Re: SUO: RE: Re: KIF & Naming Problems
Robert Kent wrote:
> It may clearer (more semantic) to define the lattice of theories as
> the concept lattice of the truth classification of a first-order
> language L, whose instances are L-structures, whose types are
> L-sentences, and whose classification relation is satisfaction. A
> formal concept in this lattice has an intent which is a closed theory
> (set of sentences) and an extent which is the collection of all models
> for that theory. The theory (intent) of the join (sup) of two concepts
> is the closure of the intersection of the theories (conceptual
> intents), and the theory (intent) of the meet (inf) of two concepts is
> the theory of the common models.
I'm not sure that this is all that much *clearer* than a lattice of
deductively closed theories, Robert, but it is certainly richer because
of the explicitly semantic element, and I like the idea a lot. One
possible technical problem is that the collection of all models of a
consistent theory (even if we just pick a single representative from
each class of isomorphic models) is a proper class. Hence, extents
cannot be formal elements of your account -- e.g., they cannot
constitute the range of a function taking formal concepts to their
extents. That might be one reason to stick with a purely proof
theoretic rendering of the lattice proper. Still, nothing prevents us
from informally associating an extent in your sense with each formal
concept, nor from talking of the inf and sup of two extents (assuming a
reasonable class theory).
Cheers,
-chris
--
Christopher Menzel # web: philebus.tamu.edu/~cmenzel
Philosophy, Texas A&M University # net: chris.menzel@tamu.edu
College Station, TX 77843-4237 # vox: (979) 845-8764