Pat Hayes wrote:
On Jan 21, 2009, at 3:03 AM, Chris Partridge wrote:
Hi Pat,
Would you mind giving us (me?) a couple of clarifications?
PH>when some basic advances in logic showed that the traditional
'layering' of descriptions into individuals/classes/properties/
metaclasses/etc. was (a) not necessary and (b) expressively very
restrictive. One can keep the categories but abandon the strict
layering - in effect, allowing a given thing to be in many 'layers' at
once - and no disasters arise, if one cleaves to a certain simple,
natural syntactic discipline (which is built into both Common Logic
and RDF). The result is greatly increased expressivity and a formalism
which 'naive' users invariably find quite natural, and which makes
perfect semantic sense.
Is there somewhere we can find more details on this 'basic advance'?
It was the development of non-well-founded set theory, written up in a
Stanford CSLI monograph by Peter Aczel. There are articles on it on
Wikipedia and other places, for a full account see
http://plato.stanford.edu/entries/nonwellfounded-set-theory/
. The key point is that set theories which do not use, and even which
explicitly deny, the axiom of foundation, are not only possible, but
can be proven to be relatively consistent with ZFC. So they are just
as 'good' as a foundation as anything else. So, there is absolutely no
reason to prohibit self-containing sets such as rdfs:Class. If you
look at classical FOL in this light, you quickly end up with Common
Logic.
I suspected you might be referring to this. I have been studying the
Aczel monograph because Jon Barwise uses it as his basis for developing
his theory of common knowledge. But....
Although they could have done, the RDF and CL model theories do not
explicitly use Aczel's set theory, essentially for pedagogical
reasons: to ask someone to swallow a new set theory is much harder
than getting them to agree on a model theory. Instead, they use an
elegant trick that I learned from Chris Menzel, of distinguishing the
class-as-individual from its set-theoretical extension, i.e. by moving
to an explicitly intensional theory of classes and relations in
general. And, it turns out, this actually has practical benefits for
knowledge modeling: intensional relations are more natural and
intuitively acceptable (and computationally more tractable) than the
classical set-theoretic, purely extensional, construction.
Can you provide a reference for this 'elegant trick'? I don't
understand how you can do the same thing with a model theory that is
done by developing set theory in the absence of the axiom. Does this
mean you use a set theory that retains the axiom of foundation, but a
model theory that does not? Also, how might this moving to an
explicitly intensional theory of classes and relations affect the use
of Chris' BORO method? Since that method is apparently based on a
'ruthlessly' extensional approach?
thanks,
John Black
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