Re: Higher Order KIF & Conceptual Graphs
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Date: Sat, 18 Dec 1993 18:22:38 +0000
To: (Fritz Lehmann),, interlingua@ISI.EDU
Subject: Re: Higher Order KIF & Conceptual Graphs
Fritz, just a few comments to keep the record straight.

>        Pat Hayes said: well, that's the price you have to pay!  (For
>completeness, that is.)  The notion of non-standard numbers lying
>"outside" the real number line is so peculiar to most people, and to
>most potential users of a Knowledge Interchage language, that I think
>users who are aware of the issue will simply refuse to go along with a
>merely First-Order semantics.  

That "outside" is rather rhetorical. Nonstandard numbers don't lie off to
the side somewhere. All this talk of 'nonstandard' means, is that one's
naive intuition about the integers (which seems to be so robust) cannot in
fact be fully captured in a computably effective framework. This was
philosphically surprising and may be of concern to those interested in the
foundations of mathematics, but it is not likely to matter much to most
users of a KI language.

>Worse, the other users will ignore it,
>only to have nonstandard numbers pop up unexpectedly during attempted
>mathematical inferences.  I Imagine their baffled expressions: "HUH?..."

And now this is silly. They won't 'pop up' at all: you wil get the very
same inferences with either interpretation. But suppose you argue that the
HO quantifiers really do pin down the notion of finiteness, then I'll just
claim you're mistaken. The consequence of accepting incompleteness is that
you will have to produce an *uncomputable* set of sentences in order to
make your case. I rest mine.
>        Even without explicit, dynamic revision of the ontology, many
>(most) static ontologies will be at least partly approximative; that is,
>there is a static underlying _assumption_ that some knowledge _could_ be
>further refined with more information (one kind of revision).  Zdzislaw
>Pawlak's Rough Sets, Jerry Hobbs' granularity theory, Antti Hautamaki's
>Point-of-View Logic, Dana Scott's approximation lattices, and common-
>sense predicate systems for the real world are examples.  Such theories
>do need to allow for undefinable properties, particularly in defining
>identity of individuals --- even properties like Pat Hayes' example
>VERY-BIG.  I may be unable to define VERY-BIG, but if Tweedledum is just
>like Tweedledee, except one might be VERY-BIG, I don't want to have to
>identify them as one individual.  (Parker-Rhodes' theory of pure
>indistinguishables is a striking consequence of assuming the opposite --
>omniscience -- in the sub-physical world.)

Of course ontologies will usually be approximative in this sense. The point
of the VERY-BIG example was not to deny this, but to illustrate the
emptiness of claiming that a symbol had a meaning simply by virtue of
claiming it had it. (Incidentally, the issue about VERY-BIG is whether it
is a predicate in any ordinary sense at all, not what its definition might

Notice also that 'undefined' properties in your usage here are not simply
properties whose definition we have not yet got around to setting down in
detail, but properties whose definition *could never* be set down in the
entire history of a steady-state universe; properties which are
*unwritable*. If I decide to place these in the same category as ghosts and
goblins, I can be sure they are not going to pop up unexpectedly.

Your worries about identity are easily resolved. If Tweedeldum and
Tweedledee are not identical, it follows that there is some property which
one has and the other lacks. I can accept this mild conclusion without
feeling obliged to consider that this property might be forever beyond the
descriptive powers of my language, especially since I am quite willing in
practice to enrich it with extra vocabulary in the way suggested by Norman

Pat Hayes

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