[SCL] Re: Observation

[Chris Menzel]

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 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?

-chris