GFO Part I Basic Principles
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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.

3.1.1 Meta-Languages

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 $(\Lang{L},\Un{W})$ 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 $\Gden(x,y)$ relating expressions of L to the objects of Sem. The denotation relation $\Gden(x,y)$ stipulates a connection between a symbol $x$ and a semantic object $y$.

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 $C$ whose instances include all categories? In this case we say that $C$ is a meta-category, and exclude that $C$ is an instance of itself. $C$ 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 $X$ be a set of entities, then every category $C$ having exactly the entities of $X$ as its instances is called a categorial abstraction of $X$. 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 $X$ of entities is specified by a condition $C(x)$, then the expression $C(x)$ expresses a category which can be understood as a categorial abstraction of $X$.

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 $X$. 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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