[om-list] is new list OK?
Tom and other Packers
TomP at Burgoyne.Com
Sat Aug 19 10:21:08 EDT 2000
Luke
I got two of these letters, one with the "omlist" signature, and one
without. Has the subscription/unsubscription/question mechanism changed
too? Chris Angell wants to join, but hasn't yet. He assumed that you had
signed him up already.
Chris, go to the address on the bottom line of this letter to join.
OneList at OneModel.Org might have been cuter. ...? But I'm not
complaining.
tomp
----- Original Message -----
From: Luke Call <lacall at onemodel.org>
To: <om-list at onemodel.org>; <general-list at onemodel.org>
Sent: Friday, August 18, 2000 9:45 PM
Subject: [om-list] is new list OK?
Is the new list concept OK w/ all? (I'm sending to both in case you
haven't handled the new subscription yet).
I may be answering email only intermittently for a week or two due to
schedule.
Luke
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From: "Tom and other Packers" <TomP at Burgoyne.Com>
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Cc: "Ron (ws) Peterson" <RPeterson at Weber.Edu>,
"Rex Butler" <rexbutler at hotmail.com>,
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Subject: [om-list] Probability, Fractional Set Membership, Metalanguage -- Re: Modeling Relations
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Dr. Peterson.
You might want to keep this letter for use in my "individual project"
class this semester, especially if you don't read it all now. (And maybe
for our honours class, too.)
Mark
Thanks for the discussion.
I mainly agree with what you say, except maybe...
Natural language is full of fuzzy sets. Natural language is the
inspiration for fuzzy set theory and fuzzy logic. Maybe you meant that
natural language has a very, very hard time defining the degree of
membership for any of these fuzzy sets on its own, which I completely agree
with.
I have a conjecture: It is possible to replace all fuzzy set membership
functions, and never have to use fuzzy set theory again, by adding one or
more non-fuzzy, ontological dimensions which describe real conditions in the
real world in a non-fuzzy, and often concrete way. The key is increased
dimensionality in the model. This could be useful, for example in solving
your disagreements with fuzzy membership values. But, for now, I don't have
a proof of this, nor a general mechanism for translating fuzzy membership
functions into measurable, real-world states, so from here down I'll be
writing as though I didn't say this.
There has to be some convention in artificial languages, such as
mathesis, for generating fuzzy membership values. I agree with you that
this number, this percentage, this fractional degree of membership, is
arbitrary, and lacks "rigour" or strict meaning, in the sense that two
randomly-chosen membership functions used to describe two different fuzzy
sets cannot be compared perfectly, e.g. two 75% values attached to two
different sets cannot be said to necessarily represent the same degree of
membership. But that doesn't mean that this number is meaningless, in
context. It has a limited but real value, or the usefulness of relative
meaning, in the sense that it can be used well within one set, for comparing
degrees of membership calculated by the same membership function.
For example, if we define an unit of time to be fractionally a member of
the set called "daytime" (as opposed to night time), then no matter what
value we gave to "noon" and "15 minute after sunset", as long as "15 minute
after sunset" had a lesser membership value than "noon", and the hours
between these two had intermediate values, they would be useful and
meaningful -- just not meaningful when compared to 'degree of membership of
a bee's ganglion to the set called "brain"'.
The interpolation may not be linear.
This is an example of where real-world metrics could possibly replace
fuzzy sets. Instead of worrying about degree of membership in the "daytime"
set, just measure the number of candelas of light.
To overcome some of the problem with the abstractness of fuzzy set
membership, I would not use something as geometric as 75%, which seems to
allows for interpolation between values in a linear, geometric way.
Instead, I would use symbolic language, just like you have done: e.g.
"slightly", "somewhat", and "mostly".
About my original request to consider probabilities and
fractional membership, (regarding attributes of a generic entity):
I think there are two or three situations of when a generic entity has
an attribute to a fractional degree (to be referred to as groups 1 through 3
below): either (1) this means that all of its sub-entities (specific
instances) have the same attribute to the same fractional degree, or (2) it
could mean that some ratio of the total number of sub-entities has that
attribute to degree 100%, where the ratio is the same as the fractional
degree of membership. Or (3) there could be some combination of the two. I
think there could be a fairly straight forward formula for calculating the
fractional degree of attribution of the generic entity based on a variety of
degrees of attribution of all sub-entities. This would be useful because of
the second of the following two uses of attribution to a generic entity:
(1) A generic entity has an attribute, *by definition*, and therefore
all sub-entities would have to share that attribute to the same degree (or
perhaps a range of degrees) as defined in the attribute of the generic
entity. This would be group 1, above.
(2) A generic entity has an attribute to some (probably incomplete)
degree, simply because those entities which are by definition sub-entities
have that attribute to various degrees. In this situation, the attribute is
not part of the definition of the generic entity. The fact that some or all
of the sub-entities share this attribute is "coincidental". This would
include groups 2 and 3 above.
This is very useful. How do we look for patterns in a body of
knowledge? One way is this way: Say we find an attribute in one of the
instances of a generic class of objects which is not by definition a part of
all instances of the generic class. We could then look for this attribute
in all the others instances. If it turns out that all instances of this
class have the same attribute, to a fairly large degree, we have just found
something significant: a pattern that was perhaps unknown before -- and I
mean "pattern" in probably the most generally applicable sense of the word.
Applications: (1) finding a better method for diagnosing a disease, such
as a non-invasive techniques in situations where traditionally only invasive
techniques could diagnose such a disease. E.g. outside observation instead
of endoscopy or incisions. Also, (2) this is the only way I can think of
that treatments for diseases are found. First you have a set of symptoms, a
"syndrome". This is the generic entity. How do you relieve the symptoms?
You find some attribute(s) of the condition which makes it possible to
relieve the symptoms. The fact that the syndrome can be treated is not part
of its definition, but it is very useful information which can be applied to
all instances of the syndrome.
After that happens, after a new attribute is found which is apparently
universal among all instances of this generic entity, there is the potential
difficulty of deciding, "Should this new attribute become part of the
definition of the generic class?". But that is a separate issue entirely.
One use of this question is the following. Actually, this question is
useful even when not all of the sub-entities share the same attribute. Here
is an example of such:
Say, only half of the total number of sub-entities share the same
attribute, and the other half don't. It's probably now time to coin a
couple of new terms. I.e. it's time to form an intermediate level of
abstraction composed of two sets. Each set is a specification of the
generic entity in question, and each set is also a generalisation of half of
the instances of the generic entity. We could say that half of the generic
entity are type A, and the other half are type B. This is very useful. For
example: the treatment of a disease example. If you can only treat half of
the instances of a disease in one way, and the other half you cannot treat
in this way, then there is potentially a lot of risk in not coming up with
subclassifications A and B. There is risk in not treating type A when you
can, and there is risk in improperly treating type B when you cannot.
Coming up with subclassifications A and B would be done by a
maximisation algorithm operating on "group 3" fuzzy-membership values of
subsets of the abstract entity. With enough empirical data collected, this
type of inference could be extremely useful, not just in medicine, but in
just about any other field.
Back to probabilities. I wasn't equating probabilities with
fuzzy-membership, I was saying that probabilities could be calculated
directly from fuzzy-membership values, as in group 2 above. If there had
been established some ratio of the number of sub-entities which have an
attribute, then if you are given a random sub-entity, and you don't know
anything else about it, and are asked whether or not it has the given
attribute, then you could say that it does, with a certainty derived
directly from the fuzzy-membership value of the parent class.
For example, 95% (a fictitious percentage) of all tigers are orange with
black stripes. If I tell you that I bought a tiger, (presumably because I
was tired of the neighbourhood dogs leaving little messes on my lawn), and I
asked you whether or not my tiger was orange with black stripes, you should
not say "yes" with absolute certainty. Instead, you should say "probably"
with 95% certainty. Then, if I told you that my tiger was actually white
(albino),
you would not hang your head in shame. My tiger was part of the 5% which
you already knew about, and took into account.
In this example, in the case of the generic parent class (tiger), the
attribute has a fuzzy membership. In the case of a specific child class (my
tiger), this same attribute has a probability of attribution. It's a
fractional membership on one level and a probability on the other.
This type of fuzzy membership (group 2) would have to be represented
differently than the normal kind (group 1), where all members have the same
attribute to an equal, fuzzy degree, because the two groups represent
different things.
By the way, I'm glad to hear that you are still part of the Enlightened.
I remember you once had the argument that fractional membership was the same
as probability, a common argument against the need for fuzzy set theory.
Then you bought a good book, (at the same time you bought that very good
computational linguistics book that I bought from you).
An example of group 1 would be: "All insects contain closed circulatory
systems, to a consistent but fractional degree.". Thus, any instance of
"insect" would have one or more tubular hearts, and limited arteries, from
which the blood escapes into body cavities and sloshes around. This
circulatory system is "closed" to a fractional degree. Another example of
group 1: "All insects contains a brain, to a fractional degree." Thus, any
bee I happen to find busily buzzing around in my alfalfa blossoms would have
three pairs of what are called "ganglia", which are not true or complete
brains in the sense of a vertebrate's brain.
In combining the two examples into a group 3 example, I believe that
there is some usefulness in a metric, i.e. measurement, describing the two
types of fuzzy sets combined, (as in the maximisation algorithm mentioned
above, which I would actually call a "semantic compartmentalisation
algorithm", by the way). If "insects account for about three-fourths of all
described species of animals" (an actual quote from "Invertebrates", by
Eugene Kosloff), and all insects, and only insects, have brains to degree
50% (not an actual figure), then (pretending for a moment that no other
animal had a brain) animals, as a generic class, would have a
"brain-containing degree" of 0.375.
This kind of fuzzy value would probably not be used for algorithmic
searches very often, when we are looking for crisp, definitive output, and
never in this case without an attached description of its derivation, i.e. a
noting of it's being derived from the two different types of fuzzy
membership values. But this kind of fuzzy value could be useful in
abstract, "intuitive" or heuristic inferences.
Regarding your example of the fractional attribute of an entity which is
a sub-entity to fractional degree: Using a "1/2 scale factor", as you
describe, is exactly what I was talking about before.
Regarding cement. Of course you would not use something as simple
(simplistic and incomplete) as, "cement is typically 90% sand", and then try
to deduce that your 10%-cement is composed of 9% sand. You would do what I
think you were suggesting, and that is to use all possible percentages of
sand-in-cement, to derive the probability range of a function (a function
whose domain is a continuous list of percentages of composition of sand in
cement), which function is perhaps based on the statistical deviation
gaussian (bell curve).
I think our inference engine could make any inference it wants (or which
we want), even before fractional membership is well defined; though at
first, until it is well proved, the probability value attached to all of
these "guess work" inferences will be quite low.
But low or not, they will have value, i.e. they will have usefulness,
because if we *need* a certain answer, and we have low probabilities, we
will simply start at the largest probability values, and verify these
manually, until we find something usable. And if we never come up with a
large degree of certainty, and an action is required, we will simply have to
use the highest of those pitifully low probabilities as the basis of
behaviour.
We humans always make decisions based on incomplete information, and
imperfect certainty. People's intuitions work the way described above
(subconsciously), and they have to. These intuitions are very useful, even
if they are wrong some of the time after verification. We have to deal with
fuzzy and incomplete information.
A good way to handle incomplete information is to make a model which can
record as much of the information as possible, (and, for now, this includes
crude measures of fractional membership) and then to add redundancies,
verifications and validations on the union of a large body of worldviews,
each of which are incomplete.
Now, about internodes: Yes, I like singly-rooted systems, too. So, I
guess we would, symbolically, define all of our symbols (nodes, internodes,
metainternodes) as instances of some more (metalinguistically) abstract
entity, but I'm not sure we have to call it "node" -- or, at least, there
should be some distinction between nodes at various levels of the linguistic
tree.
Okay, I agree that I should not prevent others from using
meta-metainternodes.
Actually, mathesis *is* a singly-rooted tree, and has been for a few
years now. (In fact, you may have been the one to get me thinking about the
need to do so, two or three years ago, the last time we had this
conversation.) This is what I meant when I said (in an earlier letter) that
all three mathetical construct were related on a metamathetical level.
Remember, my meta-metalanguage?
I guess I assumed that you were talking about flattening things into the
linguistic level *and* flattening the three things into one node construct,
which was a bad assumption on my part.
As long as we can have a geometric interpretation of the three
constructs, I'll be happy. You'll agree with this, if you've studies the
rigorous foundations of mathematics, or axiomatic (Euclidean-style)
geometry, and have agreed with the popular opinion in the mathematics
community. (I'm not talking about Cantorian set theory in particular, which
we both find unacceptable as the foundation of mathematics; I'm talking
about proofs and axiomatics in general.) We want/need as broad a potential
for interpretation as possible. I think we agree here, which ironically has
been our disagreement. We've just been looking at generalising the system
from slightly different perspectives.
Regarding self-reference. I've been using mathesis to model
self-reference (in my simple thought-experiments) since I started reading
about such things a few years ago. I think we agree here, and have agreed
all along. (I guess I continue to have communication problems.)
Except, I don't see how a sentence can be called an internode. The
sentence contains semantics which include a relation (internode) which
relates the sentence as a whole to a truth value. (E.g. the proverbial
"This sentence is false.".) We'll have to discuss this later.
Back to nodes. A node is a node, on level three, level two, and level
one. An internode is a node, but only on levels two and one, not on level
three. This has been the case in mathesis for many, many months. But I
have preferred to not call them all "nodes", I prefer to call nodes of
levels one and two "semeia" (singular: "semeion"), to force a distinction in
meaning and in usage.
Actually, there is a second way of handling this, in which my three
mathetical constructs are fixed at a specified level of the linguistic tree,
and all other additions, such as metalanguages, are added below. In this
case, mathesis would be the top metalanguage, forever, no matter how many
layers are added below. Also, all the things you talk about could be called
nodes, as long as they are outside of mathesis.
For example, nodes represent concepts which are normally described by an
object language, e.g. my sister's cat Oliver. But nodes can also represent
linguistic constructs which are normally described by a metalanguage, e.g.
the name "Oliver". A trans-linguistic internode relates the two nodes, and
says that "Oliver" represents Oliver, symbolically. Another internode would
compare a linguistic construct, like a sentence (as a node) with truth.
But we better sit down sometime, in person, because I'm sure I'm not
describing this very well. Plus, I admit to not having worked out all the
kinks in mathesis yet. There are more than one type of internode, which I
have not yet described to you. I'm a little uneasy about presenting them
yet, because they're not clearly and strictly defined yet. And I'm not yet
confident that they can be.
I enjoyed this letter. I think we are getting closer to understanding
each other. We are making explicit many implicit ideas in a way that will
make theoretical design, and the implementation thereof, possible.
And not to reopen wounds with others in this company, but I want to
point out that these past few letters have been basically good, convergent
communication, such that my ideas and Mark's ideas feel to me as though they
are converging, not diverging, the longer we talk about them. I think we
are both learning and admitting new things in the process.
It would have been a more efficient job for Mark and me to have talked
about this in person, though, (there was some miscommunication going on
which we probably could have handled better if we were sitting down talking
and drawing diagrams) so ...
I'm still wanting to get together. When is good for everybody?
ciao,
tomp
----- Original Message -----
From: Mark Butler <butlerm at middle.net>
To: One Model List <om-list at onemodel.org>
Cc: Rex Butler <rexbutler at hotmail.com>
Sent: Saturday, August 26, 2000 12:36 AM
Subject: Re: Modeling Relations -- Re: use cases
Tom and other Packers wrote:
> We need to consider fuzzy sets, too. If you change or add an
attribute
> to a class, to a partial/fractional degree, that will also affect all
> members of that class, but in an incomplete way: it will change or add the
> probability-of-fractional-degree that such an attribute applies to any
given
> member of that class.
Probability (or likelihood) is a special kind of fractional set membership
that must be carefully distinguished from the more general case.
Probability
expresses the measure of confidence that a certain proposition is correct,
based on a correct accounting of missing information. Fractional set
membership, on the other hand, expresses the general idea of degree of
membership in a set with ill defined boundaries. The latter can still exist
even in a environment of perfect information, whereas probability becomes
relatively useless.
To combine the two ideas (degree of belief with degree of membership in an
arbitrary set) requires expressing the truth value of a set membership
proposition as a probability density function over a degree of set
membership
axis. This allows ideas to be expressed like "I completely sure it is
half-tiger" or "I am convinced it either is or isn't a tiger (but it is not
a
mixture)".
You cannot combine probability and degree of set membership into the same
scalar value without throwing rigor out the window. Unfortunately doing it
correctly is computationally expensive, so there needs to be judgement
exercised when using both together.
In my opinion, probability is a much more effective concept than more
arbitrary fuzzy set membership, primarily because it is defined rigorously
in
the language of science on a class of largely man-made sets with distinct
boundaries. Scientists, as a rule, eschew fuzzy sets whenever possible
except
in cases where the degree of set membership can be given a rigorous
definition. Expressions like "78% warm" are nearly meaningless, because
there
are but the barest conventions for expressing fuzzy set membership in
natural
language.
However, there are certainly enough cases where it is extremely useful to
distinguish between "slightly", "somewhat", and "mostly" that I agree that
fractional set membership should be supported.
> Or, if a member is a member to a fractional degree,
> then any attribute added to its superclass, even one added completely to
> that superclass, will only affect that member to the degree to which it is
a
> member of the superclass. If both the attribute and the membership are
> fractional, then some convention needs to be established for combining the
> fractionality here, such as multiplication of the fractional value (a
value
> between 0 and 1).
Unfortunately there is no rigorous way that you can do this. If you say
that
the superclass has attribute of degree X, then it follows that all members
of
that class have the same attribute of degree X.
The only legitimate thing you can say is that the expected degree of the
attribute is X given a sufficiently large sampling of the members of the
class, which is short hand for specifying the expected value of the
probability distribution over the degree of attribute. You could also go on
to add the standard deviation to give a more complete picture of the
distribution.
If a certain member of the set is only a member of degree 1/2, the only
thing
you might possibly do (in desparation) is to use the 1/2 as a scale factor
for
combining the parents classes probability distribution with all other
probability distributions of sets the member might be fractionally a part
of.
What you cannot do is multiply the centroid value of the parents probability
distribution with the degree of set membership of the child. If cement is
typically 90% sand, and my mixture is 10% cement, I have no basis for
concluding that my mixture is 9% sand. What if the other component of the
mixture is 50% sand - then my mixture is actually 54% sand.
Now if I do not know what the other components of my mixture are, all I can
claim is that the probability that my mixture has less than 9% sand is
relatively low. And this is assuming that the attribute degrees combine
linearly. In the more general case that is purely guess work - unless you
explicitly specify how multiply inherited attribute degrees combine,
assuming
that they combine linearly can lead to the worst logical errors imaginable.
For an inference engine to operate safely, it must assume that it can make
no
such conclusions until specifically instructed otherwise. In any case,
statistical information, including expected values or centroids of attribute
degrees, has no necessary bearing on a sub-class even if it of degree of
superclass membership 1.0. Just because the human race is 50% male, does
not
mean a typical physics class is. And if it is only half human, the stretch
is
much worse.
>
> A fractional attribute to a class can be over-ridden by the member,
> which would say that the degree to which that attribute affects it, the
> individual member, is 100%. Considering all members'
degree-of-attribution,
> and the degree of attribution of this attribute to the general class, we
can
> achieve a useful redundancy: you can compute the degree of attribution for
> the entire class based on the degrees found in the individual members, and
> see if these two number match.
Again, you cannot override an actual attribute degree, unless the member
object is only fractionally a member. An actual attribute degree has to
apply
to all true subclasses as well. And if the attribute degree is only a
statistically expected value, then it is only in desparation that you can
assume those statistics are followed by the subclass.
> About internodes as nodes, I've known about this opinion of Mark's for
a
> while, that internodes should be considered as nodes, but I've not been
able
> to explain why I didn't like that idea until now. My explanation follows
> what I said in a previous letter about there being two interpretations of
> the three mathetical constructs: the symbolic and the geometric. If you
> looked closely at those few first paragraphs of that letter, you noticed
> that in the symbolic interpretation seems to allow for exactly what Mark
is
> suggesting. But the geometric interpretation is something different. In
> Mathetical geometry, there is a distinction between internodes and
> metainternodes. Internodes have a slope, which is measured or compared
> using metainternodes, but metainternodes do not "have" a slope in the same
> sense. They are slope. They define slope. Do you see what I mean?
My opinion is that the node / internode / meta-internode system is a special
case of the more general symbolic system. It may prove to be sufficiently
powerful and intuitive to obviate the need for being able to handle what you
would call meta-meta-internodes, and so forth, but I see no need to
unnecessarily prevent a system from being able to handle such constructs. I
beleive that a linguistic system should be "singly rooted", which means in
this case making sure that a "internode" is an instance of a node, which
automatically makes "meta-internodes" possible to the nth degree.
I do not think it is a wise idea to constrain the level of representation
the
system is capable of on the lowest possible level, rather than on a policy
basis on a much higher level -- If you have no need for
meta-meta-relationships, you are not forced to create them, but you should
not
preclude others from creating them, let alone other bizarre inter-meta-level
relationships.
A singly rooted system is simply a more general architecture than what you
are
proposing. It is capable of doing everything you propose, and then some.
Even if your theory is perfect and all knowledge can be represented with a
strict three level system, you probably shouldn't bar other knowledge
representational systems from using the core framework when doing so would
probably make the core system much easier to develop. Why should one write
code three times to handle three fundamental types of meta-objects when it
can
be written once instead?
> I want the model to be seamless, and the conversion from one
relational
> level to the other (from internodes to metainternodes) seems like an ugly
> seam to me. There may be instances when we want to compare nodes and
> metainternodes more directly, without having to go through any shifts in
> levels of relationality. And like I said, I see no reason to go beyond
> metainternodes, *and* three construct -- just three ... that's not all
that
> many, you know? It's simple, and it's powerful.
>
> Do you have any more to add to your reasons for wanting only two
> constructs?
I don't want two basic constructs, I want *ONE* basic construct. I want
everthing to be "singly rooted" and noun-ifyable, and you are fundamentally
saying that relationships are neither nouns nor first-class members of the
noosphere. I can't possibly see how you can model existing forms of human
expression when natural language is a singly rooted expressive system and
you
want to make the representation of a very large class of natural language
expressions in such a system impossibleby forcing it into a three-rooted
expressive system.
Simple examples:
"This sentence is true"
"This sentence is false"
Say we make "This sentence" a node. But unfortunately the whole sentence is
also a relationship between the sentence itself and the concept "true" or
"false", which means this sentence must be an internode. But a single
concept
cannot be both a node and an internode in your system, so it is impossible
to
represent any self referential expression.
Now these may be obscure, but how else are we to analyze expressions like
"Sam
really believes that "This sentence is false" is a consistent statement"?
- Mark
--
Mark Butler ( butlerm at middle.net )
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Epic Systems
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Subject: [om-list] Re: Modeling Relations -- Re: use cases
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Tom and other Packers wrote:
> We need to consider fuzzy sets, too. If you change or add an attribute
> to a class, to a partial/fractional degree, that will also affect all
> members of that class, but in an incomplete way: it will change or add the
> probability-of-fractional-degree that such an attribute applies to any given
> member of that class.
Probability (or likelihood) is a special kind of fractional set membership
that must be carefully distinguished from the more general case. Probability
expresses the measure of confidence that a certain proposition is correct,
based on a correct accounting of missing information. Fractional set
membership, on the other hand, expresses the general idea of degree of
membership in a set with ill defined boundaries. The latter can still exist
even in a environment of perfect information, whereas probability becomes
relatively useless.
To combine the two ideas (degree of belief with degree of membership in an
arbitrary set) requires expressing the truth value of a set membership
proposition as a probability density function over a degree of set membership
axis. This allows ideas to be expressed like "I completely sure it is
half-tiger" or "I am convinced it either is or isn't a tiger (but it is not a
mixture)".
You cannot combine probability and degree of set membership into the same
scalar value without throwing rigor out the window. Unfortunately doing it
correctly is computationally expensive, so there needs to be judgement
exercised when using both together.
In my opinion, probability is a much more effective concept than more
arbitrary fuzzy set membership, primarily because it is defined rigorously in
the language of science on a class of largely man-made sets with distinct
boundaries. Scientists, as a rule, eschew fuzzy sets whenever possible except
in cases where the degree of set membership can be given a rigorous
definition. Expressions like "78% warm" are nearly meaningless, because there
are but the barest conventions for expressing fuzzy set membership in natural
language.
However, there are certainly enough cases where it is extremely useful to
distinguish between "slightly", "somewhat", and "mostly" that I agree that
fractional set membership should be supported.
> Or, if a member is a member to a fractional degree,
> then any attribute added to its superclass, even one added completely to
> that superclass, will only affect that member to the degree to which it is a
> member of the superclass. If both the attribute and the membership are
> fractional, then some convention needs to be established for combining the
> fractionality here, such as multiplication of the fractional value (a value
> between 0 and 1).
Unfortunately there is no rigorous way that you can do this. If you say that
the superclass has attribute of degree X, then it follows that all members of
that class have the same attribute of degree X.
The only legitimate thing you can say is that the expected degree of the
attribute is X given a sufficiently large sampling of the members of the
class, which is short hand for specifying the expected value of the
probability distribution over the degree of attribute. You could also go on
to add the standard deviation to give a more complete picture of the
distribution.
If a certain member of the set is only a member of degree 1/2, the only thing
you might possibly do (in desparation) is to use the 1/2 as a scale factor for
combining the parents classes probability distribution with all other
probability distributions of sets the member might be fractionally a part of.
What you cannot do is multiply the centroid value of the parents probability
distribution with the degree of set membership of the child. If cement is
typically 90% sand, and my mixture is 10% cement, I have no basis for
concluding that my mixture is 9% sand. What if the other component of the
mixture is 50% sand - then my mixture is actually 54% sand.
Now if I do not know what the other components of my mixture are, all I can
claim is that the probability that my mixture has less than 9% sand is
relatively low. And this is assuming that the attribute degrees combine
linearly. In the more general case that is purely guess work - unless you
explicitly specify how multiply inherited attribute degrees combine, assuming
that they combine linearly can lead to the worst logical errors imaginable.
For an inference engine to operate safely, it must assume that it can make no
such conclusions until specifically instructed otherwise. In any case,
statistical information, including expected values or centroids of attribute
degrees, has no necessary bearing on a sub-class even if it of degree of
superclass membership 1.0. Just because the human race is 50% male, does not
mean a typical physics class is. And if it is only half human, the stretch is
much worse.
>
> A fractional attribute to a class can be over-ridden by the member,
> which would say that the degree to which that attribute affects it, the
> individual member, is 100%. Considering all members' degree-of-attribution,
> and the degree of attribution of this attribute to the general class, we can
> achieve a useful redundancy: you can compute the degree of attribution for
> the entire class based on the degrees found in the individual members, and
> see if these two number match.
Again, you cannot override an actual attribute degree, unless the member
object is only fractionally a member. An actual attribute degree has to apply
to all true subclasses as well. And if the attribute degree is only a
statistically expected value, then it is only in desparation that you can
assume those statistics are followed by the subclass.
> About internodes as nodes, I've known about this opinion of Mark's for a
> while, that internodes should be considered as nodes, but I've not been able
> to explain why I didn't like that idea until now. My explanation follows
> what I said in a previous letter about there being two interpretations of
> the three mathetical constructs: the symbolic and the geometric. If you
> looked closely at those few first paragraphs of that letter, you noticed
> that in the symbolic interpretation seems to allow for exactly what Mark is
> suggesting. But the geometric interpretation is something different. In
> Mathetical geometry, there is a distinction between internodes and
> metainternodes. Internodes have a slope, which is measured or compared
> using metainternodes, but metainternodes do not "have" a slope in the same
> sense. They are slope. They define slope. Do you see what I mean?
My opinion is that the node / internode / meta-internode system is a special
case of the more general symbolic system. It may prove to be sufficiently
powerful and intuitive to obviate the need for being able to handle what you
would call meta-meta-internodes, and so forth, but I see no need to
unnecessarily prevent a system from being able to handle such constructs. I
beleive that a linguistic system should be "singly rooted", which means in
this case making sure that a "internode" is an instance of a node, which
automatically makes "meta-internodes" possible to the nth degree.
I do not think it is a wise idea to constrain the level of representation the
system is capable of on the lowest possible level, rather than on a policy
basis on a much higher level -- If you have no need for
meta-meta-relationships, you are not forced to create them, but you should not
preclude others from creating them, let alone other bizarre inter-meta-level
relationships.
A singly rooted system is simply a more general architecture than what you are
proposing. It is capable of doing everything you propose, and then some.
Even if your theory is perfect and all knowledge can be represented with a
strict three level system, you probably shouldn't bar other knowledge
representational systems from using the core framework when doing so would
probably make the core system much easier to develop. Why should one write
code three times to handle three fundamental types of meta-objects when it can
be written once instead?
> I want the model to be seamless, and the conversion from one relational
> level to the other (from internodes to metainternodes) seems like an ugly
> seam to me. There may be instances when we want to compare nodes and
> metainternodes more directly, without having to go through any shifts in
> levels of relationality. And like I said, I see no reason to go beyond
> metainternodes, *and* three construct -- just three ... that's not all that
> many, you know? It's simple, and it's powerful.
>
> Do you have any more to add to your reasons for wanting only two
> constructs?
I don't want two basic constructs, I want *ONE* basic construct. I want
everthing to be "singly rooted" and noun-ifyable, and you are fundamentally
saying that relationships are neither nouns nor first-class members of the
noosphere. I can't possibly see how you can model existing forms of human
expression when natural language is a singly rooted expressive system and you
want to make the representation of a very large class of natural language
expressions in such a system impossibleby forcing it into a three-rooted
expressive system.
Simple examples:
"This sentence is true"
"This sentence is false"
Say we make "This sentence" a node. But unfortunately the whole sentence is
also a relationship between the sentence itself and the concept "true" or
"false", which means this sentence must be an internode. But a single concept
cannot be both a node and an internode in your system, so it is impossible to
represent any self referential expression.
Now these may be obscure, but how else are we to analyze expressions like "Sam
really believes that "This sentence is false" is a consistent statement"?
- Mark
--
Mark Butler ( butlerm at middle.net )
Software Engineer
Epic Systems
(801)-451-4583
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From: "Tom and other Packers" <TomP at Burgoyne.Com>
To: "Mark Butler" <butlerm at middle.net>
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Subject: [om-list] Re: meeting
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Lee et alia
Is Sept 2 good for an OM meeting? Morning, Afternoon, or over lunch? I
want everybody there who can make it, even those who may not think they
would contribute anything.
tomp
----- Original Message -----
From: Mark Butler <butlerm at middle.net>
To: Tom and other Packers <TomP at burgoyne.com>
Sent: Saturday, August 26, 2000 2:38 PM
Subject: Re: meeting
Saturday September 2 is fine with me.
- Mark
Tom and other Packers wrote:
>
> meeting
>
> Hello Om People
>
> I guess we all got busy again. I haven't heard from most of you
> recently.
>
> Some of us want to get together, in person, to discuss the OM thing,
and
> I think now is the time to start looking for days and times. Let's get as
> many people together as possible, but don't feel badly if you can't make
it.
> We'll do this again, probably many times in the more distant future.
>
> I propose September 2, the first Saturday after school starts. That
> will be a good time for me, because I am thinking of combining my work on
OM
> with my work in an "individual projects and research" CS class I will be
> taking next semester. "Feed two birds with one crumb" as my boss, Dan
Cook,
> says.
>
> tomp
--
Mark Butler ( butlerm at middle.net )
Software Engineer
Epic Systems
(801)-451-4583
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Subject: [om-list] Mark
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Mark
I forgot to mention one crucial thing: a node-internode-node is not the
same as an internode. Think about that in the context of the letter I just
responded to, and I'll think about how to explain this further. It's still
not clear in my mind. And I still haven't read your last letter.
tomp
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From: "Tom and other Packers" <TomP at Burgoyne.Com>
To: "OM List" <OM-List at OneModel.Org>
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Subject: [om-list] Re: "marketing"
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Lee
I do think that we had planned on GPL-ing it. But, I also believe that
there is one version of the GPL that allows for "selling" the software.
More about this later.
tomp
----- Original Message -----
From: Lee Howard <redder at deanox.com>
To: <general-list at onemodel.org>
Sent: Sunday, August 27, 2000 10:10 PM
Subject: "marketing"
I have a little confusion as to the OM project. (Well, I have more than *a
little* confusion with regards to it, but this is a part of it...)
Tom has spoken/written often about "marketing" the product when it's done.
Although "marketing" can mean a lot more things, my understanding has been
that specifically, this "marketing" is intended for selling it - for money
and profit.
My concern is that on the OM website, this project is said to be under the
GPL, and talk has gone on about CVS and other things (although CVS need not
be used specifically for "free" things). These two ideas seemingly are in
conflict.
In one case, (selling software) the code itself (or even the software
itself) is never sold, and in order to protect copyrights, the customer
pays for a license to use it.
In the latter case, (GPL) the code itself (and likewise the software) is
also never sold because it is essentially "free" in the terms of the GPL
(the license to use it is free, and the code is free to alter). The
customer in this case typically pays for the software's customization and
implementation.
Honestly, I think both of these cases are profitable [in most software
applications]. (Stuff like antivirus software can conceivably only
function profitably under the licensing method because it's really just a
compliation of information or "virus patterns" of which the free
distribution would undermine the profitability.) I think both of them have
their purpose and place, and I tend to favor one over the other, but could
somebody please clarify the position of this project?
Thanks.
Lee.
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Hello everybody,
Is this list alive? It seems everyone is still using
general-list at onemodel.org instead. Also, is there a performance problem with
the service om-list is on - the turnaround time seemed to be very poor the
last time I posted.
- Mark
--
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From: "Tom and other Packers" <TomP at Burgoyne.Com>
To: "Jeremy P. Almond" <jeremy at thoughtform.com>
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Subject: Re: [om-list] Re: meeting
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Jeremy and All Other OM people
We are planning on meeting at my house between 13:30 and 14:00 (on
Saturday), and then going off to eat somewhere if we have enough votes to do
so. Either which way, we are planning on answering the following questions
in fairly simple terms: What is OM? How do we go about making OM?
Here are my requirements for this first OM meeting: (1) We must have
three or more people there. (2) Two or more of the following people must be
present: Lee Howard, Mark Butler, TomP, Luke Call, (presently the four
principal interested parties, regarding the OM project).
Is any of you not going to come? (Out-of-state people are assumed to
not be coming, so don't feel bad, Luke and Chris.)
Hay, we could eat Indian food, if we can find a restaurant. That way
Liz would have an excuse to get involved ... sort of.
By the way, Luke, I think the problem with Jared Norman's registering
for the list not working was that his e-mail client was not set up as a
default "mailto" command responder, if that makes any sense.
ciao,
tomp
----- Original Message -----
From: Jeremy P. Almond <jeremy at thoughtform.com>
To: Tom and other Packers <TomP at Burgoyne.Com>
Sent: Monday, August 28, 2000 3:21 PM
Subject: Re: [om-list] Re: meeting
I think it would be fun. Just say the time and place and I shall be there.
Thanks,
Jeremy Almond
----------------------------------------------------------------------------
Have a nice day.
----------------------------------------------------------------------------
----- Original Message -----
From: "Tom and other Packers" <TomP at Burgoyne.Com>
To: "Mark Butler" <butlerm at middle.net>
Cc: "OM List" <OM-List at onemodel.org>
Sent: Saturday, August 26, 2000 4:23 PM
Subject: [om-list] Re: meeting
Lee et alia
Is Sept 2 good for an OM meeting? Morning, Afternoon, or over lunch? I
want everybody there who can make it, even those who may not think they
would contribute anything.
tomp
----- Original Message -----
From: Mark Butler <butlerm at middle.net>
To: Tom and other Packers <TomP at burgoyne.com>
Sent: Saturday, August 26, 2000 2:38 PM
Subject: Re: meeting
Saturday September 2 is fine with me.
- Mark
Tom and other Packers wrote:
>
> meeting
>
> Hello Om People
>
> I guess we all got busy again. I haven't heard from most of you
> recently.
>
> Some of us want to get together, in person, to discuss the OM thing,
and
> I think now is the time to start looking for days and times. Let's get as
> many people together as possible, but don't feel badly if you can't make
it.
> We'll do this again, probably many times in the more distant future.
>
> I propose September 2, the first Saturday after school starts. That
> will be a good time for me, because I am thinking of combining my work on
OM
> with my work in an "individual projects and research" CS class I will be
> taking next semester. "Feed two birds with one crumb" as my boss, Dan
Cook,
> says.
>
> tomp
--
Mark Butler ( butlerm at middle.net )
Software Engineer
Epic Systems
(801)-451-4583
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From: "luke call" <lacall at onemodel.org>
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Mark was right--there is a delay in getting posted messages through to the list because it is asking for my administrative approval on every one. Please bear with me as I get this straightened out.
Also, messages sent to me at lacall at onemodel.org between Monday at 7:00 am MDT and Tuesday around 4 pm MDT probably bounced to the sender due to configuration glitches (my being out of the country plus the ISP changing how forwarding is done). If you sent something to me personally then, could you resend?
I'm catching up on carefully considering and replying to many good things that have been said but it's a slow process. Unfortuately can't make it to the gathering Sept 2 or whenever it is (would really, really love to) but possibly around Thanksgiving?
Thanks everyone,
Luke
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From: "Tom and other Packers" <TomP at Burgoyne.Com>
To: "One Model List" <om-list at onemodel.org>
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Subject: Re: [om-list] Re: Modeling Relations -- Re: use cases
Date: Thu, 31 Aug 2000 12:45:24 -0600
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Mark
About "He thinks she likes him.". Assuming that I meant to make just
one geometric relation exist between every pair of nodes is correct, but not
complete. There are more than two nodes in this sentence. If we want the
strict, concretely geometric version, there might be hundreds of nodes here,
all related to the implied or assumed definitions of all the words being
used. At the very least, there would be references to a "he", a "she", a
state of "liking", and a state of belief, like a worldview, which can
contain all the other components if needs be.
By the way, in mathesis, node-internode-node pairs can be nodes.
Internodes are not nodes by themselves. I still don't understand why a
sentence is an internodes, according to any system. (A sentence is a node,
in Mathesis.)
In the more abstract symbolic version of Mathesis, I do believe that it
is necessary that the relations you have used, e.g. believes( x,y ), and
likes( x,y ), will be treated like a single vector between one pair of
nodes, and can be broken into vector components as you have mentioned, but
they won't be complete relations between the two nodes, and therefore there
can be more than one internode of this type for any given pair of notes.
And this is not the ideal situation, this is a practical situation, and I
think
we should try for the ideal as much as possible.
We're after the same thing here. I want this model to be
expandable/generalisable. Let's talk on Saturday, shall we?
tomp
----- Original Message -----
From: Mark Butler <butlerm at middle.net>
To: One Model List <om-list at onemodel.org>
Cc: Rex Butler <rexbutler at hotmail.com>
Sent: Saturday, August 26, 2000 12:36 AM
Subject: [om-list] Re: Modeling Relations -- Re: use cases
Tom and other Packers wrote:
> We need to consider fuzzy sets, too. If you change or add an
attribute
> to a class, to a partial/fractional degree, that will also affect all
> members of that class, but in an incomplete way: it will change or add the
> probability-of-fractional-degree that such an attribute applies to any
given
> member of that class.
Probability (or likelihood) is a special kind of fractional set membership
that must be carefully distinguished from the more general case.
Probability
expresses the measure of confidence that a certain proposition is correct,
based on a correct accounting of missing information. Fractional set
membership, on the other hand, expresses the general idea of degree of
membership in a set with ill defined boundaries. The latter can still exist
even in a environment of perfect information, whereas probability becomes
relatively useless.
To combine the two ideas (degree of belief with degree of membership in an
arbitrary set) requires expressing the truth value of a set membership
proposition as a probability density function over a degree of set
membership
axis. This allows ideas to be expressed like "I completely sure it is
half-tiger" or "I am convinced it either is or isn't a tiger (but it is not
a
mixture)".
You cannot combine probability and degree of set membership into the same
scalar value without throwing rigor out the window. Unfortunately doing it
correctly is computationally expensive, so there needs to be judgement
exercised when using both together.
In my opinion, probability is a much more effective concept than more
arbitrary fuzzy set membership, primarily because it is defined rigorously
in
the language of science on a class of largely man-made sets with distinct
boundaries. Scientists, as a rule, eschew fuzzy sets whenever possible
except
in cases where the degree of set membership can be given a rigorous
definition. Expressions like "78% warm" are nearly meaningless, because
there
are but the barest conventions for expressing fuzzy set membership in
natural
language.
However, there are certainly enough cases where it is extremely useful to
distinguish between "slightly", "somewhat", and "mostly" that I agree that
fractional set membership should be supported.
> Or, if a member is a member to a fractional degree,
> then any attribute added to its superclass, even one added completely to
> that superclass, will only affect that member to the degree to which it is
a
> member of the superclass. If both the attribute and the membership are
> fractional, then some convention needs to be established for combining the
> fractionality here, such as multiplication of the fractional value (a
value
> between 0 and 1).
Unfortunately there is no rigorous way that you can do this. If you say
that
the superclass has attribute of degree X, then it follows that all members
of
that class have the same attribute of degree X.
The only legitimate thing you can say is that the expected degree of the
attribute is X given a sufficiently large sampling of the members of the
class, which is short hand for specifying the expected value of the
probability distribution over the degree of attribute. You could also go on
to add the standard deviation to give a more complete picture of the
distribution.
If a certain member of the set is only a member of degree 1/2, the only
thing
you might possibly do (in desparation) is to use the 1/2 as a scale factor
for
combining the parents classes probability distribution with all other
probability distributions of sets the member might be fractionally a part
of.
What you cannot do is multiply the centroid value of the parents probability
distribution with the degree of set membership of the child. If cement is
typically 90% sand, and my mixture is 10% cement, I have no basis for
concluding that my mixture is 9% sand. What if the other component of the
mixture is 50% sand - then my mixture is actually 54% sand.
Now if I do not know what the other components of my mixture are, all I can
claim is that the probability that my mixture has less than 9% sand is
relatively low. And this is assuming that the attribute degrees combine
linearly. In the more general case that is purely guess work - unless you
explicitly specify how multiply inherited attribute degrees combine,
assuming
that they combine linearly can lead to the worst logical errors imaginable.
For an inference engine to operate safely, it must assume that it can make
no
such conclusions until specifically instructed otherwise. In any case,
statistical information, including expected values or centroids of attribute
degrees, has no necessary bearing on a sub-class even if it of degree of
superclass membership 1.0. Just because the human race is 50% male, does
not
mean a typical physics class is. And if it is only half human, the stretch
is
much worse.
>
> A fractional attribute to a class can be over-ridden by the member,
> which would say that the degree to which that attribute affects it, the
> individual member, is 100%. Considering all members'
degree-of-attribution,
> and the degree of attribution of this attribute to the general class, we
can
> achieve a useful redundancy: you can compute the degree of attribution for
> the entire class based on the degrees found in the individual members, and
> see if these two number match.
Again, you cannot override an actual attribute degree, unless the member
object is only fractionally a member. An actual attribute degree has to
apply
to all true subclasses as well. And if the attribute degree is only a
statistically expected value, then it is only in desparation that you can
assume those statistics are followed by the subclass.
> About internodes as nodes, I've known about this opinion of Mark's for
a
> while, that internodes should be considered as nodes, but I've not been
able
> to explain why I didn't like that idea until now. My explanation follows
> what I said in a previous letter about there being two interpretations of
> the three mathetical constructs: the symbolic and the geometric. If you
> looked closely at those few first paragraphs of that letter, you noticed
> that in the symbolic interpretation seems to allow for exactly what Mark
is
> suggesting. But the geometric interpretation is something different. In
> Mathetical geometry, there is a distinction between internodes and
> metainternodes. Internodes have a slope, which is measured or compared
> using metainternodes, but metainternodes do not "have" a slope in the same
> sense. They are slope. They define slope. Do you see what I mean?
My opinion is that the node / internode / meta-internode system is a special
case of the more general symbolic system. It may prove to be sufficiently
powerful and intuitive to obviate the need for being able to handle what you
would call meta-meta-internodes, and so forth, but I see no need to
unnecessarily prevent a system from being able to handle such constructs. I
beleive that a linguistic system should be "singly rooted", which means in
this case making sure that a "internode" is an instance of a node, which
automatically makes "meta-internodes" possible to the nth degree.
I do not think it is a wise idea to constrain the level of representation
the
system is capable of on the lowest possible level, rather than on a policy
basis on a much higher level -- If you have no need for
meta-meta-relationships, you are not forced to create them, but you should
not
preclude others from creating them, let alone other bizarre inter-meta-level
relationships.
A singly rooted system is simply a more general architecture than what you
are
proposing. It is capable of doing everything you propose, and then some.
Even if your theory is perfect and all knowledge can be represented with a
strict three level system, you probably shouldn't bar other knowledge
representational systems from using the core framework when doing so would
probably make the core system much easier to develop. Why should one write
code three times to handle three fundamental types of meta-objects when it
can
be written once instead?
> I want the model to be seamless, and the conversion from one
relational
> level to the other (from internodes to metainternodes) seems like an ugly
> seam to me. There may be instances when we want to compare nodes and
> metainternodes more directly, without having to go through any shifts in
> levels of relationality. And like I said, I see no reason to go beyond
> metainternodes, *and* three construct -- just three ... that's not all
that
> many, you know? It's simple, and it's powerful.
>
> Do you have any more to add to your reasons for wanting only two
> constructs?
I don't want two basic constructs, I want *ONE* basic construct. I want
everthing to be "singly rooted" and noun-ifyable, and you are fundamentally
saying that relationships are neither nouns nor first-class members of the
noosphere. I can't possibly see how you can model existing forms of human
expression when natural language is a singly rooted expressive system and
you
want to make the representation of a very large class of natural language
expressions in such a system impossibleby forcing it into a three-rooted
expressive system.
Simple examples:
"This sentence is true"
"This sentence is false"
Say we make "This sentence" a node. But unfortunately the whole sentence is
also a relationship between the sentence itself and the concept "true" or
"false", which means this sentence must be an internode. But a single
concept
cannot be both a node and an internode in your system, so it is impossible
to
represent any self referential expression.
Now these may be obscure, but how else are we to analyze expressions like
"Sam
really believes that "This sentence is false" is a consistent statement"?
- Mark
--
Mark Butler ( butlerm at middle.net )
Software Engineer
Epic Systems
(801)-451-4583
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From: "Tom and other Packers" <TomP at Burgoyne.Com>
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Subject: [om-list] Om Meeting
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OM People
How about we meet here at my house at 17:00 or 18:00?
I'm getting responses back so slowly from you people, it's hard to
co-ordinate this. If I continue to not get responses back quickly enough
(from the local people) to co-ordinate this, I will assume that no one is
really that interested (...Mark?), and will (1st) postpone it to next
Saturday, (2nd) cancel it.
So far, we have an "okay" from Jeremy and Doug for mid-afternoon. Will
later work for you two? Because then we had a "less-effective" from Lee.
Will later work better for you? I think Mark liked Saturday, but I don't
know what time(s).
tomp
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From: "Tom and other Packers" <TomP at Burgoyne.Com>
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Subject: [om-list] Meeting tomorrow
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Om people:
The meeting will be later, tomorrow. Meet at my house at 18:00.
Perhaps I didn't make myself clear, but I want to discuss what OM is,
what we should do to make OM, and this includes addressing all the major
disagreements we have encountered recently through email which we can fit
in, timewise; in fact one of the main motivations of the meeting (for me) is
to talk to Lee about what we were talking about in e-mail recently, the
things we couldn't resolve in e-mail, because the only effective way to
"resolve concerns" here is vocally.
tomp
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