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Overview
General Formal Ontology (GFO)
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Subsections
3.1 Meta-Languages and Meta-Categories
There are two kinds of (interrelated) meta-levels, one which is based
on the idea of meta-language and the other is founded on the notion of
meta-category. Both kinds of abstraction are discussed in the
following sections. The current document is mainly concerned with systems
of categories, which arise from the principle of categorial abstraction.
The architecture of meta-languages is elaborated upon and presented in Part II
(Axiomatics and Ontology Languages) of this report series.
Let W be a world of objects. A formal language L,
whose expressions refer to the objects in W, is called an
object-level language for W. In order to specify and
communicate the meaning of these expressions, a meta-language
M for the pair
 is required. That means,
M is a language whose
expressions refer to the items included in L or in both,
L and W, but which also refer to relations between L and
W. A formal language L has a semantics if there is a class
Sem of objects, and a relation  relating expressions
of L to the objects of Sem. The denotation relation
stipulates a connection between a symbol  and a
semantic object  .
Set theory is a convenient mathematical tool for describing and
modelling arbitrary
structures:
Moreover, set theory is intimately tied to logical languages because
the commonly accepted approach of Tarski-style
model-theoretic semantics (60) is based on
set-theoretical constructions. The
relationship between such languages and their meta-theoretical
treatment is well established. Hence, we adopt set theory as a
general and an abstract modelling tool.
3.1.2 Categorial Abstraction
The other type of meta-level is related to the notion of a meta-category,
which is a generalization of a meta-set or a meta-class in the set-theoretical
sense. Is there a category  whose instances include all categories? In this
case we say that is a meta-category, and exclude that is an instance of
itself. is then a meta-entity with respect to the next lower level of
abstraction. This principle can be expanded to arbitrary sets of
entities. Let  be a set of entities, then every category having
exactly the entities of as its instances is called a categorial
abstraction of . Usually, there can be several distinct
categorial abstractions over the same set of entities. It is an open question
whether sets of entities without any categorial abstraction exist.
If a set of entities is specified by a condition  , then the
expression expresses a category which can be understood as a
categorial abstraction of .
There are no well-established and
complete principles of categorial abstraction. Furthermore, a classification of
different types of categorial abstractions is needed. A categorial
similarity
abstraction tries to find properties that are common to all members of the
set . The specification of such categorical similarity abstractions
in a language
uses conjunctions of atomic sentences representing - in many cases -
perceptive properties. There are also disjunctive
conditions, for example the condition x is an ape or x is a bridge;
obviously, the set of instances of this condition cannot be captured by a
similarity category. More complicated are categorial abstractions over
categories, for example the category species in the field of biology. A
specification of the category species captures more complex
conditions that are common to all (concrete) species.
Robert Hoehndorf
2006-10-18
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