On Tue, 16 Sep 2008, John F. Sowa wrote:
> Pat,
>
> On that point I certainly agree with you:
>
> > Your construction takes a Kripke structure - no mention there
> > of a language or a signature - and begins by mentioning a set
> > of [sentences] .. And my point is simply that no such set is
> > defined by the Kripke structure. You still have not defined it.
> > Until you do, your construction isn't well-defined. (01)
Pat is careful to talk about Kripke *structures* here, as distinct from
Kripke *models*, which is what I thought you were talking about. As Pat
notes, a Kripke structure makes no mention of a language. A Kripke
*model* is a Kripke structure *plus* a mapping from a given language to
appropriate elements of the structure. These two notions are sometimes
conflated and folks can end up talking past one another. (02)
> That is closely related to something that I have been saying for
> years: A Tarski-style model is a set of entities and a set of
> relations among those entities. Those entities and relations are
> *approximate* representations of aspects of the world according to
> some ontology. (03)
Sure. Every model leaves out information that is found in the piece of
the world that the model represents. I've heard you make a stronger
claim, though, namely that Tarski-style models can't contain real world
objects like people, tables, cabbages, etc. That is completely false. (04)
> But they do not have an existence in the world that is independent of
> the ontology that we use to characterize them and identify particular
> instances. (05)
Well, yeah, I mean, you do have to have people, tables, and cabbages in
your ontology to include them in the domain of a model. You seem to be
wanting to say something deeper, but I'm not seeing it. (06)
> We have argued about that point before, and logicians and philosophers
> have taken many different positions on it. Although I do not believe
> that the real world *is* a model, it is at least conceivable that some
> people might make that claim. (07)
It might be conceivable, but I don't know how anyone could *plausibly*
make that claim. By definition a model is a mathematical entity of a
certain sort, usually an ordered n-tuple of some ilk. The real world,
whatever it is, isn't an n-tuple. (08)
> But to claim that some imaginary possible worlds have an objective
> existence that can determine truth or falsity in our formalisms is
> beyond my "Will to Believe". (09)
Whoa, complete change of subject there... (010)
> > What is the signature of the language? Does it have binary
> > relations in it? Etc. . It's not enough to say 'use FOL' or
> > some such reply. That does not determine a set of sentences.
>
> My answer is that I think of Kripke-style possible worlds as
> a collection of Tarski-style sets of entities and relations. (011)
That is a reasonable, if somewhat informal, way to think of them. (012)
> Those relations determine the signature and ontology. (013)
Hang on -- as noted already, in Kripke *structures* there are no
languages involved at all. And when you do add a language, there is
nothing about the Kripke structure that determines anything about the
its signature. (014)
> I'll confess that if you think those possible worlds have an
> independent existence in a Platonic heaven, I cannot imagine any
> formalizable way to map them into a Dunn-style model. (015)
No one has made any such claim. The discussion here about converting
between Dunn and Kripke models has been entirely mathematical in nature. (016)
> So my claim reduces to this:
>
> 1. A Kripke-style model in which each "possible world" is
> a Tarski-style model (with a set of entities D and a
> set of relations R over D) can be mapped to a unique
> Dunn model. (017)
I gave what I believe is a simple counterexample to this claim. Did you
miss that post? (018)
> 2. But I do not know how to define or even think coherently
> about a Kripke-style model that consists of possible worlds
> that have an objective existence independent of any particular
> ontology that would determine the Tarski-style pairs (D,R). (019)
There is no problem at all if you accept the objective existence of
possible worlds. Of course, perhaps you don't. (I don't, except as
mathematical entities of a certain sort that don't really deserve the
name.) (020)
> If you claim that there is a meaningful notion of a Kripke model that
> is independent of any ontology, then I'd like to know how you think it
> could be mapped to anything computable or formalizable. (021)
The notion of a Kripke model requires only that there be sets. But any
additional entities you add to your ontology beyond sets can themselves
in the domains of those models. (022)
-chris (023)
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