On Jan 21, 2009, at 3:03 AM, Chris Partridge wrote: (01)
> 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'? (02)
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. (03)
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. (04)
> Agree about the advantages of de-layering, but think that in a wider
> community we need to be careful about the senses of the terms we are
> using. (05)
Agreed. (06)
>
> For example, can you clarify what is meant by 'individual' here? (07)
I use this word strictly in the logical sense, to refer to any entity
in the universe of discourse, any thing in the set that the
quantifiers are understood to range over. The word in this sense is
most emphatically not a classifier word. Any kind of thing can be an
individual in this sense. Note that the classical approach to FOL uses
it in this sense when it prohibits relations from being "individuals",
i.e. from being in the universe of quantification. (08)
> I know
> there are a range of possible senses. I assume that here it not
> individual
> in the Aristotelian sense of primary substance (09)
Indeed not. That notion does not even make sense, IMO. However, it
should be pointed out that even someone who believes that there are
'true individuals' which are intrinsically distinct from abstracta
such as predicates and relations, can still use the expressive freedom
of CL with a clear conscience, by thinking of it as a form of punning.
When a logical name aaa is used as an individual name it refers to the
individual called 'aaa', and when it is used in a relational position,
it refers to a relation, also called 'aaa'; but (from this point of
view) this is merely a pun. The CL model theory can be read this way
(which is also the way that OWL2 is constructed), provided one
remembers that equality applies to all the different 'punning' senses
of a name simultaneously. If you like, think of the name as denoting a
'bundle' consisting of an individual, a function and a relation, with
the syntactic category of each occurrence of the name determining
which part of the bundle is denoted by that particular occurrence.
Equality then applies to 'bundles'. This is an accurate description of
the machinery of the model theory, but I find its complexity ugly and
obstructive. In actual use, I find, it is much easier to just allow
names to denote things of indeterminate 'type'. One quickly gets used
to using the logic this type-free way, and it is liberating. (010)
Pat (011)
> - something that, by
> definition, cannot have members (as "one thing can be both an
> individual and
> a class (and a property) in the very same ontology").
>
> Regards,
> Chris
>
>
>
>
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