Re: [ontolog-forum] cyclic and acyclic definitions

[Jawit Kien <jawit.kien@xxxxxxxxx>]

I'm very new to this forum, so some of my statement's below may be wrong-headed,
but I'd like to understand. Please forgive my lack of knowledge. I also don't want to sound like
I am attacking you (Bart) since I read another one of your postings and I was impressed
by how clear it was. Unfortunately, I'm not understanding this one.

On Tue, Apr 21, 2009 at 1:33 PM, Bart Gajderowicz <bgajdero@xxxxxxxxxx> wrote:

*** Originally from [ontolog-forum] web-syllogism-and-worldview ***
DC> Well this thread is veering in directions I wouldn't
> have expected
It's an interesting analysis of NL and human development.
I decided to start a new thread as to not interrupt the conversion in
*web-syllogism-and-worldview*

Let me state that my original intention was to find a way to model
cyclic definitions in a complete system. My approach was to convert
cyclic to acyclic definitions, via context. My envisioned result is
purely hierarchical, for the purpose of comparing 2 hierarchies, not
their instances.

Your result is hierarchical or a hierarchy?  You defining a function that
has a result which is a hiearchy, or are you describing the result (presumably
of the comparison operation) as being able to be described as a matching
some defintion of "hierarchy", hence able to use the adjective "hierarchical" to
describe it.

I assume a cyclic definition is one that uses the defined term in the definition
of that term. like saying a cat is a cat with black fur. Or, a #$Thing is an instance
of a #$Collection and a #$Collection is an instance of a #$Thing.
 
I've been told that Set theory does allow self-references in definitions but
Mereology Theory does not. Which are you using?

AA> There is much bigger problem here, the semantic one.
> Nobody can describe natural phenomena with logical syntax. Otherwise science
> would be redundant.

Don't you agree that logical syntax (and the meanings from FOPL) just provide a
different tool to describe phenomena? Isn't the point of science to describe things,
so a more precise tool to describe it would be better?

> Broadly, there are two types of conceptual systems:
> logical semantic systems (LOS), involving the so-called formal semantics,
> real semantic systems (RES), involving the real semantics, as in:

If semantics is just a fancy word for meaning, what do the people who use the
term RES mean by the phrase "real meaning" ?  I can see formal semantics as
attaching some formal mathematical model to a formal syntax defined by a grammar
of some sort, but what other kind of meaning can they mean? I'm thinking of some
kind of rigorous definition, I'm sure anyone can come up with a definition that doesn't
have a rigor attached to it. "It means what I say it means" or summat like that.
 

AA> 1. a denotation rule (reference), indicating the physical referent p;

I don't have to be concerned with indexicality of physical objects,
only classification through the fact that at some point, a particle
with similar attributes existed, and had a certain relationship to
something else, and was classified as type of X particle.

You said "at some point" is that a physical location, or a point in time?
or at some step in a logical inference? When you say "particle" are you
just trying to be more general than just saying "object" or "thing" or are
you thinking of sub atomic particles, or particles in a language?
I'm sorry, even when I try to state what you just said formally, I get too
many questions.
      classification - the act of assigning something a class
or   classification - the hierarchy of classes and subclasses that could be
                        used assign some thing a class.

similar attributes - how is this different that having a "certain relationship"
aren't attributes relationships too? what makes the "certain relationship"
special enough? and don't attributes refer to something else as well?
I assume "type of X particle"  is another way of saying the particle can be
assigned to the class X.  Why do you use the word "type" here instead of
class?

AA> 2. a representation rule (semantic assumption), measuring the position of
> the particle x(p, u,t) re. to some reference frame of units u at some moment
> of time t.
> With the RES you also find out that m(p) represents the inertia of particle
> (semantic axiom), and that it is subject to the conservation laws and
> Newton's law of motion (factual axioms).

This stuff  sounds like a mix between physics  and some kind of  knowledge
representation to me, hence I am very confused.

I wouldn't use temporal information to identify a particular particle,
but I could use the recorded speed of some particle, to identify
others as belonging to a group that behave in a similar manner.

If we are talking about physics and atomic particles, isn't there some
kind of law that says you can't know location and some other quality at the
same time?

If I implement some uncertainty measures, something like fuzzy logic,
at some level in the system, most likely at the last steps of
inference, I could dynamically calculate class membership or inclusion
based on empirical data.

I assume you mean class membership in the particles that are acting in
a certain way historically, which is what you were talking about before.
Why would inference, fuzzy logic or uncertainty even come up?
 

 This may be used to either reach a new level
of granularity, or resolve conflicting definitions, that are not
available through reasoning alone. I can equate this to defining a
structure for 2 molecules, X and Z, but resorting to experiments to
see how they interact. If up to that point, X and a new molecule Y
were structurally equivalent, but reacted differently to Z, I could
say that structurally X and Y are the same, but they are different in
the context of "reaction with Z".

You  seem to be using "reaction with Z" as a way of classifying X & Y.
how is this different from the attributes and relationships  you talked
about earlier?

By the way, now you seem to be talking about molecules instead
of particles. I'm very confused about the actual things you are trying
to classify.  Unless this is some metaphor that I don't understand.

JSF> Formalization is not some kind of ultimate goal that we're
> striving to achieve.  It is a very mundane, everyday process
> that children learn in elementary arithmetic.

John, what I meant by formalizing, and I thought you meant as well,
was agreeing on some standard to create and interpret ontologies, the
topic of the Summit.  Could you please clarify what you meant.

 

BG>> Fields such as fuzzy logic and probability theory let us
>> make statements about the world based on empirical data.

JSF> Not quite.  Those two systems, fuzzy logic and probability
> theory, are defined formally by patterns.

If we're trying to model the world, wouldn't this be an appropriate
way of doing it, or at least approximating it? If we can't define it
precisely due to ambiguity, couldn't corpora in the form of
statistically built hierarchies, help us here? If a class is
structurally the same in two different systems, and we find, through
empirical means, 2 contradictory instances in the two systems, which
differ in some attribute we didn't consider, then hierarchical
information would not tell them apart.


JSF> Again, you have to recognize that the C4.5 procedure is purely
> formal....  The only things that they process are
> patterns of character strings that represent somebody's best
> guess about reality.
> The GIGO principle still holds:  Garbage In, Garbage Out.

I agree completely.

> BG> I'm wondering whether there is a point where statistical
>  > analysis can take over where we simply don't know enough
>  > about a topic to infer information using logic alone.
>
JSF> There is no difference in principle... Logic can be used to
> define what is an A, what is a B, and what is a C.
> Statistics counts how many As, how many Bs, and how many Cs.
>
> As for the C4.5 algorithm, it is more closely related to
> logic than it is to statistics.  It's a kind of learning
> algorithm, but what it learns is a *decision tree* that
> can be expressed as a very large nest of if-then-else
> statements.  That tree can be mapped to a program in any
> language that supports if-then-else statements, such as
> C or Java.  It can also be mapped to first-order logic.


Again, I completely agree.
Perhaps I envision it differently, because I'm not concerned with how
many A's, B's or C's, but with classifying where instances lie based
on their attributes, adding to the structural definition. C4.5 would
give me that. For example:

In addition to structural information:
(forall x,z (R(x,z) iff exists y, s.t. R(x,y) ^ R(y,z)))

... I add something along the lines of:
(forall x,z (R1(x,z) iff exists y, s.t. R(x,y) ^ R(y,z)) ^ R2(x,y) ^ R2(y,z) )

Where R2(a,b) could be:
R2(a,b) => a.id < b.id
 OR
R2(a,b) => a.country is part of b.continent

Does you added axiom mean that
 you basically add a transitive axiom for R  & R2 at the same time?
 doesn't the original axion mean that the relation R obeys a  transitive axiom ?

You can define ranges like this for error checking in XML,
specifically ranges in OWL to define variations in objects. So that's
nothing new.

Back to my proposed way of handling cyclic definitions, by changing
the attributes and properties to acyclic definitions:
Following Azamat's questions:
I could perhaps define relationships this way to determine if A and B
are transitive hierarchies with R10(A,B) (see below):

so A is a transitive hierarchy, and B is a transitive hierarchy.
( actually, I presume they are actually variables  that stand for two
specific transitive hierarchies )  I think of a hierarchy as being
iso-morphic to a tree of nodes.  Thus R10 is a relationship
between trees.

1. restrained by the class inclusion relationships
K-9 is a Species is-a Animal

so K-9 is a class,  Species is a class,  Animal is a Class,
Species is a  specialized class of Animal
and K-9 is an instance of the class Species ?
presumably both A & B are each a hierarchy of classes?
so my tree-of-nodes has each node as a class?

2. restrained by the the class membership relationship
Fido is a K-9

Fido is an instance of the class K-9 ?
What is the tie to A and B? Is Fido an instance of A and
at the same time an instance of B ?  But what does it
mean to be an instance of a hierarchy? I would have thought
an instance of a hierarchy was a tree, but now you seem
to be defining instance as a relationship between some
other group of things to the classes that are the nodes
in the trees. so Fido can't be an instance of A or B,
I guess, unless Fido is the name for some tree which
can be classified as a "K-9".

In the context of **structural categories**:
R10(K-9, Animal) holds
R10(Fido, Animal) doesn't hold, because Fido is not a subcategory of
Animal, only an instance of it.

R10 is an un-named relation that holds between the class K-9 and the class Animal
? But where do A and B come into this? I thought R10 was taking hierarchies,
not elements of a hierarchy.

In the context of **instances**:
R10(K-9, Animal) does not hold because you need an instance attributes
to start with.

What does this mean? an instance attribute to start with? Are we following
some path, that I haven't seen explicated yet, which has an instance attribute
(presumably a relation on instances) as the "starting point" of the path?
 

Both of these hold because Fido is an instance of K-9, which is a
subclassOf Animal.
R10(Fido, K-9) hold
R10(Fido, Animal) holds

whoa, didn't you just say a paragraph ago that F10(Fido,Animal) does NOT hold?
 

I'm not sure how this distinction would work, but one possibility is types:
(forall x,y R10(x,y) iff (x.type == y.type))
... where R is a subclassOf relation, and R1 is a special R relation
with type check for transitivity using R10:
(forall x,z (R1(x,z) iff exists y, s.t. R(x,y) ^ R(y,z)) ^ R10(x,z))


Fido.type => 'instance'
K-9.type => 'category'
Species.type => 'category'
Animal.type => 'category'

Note that Fido could never be of type 'category' with Animal, only instance.
If we wanted to look at Fido's individual parts, they would be Fido's
paw, Fido's tail, etc, transitive to K-9's-paw, K-9's-tail. I'd have
to prevent Animals from having paws.  Perhaps through empirical means,
where I find an instance Giraffe named Mindy which is an animal but
does not have paws.

It seems that you are engaging in some logical conclusion that almost makes
sense, but not quite there. If Animal is a category, which I assume is something
like a #$Collection, it is not of the same natural kind as Fido, so you can't
infer along an instance-of chain which has two categories in it.

in other words, since Fido is an instance of K-9 then the instances of K-9
have the same predicates as Fido, but K-9 is a collection, so it will have
the predicates for collections. not the predicates for instances of K-9.

by transitivity, I would assume that since K-9 is a specialization of Animal,
then if Fido is an instance of K-9, then Fido is an instance of Animal.
hence the the predicates that apply to instances of Animal would also
apply to Fido as well.  I would NOT assume that the predicates that apply
to K-9 also apply to the instances of Animal, as presumably there could
be other specializations of Animal which are disjoint from K-9.  The predicates
that applied to the instances of Animal would apply to the instances of some
other specialization of Animal (call it Feline), as well as K-9.  That is what it
mans to say that K-9 is a specialization of Animal. (or that Feline is a
specialization of Animal)


JK

--
Bart Gajderowicz
MSc Candidate, '10
Dept. of Computer Science
Ryerson University
http://www.scs.ryerson.ca/~bgajdero

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