Cathy, (01)
In effect, that's exactly what I would say any truly general
upper ontology should be: (02)
CL> ... one might substitute a mapping between upper ontologies
> for an upper ontology proper. (03)
In fact, let's just call it a hierarchy of names (HoN). It would
contain all the names of all the types and relations that occur
in any theory of the lattice. But HoN itself could be considered
as a theory with a very large number of names and very few
axioms. In that sense, HoN would also occur in the lattice. (04)
CL> One thing my time working on CYC really brought home to me was
> what a holistic entity such an elaborate and richly axiomatised
> ontology is (by which I mean that a small change, such as a new
> assertion, particularly at the upper level, has the potential
> to alter any number of inferences in previously unanticipated
> ways). (05)
That is exactly why I don't want to have detailed axioms in HoN.
They inevitably create unexpected and confusing implications
that cause the entire ontology to be fragile. That's the point
of my motto about axioms in HoN: When in doubt, leave it out. (06)
CL> Thus I don’t see how any kind of ‘mapping’ is going to account
> for the semantics of a system of such complexity without
> reproducing some kind of structure of equal complexity. (07)
I definitely would not want HoN to have anything as complex as
the upper levels of Cyc, SUMO, DOLCE, or many others. I want
to keep HoN simple by leaving out almost all the axioms except
for type-subtype relations that are non-controversial. (08)
Since the lattice would contain all possible theories that use
names in HoN (and possibly additional names that are not in HoN),
all the conventional upper-level-style ontologies -- Cyc, SUMO,
DOLCE, etc. -- would be in the lattice of theories. HoN would
be much simpler than any of them. (If any of them use names
that conflict with HoN, then some relabeling would be necessary.
For example, the HoN process might map to a supertype of CycProcess
or SumoProcess, etc.) (09)
HoN would only impose the following constraint: (010)
If name N1 is a supertype of N2 in HoN, then for any theory T
in the lattice in which both N1 and N2 occur, the following
assertion must be an axiom or a theorem of T: (011)
For all ...x, N2(...x) implies N1(...x). (012)
The notation ...x is the representation for sequence markers
in Common Logic. It represents an arbitrary sequence of n
variables, where n=0, 1, 2, ... This would allow HoN to
contain a hierarchy of all the type and relation names that
may be used in any of the theories. In other words, HoN
would contain subhierarchies for relations with any number
of arguments (including propositional constants, if n=0). (013)
CL> I don’t see how you can simultaneously claim to be taking
> on board Quine’s criticisms of Carnap (i.e. effectively
> destroying the concept of an analytic truth) AND claim that
> you are committed to upper ontologies which contain only
> analytic truths. (014)
Carnap was very careful in *not* defining the notion of
analyticity for natural languages. He always said "analytic
in L" where L was some formal language. Quine did not refer
to Carnap in his essay "Two Dogmas". Instead, he showed
that it was not possible to define a notion of analyticity
that covered natural languages. (015)
Although I would like to use HoN as an intermediate stage in
the mapping from natural languages to logic, I would put HoN
on the logic side of the fence -- it would, in effect, be a
very general language L, which would contain analytic statements
about the way the names were used in the other theories of the
lattice. (016)
CL> Give me an example of something you _would_ put in there (017)
For starters, there would be the label Entity as a supertype
of all types. And for any n=0, 1, 2,..., there would be a
label Relation-n as a supertype of all relations with n arguments.
(Since types are defined by monadic relations, there would be
the axiom Entity < Relation-1 -- that leaves open the question
of whether some monadic relations do not represent types.) (018)
I'd also have the type LivingOrganism as a subtype of Entity,
and with the axioms (or theorems) Plant < LivingOrganism,
Animal < LivingOrganism, HumanBeing < Animal, etc. (019)
But I'd leave out any claim about the taxonomy of living beings
that was in any way controversial or undecided. For example,
the theory of cladistics would say that since birds and
mammals evolved from reptiles, one could say that birds and
mammals are special cases of reptiles. However, that is not
a general assumption about the notion of reptile. Therefore,
I would not put it in HoN, but it could be included in some
more specialized theory. (020)
This last point illustrates the differences between Quine and
Carnap. It vindicates Quine's position that no sharp line can
be drawn between the set of analytic and synthetic statements
in any natural language. But Carnap's position could be
saved for any formal language L just by omitting all the
axioms that cause trouble. (The more specialized theories in
the lattice could add any special cases that seem to be useful.) (021)
John (022)
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