hidden FO MT claim (was: [SCL] 'moderately' folded semantics)

[Pat Hayes]

At 10:35 AM -0700 4/15/04, Robert E. Kent wrote:
>  > This does not extend the universe. The 'harm' is done (i.e. the MT
>>  fails to correspond to a conventional FO MT) in the case where a
>>  relation name used purely relationally forces the universe to contain
>>  entities that it would not otherwise contain, as in Horrock's example:
>>
>>  (forall (x) (= x a))
>>  (P a)
>>  (not (Q a))
>>
>>  which is FO satisfiable, and SCL satisfiable (with an identical
>>  model) but does not have a satisfying moderately folded
>>  interpretation.
>
>On second thought, do you not think that there is a "hidden claim" in the FO
>MT that there exists a class of individuals -- denote this by ' I '. Then
>the SCL text corresponding to the above FOL text might be either
>
>(forall (x I(x)) (= x a))
>(P a)
>(not (Q a))
>
>or the more complete SCL text
>
>(forall (x I(x)) (= x a))
>(P a)
>(not (Q a))
>(not (I P))
>(not (I Q))
>(not (I I))
>
>The relation ' I ' is being used to define the range of FO quantification,
>and in the translation would "guard" all quantifications.

Yes, I agree this would be intuitive, but the problem is that it 
really does not make sense when stated in this straightforward way, 
since the FO model of this latter would have a larger universe than 
the FO model of the example, which kind of therefore begs the 
question (and does not honestly overcome the Horrocks objection: 
remember that part of our aim here is to reproduce conventional FO 
interpretations unchanged, not to redefine FOL). But yes, this 
intuition is very much what I had in mind and was trying to make 
precise with the notion of a 'header' which would allow an extended 
universe of discourse and have an explicit name for the 'real' 
universe of discourse, like your I.

I now think that SCL cannot tolerate the delay involved in getting 
this worked out in detail, however, so propose to remove this from 
the draft document (which will therefore simply have optionally named 
pieces of SCL text as the topmost structure) and get that out ASAP, 
and then return to this issue in a later revision/extension. 
Unfortunately we are now well behind schedule and everyone involved 
seems to have less and less time free to spend on SCL work.

Pat



>Robert E. Kent
>rekent at ontologos.org


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