Re: SUO: RE: Reflections on SUO-KIF Ontology
>
> On Wed, Aug 16, 2000 at 03:56:17PM -0700, David Whitten wrote:
> > ...
> > Your question did bring up one issue for me.
> > I notice that KIF, and the restricted
> > subset of KIF that is in the document at
> >
> > http://ltsc.ieee.org/suo/suo-kif.html
> >
> > does not support the counted versions
> > of thereexists. ie: there-exists-at-least
> > or there-exists-at-most or there-exists-exactly.
Chris Menzel kindly replied:
>
> All are expressible in SUO-KIF by means of existential quantification
> and identity, e.g:
> (exists-at-least-3 (?x) (P ?x))
>
> can be defined as
>
> (exists (?x1 ?x2 ?x3)
> (and (P x1?) (P ?x2) (P ?x3)
> (not (or (= ?x1 ?x2) (= ?x1 ?x3) (= ?x2 ?x3))))).
>
<SNIP>
> Such quantifiers can be defined in general for all natural numbers n by
> means of a schema (i.e., in the metalanguage).
> -chris
Thank you chris for taking the time to show me
these expansions.
I was actually thinking of the quantifiers:
(exists-at-least-n ?n (?x) (P ?x))
(exists-at-most-n ?n (?x) (P ?x))
(exists-exactly-n ?n (?x) (P ?x))
(exists-range ?n ?m (?x) (P ?x))
Which involve the ?n value as an input.
For the case of ?n == 3, it should generate expansions
similar to those you so kindly provided.
ie: (exists-at-least-n 3 (?x) (P ?x)) should have
the meaning of (exists-at-least-3 (?x) (P ?x))
Re-reading your comment, I realize you may indeed be thinking
of the same kinds of quantifiers as I was, but limiting the
statements that use them to a language above the first
language in the meta-language hierarchy.
I'm not familiar enough with Tarski's stratified metalevels,
but I know it has the levels (loosely derived from
Chapter 5 of John Sowa's KR book):
0) Object Language (L-0) has a Universe of Discourse (D-0)
but cannot refer to its own symbols or truth values of its own
statements.
1) Meta Language (L-1) is a first order language that
has a Universe of Discourse (D-1)
including D-0, plus the symbols of L-0,
the truth value statements of L-0
and the relationships between the language L-0 and and D-0.
2) Meta Meta Language (L-2) is also first order but
including a Universe of Discourse (D-2) that is
the union of L-1, L-0 and D-0.
n) the nth metalanguage is constructed by induction on n.
I assume the schema you are refering to is a statement
in language L-1. I expect that the suggested nth-subclass
and nth-domain relations are in L-1 as well.
The instance-of and subclass-of relations seem to exist
in every level, since every level includes more Classes
and Instances in the Universe of Discourse of that
level. Of course, I may be confused, and it is only
necessary that these relations occur in language L-0.
If my analysis is accurate, then it may be useful to
tag the relations in our ontology with the language
level that they occur as a part of.
David