Nichael, (01)
I'm with Pat that intended models have a "deja vu" effect, and it's
unclear at first why there is any benefit to relating axioms to intended
models, when intended models are necessarily less formal than axioms.
You can't really prove things about this relation in the strict sense of
showing an axiomatic proof, because the representation of intended
models isn't itself a set of axioms. So model theory can't be a
completely rigorous methodology for determining ontological correctness. (02)
For example, the intended models you (Michael) use for PSL are notated
in a mathematical notation that is not itself an axiomitization. To me
the mathematical notations, and the even more informal pictures that go
with them (I know you draw these!), are an intermediate form between the
fully formal axiomitization and the vagua meanderings of thought. (03)
The only point of establishing the relation of the axioms to these less
formal representations is for your own assurance, and others who know
those less representations. Mathematics happens to be designed to make
it easier for people to reach agreement about it, so is a good candidate
for representing intended models. However, for people who don't know
it, it isn't much assurance that a set axioms can be mapped to it. (04)
If it happened that people found axiomitizations perfectly perspicuous,
then intended models would not be necessary. Perhaps Pat is one of
those people. For me, I find the intended models of PSL, for example,
very useful it understanding the axioms and checking that they are what
they should be. Can we measure the amount of this "assurance" provided
by a model theoretic semantics ? Not so clear how to do that. (05)
Conrad (06)
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