PatH,
Continuing the discussion about the number of primitives
and the (in)compatibility of perdurantism and endurantism:
(the most relevant parts are extracted from the note of
PatH 6:15 PM Friday 20090109)
(1) the number of primitives and uniqueness of the set
[PC] >> Yes, it is true that different
independent developers could use different combinations of modules, provided
that those modules were known to be (or not disproven to be) logically
consistent, and still achieve optimal interoperability. But this still
requires that the chosen modules have all of the basic concepts that allow all
of the newly created domain terms to be ‘expressed in terms of’ (I
am using your phrase) those terms in the modules chosen. This means that
*somewhere* in the collection of mutually consistent modules there have
to be enough of the basic terms that are used to express the meanings of newly
created domain terms, so that any domain can be represented.
[PH] > OK, lets agree on this for the sake of argument.
But note, it does not follow that there is a *single* such set. That is the
point.
[PC] >>That set of modules would then contain what I
think of as the minimum set of primitives
[PH] > By using the word "the" here, you
beg the central question: whether there is a *single* set of primitives out of
which all other meanings can be formed.
Ah so. This is an objection I hadn’t gleaned from the earlier notes.
Interesting point. I thought the main objection was about the finite
number of primitives, not their identity. I confess it has been an assumption
of mine that, from the point of view of a “primitive” concept being
indivisible into smaller primitive concepts (crudely analogous to an atom, in
its role as a component of molecules), that a finite inventory of such
primitive-atoms (if such exists) would be unique. I have never seen this
point raised, and it may be significant, but right now I can’t think of
any way a set of indivisible primitives could be anything but unique. Limited
imagination, perhaps.
Now I have noticed that there are mathematical
objects that have very non-intuitive properties, and perhaps a mathematician
can provide some examples of how a set of primitive concepts (not divisible
into component primitive concepts) might be able to be formulated in more than
one way. I am genuinely curious about this suggestion.
(2) are primitives finite in number?
[PC] >> , though the modules can also have other ontology
elements in them. It is not clear to me that we can have a *stable*
set of modules to serve the purpose of interoperability unless we have some
confidence that all or most of the terms needed to express the meanings of
domain terms are there at the start.
[PH] > Seems clear to me that if we impose this condition
as a requirement, we have shot ourselves in the foot. And why should the set of
concepts be "stable" ? Isn't it more realistic to assume that new
concepts will always be being constructed, that knowledge is always
open-ended?
Well, knowledge is certainly open-ended, but if new concepts
(or terms labeling those concepts) are always composed from some combination of
pre-existing concepts, then the number of primitives need not increase as knowledge
increases. This may depend on what you mean by ‘knowledge’.
With a set of basic concepts we can predict an infinite number of things that *might*
exist in the real world, but I think of knowledge as knowing what actually does
exist and what doesn’t. Learning about the laws of physics (and
chemistry and biology) that exist in our real world allows us to know some of
the things that can’t exist. That’s knowledge, in my lexicon,
but it doesn’t necessarily require new primitives.
Is
it possible that learning *some* new things always requires creating new
primitives? Possibly, but that is the question that I have suggested
can be investigated by the process of creating a plausible set of ontology elements
based on primitives (as best we can discern them) and then seeing how quickly
that inventory in the FO must increase to allow the elements in each new additional
domain ontology to be expressed in terms of the primitive set. If there is a
finite set, then from the increase in required primitives for each new block of
domain concepts, it will be possible to assign a probability that the total
inventory will reach an asymptote at infinity. The alternative is that no
limit is indicated, and the number of primitives behaves, say, like the number
of prime numbers as the number of integers increase. I do not know for sure
which is more ‘realistic’, but my suspicion lies with the finite
number, based on evidence from linguistic usage, such as the Longman defining
vocabulary and the relatively small number (2000 – 5000) in AMESLAN sign
language dictionaries. There are other suggestive kinds of evidence from
language, but what needs to be determined experimentally is whether the finite defining
vocabularies in languages are in fact analogous (at least on the point of the
number of primitives) to the task of creating ontology elements as combinations
of other ontology elements.
(3) [PC] >> If it turns out that there is no such thing as
a finite set of primitives, and it becomes necessary to continue adding new
primitives indefinitely as new domains are derived from the FO, then the FO
will not be fully stable.
(4) On the question of whether 3D and 4D entities are logically
compatible.
One point I did not make in the discussion of how this could be
represented is that allowing a time slice of a non-4D ‘dimension neutral’
Object in the OWL version of an ontology (which I do in the current OWL version
of COSMO) does not mean that the CL-compliant version would that syntactic
device. It is used in the OWL version to reduce time-dependent assertions
to binary, to fit into the OWL syntax. When the OWL version is translated
into CL, every instance of an Object that is a time slice would be translated
as a time slice of ‘Object4D’. Then Object and Object4D could
be disjoint , making it easier to include a continuant/occurrent view into the CL
version of the FO.
Regarding some specific comments on this point:
[PC]
>> My idea of a ‘unified’ 3D-4D ontology would permit (among
other things) the assertions (I hope the meaning is clear from the labels):
>> {PH
isanInstanceOf Object}
>> {PH4D
isanInstanceOf Object4D}
[PH > Can you explain the difference between Object and
Object4D? Intuitively, that is. What criteria are there for deciding whether a
given thing is in one category or the other, that we can explain to knowledge
modelers in the user handbook? Can there be one of these without the other also
existing, such as a PH without a PH4D, or vice versa? (Why not?)
In an assertion containing an instance of dimension-neutral ‘Object’
that is not also an instance of TimeSlice (having begin and end time points) ,
if the assertion does not have an explicit time index, it must be interpreted
as meaning that the relation holds throughout the lifetime of the Object; for
the Object to have an incompatible property at any time would be a
contradiction. So the dimension-neutral Object behaves in some ways like
an Object4D, except that it can also be used in explicitly time-indexed
assertions. If one decides, in a CL version, to make Object and Object4D
disjoint, then in the CL version, there cannot be any time-slices of an
Object. In that case, an assertion on an Object that is not explicitly
time-indexed can still be interpreted as holding throughout the lifetime of the
Object, or it may be forbidden as a syntactic error. That option would be
decided by the committed creating the FO.
There may be better ways to accommodate both endurantist and
perdurantist representations in the same ontology. The point I am
trying to make here is that, although the philosophical stance of an
endurantist and of a perdurantist may rest on incompatible models, *ontological
assertions* about either such entity can be accurately translated into
assertions about the other representation. I thought that your note from
March made much the same point.
(5) [PC] >> {PH4D
isTheWholeLife4dVersionOf PH}
{PHt1t2 isaTimeSliceOf PH4D from
t1 to t2}
[PH] > What relationship , if any, is there between
PHt1t2 and PH?
The relationship is a combination of the relations between PH and PH4D, and the
relation between PH4D and PHt1t2. It could be compressed into one
relation “isaTimeSliceOfThe4D”, if
that is useful:
{ {PHt1t2 isaTimeSliceOfThe4D PH from
t1 to t2} iff
{ {PH4D
isTheWholeLife4dVersionOf PH}
and
{PHt1t2 isaTimeSliceOf PH4D from t1 to t2}}
}
(6) [PC] >> If I
included a ‘during’ similar to the one PH uses, it might look like:
>> {(PH during
t1t2) isIdenticalTo PHt1t2}
[PH] > ?? So (PH during BirthDeath) is identical to
PH4D ?
Yes, though I would use one of these alternative
syntactical constructions, which are equivalent to each other:
(during PH BirthDeath)
{PH during
BirthDeath}
. . . where ‘during’ is interpreted as a function
generating a time-slice of a dimension-neutral Object - the functional alternation
of the concatenated relation ‘isaTimeSliceOfThe4D’,
described above, though syntactically different.
One caveat, to be precise: This would be true only if
PH were interpreted as ‘PH-while-alive’. I do prefer to allow
dead and unborn people to be represented as a ‘Person’, distinguished
from living people by an ‘alive’ (dead/notBorn) attribute.
This requires a careful definition of what a ‘whole-life’ interval
is, for each category of Object.
(7)[PH]
> If you are willing to allow dimension-neutral objects to have
time-slices, you have completely abandoned the foundational ideas behind continuants.
Whatever these PH things are, they aren't continuants. I therefore see no
purpose in having them. Why not simply identify PH with PH4D?
Because this allows the use of syntactic structures that are
used with 3D objects, and serves the purpose of allowing a syntax that is more
congenial to those who prefer to work in that paradigm. The point is to
try to create a foundation ontology that can be agreed to by the largest number
of people, so as to serve the purpose of creating the largest possible user community
– which is the whole purpose of the proposed FO project. Where
there are irreconcilable differences, these can be encoded in extensions,
provided that the differences are not contradictory to the logic of the base
FO.
Allowing time-slices of an Object is not required, Object and
Object4D can be disjoint, related by the axioms above. In the OWL version
it is a convenience, but I believe that conversion of the OWL version to a
corresponding CL-compliant version where Object and Object4D can be disjoint is
possible (though I haven’t yet written the program). The
point of this discussion, from my side, is to emphasize that, though
philosophical views can involve incompatible models, when we are concerned with
the practical task of creating an ontology, assertions syntactically reflecting
on one view can be accurately translated into assertions syntactically
reflecting the other view. This is the sense in which I consider the
alternative views to *not* be ‘logically incompatible’ –
the practical consequences of both can be represented in a single consistent ontology.
If there are logical inconsistencies that must be represented by something other
than belief systems, it may be necessary to include them in extensions, or in FO-described
ontologies that are not part of the hierarchy of ontologies (which can be
viewed as a constrained ‘lattice of theories’).
(8) [PC] >> The axioms
seemed to be consistent with that, though the (x during t) structure was not
repeated in the later axiom set. Perhaps you meant something else by
‘unified’?
[PH] > I was using the term optimistically, but subsequent
discussions have made me less optimistic than I had become.
In your note back in March, I got the (erroneous?) impression
that your axioms described a method of syntactically accommodating both perdurantist
and endurantist views (or at least their _expression_ in assertions about the real
world). You think now that there cannot be such a combination?
If it turns out that there are important (used by more than two
groups) ontological structures that are primitive but incompatible, it will be necessary
to choose one. I am not sure that that will be necessary, because I still
expect that incompatible theories will almost always be expressible as
combinations of primitives – but it may happen. Even so, I
expect that the number of potential users and developers that are willing to
adopt the result of a collaborative FO development effort will be sufficiently
large to form a user community that can maintain and evolve the starting
product. The goal is to create a user community large enough so
that anyone who wants to interoperate will have *some* widely-used FO to
use for that purpose, with enough public examples of use to provide confidence
that it will serve the local needs. Right now I think that the existing
FOs are not used widely enough, and that a collaborative project can create one
used more widely than any existing one. If it can be based on a finite
number of primitives, that will be helpful, but not critical. There may
well be some ontologists who simply won’t accept the result of the
project; as long as there are enough users to form a large community that helps
to evolve the FO and its extensions, that won’t prevent the project from
achieving its goal.
PatC
Patrick Cassidy
MICRA, Inc.
908-561-3416
cell: 908-565-4053
cassidy@xxxxxxxxx