OK, let's be careful. (01)
There is a mathematical set theory called "rough set theory", that was
introduced in the 1980's (02)
PAWLAK, Z. 1982. ROUGH SETS. International Journal of Computer &
Information Sciences, 11(5), pp.341-356. (03)
and has been studied quite a bit since. Here are some more recent articles: (04)
PAWLAK, Z. and A. SKOWRON. 2007. Rudiments of rough sets. Information
Sciences, 177(1), pp.3-27.
PAWLAK, Z. and A. SKOWRON. 2007. Rough sets and Boolean reasoning.
Information Sciences, 177(1), pp.41-73.
PAWLAK, Z. and A. SKOWRON. 2007. Rough sets: Some extensions.
Information Sciences, 177(1), pp.28-40. (05)
It might be a good idea to emphasize at this point that sets and
concepts are not the same thing. A set is determined by its membership,
a concept is determined by its definition. The extension of a concept is
a set, but the same concept can have extensions with different
membership in different contexts, such as at different times (this may
be an abuse of the term "context", I'm not sure.) However, if we,
implicitly or explicitly, specify the context so that a concept has a
unique extension, then we can work with sets through their defining
Rough set theory uses the notion of "indiscernability", where there is
an attribute which can be measured to some precision or "granularity"
and this precision does not fully describe reality. The universe of
discourse is clustered into groups that indiscernible with respect to a
set of attributes.
"Crisp" sets consist of unions of these clusters. But it is also
possible to have sets, as determined by listing their membership, which
cannot be defined precisely by specifying regions in the quality spaces
of the available attributes because there are indiscernible clusters
that fall partly within and partly outside of the set. (07)
Such sets are rough sets, and they have two approximations:
an lower approximation, which consists of the greatest crisp subset,
a upper approximation, which consists of the smallest crisp superset. (08)
Given only the available attributes of an individual x, and the upper
and lower approximations of a rough set, then the question "Is x a
member of rough set B" has three possible answers, definitely yes,
definitely no, or maybe. Thus rough sets intuitively carry a 3- valued
logic, although the mathematics is more nuanced - see
POLKOWSKI, L. 2003. A note on 3-valued rough logic accepting decision
rules. Fundam. Inf., 61(1), pp.37-45. (09)
So we can also wonder about the extension of the "rough" notion to
That approach has been explored:
INUKAI, Y., A. GEHRMANN, Y. NAGAI and S. ISHIZU. 2007. ROUGH SET THEORY
USING SIMILARITY OF OBJECTS DESCRIBED BY ONTOLOGY [online]. [Accessed
Jan 13, 2011]. Available from:
Or we may ask a different but related question: Instead of an atomic
concept being a Boolean unary predicate, what if it is a 3-valued unary
predicate? I'm not sure if there are published papers addressing this
If there are other perspectives on this issue, it would be interesting
to hear about it from other list members. (013)
John F. Sowa wrote:
> We have to make a clear distinction between ontology and the tools,
> languages, logics, and reasoning methods used with any ontology.
> The subject line of this thread could be very misleading.
> Ever since Aristotle, categories and hierarchies of categories
> have been useful for ontology -- primarily because the study
> of existence leads to a study of what kinds of things exist.
> A's syllogisms and his method of definition by genus and
> differentiae have also been useful. But many people (starting
> with Aristotle himself) have noted that prototypes rather
> than strict definitions are better for some applications.
> In general, there is *no* specific logic or reasoning method
> that is either essential or irrelevant to ontology. The
> choice depends entirely on specific applications -- or even
> on very narrow questions or problems about an application.
> Re rough sets: This is an important topic, but I would be
> very cautious about any way of thinking that combines the
> word 'ontology' with any particular reasoning method.
> That is in fact why I have been unhappy with the phrase
> "Web Ontology Language" used as a scrambled acronym for OWL.
> It suggests to many novices, that the word 'the' belongs in
> front of that phrase -- but that idea is hopelessly misguided.
> Ontology is *orthogonal* to any and all versions of logic,
> reasoning methods, and implementation tools. Any specific
> language or tool tends to channel thinking into certain paths
> that may be useful for some applications. But ways of thinking
> that are specialized for one kind of application can often be
> inappropriate or even misguided for other applications.
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