SUO: Re: Lattices, Objects, Signs
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
LOS. Note 2
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
I will try to explain my claim that a sign relational framework
is the minimal adequate framework for thinking about lattices of
models and theories with any degree of computational practicality.
One of our continuing problems -- a big share of the fuss about
intercommunicability and interoperability -- is the question of
whether and just how far two different-looking theories might
really be describing the same objective domain.
The relationship between theories and models is analogous in some
general sense to the relationship between signs and their objects.
The analogy would be tighter, or more effective in computational
terms, if we restricted the discussion to finitely axiomatizable
theories, and replace the theories with the finite axiom sets
that generate the theories proper.
At any rate, the question as to whether two theories refer to the same models
can be recognized as a question about a certain type of co-reference relation.
Indeed, answering this bigger question is a recursive process that will recur
to some degree down to questions about the co-reference relations that obtain
at the "micro-semiotic" level, to coin a phrase.
The kicker is: Co-reference relations are 3-adic relations.
They cannot be analyzed in terms of, or defined by means of,
anything less complicated than other 3-adic relations.
Thus we have the following sort of picture:
o--------------------------o---------------o--------------------------o
| Syntactic Domain 1 | Object Domain | Syntactic Domain 2 |
o--------------------------o---------------o--------------------------o
| |
| o-----------o o-----------o |
| /| s s s ... |\~~~~~~~~~~~~~o~~~~~~~~~~~~~/| s s s ... |\ |
| / o-----------o \ / \ / o-----------o \ |
| / \ / \ / \ |
| o-----------o \ / \ o-----------o \ |
| | s s s ... |~~~~~~~~\~~~~~o~~~~~~~\~~~~~~| s s s ... | \ |
| o-----------o \ \ \ o-----------o \ |
| \ \ \ \ \ \ |
| \ o-----------o \ \ \ o-----------o |
| \ | s s s ... |~~~~~~\~~~~~~~o~~~~~\~~~~~~~~| s s s ... | |
| \ o-----------o \ / \ o-----------o |
| \ / \ / \ / |
| \ o-----------o / \ / \ o-----------o / |
| \| s s s ... |/~~~~~~~~~~~~~o~~~~~~~~~~~~~\| s s s ... |/ |
| o-----------o o-----------o |
| |
o---------------------------------------------------------------------o
Figure 1. Lattice of Objects Inducing a Diversity of Sign Partitions
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o