John, I need to get a little more background on Kripke. Is Naming and
Necessity the publication to get the right background on Kripke semantics ? (01)
John F. Sowa wrote:
> Pat and Chris,
>
> CM> Pat is careful to talk about Kripke *structures* here, as
> > distinct from Kripke *models*, which is what I thought you
> > were talking about.
>
> In my first note in this thread, I explicitly mentioned the Kripke
> triple (K,R,Phi) of the set of worlds K, the accessibility relation R,
> and the evaluation function Phi. When I spoke about a mapping from
> Kripke semantics to Dunn semantics, I meant that for every choice of
> (K,R,Phi) and for every world w in K, there is a unique pair (M,L) of
> facts and laws in Dunn's sense. I apologize if any of my statements
> weren't sufficiently qualified to be clear.
>
> JFS>> 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.
>
> CM> 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.
>
> I'm glad that you gave some examples, which on the surface seem to be
> relatively straightforward: "people, tables, and cabbages", but as we
> have seen there are hotly debated questions about when an egg becomes
> a person and where is the borderline between the species Homo sapiens
> and the genus Homo or the broader categories of hominid vs. hominin.
> Many other issues arise with birth defects, Siamese twins, people
> with no discernible brain function, etc.
>
> There are equally thorny questions about the borderline between tables
> and other flat surfaces. And the single species Brassica oleracea has
> been bred into cabbages, cauliflower, broccoli, and other varieties
> that a nonspecialist wouldn't even recognize as a member of the genus
> Brassica. When you get to relations that are commonly expressed as
> prepositions, verbs, adverbs, and abstract nouns, all bets are off.
>
> PH> Actually there seems to be a fairly robust and widely accepted
> > consensus that semantic structures can be parts of reality.
>
> CM> 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.
>
> The professor of statistics and industrial engineering, George Box,
> made the very widely quoted statement "All models are wrong, but
> some are useful." I can imagine armchair logicians and philosophers
> who might say "Semantic structures can be parts of reality", but I
> can't imagine Box or anybody who quotes him accepting that statement
> with a straight face.
>
> PH> 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.
>
> If we consider the triple (K,R,Phi), the evaluation function Phi
> is defined in terms of the syntax (or at least the abstract syntax)
> of some specific language L. That language L would be the one to
> use in order to define the mapping from a possible world w in K to
> a Dunn-style pair of laws and facts.
>
> PH> ... your construction is under-defined. You have to specify how
> > to determine the set of sentences.
>
> Given a Kripke triple (K,R,Phi), for each world w in K, the set of
> all sentences of L that are true in w constitute the facts of w:
>
> Facts of w =df {s in L | Phi(s,w)}
>
> The set of all sentences of L that are necessarily true in w
> constitute the laws of w. According to Kripke, those are the
> sentences that are true in every world w' accessible from w:
>
> Laws of w =df {s in L | forall w', R(w,w') implies Phi(s,w')}
>
> PH> ... there is a meaningful notion of relational structure
> > independent of any ontology. Kripke's own definitions are
> > independent of ontology.
>
> I agree that the notion of 'Kripke structure' is independent of
> any ontology. But any specific set of Kripke worlds K uniquely
> determines everything that exists or can exist in any world and
> all the possible relationships presupposed by K.
>
> For any world w in K, the relational structure (D,R) of w determines
> the set of true ground-level sentences about w in any given syntax.
> The collection of all those ground-level statements for every w in K
> determines all the entities that exist or can exist in any world
> of K and all the possible relationships among them.
>
> What would you call that other than an ontology?
>
> John
>
>
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> (02)
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