[SCL] Re: Observation

[pat hayes]

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? GOFOL = Good 
Old-fashioned FOL, right?)


>
>>  BTW, all this makes me think that it might be worth stating the MT in
>>  'conventional' terms ( ie without an explicit extension mapping) but
>>  being explicit that we are using Aczel's set theory rather than Z_F:
>>  what do you think?
>
>Surely wouldn't hurt to provide it as an alternative formalization.

I was thinking more that it would be the primary one, and the 
explicit-ext-mapping version introduced only as a way of rendering 
the 'real' MT into Z-F. But I am not sure enough of my Aczelian 
ground to know if this would be exactly true. For example, the exact 
version supports intensional relational reasoning: does Aczel's set 
theory allow that? If we want to have intensional relations and need 
ext to do that, then there seems to be little utility in mapping to 
Aczel-plus-ext from Z_F-plus-ext.

Pat

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