On Sep 16, 2008, at 10:30 AM, John F. Sowa wrote: (01)
> 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.
>
> 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. (02)
I'm glad you have been saying this for years, as it is true. But that
is entirely irrelevant to what we were discussing. (03)
> Those entities and relations
> are *approximate* representations of aspects of the world
> according to some ontology. (04)
Well, as an aside, they can be approximate, and they can also be quite
accurate with no approximation at all. Approximation is not inherent
or necessary. And they are not (or rather, need not be)
*representations*. But lets leave this debate aside, as we have had
it before many times. I will just note that almost all writers on
formal semantics (including Quine, Russell and Carnap, and indeed
every written authority I have ever consulted on the topic) disagrees
with you. (05)
> 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. (06)
Well, you know we disagree on this issue and have argued it over and
over in many public forums. But it is completely irrelevant to what we
had, until this message, been talking about in this thread. (07)
> We have argued about that point before, and logicians and
> philosophers have taken many different positions on it. (08)
Actually there seems to be a fairly robust and widely accepted
consensus that semantic structures can be parts of reality. (09)
>
> Although I do not believe that the real world *is* a model,
> it is at least conceivable that some people might make that
> claim. (010)
It is simply true that many people have made that claim: or more
often, made that assumption, taking it as obvious. (011)
> 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". (012)
OK, I'm happy to not argue with you about the metaphysical reality of
possible worlds. However, that has nothing to do with what we were
talking about. (013)
>
>> 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. (014)
So do I: that is how they are defined. (015)
> Those relations determine the signature and ontology. (016)
NO THEY DON'T. That is my point. As you point out above, a Kripke/
Tarski (might as well just say model-theoretic) structure consists of
entities and relationships, understood mathematically. It does NOT
determine the signature of a language. A given structure can be used
to interpret a wide range of languages, and even the same language in
a wide range of ways. So to simply speak of "the set of sentences true
in" such a structure does not specify anything. That could be just
about any set. (017)
This point has nothing to do with metaphysics or the nature of
reality. it is a boring point about the use of mathematical language
to define a mathematical construction. (018)
> 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.
>
> 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. (019)
And that is what I claim you have not demonstrated, as your
construction is under-defined. You have to specify how to determine
the set of sentences. (020)
> 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). (021)
Im not sure I follow you. These constructions - Kripke's and Dunns -
are defined purely mathematically. Chris gives an example of a Kripke
structure in an earlier message in this thread. The metaphysical
'nature' of the things called possible worlds isn't germane to
constructions like this, nor to issues of conversion between them. (022)
>
>
> If you claim that there is a meaningful notion of a Kripke model
> that is independent of any ontology, (023)
Of course there is, just as there is a meaningful notion of relational
structure independent of any ontology. Kripke's own definitions are
independent of ontology. This is standard textbook stuff. (024)
> then I'd like to know how you
> think it could be mapped to anything computable or formalizable. (025)
"Mapped to"? It is a formal description of a semantic interpretation.
Semantics isn't computable, by and large. Tarski structures need not
be computable. The link to computablity is through metatheorems like
the completeness theorem. The 'methodology' is called formal logic. (026)
Pat (027)
>
> If you can do that, I suspect that the same methodology would
> uniquely determine a Dunn-style model.
>
> John
>
>
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