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