On Sep 16, 2008, at 4:23 PM, Rick Murphy wrote: (01)
> 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 ? (02)
No, thats on another topic (though not unrelated). The 1963 paper
cited by Chris a while back (03)
Kripke, Saul, "Semantical Considerations on Modal Logic," Acta
Philosophica Fennica, 16, (1963): 83-94 (04)
is the original, but there are many expositions available now. These
aren't bad as places to start: (05)
http://www.informatik.uni-leipzig.de/~duc/Thesis/node50.html
http://en.wikipedia.org/wiki/Kripke_semantics
http://plato.stanford.edu/archives/win2001/entries/logic-modal/ (06)
Pat (07)
>
>
> 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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