates a bell-shaped fuzzy
membership function centered on this
concept and affecting all its close
neighbors. After this, the system enters an
interactive question-answering stage to try
to reduce the fuzziness of this fuzzy set
referred to as a knowledge space. Concepts
with high degree of fuzziness are selected,
and the user is asked if she is interested in
them. If the answer is "YES" another bell-shaped membership function is created over this new
and fuzzy union is performed with the previous state of the knowledge space. If the answer is
inverse of the bell shaped membership function is created, and fuzzy union is performed. After a
steps the system reaches a less fuzzy state of the knowledge space which captures the users
Associated records are then retrieved (figure 3).
extending Nakamura and Iwai's scheme to evidence sets, more forms of uncertainty can be
and several contexts can be treated simultaneously. Consider that instead of one single database
concepts and properties) we have several databases which share at least some of their concepts.
they have different properties, the similarity between concepts is different from database to
instance, key-word "fuzzy logic" will be differently related to key-word "logic" in the library
of a control engineering research laboratory or a philosophy academic department. We may
however to search for materials in both of these contexts. Evidence sets can be used to quantify
relative interest in each of these contexts (the probability restriction from DST), and the extended
approximate reasoning operations of intersection and union can be used in very much the same
the one for the fuzzy set case. Instead of reducing only the fuzziness of the knowledge space, this
aims at reducing all the three main forms of
uncertainty discussed earlier.
In figure 4 the bell shaped membership function
is depicted for two different databases with their
probabilistic weighting of 0.3 and 0.7
respectively. Notice that concepts that are close
to the central concept x in one context may not
be close in the other context, thus the
introduction of contextual conflict. Figure 4
shows the membership functions as specific and
not as interval valued, this is just to facilitate
understanding. In reality, membership functions
would be created from the 2 different database
metrics by utilizing Turksen's DNFCNF
interval-valued relationships. Such a database
retrieval system has been developed and
discussed in deatil in Rocha [1997c].
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1. An example of a radial category [after Lakoff, 1987] is the category of mother. A listing of core properties, coherent in the context of birth, would be, for instance: woman who gives birth, raises, nurtures, educates a child. However, members of the category of mother exist which do no obey such listing: adoptive mother, surrogate mother, etc. These members do not obey the entire list; however, they are elements of the category mother. They are also not random elements, but are unpredictable until a different context is introduced.
2. This idea of interpreting bodies of evidence as perspectives, spins off from a generalization of Gordon Pask's  Conversation Theory which I have proposed with the construction of a data-retrieval system [Rocha, 1991, Medina-Martins and Rocha, 1992].