|From:||Jawit Kien <jawit.kien@xxxxxxxxx>|
|Date:||Wed, 22 Apr 2009 14:28:51 -0500|
On Wed, Apr 22, 2009 at 12:37 AM, Bart Gajderowicz <bgajdero@xxxxxxxxxx> wrote:
How does a cyclic definition introduce incompleteness?
Maybe I don't know what incompleteness means for a logical theory.
Does "incompleteness" mean that you can prove a contradiction,
and then since you are saying a contradiction is true, then anything
possible you want to say is true as well?
Good. At least we have a goal that I agree with.
I seem to recall that if you are not explicitly using time as a variable in a logical
system, then you have to deal with "timeless" things that are always true rather
than being concerned with what used to exist or what could exist.
The first idea I thought of as timeless was "all apples are red" but then I remember
that apples start off green, and then turn red, and then I guess, turn brown and mushy,
ending with it black and gnarly. So I can't really say that is timeless.
My next reaction was that I could use a 4-d representation like Matthew West
suggests sometimes. But then I realised that if any particular apple over time
has several colors, then I guess the four-dimensional apple really has a series
of colors (indexed by the "time-slice" you are interested in)
This would work, and would let "colorOf" be a predicate that only makes sense for
a timeslice rather than for the 4 dimensional apple. The thing is, that I can easily
imagine this "colorOf" being a relation between a 4-dimensional apple and a
green-red-brown-black "color column" because really, the apply isn't green and then
red, it actually becomes less green and more red over a time range, likewise for
each of the other color changes. Likewise I can imagine that the "sizeof" predicate
when a 4-dim predicate, would connect to a column of numbers+units, kindof
like a function mapping a time-marker to a size, I guess, because at any particular
time the apple has only one particular size. I said a time-marker, because, while
a universal time since the beginning of the universe, so to speak, could be the
input, I think it more general to talk of a time since the apple was plucked from
the tree, or summat like that. A relative time like that would let you use the same
function over and over again for different apples, since presumably apples that
are sufficiently alike will change color at about the same rate.
Of course, I might be confused.
Okay, so the context is some kind of group of true statements which
are used to make sense of what happens when you do the activity of
classifying something ("assigning something a class"). I guess this
is indistinguishable from a microtheory, but someone who knows more
about will have to weigh in to see if I am guess right.
By the way, I assume you made a typo when you said you would identify
a Cat as an instance named Fido. and really meant that you would classify
an instance named Fido as an instance of the class named Cat.
Thinking of this, I guess when I said "an instance named Fido", I must
be thinking of something like "an instance of #$Thing named Fido"
where #$Thing is a generalization of Cat.
I guess I'm thinking of databases where a "colorOf" column would exist
for each instance. But what if you add Colour as part of your classification
but you don't know what color a particular instance might happen to be?
Would the value of Colour attached to the colorOf predicate be in
some kind of vague "super-position" where it is all possible colours at the
same time? And if so, what about Classifications that REQUIRE the
colorOf predicate to have some particular value, like I assume a BlackCat
classification would require. Is the instance both a BlackCat and a
TabbyCat and a GreyFn(Cat) at the same time?
Or do you have some way of adding a "Colour" that isn't actually a color
but rather the "Unknown" Color, which is kind of default true until you
actually know what color that particular Instance may happen to be?
Then does BlackCat have some kind of exception rule that says all
cats that have UnknownColor as the "colorOf" are possibly instances?
I guess a Fuzzy Logic value of a probability of Black colour makes sense.
something like if you know apriori that 25% of all cats are black then
the value of colorOf when Unknown has a 25% blackness. If all the other
colors have less than 25% probability of being true then the system
could do some kind of default logic where it assumes that the cat is
a BlackCat until it has information to the contrary. This would necessitate
that the system was a non-monotonic reasoner I guess, which I have been
told is actually quite hard to implement.
Or am I confusing the probability of something being true with the percentage
of certainty that something is true?
Does assigning a degree of truth to a class mean anything? How can a class
be true or false? I thought only sentences could be true and false. Or are you
using some shorthand (metonymy?) to mean that the "truth" of a class is the
same as the truth of some sentence that holds for every instance of the class?
I can imagine a pool of sentences that are all true of every instance, a pool
of sentences that are thought to be true of every instance because no counter
example has yet been found, as well as two corresponding pools of sentences
that are all false or are thought to be false. These would correspond to
"sufficient" true sentences, I guess for some kind of modal logic instead of
a First Order Predicate Logic (FOPL). These sentences could be tied together
to make some kind of grand single sentence with logical AND between each
one. I don't really know what you would do for a more fuzzy kind of logic.
Would you have pools of sentences that are only thought to be true if some
kind of guard condition is true? maybe like an IF-THEN statement or a WHEN
statement? But this seems to be just a way to keep a single class when the
natural thing would be just to make sub-classes where the guard condition
is known to be true, and then put those dependent pools in as the known-true
and the thought-to-be-true pools of the sub-class.
I guess if you had a pool of sentences that could be combined with a logical
OR between each one, that would be something like a "necessarily" true
condition for the class using modal logic. There would be similary necessarily
thought-to-be-true, "necessarily false" and "necessarily-thought-to-be-false"
pools as well.
If I don't really understand how "necessarily true" and "sufficiently true" work
for modal logic, could someone tell me where I am wrong?
The reason I think that these would be useful in an ontology is that if you have
a predicate like "weightOf" which is used in one of these 8? pools about
instances of a class, and you have some instance of #$Thing which you don't
know if it is an instance of a class, but you do know that the predicate doesn't
make sense for the instance, you can immediately reject it as being an instance
of the class. And I guess you could say you know that it is an instance of the
NOT-class (is that called a ~Cat in this example?) that John Sowa talks
about in some of his writings.
If you don't actually have a true or false value for each sentence, (and recursively
for each predicate-on-some-values ) but you have some kind of fuzzy truth number
instead, I guess you can use a FUZZY-AND and a FUZZY-OR logical connector
above on the sentences. The thing I don't know is how do you know what
fuzzy-truth number to attach to a particular sentence or predicate? It seems that
the fuzzy-truth operations sound good on a slide show, but aren't really useful
in real knowledge base.
Could someone enlighten me please about how you actually do this fuzzy logic
so relationships refer to "external" entities and properties refer to "internal" entities?
Does each class have a list of each so you can tell the difference? Is a property just
a monadic predicate? What kind of structure are you thinking of? A C struct?
a list of properties and methods like some object oriented languages use?
a mathematical structure like a ring or field or partial order or something more exotic?
I'm still confused, could you elaborate a little more?
That's the principle I was thinking of. So even if you aren't really talking
about particles, is there a similar principle that holds between some set
of attributes/qualities/properties/markers/relationships/truth-valued-connectors ?
I guess this would have to be a discussion in its own right...
Do you envision associating with each individual thing in the computer
the reasons why that individual is an instance of a particular class?
I assume you are allowing an individual to be an instance of more than one
class. I guess if you don't, you could identify an "individual" as a group of instances
all of which are known to be the "same thing" but which really is the
instance of the thing as one class or are the instance of the thing as some other class.
Like if Fido was a three-legged talking dog, then the label "Fido" would get you to
the group of items, and one item in the group is "Fido as an instance of the class Dog"
and another item is "Fido as an instance of three-legged Things" and another item
is "Fido as a talking thing", but since you have a single inheritance hierarchy, each
item has just a single path up to the root. As long as you have control over what happens
when a word is used to describe something in the computer, you could do this kind
of slight-of-hand and no one could tell if you had a single-hierarchy or a multiple-hierarchy.
but maybe it would make inference more efficient or something.
That kind of makes sense. I don't know what statistical inference really
means. Have you got a link I could follow to read more about it?
Is this the same thing as Bayesian probablilties where you can tell
the difference between whether something is an actual conclustion
or just a coincidence? I seem to recall something like that in school.
so I guess reaction as you use it here is a metaphor based on a chemical
reaction? Does the metaphor allow for things that are catalysts in that they
aren't actually part of the reaction, but when they are around they make the
reaction happen faster? I guess if you are using a reaction as a metaphor
for the logical inferencing process, there could be some facts that make
the inferencing happen quicker.
Such as if you know who someone's father is and who that person's
father is, you could conclude who the original person's grandfather might be.
Of course if you have a candidate grandfather, you still have to find out
if the father's father is the same person as the the candidate grandfather.
It could be that you have two aliases for the same person, or that the
candidate grandfather is actually not the paternal grandfather at all, but
is the maternal grandfather.
But maybe you aren't using a chemical reaction as the basis for your
metaphor and you mean something else like an allergic reaction or
a physics action-reaction as the basis for what you are trying to say?
Nah, numbers are boring unless you have a mathematical ontology.
Use cats and dogs, and maybe even mooses. (moosen? meese?)
Is this definition of context fundamentally a different approach than attaching the
pools of statements to a class like I outlined above? It feels like context-sensitive
in the same way that context-sensitive left-hand-sides exist in a context-sensitive
grammar, but which kind of left-hand-sides are disallowed in context-free grammars.
Whoa. Fido would have its own tree? Are you saying Fido is a tree?
I thought it was a node in a tree, or actually the name of two nodes,
one in each tree?
You also just threw in a hierachy of classification rules rather
than a hierarchy of nodes. This is moving from a taxonomy to a formal
ontology, I think. Are these classification rules generalizations of each
other, or are they independent of each other? Are these classification
rules the pools that I talked about earlier? or some other beastie?
um. did you mean to say that the nodes (classes) of A have a transitive relationship between
each other, and there is a transitive relationship between the nodes (classes) of B?
or did you mean there is some kind of tree that has trees (A & B) as nodes, and there is some
kind of relationship between those trees that is transitive?
To my way of thinking, Fido can't "exist as an instance". Fido has to
exist as an instance of Some Class. I thought K-9 and Animal have
a subclass/superclass (class-generalization) relationship between them.
In my mind, if X is a subclass of Y, then it doesn't make sense to say
that X is an instance of Y. Hmm. maybe I need to take that back. I
can imagine saying that #$Dog is a collection of individuals, thus each
of those individuals are instances of the class #$Dog. So since each
of those individuals are also instances of the class/collection #$Thing,
then #$Dog is a subclass of #$Thing. But #$Dog is an individual as
well as being a class/collection, so it is a an instance of #$Thing as
well. Is there any other counter example? My gut says that #$Thing
as the Universal Set gets a special case that other class/collections
can't get, notably because #$Thing is an individual as well, and is
an instance of itself. (That recursive kind of relationship that started
this whole thread)
This mostly makes sense. But one question...
How do I know what context I'm in? Is this where Situational Logic comes in?
I wasn't too sure what it was and what people mean by it.
So this "type" property is some kind of property whose value is the name
of another property? Is it a single valued property? or is it possible that
the sentence is true where
ThereExists FOO such that (FOO.type='instance') Logical-AND (FOO.type='category')
Hey, I'm getting some of what you are saying...
Specialization of the predicates or specialization of the classes?
I think I was using generalization/specialization to talk about the relationship
between the classes. I recall that there is also something like
specPreds and genlPreds that is defined over predicates too.
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