Many thanks for the pointers, Pat. This excellent discussion has
inspired me read these and track down the source material ! (01)
BTW - do you share any of your bookmarks on del.icio.us ? (02)
Pat Hayes wrote:
>
> On Sep 16, 2008, at 4:23 PM, Rick Murphy wrote:
>
>> 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 ?
>
> No, thats on another topic (though not unrelated). The 1963 paper cited
> by Chris a while back
>
> Kripke, Saul, "Semantical Considerations on Modal Logic," Acta
> Philosophica Fennica, 16, (1963): 83-94
>
> is the original, but there are many expositions available now. These
> aren't bad as places to start:
>
> 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/
>
> Pat
>
>
>>
>>
>> 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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