SUO: Re: mathematical definitions for monocosmic and polycosmic
As an addendum:
I forgot to relate this to the colimit of a diagram of theories. The colimit
of a diagram of theories resolves into two steps, direct flow and meet. The
colimit for a diagram of theories
D : G --> |Theory|
can be expressed as the meet in the lattice of theories over the colimit of
the base language diagram base(D) of the direct information flow of D along
the colimit injections of the base language diagram
col(D) = meet(flow(D)).
Hence, D is *monocosmic* when there is a model theory below the colimit
th(M) <= col(D).
for some model M with underlying type language lang(M) = col(base(D)).
In other words, D is monocosmic when col(D) is consistent.
Also, for the definition of polycosmic we probably do not want to consider
the trivial cases when one of the theories in a diagram of theories is
inconsistent. Thus, we define a diagram of theories D to be *pointwise
consistent* when all the theories in flow(D) are consistent. Of course, a
monocosmic diagram of theories D has this property by default. Hence, we can
say that D is *polycosmic* when D is pointwise consistent, but not
monocosmic. This means that all the theories in flow(D) are strictly above
the bottom inconsistent theory, but meet(flow(D)) = col(D) is equivalent to
the bottom theory.
In summary,
D is *monocosmic* when col(D) is consistent, and
D is *polycosmic* when D is pointwise consistent, but col(D) is
inconsistent.
Robert E. Kent
rekent@ontologos.org
----- Original Message -----
From: "Robert E. Kent" <rekent@ontologos.org>
To: "SUO" <standard-upper-ontology@ieee.org>
Sent: Saturday, June 14, 2003 12:05 PM
Subject: SUO: mathematical definitions for monocosmic and polycosmic
>
> All,
>
> We can mathematically define the notions of monocosmic and polycosmic
> libraries of modules, and we do so in the following series of definitions.
>
> A library of modules is conceptually situated within the context of a
> lattice of theories (generalization/specialization hierarchy) and its
> correlated structure known as the truth concept lattice. In the IFF an
> unpopulated monolithic object-level ontology is represented as an IFF
> theory. The IFF regards a library of modules as an unpopulated modularized
> object-level ontology. In the IFF an unpopulated modularized object-level
> ontology is represented by a diagram of theories and theory morphisms.
>
> A diagram of theories and theory morphisms of shape G for some graph G is
a
> graph morphism
> D : G --> |Theory|
> from the graph G to the (set-theoretically large) graph |Theory| whose
nodes
> are theories and whose edges are theory morphisms. The diagram is
*discrete*
> when the shape graph is discrete (that is, has only nodes but no edges).
Any
> diagram of theories
> D : G --> |Theory|
> has an underlying base diagram of languages of the same shape
> base(D) : G --> |Language|,
> where the language (language morphisms) at any indexing node (edge) of
graph
> G is the underlying base language (language morphism) of the theory
(theory
> morphism) at that node (edge).
>
> It is important to note that the theories within a diagram of theories D
do
> not necessarily have the same underlying language. To semantically compare
> these theories and to conceptually situate them within the context of a
> lattice of theories, we move them to the lattice of theories over the
> colimit of the base: that is, to the lattice of theories over the language
L
> = col(base(D)). For any diagram of theories D there is a *direct
information
> flow* diagram
> flow(D) : G --> |LOT(L)|
> in the lattice of theories over the colimit language L = col(base(D)).
This
> is the direct flow along the colimit projections; flow(D) is a diagram of
> theories that all have the same underlying base language L and hence can
be
> semantically compared.
>
> The L-theory th(M) associated with a model(-theoretic structure) M having
> the underlying type language L = lang(M) consists of all L-expressions
> satisfied by the model M. A *model theory* is defined as any theory of the
> form th(M) for some model M.
>
> A diagram of theories D is *monocosmic* when there is a model M with
> underlying type language L = lang(M) = col(base(D)) which satisfies all
the
> theories in flow(D); that is, when flow(D) lies within the principle
filter
> of the model theory th(M). A diagram of theories D is *polycosmic* when it
> is not monocosmic; that is, when there is no model which satisfies all the
> theories in flow(D); that is, when there are two mutually inconsistent
> theories in flow(D). In the IFF there are some extreme polycosmic diagrams
> of theories, where any two theories are either equivalent or mutually
> inconsistent (each of these theories lies at the lowest level in the
lattice
> of theories strictly above the bottom inconsistent theory containing all
> expressions).
>
> Bottom line: we can mathematically define the notions of monocosmic and
> polycosmic; and from the metatheoretic standpoint the notion of a
polycosmic
> diagram of theories is very natural.
>
> Robert E. Kent
> rekent@ontologos.org
>
>
>
>
>