[SCL] Re: Observation

[pat hayes]

>On Tue, Jun 03, 2003 at 03:00:51PM -0500, Pat Hayes wrote:
>>  Two other points.  Only the last one is really significant.
>>
>>  > > >> In GOFOL there aren't any relations which are individuals, by
>>  >> >> definition.
>>  >> >
>>  >> >That is false.  The domain of quantification in an first-order
>>  >> >interpretation is any nonempty set of things.  Those things might be
>>  >> >relations, either extensional or intensional.
>>  >>
>>  >> BUt the conventional MT for FOL assumes a segregated vocabulary, maps
>>  >> relation symbols to relations which are *identified* as sets of
>>  >> tuples of members of I, and quantifies over I; and is usually
>>  >> interpreted relative to Z_F set theory which has the axiom of
>>  >> foundation.
>>  >
>>  >Well, of course.  But that is neither here nor there.  Your claim was
>>  >that the extension of "Rel" would be empty.  That is false.  It's
>>  >extension would be I intersect R.
>>
>>  In a GOFOL interpretation, I intersect R *is* always empty. for the
>>  reasons outlined above. (??This seems obvious. Why are you arguing
>>  against it? Are we failing to communicate somehow? ...)
>
>I suspect so.  What I'm saying is trivially true.  Here's a simple
>example. 
>
>LEXICON
>
>Predicates: P, Q, Rel
>Indcons: P, a
>
>INTERPRETATION
>
>I = {1,2,3,4,5}
>R = {5,10,15}
>
>ext(5)  = {1,3,5)
>ext(10) = {<1,2>, <4,5>)
>ext(15) = {5}
>
>V("P") = 5
>V("Q") = 10
>V("Rel") = 15
>V("a") = 1
>
>Note the interpretation of "Rel" is I intersect R = {5}, 5 being the
>lone "reified" property or relation in this interpretation.  All I had
>in mind as the GOFOL counterpart was an App/Holds transmogrification

Ah, this is the communication breakdown. By 'GOFOL' I mean, the 
traditional FOL subset of the syntax *with the traditional FOL model 
theory applied to it*, not the holds/app translation of this 
fragment. Good Old-Fashioned First Order Logic.  What Ian (and many 
others) would like is that SCL contains a sublanguage which is 
syntactically and satisfiably indistinguishable from GOFOL. Nobody 
has ever said that SCL cannot be transmogrified into TFOL: that has 
never been the contentious issue.

>of
>the language, in which predicate-only SCL predicates would remain
>predicates and receive the extensions of their denotations as semantic
>values, whereas SCL predicates designating individuals become GOFOL
>indcons.  Thus:
>
>TRANSMOG_LEXICON
>
>Preds:  Q, Rel, Holds
>Indcons: P, a
>
>TRANSMOG_INTEPRETATION
>
>I as above; R drops away (though its overlap with I remains in I).
>
>V*("P") = 5
>V*("a") = 1
>V*("Q") = {<1,2>, <4,5>}
>V*("Rel") = {5}
>V*("Holds") = {<5,1>, <5,3>, <5,5>}
>
>So "Rel" has exactly the same extension that it had before -- a
>"reified" property that holds of 1, 3, and 5.
>
>Like I said, trivial.  Where were we miscommunicating?

See above. In the traditional FO model theory,  there is no set 
corresponding to our R: the MT maps relation symbols directly to sets 
of Ntuples over I, and relations are never reified.

Pat

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