John,
Thanks for the comments. Those are useful distinctions and useful
terminology. If you can find some more time (at some point) for this line,
there are a few additional details on which I could use some more
explication.
There is also a reply a the bottom here to a point in John's most recent
note. (01)
> JFS>> Adding more axioms is a change, and it can create inconsistencies
> when different people add different axioms.
>
> JFS>> As soon as you add more axioms to a theory, the "meaning" of the
> >> so-called "primitives" changes.
>
> PC> I am not certain that that is true. If one adds subtypes to the
> > types of an ontology, and each subtype has some properties or
> > restrictions not applying to the parent, then it does not seem to me
> > that the *meaning* of any of the parents changes, though we are
> > asserting more information about the properties of the parents
> > (i.e. that some instances have or may have
> > certain properties).
>
> There are many ways of extending a theory. What you are describing
> is a "conservative" extension, which merely assigns names to
> "closed-form expressions".
>
[[PC]] Yes, that would seem to be the effect of just creating new types.
Would it also apply to new relations - where a relation term is created and
the meaning specified by asserting some set of logical consequences if that
relation were asserted? (02)
> JFS>> Since you haven't formalized your system in full detail, it's hard
to
> say exactly what it would be if fully defined. But I suspect that
> what you have in mind is a closed system: a finite set of relations
> (i.e. primitives) that are fully specified by necessary and sufficient
> conditions. The semantics of those primitives would never change in
> any way.
> (03)
I am not sure that the primitives as I imagine them in an FO would *all*
be specified by necessary and sufficient conditions - there may be some
(most?) that are only specified by necessary conditions, plus the linguistic
documentation. In the existing COSMO, few of the classes are defined as
N&S, but I haven't yet made a systematic effort to determine whether that is
possible in most cases.
There is a difference between the primitives as I imagine they would be
represented at some time in the future, and how they would be represented
after the initial demos of the FO were completed. For the near term,
assuring correct use of the intended meanings will depend to some extent
(not sure how much) on users reading the natural language documentation,
because at present only people can supply the physical and analogical
grounding that gives full meaning to the terms. At some point, when (I
presume) computers have more sophisticated perception and robotic
capabilities, those procedures could supply much or perhaps all of the
physical grounding that would make the meanings independent of human
interpretation. There would, of course, be a need for an NL interface for
people to interact efficiently with such systems. But that's a long way
off. For now, I think it is likely that the logical descriptions will still
need supplementation from the documentation, so as to assure that the
programs that use the FO elements do so consistently. Of course the
'meaning" as far as the computer itself is concerned resides only in the
ontological structure - it is the proper interpretation by people (including
programmers) that may require good documentation. (04)
> JFS> Then all possible terms that could be defined in your system would
> consist of closed-form definitions of the following kind:
>
> Every term T is either a primitive or it is defined as synonymous
> to an expression that is composed only of primitives.
>
That seems right - I haven't yet attempted to analyze the COSMO in those
terms. It looks like a good goal to aim at. (05)
> JFS> Most systems of mathematics are far more complex than that, and
> you couldn't even define the full range of useful physics with such
> a limited framework.
>
At this point I have lost all ability to visualize the problem you are
referring to. Perhaps elaboration of this point can be deferred until some
kind of agreement on other points under discussion is reached? (06)
> JFS> Longman's dictionary seems to work, but only because the definitions
> are so vague that they don't constrain the subject matter very much.
>
Yes, some are, but I believe that one can make them as specific as one
wants, still using only the defining vocabulary. The ambiguity of the
individual words is not that great - Guo's work estimates an average of
fewer than 2 senses per word in the Longman definitions. If, in a logical
system, one needs even twice as many senses to logically specify a large
vocabulary, that would still be fewer than 7000 terms. It's a possibility
that I think is important enough to justify a project of the proposed size. (07)
> JFS> Pat has not shown any credible evidence for the claim that an FO
based on such primitives would help support interoperability among practical
computer systems.
No, and neither has anyone else shown "credible evidence" that their
approaches would support accurate general interoperability. The only
convincing "evidence" that I can imagine is to conduct a project with a
large number of independent program developers and see if the
interoperability tactic works. Meanwhile, in the last few days, the
country has wasted another billion dollars in lost efficiency. The problem
is immensely practical and large enough that every credible approach should
be funded. It also seems imperative to me that the approaches that promise
the fastest success should get priority. The take-it-slow let's accumulate
enough tidbits of evidence to convince even the most recalcitrant of
skeptics approach is guaranteed to be extremely wasteful. (08)
> JFS> On the contrary, its closed nature would probably create more
obstacles than it could eliminate.
I don't think I grasp the meaning of 'closed' as it is raised here and is
applied to the FO. As I mentioned, one of the types of 'primitives' would
be those whose meaning can only be recognized from a knowledge of instances.
This is where the human interpretation would be required for proper use of
the FO in the near term. This is not just a closed system of symbols.
Other forms of grounding, such as accessing the internet, also go beyond
what I think is meant by 'closed' in this sense. (09)
About Wierzbicka's 'primitives':
> JFS> But I must emphasize that the so-called "primitives" that Anna W.
and others have proposed are most definitely *not* primitives in the sense
that mathematicians use.
I agree that Wierzbicka's linguistic work has little relevance to the
issue of logical description of primitives, she was concerned with concepts
lexicalized in multiple languages. Her set is the least relevant of the
analogies cited, and was included only to add some historical perspective to
the projects investigating linguistic primitives. That was one-third of one
of the eight slides discussing primitives. If it serves more to mislead
than enlighten, I probably should delete it. The more relevant analogies
are the Longman, the sign language vocabulary, and the Chinese character
inventory. (010)
Pat (011)
Patrick Cassidy
MICRA, Inc.
908-561-3416
cell: 908-565-4053
cassidy@xxxxxxxxx (012)
> But Chris M. used the example of set theory, all versions of which
> have the same two "primitives" -- subsetOf and elementOf. Any
> extension that merely gave names to various combinations of those
> operators without adding new axioms would be a conservative extension.
>
> However, the many different versions of set theory have added different
> axioms. All the main theories include finite sets and even countable
> sets in compatible ways. But they differ on the extensions to higher
> orders of infinity. The two basic primitives differ from one version
> to the next on those extended branches.
> (013)
[[PC]] (014)
> Most systems of mathematics are far more complex than that, and
> you couldn't even define the full range of useful physics with such
> a limited framework.
>
> Longman's dictionary seems to work, but only because the definitions
> are so vague that they don't constrain the subject matter very much.
>
> John
>
>
>
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