Re: looking for a predicate hierarchy

On Mon, 8 Jan 2007 06:19:48 +0100 (CET), V.J. Kumar wrote:

"Dmitry A. Kazakov" <mailbox@xxxxxxxxxxxxxxxxx> wrote in
news:c4pg3bzabsq7$.16wavw6in6pvv$.dlg@xxxxxxxxxx:

On Fri, 5 Jan 2007 17:54:56 +0100 (CET), V.J. Kumar wrote:

Various interval arithmetics can be built on top of
the floating point rational endpoints. However, this stuff is not
really relevant to whetherfuzzy logic/posibility theory is a good
reasoning system.

Though interval computations and fuzzy had developed independently,
presently they are considered closely related. If you look at the
scope of interval computation conferences, you will see a lot of
"fuzzy content" there. This is a field where snake oil would not sell
any well.

I am not sure whether conflating methods to deal with real number
modelling system (f.p.) inadequacies with f.l./pos.t. suitability for
resoning is very productive. Is uncertainty of the kind 'the man is 6
foot tall' of the same kind as the floating point rounding problems?

Yes it is. Clearly rounding direction is not a random variable. Similarly
the height of a man when estimated by a human expert.

All rules of interval computations in terms of their bounds can be obtained
directly from possibility:

X@Y(r) =def=
= pos(X@Y|{r}) =
= Sup min { pos(X|{x}), pos(Y|{y}) } =
x@y=r
= Sup min { X(x), Y(y) }
x@y=r

Here @ denotes +,-,*,/.

I did not say that. I said that if the fuzzy crowd thinks that one
crisp number is not good enough to approximate vagueness, why two
crisp numbers are radically better ?

Because they define an interval.

How can the interval be defined with crisp boundaries ? The boundaries
themselves cannot be crisp by the same reasoning a sinle number cannot.

By which reasoning?

A counterexample: consider 0 and 1 as the lower and upper boundaries
correspondingly. These are A) true boundaries, B) crisp.

According to this, for example, "pi is about 3" is not vague,
because the ordered set of symbols ("pi", "is", "about","3") is
crisp.

I do not understand what pi has got to do with vagueness. It's a
crisp point on the real line, and the way we represent it is
immaterial. The 'vageness' in your example is contrived.

In what sense? How does "pi is about 3" define pi?

You confusing vageness with ignorance or some other intent.

Does it make any difference? If it exists then only in motives, which are
outside the scope.

pi or sqrt (2) are as vague natural numbers 1 or 2.

Yes.

Saying that pi is about 3 means
that the person who says that either has no clue what pi is or does not
care about a more precise value, although he can present such value if
needed.

Or, that the model is incapable to represent pi. But what about "pi is
3.14"? Is it less ignorant? How do we measure ignorance? How do we reason
from ignorant statements?

Because you don't like purely symbolic systems,

Where did I say that "don't like purely symbolic systems" ? I do not
even understand what you mean by it.

I mean you are keep on asking for "substances."

That's a legitimate request for a theory that claims to solve practical
problems.

Why solving practical problems should be related to substances? This again
goes into philosophy. Why arithmetic is applicable to counting apples?
Because it has a substance? Come on, nobody knows why it does.

It must be a misunderstanding. f(...) I described is just |= treated
as function of the arguments:

1. facts (the truth value of A)
2. rules set (the truth value of A=>B)

Does |= matter to inference?

|= does, but what you've described as f(...) is not a logical
consequence relation by any stretch of imagination.
[...]
So my question goes. What defines the scope of logic? Why
truth-functionality of |= is out of scope, but one of V is in? To me V
is not a big loss, when |= is lost.

You've never used |=, you did introduce f(..) and said above that it is
|=. Unfortunately, it is not. What do think |= is ?

I may have misunderstood what you meant, in which case please restate
your point.

The point is that if |= cannot be defined as a truth-valued function, why
then it were so important to have ^, /\, V such?

Consider the second red ball example where (red, small) has (0.5,
0.5) and (0.5, 0) membership degrees for y and z repspectively
(see the earlier message). Now, according to Zadeh and others,
one can use membership function as possibility distribution. So
since you agree that Pos(Red(x) or Small(x)) = max(Pos(Red(x),
Pos(Small(x)), then we have the same problem as when using the
classical fuzzy logic, namely that the possibility of y being
red OR small is the same as z being red OR small. It looks
counterintuitive. How do you explain that and in what way is
possibility theory useful in making a selection decision between
y and z ?

Let's replace it by an equivalent crisp example:

x is in the set Red and in the set Small (1, 1)
y is in Red and not in Small (1, 0)

then

both x and y are in the set Red U Small

Does this render set theory counterintuitive?

You are missing the point. The classical logic does not
distinguish between degrees of membership,

How so? 0 and 1 are two different degrees of membership.

That's coming from the fuzzy logic vocabulary. The classical logic
has truth values not degrees, it's odd to talk about degrees of
truth if you have only two truth values.

This cannot be a serious argument. The criteria in "intuitiveness" you
are using to beat fuzzy framework, must be universally applicable to
all systems. Otherwise, it becomes apples and oranges. Your example
should equally disqualify both or else be irrelevant.

It's a very minor point. If you like, let's call the classical logic 0
and 1 degrees of membership rather than, as is the custom, truth values,
it does not change anything.

That was not the point. The point was, _according to your reasoning_, x
must be STRICTLY more in Red U Small than y, BECAUSE some people on the
street would prefer x.

For inclusion in the standard set theory, the only possibility to have a
strict more-relation is to be IN vs. to be NOT-IN. So, immediately, the
only correct answer were x is in Red U Small, y is not in Red U Small. Thus
either, the set theory is wrong or people on the street are. What is your
choice?

so its answer is as expected when posed
like that: is a ball in Red or is it in Small (or both)?

and this should be counterintuitive to you, because a higher
smallness of x should make it also more reddish-or-smallish than y.

Why is it counterintuitive ? The question was whether the ball is
red OR small, since there are no degrees of truth in if posed in the
classical logic.

The answer given by both logics is *same*. Why this answer is
intuitive in classical logic, but not in the fuzzy one?

Model is adequate
Framework is consistent
-----------------------
Answer is adequate

The answer is the same, that's true, but as was said before, f.l.
should be an elaboration, rather than a simple restatement of what could
be obtained by the old classical logic.
[...]
So why use it if it's no 'better' than the classical one ?

Because:

1. the inputs might be uncertain (= data, like A, B)
2. the rules (= knowledge, like A=>B) might be uncertain

So what ? With all this stuff we do nothing, but arrive at the same
conclusion as the classical reasoning would. It's hadly much of a gain.

The gain is that we cannot arrive at any conclusion using classical
reasoning. It does not deal with either 1. or 2. We need some substitutes,
models of uncertainty to bring it back to the classical framework. This
substitute is fuzzy truth value. Any theory of uncertainty is all about
this.

Doing so, you shall *not* query for what truth value is [within the same
framework]. Because that would bring you back to the starting point, to
classical reasoning without uncertainty. You also cannot claim its
equivalence to classical reasoning without truth values on the basis that
the reasoning might be same. It is not, intrinsic truth values make the
difference.

So what model would you suggest for this trivial puzzle ? I am all
ears ;)

Trivial? Tell me first if you'd still prefer a ball of 0.2 yoctometer
diameter, and how are you going to determine its color!

You are evading the answer. Let's try it again:

1. What model would you suggest to solve the problem in a better way than
the classical reasoning.

None. I don't propose a replacement for. I do use of an uncertainty
measure.

2. How would you obtain possibility/necessity distributions.

By postulating them.

3. How would using interval possibilities improve your reasoning in
comparison to crisp real numbers ?

By using necessities. People prefer reddish-smallish balls and dislike
balls which aren't red or small. When you model both attractiveness of a
ball (as Red/\Small) and its unattractiveness (as ~Red/\~Small), then for

Red|x = [1, 1], Red|y = [0.5, 0.5]
Small|x = Small|y = [0.5, 0.5]

You'll get

Attractive: pos(Red/\Small|x)<=0.5, same for y.
Unattractive: pos(~Red/\~Small|x)=0, pos(~Red/\~Small|y)<=0.5

i.e. y is more unattractive.

But, again /\ is a poor connective to model preferences. And preferences
are unrelated to uncertainty. You should first get a preference model
working and then consider how it would perform with uncertain input data.
The example, as I showed above you proposed a wrong preference model. This
model does not work in crisp case. Why should it suddenly start working in
a fuzzy case?

No, I am asking how do you determine pos(~Red(x)) ? Where does the
number come from ? Suppose we know that Pos(Red(x)) equals 0.5,
what is Pos(~Red (x)) ? When you answer that, we can discuss the
min(Nec(Red(x)), Nec(Small (x)) problem ;)

[Putting my intuitinistic cap on]

You cannot obtain pos(~Red|x). You should have known it from the
same source as pos(Red|x). When x is singleton, then
pos(Red|x)=1-pos(~Red|x). There exist also next two fundamental
values: pos(Red|~x) and pos(~Red|~x). How is it possible that

1. given x is red
2. given x is anything but red
3. anything but x is red
4. anything but x is not red

That's cool, but all the talk does not tell us *what* numeric value
of Nec(Red(x)) is. Take for example, a biased coin toss. One can
throw the coin, say, a hundred times and arrive at the experimental
frequency interpreted as probability Probabiliy(heads)= 0.7.

Nope. That's wrong. What you could get is at best

Pr (0.69 <= Pr (Heads) <= 0.71) | H) >= 0.8

Whether interval valued probabilities are 'better' than the traditional
ones is not under discussion here.

No, this is exactly how "traditional" statistics works. You cannot
determine a distribution of probabilities from samples. You can only get
the probability of a random variable being in a confidence interval. So you
could say that, provided the hypothesis H, the probability of heads were
80% in [0.69, 0.71]. No more than that. This does not give you any chance
to determine the probability of Heads to any certain degree. Because all
statements about probabilities are like above, self-recursive.

H is the hypothesis. All answers mathematical statistics yields are in
terms of probability. Otherwise, you could unwind it back to a
deterministic set-up.

Forget about probabilities, they do not exist in your world., talk in
terms of membership functions, possibilities, necessities.

Probability gives an explanation why the very question is illegal. That
dispute is 200 years old. We need not to go into it once more for
necessities or possibilities.

--
Regards,
Dmitry A. Kazakov
http://www.dmitry-kazakov.de
.

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