John,
Thanks for your detailed clarification. I think we do still have some
disagreements.
First, I agree with your comments about the difference between dictionary
and mathematical definitions. I also believe that many of the definitions
used in Longman would not be adequate to specify the meanings of terms to
the required level of detail. But my own tests suggest that the same set of
words would do that job, possibly needing a little additional
supplementation (as in the case of 'dimension' mentioned earlier). But I do
not believe that the meanings of terms as specified in an ontology follow
the mathematical method of creating mathematical concepts. In most cases,
we can only specify necessary conditions for membership in a type. There is
simply no way to reduce the meanings to some very small set of primitives.
I don't know what the number is, but I feel that the number of necessary
primitives is an intensely important question to answer if we are ever to
put the study of computational ontology on a scientific basis. You (and
others) may well not agree with this. I think that the difference between
us may hinge very much on the degree of accuracy that we consider necessary
for interoperability. I am focused on the needs of mission-critical
applications where the machines will be making important decisions
automatically - where very high accuracy is required. I am not sure what
applications you have worked on for which analogical reasoning is adequate.
Searching for information, where people are presented with a list of
possibly relevant documents and they make the final relevance judgment, does
not require high accuracy - though higher accuracy is better. (01)
> PC> My point is that it is an important enough issue to warrant
> > the effort required to discover that number -- a project adequately
> > funded to support at least 50 people half time for a couple of
> years.
>
> But I don't believe that there is any number to be "discovered".
> If somebody set out to do the project with 10 primitives, they
> could probably succeed. But then somebody else could add one more
> axiom and reduce the number of primitives to 9. And somebody else
> would add 8 more axioms and reduce the number of primitives to 1.
>
> If you completed such a project, it would be pointless, since it
> wouldn't prove anything.
>
Well, if we found that 6000 basic concept representations are necessary
and adequate to specify the meanings of the 100,000+ terms in WordNet, it
would prove that those who believe there is no such thing as a 'Conceptual
Defining Vocabulary' are wrong, and may be pursuing alternatives that are
less effective at achieving interoperability. It would also provide a means
of enabling accurate interoperability among a very diverse set of
ontology-based applications. If it turns out that that number can be reduced
by further analysis to 4000, so much the better. (02)
Since you think that 10 primitives may be enough, it is now worthwhile
asking just what the criterion is for a concept representation to be
considered one of the 'primitives' suitable for inclusion in the foundation
ontology (PatH asked that a few posts ago, but it is very relevant to your
comment that we might need only 10 primitives). (03)
For the present, I use a *tentative* set of criteria (below) to decide
whether a type or relation is 'primitive' in the sense of not being able to
be adequately specified by pre-existing ontology elements. If anyone else
who thinks that finding a list of primitives is worthwhile believes that
these criteria should be modified, I will be very interested in such
thoughts. (04)
****************************************************************************
**********************************
What is a primitive concept? (05)
A concept that cannot have its meaning specified solely by use of
preexisting elements in the foundation ontology is a primitive concept, and
should be included in the foundation ontology. These criteria assume that
the meaning of every type is specified by describing necessary conditions
for being an instance of that type. When this is not possible (cases 3 and
4 ), there are other considerations. This also assumes that every relation
has logical consequences specified for the case when that relation holds. (06)
(1) if the meanings of two or more new concepts to be represented can only
be specified by reference to each other, and none can be specified without
reference to one of the others, all of these would have to be added to the
foundation ontology as new primitives.
(2) if the only way to distinguish a new concept from others is to declare
that it is disjoint with other concepts, that new concept will be a
primitive.
(3) if the meaning of a new type (class) cannot be specified by necessary
conditions for membership, but must be explained by pointing to real-world
instances, then that concept may also be a primitive. (But see *3 below for
further discussion)
(4) if a type does not have any uninherited necessary conditions, but is
specified solely as a union of some set of subtypes, it is not necessarily
primitive and is not required as a part of the foundation ontology. It may
nevertheless be convenient to include it in the foundation ontology, to
allow assertions that efficiently refer to all of the subtypes. See *3
below.
(5) a relation that is not a subrelation of another relation, and is not the
logical consequence of some other relation(s) is a primitive.
(6) if two relations necessarily imply each other, and neither is a
subrelation of another relation, those relations are both primitives.
================== (07)
*3 These kinds of primitive can present special problems.
To specify a meaning of some type A by pointing to instances or subtypes
is to specify the meaning by using sufficient conditions, not necessary
conditions
*3a . If a type is specified as a union of subtypes, and each of the
subtypes is specified by necessary conditions, then the meaning of type A is
also by implication specified by necessary conditions, as being an inclusive
disjunction of the necessary conditions of the subtypes, and this is
acceptable.
*3b If a type is specified by an exhaustive list of instances, then it is
adequately specified and acceptable as a primitive.
*3c If a type is specified by pointing to only some, but not all
instances, then it is problematic, since the logic will not be able to
relate it to other concept representations. It may be accepted, but effort
should be made to specify the meaning in more detail. At a low (leaf node)
level (a 1958 Ford Edsel), such primitives may be useful and harmless. At a
high level, if highly abstract concepts (such as continuant and occurrent)
are specified only as the union of some set of subtypes, the meaning will
depend only on the subtypes. Though logically acceptable, the ability of
the human developers to understand the intended meanings of types defined
that way can be seriously diminished. (08)
****************************************************************************
*********** (09)
Back to John and the 10 primitives. (010)
What ten primitive concepts do you think will be adequate to specify the
meanings of other concepts by necessary conditions? If there are 13
relations in the interval calculus alone, does this not suggest a much
larger number of primitives? (011)
Pat (012)
Patrick Cassidy
MICRA, Inc.
908-561-3416
cell: 908-565-4053
cassidy@xxxxxxxxx (013)
> -----Original Message-----
> From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-
> bounces@xxxxxxxxxxxxxxxx] On Behalf Of John F. Sowa
> Sent: Wednesday, March 12, 2008 2:22 AM
> To: [ontolog-forum]
> Subject: Re: [ontolog-forum] Ontology similarity and accurate
> communication
>
> Pat H., Mala, Chris, and Pat C.,
>
> JFS>> Please don't confuse me with Pat C. I do not believe that any
> >> of the current schemes is going to achieve any kind of consensus.
>
> PH> OK, sorry if I misunderstood. You seemed to be agreeing with him.
>
> My claim is the following:
>
> 1. We should use the words of a natural language as a way of indexing
> various ontological theories -- more or less what people do with
> WordNet. But I would drop the WordNet upper levels (which very
> few people actually use) and treat it (along with other additions)
> as a valuable lexical resource.
>
> 2. The detailed reasoning would be done with the axioms of various
> specialized theories. For example, a theory of skiing might
> treat a glacier as a continuant, but a theory of geology would
> treat it as an occurrent.
>
> In my note to Pat C., I said that if all he was claiming is a very
> sparsely axiomatized hierarchy, that would be close to my position.
> However, he has now added a further comment, which I'll discuss
> below.
>
> I agree with Lenat that for any serious reasoning, the middle and
> lower levels are much more important than the upper levels. In fact,
> I would recommend that any widely used axioms in the upper level be
> moved into a library of general purpose modules that could be used
> in a mix-and-match way for the various specialized ontologies.
>
> MM> However, I often find that the relationships do not fall into
> > the five neat categories you have enumerated. They are showing
> > more fuzzy relationships - such as similar contexts that have
> > been used for defining two classes (by way of similar restrictions,
> > etc.) or prototypical patterns of defining certain classes of
> > axioms. The axioms are related, but in looser or more subtle senses.
>
> I agree that those relations don't cover all the interesting relations
> among theories. It's possible to use those relations in defining a
> lattice of theories, but a lattice has multiple branches which may
> have similarities that are not captured by those operators.
>
> For example, we could have a general theory of cooking, which might
> have some very general axioms about preparing food. But then it
> could have many specialized branches for different cuisines, such
> as French, Italian, Chinese, Indian, Mexican, etc. None of those
> branches would be generalizations or specializations of one another,
> but they might have common structures that are not shown in the
> lattice.
>
> Sometimes those common structures are obscured by the choice of
> names for the types and relations. For example, there might be
> specialized ways of frying food in different cuisines. That
> would lead to two possibilities:
>
> 1. If the ontology had some basic primitives for defining each
> of the variations, then the similarities would appear in a
> least common generalization of two or more cuisines.
>
> 2. But if the different branches used different names for the
> types and relations, those commonalities would be hidden.
>
> MM> Is there a way to fit these types of observations as formal
> > relationships into your lattice theory? I wonder if you have had
> > such experiences with your graph matching algorithms.
>
> I believe that a lot more can be done with the analogy finding
> methods, and that is a topic that we (at VivoMind) are eager to
> pursue. However, it does take time and effort to do so, and we
> currently have just a small group and not enough resources to
> do the necessary work right now.
>
> For an example of how the analogy methods can be used to align
> ontologies and find hidden patterns between different branches,
> see slides 9 to 16 of the following talk:
>
> http://www.jfsowa.com/talks/semtech3.pdf
> The Goal of Language Understanding
>
> JFS>>> My ideal for an upper-level ontology would be the barest minimum
> >>> number of axioms -- and the ideal number is 0.
> >>
> PH>> ... I guess I don't understand this. All an ontology is, is
> >> axioms. What does it mean to have zero axioms?
> >
> CM> I took him to mean no *proper* axioms, i.e., the ideal upper-level
> > ontology is predicate logic (or whatever your base logic is). I'm
> > not really sure what it means to say that.
>
> I should have been clearer about that. Earlier I had mentioned that
> WordNet and related terminologies have very few axioms beyond just
> the statements of is-a and part-of. Each is-a link or part-of link
> would count as one axiom. I should have said that I don't believe
> that the upper levels should have any additional axioms beyond those
> very minimal statements.
>
> All the more detailed axioms should be put in optional modules that
> are used for more detailed specialties. For example, the axioms
> about the heart for a heart specialist, a family doctor, a nurse,
> or a patient would get progressive less detailed, more simplified,
> and less rigorous.
>
> PC> I think we do need a common basis of at least a few thousand
> > concepts. I think we differ only on the number of concepts
> > about which we can find a large body of agreement (which will
> > not be not universal, as PatH correctly asserts).
>
> OK. That clarifies the issue. The Longman's dictionary had a
> basic defining vocabulary of 2,000 words, and I think you based
> your estimates on that book. But I think that number is misleading
> for several reasons:
>
> 1. Longman's has done their best to reduce their defining vocabulary
> in order to make the dictionary more accessible to readers who are
> just learning English. That is indeed a useful goal.
>
> 2. But they don't attempt to itemize all the knowledge that a child
> would learn about the world in just trying to move around and
> interact with things. Most of that knowledge is never verbalized,
> but it is a prerequisite for all higher-level knowledge.
>
> 3. Most dictionaries try to state definitions in the Aristotleian
> style of genus + differentiae. They say "An X is a Y such
> that..."
> That implies that Y is the genus or supertype or hypernym and the
> three dots are the differentiae that distinguish X from other
> things of type Y.
>
> 4. However, most axioms in mathematics do not use that style.
> Peano's axioms for integers, for example, don't say "An integer
> is a Y such that..." They just say "There is an integer 0,
> there is a function S, and and any N satisfying the following
> axioms is also an integer."
>
> Because of these points, there is no hard and fast minimum number
> of primitives. In fact, would probably be possible to design a
> method of distinguishing thousands of different English words with
> just a few abstract primitives like 0 and S that could be used to
> define relational structures that could define a mechanism for creating
> abstract definitions for all the words of English.
>
> In effect, you would only need perhaps 10 primitives and one axiom
> for every word (which might be arbitrarily long). But those
> definitions would probably be very hard to read.
>
> PC> My point is that it is an important enough issue to warrant
> > the effort required to discover that number -- a project adequately
> > funded to support at least 50 people half time for a couple of
> years.
>
> But I don't believe that there is any number to be "discovered".
> If somebody set out to do the project with 10 primitives, they
> could probably succeed. But then somebody else could add one more
> axiom and reduce the number of primitives to 9. And somebody else
> would add 8 more axioms and reduce the number of primitives to 1.
>
> If you completed such a project, it would be pointless, since it
> wouldn't prove anything.
>
> John
>
>
>
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